microtechnology and mems
microtechnology and mems
Series Editor: H. Fujita D. Liepmann
The series Microtechnology and MEMS comprises text books, monographs, and
state-of-the-art reports in the very active field of microsystems and microtechnology. Written by leading physicists and engineers, the books describe the basic
science, device design, and applications. They will appeal to researchers, engineers,
and advanced students.
Mechanical Microsensors
By M. Elwenspoek and R. Wiegerink
CMOS Cantilever Sensor Systems
Atomic Force Microscopy
and Gas Sensing Applications
By D. Lange, O. Brand, and H. Baltes
Modelling
of Microfabrication Systems
By R. Nassar and W. Dai
Micromachines as Tools
for Nanotechnology
Editor: H. Fujita
Laser Diode Microsystems
By H. Zappe
Silicon Microchannel Heat Sinks
Theories and Phenomena
By L. Zhang, K.E. Goodson,
and T.W. Kenny
Shape Memory Microactuators
By M. Kohl
Force Sensors for Microelectronic
Packaging Applications
By J. Schwizer, M. Mayer and O. Brand
Integrated Chemical Microsensor
Systems in CMOS Technology
By A. Hierlemann
CCD Image Sensors
in Deep-Ultraviolet
Degradation Behavior
and Damage Mechanisms
By F.M. Li and A. Nathan
Micromechanical Photonics
By H. Ukita
Fast Simulation
of Electro-Thermal MEMS
Efficient Dynamic Compact Models
By T. Bechtold, E.B. Rudnyi,
and J.G. Korvink
Piezoelectric Multilayer
Beam-Bending Actuators
Static and Dynamic Behavior
and Aspects of Sensor Integration
By R. Ballas
CMOS Hotplate
Chemical Microsensors
By M. Graf, D. Barrettino,
A. Hierlemann, and H.P. Baltes
Capillary Forces in Microassembly
Modeling, Simulation, Experiments,
and Case Study
By P. Lambert
P. Lambert
Capillary Forces
in Microassembly
Modeling, Simulation, Experiments,
and Case Study
123
Professor Pierre Lambert
Université Libre de Bruxelles (ULB)
BEAMS Department (CP165/14)
Avenue F.D. Roosevelt 50
1050 Bruxelles, Belgium
Series Editors:
Professor Dr. Hiroyuki Fujita
University of Tokyo
Institute of Industrial Science
4-6-1 Komaba, Meguro-ku
Tokyo 153-8505, Japan
Professor Dr. Dorian Liepmann
University of California
Department of Bioengineering
6117 Echteverry Hall
Berkeley, CA 94720-1740, USA
Library of Congress Control Number: 2007927260
ISBN 978-0-387-71088-4
e-ISBN 978-0-387-71089-1
Printed on acid-free paper.
© 2007 Springer Science+Business Media, LLC
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY
10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
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or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this
publication of trade names, trademarks, service marks and similar terms, even if they are not identified
as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary
rights.
9 8 7 6 5 4 3 2 1
springer.com
I dedicate this book to those whose time I devoted to writing it.
Je dédie ce livre à ceux à qui j’ai pris le temps de l’écrire.
Foreword
Within the field of microassembly, this book crosses a bridge between the
world of surface science and chemistry on the one hand and the world of
mechanical engineering on the other hand.
Indeed, the mechanical devices produced at a scale ranging from a few
micrometer up to a few millimeter are brought face to face with the effects
of downscaling, and in particular with the predominance of surface tension
effects over the gravity effects. Many illustrations of this trend can be found
in the literature and in emerging industrial products based on surface tension
effects such as the fluid lens patented by B. Berge and produced by Philips,
the emergence of capillary stop drives or, with other words, surface tension
based micro-valves, the use of surface tension combined with electrostatic
effects in the manufacturing of liquid handling systems such as the EWOD
(i.e., electro-wetting on dielectric) devices, and so on.
To focus on microassembly, two approaches are currently considered.
The self-assembly paradigm, in which surface effects are used to organize
and assemble micrometric structures (mainly up to a few micrometer), and
the microrobotic assembly, based on the miniaturization of the actuation,
high resolution micromanipulators, and gripping devices, more dedicated to
mesoscopic sized components (mainly down to about 10 µm). Self-assembly
is clearly not the subject of this book, even if some obvious links relate the
proposed models to this field.
As a scientific knowledge, microrobotics focuses on active structures, able
to produce motions and to interact mechanically, i.e., produce efforts, with
their environment at the microscale (between a few micrometer and a few
millimeter). One of the main challenging issues of it concerns the handling
of small components, in order to precisely position, assemble, characterize,
or modify them. The research in this field covers a wide area of interesting
topics, including the exploration of new phenomena (i.e., which are new from
the point of view of microrobotics) and the development of an adequate scientific background (step 1), the development of demonstrators illustrating new
strategies to pick up, to handle, and to release microcomponents, and which
VIII
Foreword
try to minimize or take benefit from the new physical effects of the miniaturization (step 2), and finally, the set up of efficient and reliable industrial
products addressing specific needs (step 3). Step 1 is fundamental in that sense
that new efficient micromanipulation systems can only be developed bearing
in mind the specificities of the micro-world and take advantage of it through
new approaches.
Precisely, this book proposes a physical understanding of the surface tension phenomena, builds models that can be used in simulations and in the
design of a surface tension based gripping demonstrator. The author uses wellknown concepts from surface science (like surface tension, capillary effects,
wettability, contact angles) and efficiently uses them as outputs of chemists
models (which explain whether a liquid will wet or not a surface), but as inputs of mechanical models predicting the amount of effort that can be used
to handle microcomponents.
The book is unique in that sense that this is the first in this direction and
it proves that the microrobotic approach can lead to very efficient systems. It
is very well organized and the content is presented in a very rigorous, pleasant,
and pedagogical manner by a real expert of the addressed issues.
We strongly recommend to all persons, students, engineers, researchers
who are interested in micromanipulation and microassembly to read it.
Besançon
Brussels
Lausanne
April 2007
Prof. N. Chaillet
Prof. A. Delchambre
Prof. J. Jacot
Preface
0.1 Context
In the current context of trend to miniaturization, the main goal defined at
the very beginning of this work was to study the influence of miniaturization
on the manipulation tasks performed in microassembly, because for a few
years, most papers dealing with microassembly have referred to overviews that
mentioned the importance of forces related to the microworld. The reader can
have a quick overview on the scales covered by the term microworld in Fig. 0.1.
In this figure, several domains can be distinguished:
1. The “macro” domain, related to conventional manufacturing and assembly
technologies
2. The “micro” domain where the limits of conventional means can be undergone and new strategies arise. Sometimes the upper area of the micro
domain is called “meso” domain
3. The “nano” domain fills the gap between the micro domain and the atoms
and molecules world. It is the ultimate domain of mechanical engineers
As a comparison, the accuracy of conventional manufacturing is about 10 µm
and the size of hair is between 10 and 100 µm. This book deals with components
meso
nano
micro
macro
L(m)
10-9
10-6
µ-accuracy
Fig. 0.1. Sizes and scales
10-3
µ-components
1
X
Preface
ranging from 10 µm to a few millimeter, with part features that can reach the
micron: The chosen case study consists in a watch ball bearing with 0.3 and
0.5 mm diameter balls.
More generally, the current breakthrough of the miniaturization of electronic components and the development of their related production equipment make it possible today to produce cheap components integrating a lot
of functionalities. These production techniques allow the 2D manufacturing
to use several materials: glass, silicon, metals. Beside these applications from
the semiconductor industry, the conventional mechanical design also tries to
reduce the size of the products and the emergence of micromechatronics develops new miniaturized robots with a lot of functionalities (sensing, actuation, guiding). This trend does not spare assembly and the products are not
only reduced in size but also the assembly and production equipment are
downscaled, giving rise to several concepts like microfactory or new assembly
strategies such as parallel assembly. The pieces of equipment and especially
the grippers are downscaled, but new grippers based on microworld related
physics are now commercially offered by a lot of industries and laboratories.
The first representation that crosses the mind when talking about micro is
that it surely must be “small.” The prefix micro can of course be understood
as defining the size of a component (10−6 m), but a microproduct has not to be
understood as a product with a size of a few microns. Let us give an overview
of some definitions that can help us better define the concepts of micropart,
microcomponent, microproduct, microsystem, microassembly. Benmayour [19]
proposes a general definition of a microproduct using an analogy with the
term “microscopic” object. In the same way as a microscopic object cannot
be seen with bare eye, a microproduct is a product that can neither be manufactured nor assembled with bare hand: The production of a microproduct
requires adapted manufacturing and assembly equipment. Unfortunately, this
definition is quite general and some conventional products like cars cannot be
considered as microproducts even when assembled with dedicated equipment.
Moreover, this definition can give us an upper boundary but cannot provide
any indications about the lower limit of a microproduct. However, it conveys
the idea that the size criterion alone cannot be taken into account.
We consider in this book microproducts like a watch ball bearing made of
microparts or microcomponents (like balls). Roughly speaking, we will consider that microproducts have sizes ranging from a few cm3 to a few dm3 . For
example, we use to speak about a micropump for a product that has external
dimensions of a cylinder with a 8 cm diameter and 2 cm height.
These microproducts are made of several microparts or microcomponents
that have a size ranging from 10 µm to a few millimeter, but they can have
some features with a size reaching 1 µm. For example, the pumping mechanism
of a micropump can be smaller than a cube with 10 mm edge, having at least
one dimension smaller than 100 µm. Nelson [130] generally refers to 1 µm–
100 µm as “microscale” and 100 µm to 1 mm as “mesoscale.”
0.1 Context
XI
As far as assembly equipment is concerned, most microfactories are actually
desktop factories, that is having external dimensions of 1 m2 ×40 cm. Bohringer
et al. [22] locates the field of microassembly between conventional assembly,
dealing with part dimensions higher than 1 mm and what they call “the emerging field of nanoassembly” (with part dimensions ≤1 µm).
A microgripper can be a gripper to handle microcomponents, even if the
whole gripping mechanism is still quite big compared to the handled part, or
it can refer to the terminal tip(s) of the gripper that is(are) in contact with the
microcomponent (for example, a particular kind of micromanipulation tool is
the Atomic Force Microscope (AFM): This equipment is not designed like a
gripper but several laboratories try to use it to push microcomponents. In this
case, the AFM tip can be considered as a gripper, made of a cantilever (100 ×
10 × 2 µm3 ) with a tip of conical or pyramidal shape of 10 µm height and a
tip radius of about 10 nm). Other criteria can be considered to characterize
microcomponents, such as, for example, the required tolerances and clearances
in order to ensure the function (the pumping mechanism of the micropump
cannot show clearances bigger than a few micron in order to guarantee that
drug can be transferred from the tank to the patient). A less quantifiable way
to define a micropart is to verify whether the models and the techniques used
in the macroworld are still valid. For example, macroassembly is clearly based
on the mechanical grip force to pick up and the own weight of the component
to release, while microassembly has to turn to other techniques due to relative
decrease of the gravity force compared to surface forces (see Fig. 0.2). As the
main goal of this work is to consider the modeling of the forces acting in the
manipulation of a micropart, we consider that the use of these forces make
sense in our microcomponents. We prefer to refer to model assumptions and
compare the sizes of a part or the roughness of a component with several
cut-off lengths arising from model assumptions. We consequently identify a
domain between a “van der Waals” cut-off length of a few tens of nanometer
Forces exerted on the component [N]
10
10
Classical gripping
Capillary gripping
5
10
Vacuum gripping
A
0
10
C
-5
B
10
-10
10
Weight ~ L3
10
-15
Vacuum force ~ L2
Capillary force ~ L
-20
10
-8
10
-6
10
-4
-2
10
10
Size of the component [m]
Fig. 0.2. Scaling laws and micromanipulation
0
10
XII
Preface
Table 0.1. Comparison between micro and macroproducts
Criterion
Size
Macroproduct
Microproduct
Below 1 mm
Below 500 µm
Accuracy
0.1–10 µm
5 µm
Clearances
Very small
Complexity
Made of several
Multifunctional, complex
elementary components
products, few components
Compact design products
Maintenance
Maintenance and replacement No maintenance, replacement
of the defective components
of the product in case of failure
Heterogeneousness
Several parts from different
technological domains involving
new joining techniques
Ref.
[166]
[41]
[166]
[91]
[166]
[154]
[166]
Table 0.2. Comparison between micro and macroassembly
Criterion
Automation
Batch size
Resource
consumption
Response time
Macroassembly Microassembly
Automatic
Manual and semiautomatic,
to be automated.
Single parts,
Batches of parts,
serial assembly parallel assembly
Expected to be lower
Ref.
[65, 166],
[159, 185]
[6]
Expected to be shorter
because of lower inertia
(limit of the nonretardated van der Waals forces, see page 10) and a capillary
cut-off length of a few millimeter (see (8.1)): This domain will be considered
as our microworld.
To give the reader a broader overview, we summarize some criteria related
to micro/macroproducts and to micro/macroassembly (Tables 0.1 and 0.2).
0.2 Contributions of this Book
This book falls into five parts whose main contributions are summarized in
Fig. 0.3 (the fifth part containing the appendices is not shown in this figure).
The first part introduces the concept of microassembly (Chap. 1), proposes in Chap. 2 a classification of the forces acting in microworld (which has
been defined in the previous section), and summaries in Chap. 3 the numerous
gripping principles proposed in the scientific literature. This summary (which
is essentially a review of the literature) serves as a basis for a gripping principles classification from which it turns out that the forces generated by surface
tension can suit the microgripping task.
0.2 Contributions of this Book
XIII
Part I: Microassembly Specificities
• Different kinds of microassembly
• What are the forces in action
• What are the possible handling principles
-- classification of the handling principles
-- proposal: the capillary gripper
Part II: Modeling and simulation of Capillary Forces
• Parameters involved in a gripping based on surface tension
• Classical methods for capillary forces computing: energy derivation method, geometrical
approximations, resolution of the Laplace equation at equilibrium
-- Proof of equivalence between the energy derivation and the Laplace equation based
methods
-- Implementation of a double iterative numerical scheme to compute forces in the axially
symmetric case, based on the solving of the Laplace equation
-- Determination of the limits of this static simulation
-- Determination of approaching contact distance, rupture distance and residual volumes after
rupture
-- Approximation of cycle times
-- Application to the watch ball bearing case study
Part III: Experimental Aspects
Testbench:
-- Set up of a force measurement testbench (from 10µN to 10mN)
-- Set up of a contact angles measurement testbench
-- Tested liquid: water, isopropanol and silicone oil, from 0.1µL to 1µL
-- Tested materials: steel, silicon, PTFE, zirconium
-- Tested geometries: concave and convex cones, spheres, cylinders
Studied parameters and phenomena:
-- Inputs: gap, geometries, contact angles, surface tension, dynamic release, volume, relative
orientation, evaporation
-- Outputs: forces and liquid bridges profiles
Watch ball bearing case study:
-- Study of the picking errors and solutions
-- Study of the releasing reliability
-- Measurement of the picking force and reliability study
Answered questions:
-- Advancing vs receding contact angle, tension term vs. Laplace term
-- Quantified comparison between picking principles
-- Quantified comparison between releasing strategies
-- Design rules for a surface tension based gripper
Part IV: Perspectives
Modelling and Simulation
- Dynamic simulation
- Capillary condensation simulation
Design and manufacturing perspectives
- Surface tension control (i.e. electrowetting)
- Design and manufacturing of a surface tension based gripper prototype for SMD components
Fig. 0.3. Contributions of this book
XIV
Preface
The second part concerns the modeling aspects. Therefore, Chap. 6
presents the underlying parameters (such as surface tension and contact angles) and models (Young-Dupré and Laplace equations), which rule the surface
tension forces (also called capillary forces). This chapter explains the action of
these forces on a solid, thanks to two terms: the so-called “Laplace” or pressure term and the so-called “interfacial tension” term (see Sect. 6.5). Based on
these parameters, Chap. 7 reviews some approximations to compute capillary
forces at equilibrium: energy differentiation methods, geometrical methods
assuming a given shape of the meniscus (typically arc or parabola). Chapter
8 details how to implement a numerical resolution of the so-called Laplace
equation to determine the meniscus shape in axially symmetric cases. This
allows the computation of the capillary forces linking a component and a
gripper, relying on the following assumptions: equilibrium, vanishing Bond
number (i.e., gravity is neglected), axial symmetry, constant contact angles,
constant volume of liquid. The originality of this model relies on the fact that
the volume of liquid can be imposed, which leads to a double iterative scheme
for the resolution. Another contribution of this book is to prove analytically
the equivalence of this approach and the energy minimization method (in the
case of a prism–plane interaction, see Chap. 9). The proposed model is applied
to the case study of a watch ball bearing, showing the interest for a gripper
geometry conforming with that of the component (Chap. 10). This model is
then enriched, thanks to a second set of parameters (Chap. 11), showing how
surface roughness and surface impurities can be included in the model through
the value of the contact angle. The contact angle hysteresis is introduced in
this chapter; however, it will be shown (thanks to experiment) how to chose
between both. Finally, this chapter illustrates with a figure from the literature
an interesting damping effect, which prevents high contact forces. The limits
of the proposed model are discussed in Chap. 12, showing the suitability of
this model even in the case of highly accelerated components. This chapter
provides some approximations of the damping time of the oscillations of the
meniscus, which indicates a first order of magnitude of the cycle time of a
surface tension based picking task. Some conditions of meniscus rupture are
given in Chap. 13. To conclude this second part, a detailed implementation
of the proposed models is given in Chap. 14.
The third part of this book focuses on experimental aspects. First, we
detail in Chap. 17 the set up of an experimental test bed allowing the measure
of the models inputs (contact angles, volumes of liquid) and outputs (forces
and meniscus shapes). Then, Chap. 18 provides numerous model validation
and exhaustive results concerning the influence of the gap, the gripper geometry, the surface tension, the contact angles (including the choice between the
advancing and the receding contact angles), the relative orientation of the
gripper with respect to the component, the conditions of dynamical release,
and the rupture distance of the meniscus. Theses results are discussed in
Chap. 21 in terms of picking and releasing strategies; therefore, we introduce
the concept of adhesion ratio φ:
0.3 What this Book Does Not Tell
φ=
Fmin
,
Fmax
XV
(0.1)
where Fmin and Fmax are, respectively, the minimal and the maximal values
of the capillary force, which is assumed to be tuned between the picking stage
(Fmax ) and the releasing stage (Fmin ). Ratios tending to zero indicate a very
flexible gripping strategy (components with a large mass range can be picked),
while a ratio tending to 1 indicates a nonsuitable gripping strategy. These
results have been then applied in a final illustration of the surface tension
gripping based on a watch ball bearing case study. The characterization of
the underlying parameters is led in Chap. 19 while Chap. 20 presents the
results of picking and releasing tasks of the 0.3 and 0.5 mm diameter balls
of this bearing. The conclusions presented in Chap. 21 discuss the results of
Chaps. 18, 19, and 20.
The fourth part contains the general conclusions and the perspectives of
this work (Chap. 22).
Finally, the fifth part contains the appendices, which includes modeling
and geometry complements, some elements of the proof of equivalence of both
capillary force models, some tracks toward a dynamical simulation, and finally,
a list of the main symbols and abbreviations used in this book.
The book is ended by a list of references and an index.
0.3 What this Book Does Not Tell
This book is an attempt to present on a comprehensive way the elements
ruling a reliable surface tension based gripping of a small component with a
gripper (typically a sub-millimeter sized component), in gaseous environment
(typically ambient atmosphere). However, the analysis proposed to understand the role of the underlying parameters ruling capillary forces is very
general, and the proposed model is only valid for axially symmetric cases.
In a whatever geometrical configuration, the reader will have to turn himself (herself) toward an energy minimization tool such as, for example, the
well-known Surface Evolver software. The case of lateral capillary forces is
hardly treated in the experimental part, and we refer the interested reader
to the work of Peter A. Kralchevsky [105]. On the same way, the so-called
self-assembly or auto-assembly is not treated in this book: These aspects of
self-assembly, which are not restricted to capillary forces, are presented, for
example, in the work of Karl F. Böhringer. It will be shown that a static
modeling is quite sufficient for our purpose; nevertheless, the reader will find
additional information concerning dynamical simulation in [156]. Finally, the
case of immersed environments is treated in [64].
Let us note that the example treated in this book concerns the case of
watch bearing balls with a diameter ranging from 0.3 to 0.5 mm. The use of
surface tension has an upper limit (the so-called capillary length equal to a
XVI
Preface
few millimeter for water), it is not limited in terms of miniaturization. Nevertheless, the manufacturing of micron-sized grippers would require adapted
manufacturing techniques that have not been considered in this book, but this
is more a perspective than a limitation.
0.4 Reading Suggestion
For a quick reading, the chapters and sections listed in Table 0.3 are essential
for a good understanding of this book. Let us emphasize the presentation of
four examples (Table 0.4).
Table 0.3. Quick reading suggestions
Chapter/Section Title
Page
Preface
3
Handling Principles for Microassembly
13
6
First Set of Parameters
41
7.1
Introduction to the State of the Art
51
on the Capillary Forces Models
8
Static Simulation at Constant Volume of Liquid 65
17
Test bed and Characterization
143
21
Final discussion of Part III
211
22
Conclusions and Perspectives
221
Appendix D
List of symbols
247
Table 0.4. Examples
Chapter
10
14
19
20
Title
Page
Application to the Modeling of Microgripper for Watch Bearings 83
Numerical Implementation of the Proposed Models
127
Watch Bearing Case Study: Characterization
189
Watch Bearing Case Study: Results
199
Brussels
April 2007
P. Lambert
Contents
Preface
0.1
0.2
0.3
0.4
.....................................................
Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contributions of this Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
What this Book Does Not Tell . . . . . . . . . . . . . . . . . . . . . . . . .
Reading Suggestion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IX
IX
XII
XV
XVI
Part I Microassembly Specificities
1
From Conventional Assembly to Microassembly . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Design of Monolithic Products for Microassembly . . . . . . . . .
1.3 Combined Part Manufacturing and Assembly . . . . . . . . . . . .
1.4 Product External Assembly Functions . . . . . . . . . . . . . . . . . . .
1.5 Product Internal Assembly Functions . . . . . . . . . . . . . . . . . . .
1.6 Stochastic or Self-Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Parallel Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
6
6
6
7
8
8
2
Classification of Forces Acting in the Microworld . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Classification Schemes of the Forces . . . . . . . . . . . . . . . . . . . . .
2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
9
10
12
3
Handling Principles for Microassembly . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Presentation of Gripping Principles . . . . . . . . . . . . . . . . . . . . .
3.3 Classification of Gripping Principles . . . . . . . . . . . . . . . . . . . .
3.4 Comparison between Gripping Principles . . . . . . . . . . . . . . . .
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
13
13
25
28
29
4
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
XVIII Contents
Part II Modeling and Simulation of Capillary Forces
5
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
6
First Set of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Young–Dupré Equation and Static Contact Angle . . . . . . . .
6.4 Laplace Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Effects of a Liquid Bridge on the Adhesion
Between Two Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 A Priori Justification of a Capillary Gripper . . . . . . . . . . . . .
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
41
41
42
43
7
45
47
49
State of the Art on the Capillary Force Models
at Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Energetic Approach: Interaction
Between Two Parallel Plates . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Energetic Approach: Other Configurations . . . . . . . . . . . . . . .
7.4 Geometrical Approach: Circle Approximation . . . . . . . . . . . .
7.5 Geometrical Approach: Parabolic Approximation . . . . . . . . .
7.6 Comparisons and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
55
57
61
61
8
Static Simulation at Constant Volume of Liquid . . . . . . . .
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Description of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Equations and Numerical Simulation . . . . . . . . . . . . . . . . . . . .
8.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
65
65
66
67
71
9
Comparisons Between the Capillary Force Models . . . . . .
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Qualitative Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Analytical Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.1 Definition of the Case Study . . . . . . . . . . . . . . . . . . . . .
9.3.2 Preliminary Computations . . . . . . . . . . . . . . . . . . . . . . .
9.3.3 Determination of the Immersion Height h . . . . . . . . .
9.3.4 Laplace Equation Based Formulation
of the Capillary Force . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.5 Energetic Formulation of the Capillary Force . . . . . . .
9.3.6 Equivalence of Both Formulations . . . . . . . . . . . . . . . .
9.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
73
73
75
75
76
77
51
51
79
79
80
81
Contents
10 Example 1: Application to the Modeling of a Microgripper
for Watch Bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Presentation of the Case Study . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 Analytical Model Based on the Circle Approximation . . . . .
10.4 Numerical Model Based on the Laplace Equation . . . . . . . . .
10.5 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Pressure Difference Saturation . . . . . . . . . . . . . . . . . . . . . . . . .
10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XIX
83
83
83
86
89
93
94
96
11 Second Set of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Surface Heterogeneities and Surface Impurities . . . . . . . . . . .
11.3 Surface Roughness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Static Contact Angle Hysteresis . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Dynamic Spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97
97
97
98
99
100
101
12 Limits of the Static Simulation . . . . . . . . . . . . . . . . . . . . . . . . .
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Performances of the Assembly Machines . . . . . . . . . . . . . . . . .
12.3 Nondimensional Numbers and Buckingham π Theorem . . . .
12.4 Another Approach: Use of a 1D Analytical Model . . . . . . . .
12.5 Limitations of the Static Model . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103
103
103
103
106
108
110
13 Approaching and Rupture Distances . . . . . . . . . . . . . . . . . . . .
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Approaching Contact Distance . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Rupture Distance and Residual Volume of Liquid . . . . . . . . .
13.4 Mathematical and Notation Preliminaries . . . . . . . . . . . . . . . .
13.5 Volume Repartition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.6 Rupture Condition and Rupture Gap . . . . . . . . . . . . . . . . . . .
13.7 Analytical Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8 Summary of the Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.9 Comparison between the Methods . . . . . . . . . . . . . . . . . . . . . .
13.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
111
111
111
113
114
115
117
119
120
122
124
14 Example 2: Numerical Implementation
of the Proposed Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.2 Liquid Bridge Simulation for the Analysis of a Meniscus . . .
14.3 Evaluation of the Double Iterative Scheme . . . . . . . . . . . . . . .
14.4 Pseudodynamic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
127
127
127
131
133
135
XX
Contents
15 Conclusions of the Theoretical Study of Capillary Forces
137
Part III Experimental Aspects
16 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141
17 Test Bed and Characterization . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.2 Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3 Test Bed Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.1 Force Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.2 Drop Dispensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.3.3 Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.4 CAD Model and Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.5 Characteristics of the Force Measurement Set Up . . . . . . . . .
17.5.1 Typical Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.5.2 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.5.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.5.4 Influence of a Misalignment on the Force
Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.6 Characteristics of the Contact Angles Measurements . . . . . .
17.7 Surface Tension Measurement . . . . . . . . . . . . . . . . . . . . . . . . . .
17.8 Modus Operandi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.9 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.9.1 Set of Available Grippers . . . . . . . . . . . . . . . . . . . . . . . .
17.9.2 Set of Available Components . . . . . . . . . . . . . . . . . . . .
17.9.3 Set of Available Blades . . . . . . . . . . . . . . . . . . . . . . . . . .
17.9.4 Available Liquids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.9.5 Contact Angles Characterization . . . . . . . . . . . . . . . . .
17.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
143
143
143
145
145
146
148
148
151
151
151
152
152
154
155
155
158
158
159
160
161
161
162
18 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 Preliminary Results: Validation of the Simulation Code . . .
18.2.1 Meniscus Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2.2 Comparison with the Analytical Expressions . . . . . . .
18.2.3 Experimental Validation . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Advancing vs Receding Contact Angle . . . . . . . . . . . . . . . . . .
18.4 Influence of the Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.1 Force–Distance Curve . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4.2 Tension Force vs. Laplace Force . . . . . . . . . . . . . . . . . .
18.5 Influence of the Gripper Geometry . . . . . . . . . . . . . . . . . . . . . .
18.6 Influence of the Surface Tension . . . . . . . . . . . . . . . . . . . . . . . .
18.7 Influence of the Contact Angle θ1 . . . . . . . . . . . . . . . . . . . . . . .
163
163
163
163
164
166
168
170
170
171
171
172
174
Contents
XXI
18.8 Influence of the Relative Orientation . . . . . . . . . . . . . . . . . . . .
18.9 Auxiliary PTFE Tip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.10 Dynamical Release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.10.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.10.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . .
18.11 Approaching Contact and Rupture Distances . . . . . . . . . . . .
18.12 Shear Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.13 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
174
176
177
177
182
185
186
187
19 Example 3: Application to the Watch Bearing Case
Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2 Available Grippers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3 Available Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.4 Liquid Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.5 Liquid Dispensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.6 Contact Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189
189
189
191
191
192
195
20 Example 4: Application to the Watch Bearing Case
Study: Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2 Picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2.2 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.2.4 Automated Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.3 Placing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.4 Compliance Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.5 Force Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20.5.2 Modification of the Force Measurement Test Bed . . .
20.5.3 Comparison Between Models and Experiments . . . . .
20.5.4 Ongoing Experimental Study . . . . . . . . . . . . . . . . . . . .
20.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199
199
199
199
200
201
202
204
205
206
206
206
206
208
209
21 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.2 Picking Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.3 Releasing Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.4 Design Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
211
211
211
213
215
XXII
Contents
Part IV General Conclusions and Perspectives
22 Conclusions and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
221
221
223
Part V Appendices
A
Modeling Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 Analytical Approximations of the Capillary Forces . . . . . . . .
A.1.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1.2 Between a Sphere and a Plane . . . . . . . . . . . . . . . . . . .
A.1.3 Between Two Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Volume Repartition by the Energetic Approach . . . . . . . . . .
A.2.1 Assumptions, Notations, and Mathematical
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2.2 L–V Interfacial Energy . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2.3 Total Interfacial Energy . . . . . . . . . . . . . . . . . . . . . . . . .
227
227
227
228
230
233
B
Geometry Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.1 Area and Volume of a Spherical Cap . . . . . . . . . . . . . . . . . . . .
B.2 Differential Geometry of Surfaces . . . . . . . . . . . . . . . . . . . . . . .
B.2.1 Mean Curvature of a Surface . . . . . . . . . . . . . . . . . . . . .
B.2.2 Mean Curvature of an Axially Symmetric Surface . .
B.3 Catenary Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
237
237
238
238
239
240
C
Comparison Between Both Approaches . . . . . . . . . . . . . . . . .
243
D
Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
247
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
251
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
261
233
234
235
Part I
Microassembly Specificities
1
From Conventional Assembly to Microassembly
1.1 Introduction
The goal of this chapter is to give an overview of different assembly strategies
that can be used at the considered scale from 10 µm to 10 mm. Indeed, even in
the field of microproducts, components have to be assembled. The production
of microsystems integrating many functionalities, many components made of
different materials require flexible, modular, accurate mechanisms, which can
finely feed, pick, orientate, move, and release different types of objects at the
right place.
The assembling and packaging operations that achieve the microcomponents’ fusion into a hybrid microsystem is usually considered a bottleneck
in the manufacturing process more than the manufacturing of components
itself. This is particularly true for very small components that require high
positioning tolerances leading to high manufacturing cost. High cost gripping
solutions for various applications concerning the handling and the assembling
of microcomponents have been developed but they do not offer satisfying
economical solutions yet. According to Breguet et al. [30], the main three
challenges characterizing microassembly are the following:
•
•
•
Precise alignment (submicron) of the components in several degrees of
freedom and in a large workspace (a few cm3 )
Grasping and releasing of these delicate components
Attaching them together
We present a taxonomy of microassembly in this chapter. To produce a miniaturized multifunctional system, we distinguish the following criteria:
•
•
Do we have to assemble a composed product or can we design it to avoid
(or at least reduce) assembly tasks?
Do we assemble a lot of loose components or can we combine assembly
and manufacturing in situ? [167]
4
1 From Conventional Assembly to Microassembly
Multifunctional product
Monolithic product
Composed product
Combined part
manufacturing and assembly
Product external
assembly functions
Assembly of loose
components
Self-Assembly or
stochastic assembly
Product internal
assembly functions
Fig. 1.1. Taxonomy of microassembly
•
•
Is the assembly equipment inside or outside the product? Can we use selfassembly (also called stochastic assembly)? [167]
Finally, is the assembly required to be serial or can the throughput be
increased by using parallel assembly? [22]
This classification is shown in Fig. 1.1.
1.2 Design of Monolithic Products for Microassembly
The first and most basic approach for microassembly consists in downscaling
the conventional approach. The use of miniaturized grippers (mainly downscaled tweezers or vacuum grippers, but an exhaustive description of the
suitable gripping principles is presented in Chap. 3) allows to pick, move,
orientate, and release microcomponents; however, the word gripper explicitly
refers to a two finger tool used to grip an object, it must be understood here
as any device allowing to pick a component, such as, for example, a vacuum
gripper with only one finger. The most often associated strategy consists in serial pick and place of components. The main drawbacks of such an approach
consist in physical limits (sticking problems at release) and in nonoptimal
solutions (all efforts for accurate positioning must be repeated for each component). Some authors [65] propose to improve this situation by combining
design for both microassembly and microworld adapted assembly equipment.
The design for microassembly is supposed to reduce the number of assembly
tasks or at least to improve the suitability of design for automated assembly.
To illustrate this, let us consider the design of a rotational joint with a one-way
actuation and an elastic force to get the system back to the equilibrium. This
example is illustrated in Fig. 1.2. In the conventional design, the rotational
joint is made of a small ball bearing (SKF produces reduced ball bearing with
1.2 Design of Monolithic Products for Microassembly
5
Spring
Moving part
Moving part
Notch hinge
Ball bearing
Fig. 1.2. Conventional design vs micro-driven design: case of a rotational joint (the
actuator is not shown)
outer diameter of about 2 mm. As a comparison, one of the smallest ball bearing with an outer diameter of 0.9 mm has been assembled by the researchers
of the MEL1 (Japan) with their microfactory [6, 135]. Once both parts and
the ball bearing are assembled, they still need to be put together with the
actuator and the elastic element allowing the backward motion. Several elements have to be manufactured and assembled. Besides the hardness of the
task, the different tolerances lead to a low-effective system with clearances.
Moreover, if the moving part has to guarantee watertightness with an antagonist counterpart, the system will probably not meet the requirements. An
alternative design could replace the ball bearing and the elastic element by an
elastic flexure hinge, combining both functionalities in one component. The
maximal deflection of the notch hinge – a notch hinge is a flexure hinge with
a circular profile – depends on the Young’s modulus and the elastic limit of
the material, and on the width and thickness of the hinge [35]. With titanium and wire electro-discharge machined2 hinges with 5 mm thickness and
100 µm width, the angular range can reach 15◦ . This new design reduces the
number of parts, highly simplifies the assembly task, and enhances the functionality of the system: no clearance, no friction, no wear, and consequently
no scraps, making this kind of design particularly suitable for biocompatible
applications.
1
2
MEL = Mechanical Engineering Laboratory, see National Institute of Advanced
Industrial Science and Technology http://www.aist.go.jp.
EDM = Electro Discharge Machining is “a machining method using a free electrical discharge between an electrode and a workpiece to generate heat flow with
high energy density, so that contact force and chatter vibration can be avoided
during machining” [87].
6
1 From Conventional Assembly to Microassembly
Micro electro-discharge
machining (drilling)
Insert of the pin
Twist for breakage
Removal of neck material by
of the neck
micro-electro discharge machining
Workpiece
Pin produced with wire
electro discharge grinding
Ultrasonic vibration
of the worktable
Fig. 1.3. Pin-in-hole combination produced by part manufacturing and assembly
steps [115] (Copyrights CIRP.)
1.3 Combined Part Manufacturing and Assembly
According to Tichem and Karpuschewski [166], “the goal of this method is to
minimize the assembly content of composed products by creating products on
basis of a combination of part manufacturing and assembly operations. This
reduces the amount of part handling operations and delicate joining operations. Clearly, this method is not a pure assembly method. It recognises the
fact that, in the microdomain, part manufacturing and assembly can in certain
cases be integrated. The separation between part manufacturing and assembly
as visible in the macrodomain vanishes.” An example of this approach can be
found in [115], dealing with a pin-in-hole combination performed by part manufacturing and assembly steps. More recently, Jing-Dae Huang and Chia-Lung
Kuo[87] have improved this method combining micro-EDM and laser welding
to manufacture and achieve pin-plate assembly of 50 µm diameter pins with
a large aspect ratio.
1.4 Product External Assembly Functions
This approach is the most conventional one, often based on the use of accuracy positioning systems and microgrippers. This method can be improved
by adding visualization systems and imaging processing. We will not focus on
this method because it is not micro-oriented. Nevertheless, it has to be mentioned because it can lead to a micro-specific assembly method when coupled
with a parallel assembly approach, which is also presented in this section.
1.5 Product Internal Assembly Functions
The principle of internal assembly is to provide a microproduct with additional functionalities such as, for example, internal actuators. The assem-
1.6 Stochastic or Self-Assembly
7
Fiber holding
groove
Bimorph
Passive spring
Silicon wafer
surface
Fiber
V-beam (thermal
actuation)
Fig. 1.4. Example of internal adjustment of optic fibers (Reprinted with permission
from [82]. Copyrights 2006 Institute of Physics.)
bly of this product with another component can then be performed in two
steps: A first coarse positioning of the component on the product is performed
with a conventional, whether miniaturized or not, handling tool. The ultimate
positioning with the required accuracy is performed inside the product, thanks
to the internal actuators. This way to perform assembly provides final microproducts with a higher complexity and more internal functionalities. The cost
aspects of such a method must be analyzed carefully. The example of internal
assembly given in [175] includes the following functions for a self-adjusting
microsystem: a controlled actuation of the component, the sensing of the
position of the component, and the freezing of the component in the final
position. The studied example consists in interconnecting optical fibers with
each other. The fine positioning is performed by using a piezoelectric plate
glued on a passive silicon layer, allowing a bending motion of the actuator
when voltage is supplied to the piezoelectric electrodes. This bending motion
is used for the fine positioning of the fibers. Two ways can be used to keep
the alignment: The first way is to use active control, the second one consists
in permanently freezing the alignment, but nowadays, no technical solution
has been proposed yet for this application. Recently, an adaptation of this
principle has been proposed in [82], which is illustrated in Fig. 1.4.
1.6 Stochastic or Self-Assembly
The underlying idea of stochastic assembly is to avoid any deterministic
interaction and control of the part position during the assembly task. The
components to be assembled are jumbled in close distance before applying a
force field that will perform the assembly. An example of stochastic assembly
8
1 From Conventional Assembly to Microassembly
Fig. 1.5. Example of stochastic assembly illustrated in [23]: capillary forces in the
adhesive (black) cause self-assembly
is cited in [166] and consists in using the capillary force of an adhesive drop to
align two parts (information on lateral forces modeling can be found in [105]).
This principle taken from [23] is illustrated in Fig. 1.5.
Other effects are cited in [22] and have been applied by several authors to
proceed stochastic assembly:
•
•
•
•
Fluidic agitation and mating part shapes
Vibratory agitation and electrostatics force fields
Vibratory agitation and mating part shapes
Colloidal self-assembly [106]
1.7 Parallel Assembly
Unlike the sequential process, parallel assembly is the assembly of more than
two devices at the same time. As a comparison, batch fabrication is widely
used in microelectronic where the same parallel concept is applied for silicon
chip fabrications. Parallel assembly avoids the drawbacks of a sequential one
for small devices: time consuming and low throughput. The drawback of parallel assembly is less flexibility compared to sequential assembly. An example
of parallel microassembly with electrostatic force fields is given in [22].
1.8 Conclusions
These different strategies for microassembly are logistic approaches for assembly
(we have not discussed technology yet). It is now essential to focus on
the ways to perform the assembly tasks: feeding, positioning, pick, and
place. More specifically, we intend to focus on microgripping. Therefore, the
following chapters are devoted to the forces acting at the envisaged scale
(10 µm to 10 mm) and the related handling principles.
2
Classification of Forces Acting
in the Microworld
2.1 Introduction
When downscaled, volumic forces (e.g., the gravity1 ) tend to decrease faster
than other kinds of forces such as the capillary force or the viscous force.
Although they still exist on a macroscopic scale, these forces are often negligible (and neglected) in macroscopic assembly. A reduced system is consequently
brought face-to-face with the relative increase of these so-called surface forces.
According to the literature on microassembly, these forces are mainly the electrostatic forces, the van der Waals forces, the liquid bridge (also called capillary or surface tension) forces, the forces due to the mechanical clamping
(contact forces) and deformation (pull-off forces), and viscous drag. The term
surface force is misleading since all these forces does not really depend on the
square of the characteristic length. Nevertheless, this term conveys the idea
that these forces decrease more slowly than the weight, which leads to some
cut-off sizes below which these forces disturb the handling task because they
generate the sticking of the microcomponent to the tip of the gripper (the
weight is no longer sufficient to overcome them and ensure release). There are
several ways to tackle this problem: These forces can be reduced, overcome, or
exploited as a gripping principle. The choice will be different according to the
manipulation strategy (see Fig. 2.1): The parameters (materials, environment,
geometries) will be chosen to maximize the force used as a gripping principle
(for example by choosing hydrophilic materials in a manipulation based on
the capillary force) and to minimize the disturbing forces (use of hydrophobic materials in a manipulation based on a mechanical gripper). This chapter
presents some general classifications of the forces according to their range and
introduces the most often cited forces in microassembly literature.
1
From one point of view, inertia forces also involve the mass of the component, but
the possible high dynamics at small scales compensate this effect, as illustrated
by the dynamical release proposed in [77].
10
2 Classification of Forces Acting in the Microworld
Forces
Gripping principles
Fig. 2.1. Forces and gripping principles: The force underlying in a gripping principle
must be maximized, all the others should be decreased
2.2 Classification Schemes of the Forces
According to Lee [118], we go over the first simplified classification of the
different forces in four main categories:
•
•
•
•
Gravity, with an infinite range
Electromagnetic force, with an infinite range
Weak force, with a range smaller than 10−18 m
Strong force, with a range smaller than 10−15 m
These last two forces are outside the scope of this work due to their very
short range (inside the nucleus). Electromagnetic forces represent the source
of all intermolecular interactions and their influence can be combined to that
of gravity in some phenomena such as the rise of a liquid in small capillaries.
The interaction between atoms, molecules, and solids is characterized by
the following:
•
•
Chemical forces and covalent bonding, with a range over the order of an
interatomic separation (typically 0.1–0.2 nm)
Coulomb force and ionic (or partially ionic) bond
Moreover, the interaction between microscopic bodies also depends on the
Lifshitz–van der Waals (VDW) forces, which can be classified into four categories:
•
Dispersion forces, also called London forces [120], are due to a Coulomb
interaction. They represent one third of the Lifshitz–van der Waals forces,
are long range (more than 10 nm), can be attractive or repulsive, and act
between all atoms and molecules, even between neutral ones. These forces
are nonadditive, which means that the interaction between two molecules
is affected by the presence of other bodies. The interaction energy of the
dispersion forces decreases as a function of the separation distance to the
sixth power ( r16 )
2.2 Classification Schemes of the Forces
•
•
11
Orientation forces, also called Keesom forces, coming from the interaction between two permanent dipoles. Their energy also depends on the
separation distance as r16
Induction forces, also called Debye forces, due to the interaction between
a permanent dipole and an induced dipole, with an energy decreasing as
1
r6
•
Retardation forces, described by Casimir and Polder, due to the nonnegligible propagation time of the electromagnetic wave between the
dipoles when their separation distance becomes higher than typically
10 nm. Because of this propagation time, the relative orientation of the
dipoles are less favorable and the interaction energy decreases faster than
for the other terms ( r17 )
A detailed description of these four terms can be found in [118] (Table 3, p10)
or in [88]. At this stage of reading, it seems that the fast decrease of the van der
Waals with the separation distance put them aside as far as microsystems are
concerned. However, a more subtle investigation shows that this decrease complies with another power law in the case of two macroscopic bodies interacting
with each other [89]. Still mentioned in [118], the Coulomb and Lifshitz–van
der Waals forces are not sufficient to explain the adhesion between two solids:
the molecular interactions (also called donor–acceptor interactions by physicists or acid–base interactions by chemists) also play a role in adhesion, but
as their range is limited to the interatomic separation (typically smaller than
0.3 nm), we will not consider them in what follows even if a more detailed
study concerning the close contact of two bodies should probably involve
their effects. Finally, we cannot conclude this section without mentioning the
role of capillary forces [48], [80]. These forces play an important role in a lot
of surrounding phenomena and applications: They allow children to build up
sand castles and everyone to collect the crumbs more easily, provoke adhesion
between microcomponents, cause reliability failure in MEMS2 applications
[104, 125, 176, 179, 184], and are of the utmost importance in microassembly. As a first conclusion, we propose the schematic summary presented in
Table 2.1.
Table 2.1. Forces summary according to the interaction distance
Interaction distance
Up to infinite range
From a few nm up to 1 mm
>0.3 nm
0.3 nm < separation distance <100 nm
<0.3 nm
0.1–0.2 nm
2
Micro Electro Mechanical System.
Predominant force
Gravity
Capillary forces
Coulomb (electrostatic) forces
Lifshitz–van der Waals
Molecular interactions
Chemical interactions
12
2 Classification of Forces Acting in the Microworld
To make this first classification easier to use from a mechanical point of
view, we have proposed [112] a different classification, making the distinction
between forces at contact (forces including deformations – JKR, DMT, and
related indicators,3 interaction energy of two bodies, and friction) and forces
at distance (surface forces including van der Waals forces, electrostatic forces,
and capillary forces). This classification is valid for gaseous environment. On
the other hand, the case of immersed environments is tackled in [64].
2.3 Conclusions
The problematics of microforces has already been described by several authors.
Maybe the most cited surveys in the microassembly literature are the works
given in [28], [58], and [89], which summarize the most important forces acting when dealing with microparts: the electrostatic forces, the van der Waals
forces, the capillary forces, the gravity forces, and the viscous forces. Although
the way these forces are involved in microassembly is not completely understood yet, it is now well established by the scientific community that these
forces are no longer negligible when manipulating and assembling parts within
the size of 0.1 mm and smaller. These effects are also experimented by a lot
of industrials involved in the handling of components of watches or mobile
phones [136]. Let us note that these disturbing side-effects are not limited
to assembly but are also encountered in manufacturing by microstereolithography: The breakdown of small mechanical structures due to the collapsing
induced by the capillary forces (the forces arise from the presence of a rinsing
liquid after the polymerization phase of the process) is reported in [184].
Some authors suggest to assemble component in a liquid environment:
Gauthier [64] has recently explained how the above mentioned forces were
decreased in water4 . This current research field on working in liquid media is
beyond the scope of this book. In Chap. 3, it will be focused on the way these
forces have been used through the literature as handling principles.
3
4
These models are extensions of the Hertz model, which takes adhesion into consideration. The Johnson–Kendall–Roberts model is more adapted for high adhesive or low stiffness contacts while the Derjagin–Muller–Toporov model is more
adapted to low adhesive or stiff contacts.
Capillary forces are totally suppressed since there is no longer a liquid/gas interface, van der Waals forces are decreased because the so-called Hamaker constant
(which is proportional to the force) is usually smaller in liquid environments, and
electrostatic forces are decreased since the dielectric constant is 80 times larger
for water than for vacuum.
3
Handling Principles for Microassembly
3.1 Introduction
Theoretical classifications of forces acting at the considered scale have been
presented in the previous chapter. Now, we summarize the handling principles
which have been proposed in literature in order to transform these forces in
technological solutions. As a lot of different gripping principles exist, it has
been decided to put forward (1) a first overview (based on a compilation of
existing [2, 141] and own [113, 177] states of the art), (2) an own classification
scheme, and (3) the comparison scheme proposed by Tichem et al. [169], used
at the end of the chapter.
3.2 Presentation of Gripping Principles
A first overview of the scientific literature on the gripping principles (not
particularly the “micro” ones) leads to Fig. 3.1.
Let us now enumerate the most common principles:
1. The friction based gripping using miniaturized tweezers [2, 102] (this is
probably the most widespread gripper in industry, together with the vacuum gripper presented in what follows); to illustrate the mechanical gripper based on the tweezers principle, we present a microgripper with two
fingers actuated by piezoelectric bimorphs (Fig. 3.2). The whole gripper
is packaged like an electronic chip. This is an example of “plug and use”
concept developed by the LAB (Laboratoire d’Automatique de Besançon,
[2]). The fingers have been manufactured by LIGA process1 . Figure 3.2b
1
“The LIGA process was developed at the IMT (Institute of Microstructure Technology), in the early eighties under the leadership of Dr. W. Ehrfeld. LIGA
is an acronym standing for the main steps of the process, i.e., deep X-ray
lithography, electroforming, and plastic molding (LIGA means Lithographie–
Galvanoformung–Abformung). These three steps make it possible to massproduce microcomponents at a low-cost,” Source: http://www.fzk.de.
14
3 Handling Principles for Microassembly
+++
----(a)
(b)
(f)
(c)
(g)
(k)
(h)
(l)
(d)
(i)
(e)
(j)
(m)
(n)
Fig. 3.1. Several gripping principles: (a) Tweezer or friction based gripper; (b)
Form closure gripper; (c) Vacuum gripper; (d) Magnetic gripper; (e) Electrostatic
gripper, (f ) Push–pull gripper; (g) Capillary or surface tension gripper; (h) Ice
or cryogenic gripper; (i) Bernoulli gripper; (j) Air cushion handling system; (k)
Standing waves gripper; (l) Squeeze film gripper; (m) Optical gripper, and (n) van
der Waals gripper. (Drawings a,b,c,e, and i are taken from [168], Copyrights CIRP.)
(a)
(b)
Fig. 3.2. The plug and produce concept developed by the LAB: the microrobot on
chip (MOC). (a) The two-finger gripper on chip; (b) gripping of Φ 0.2 mm watch
components (Courtesy of Joël Agnus, Laboratoire d’automatique de Besançon.)
3.2 Presentation of Gripping Principles
15
Table 3.1. Properties of the two finger grippers: the gripping amplitude is the
amplitude of the motion in the plane of the fingers, the insertion amplitude is that
of the motion perpendicular to this plane
Gripper
Actuation
Fingers
Gripping
Tip
Parameter
Piezoelectric bimorphs
Length
Material
Gripping amplitude
Insertion amplitude
Gripping force
Insertion force
Length
Height
Value
Unit
25
Ni
320
200
80
30
1
0.3
mm
10−6 m
10−6 m
10−3 N
10−3 N
10−3 m
10−3 m
Base cantilever
Open
P+dP
200 – 400 µm
Close
Micro container
(a)
Rubber
Silicon
P
(b)
Fig. 3.3. Example of form-closure gripping of a microbe [134]. (a) Schematic view
of cage operation: The cage opens and approaches the microbe, then closes and
captures the microbe; (b) Pneumatic actuation: Flexure of the rubber membrane
by pressure causes the fingers to tilt, creating an opening for object entry ([134],
©1999 IEEE)
shows the gripper handling small watch components (the typical diameter
of the gear shaft is about 0.2 mm). The parameters of the gripper shown
in Fig. 3.2a are summarized in Table 3.1.
As a new development of this kind of gripper, an automatic tools changer
has been proposed in [39] order to make this gripper suitable for microassembly in SEM2 environment.
2. The form closure gripping This principle is presented in [134], who uses
it to handle sensitive elements like microbes. The working principle is
illustrated in Fig. 3.3.
3. The vacuum gripper [45, 51, 163, 143, 157, 189, 190]. The pressure difference between the ambient atmosphere and the “vacuum” generated inside
2
SEM: scanning electron microscope (the sample has to be put in a vacuum chamber).
16
3 Handling Principles for Microassembly
Hollow-bored sonotrode
Vacuum
Fv
Fsf
Fsf
Levitated component
Fig. 3.4. Example of vacuum gripper: A hollow-bored sonotrode pulls the component upwards (thanks to the vacuum suction) and repels it downwards (thanks to
the squeeze film effect) [189] (Courtesy of Fraunhofer IPT.)
the gripper can be used to pick up microcomponents. Such tools are widespread in industry and an example of vacuum gripping tool can be found
in [190]. It consists in a glass pipette and a computer-controlled vacuum
supply. Because of the adhesion forces, pick operation and place operation have antagonist demands: The first one requires a large tip diameter
while the latter needs a small one. Thus, for each size of component, there
is an optimal diameter for the pipette tip. For instance, when handling
a 80–150 µm sized object, the best results were obtained with a tip size
ranging from 25 to 50 µm, which is about 25–50% of the object size. This
glass pipette is able to perform pick and place operations of 50–300 µm
sized metallic and nonmetallic particles with a success rate of about 75%.
The pump works with a 6 bar pressure supply and a voltage of 6 V. The
maximum output vacuum is −0.86 bar and the maximum output pressure
is 6 bar. The vacuum gripper presented in Fig. 3.4 is a combination between an attractive suction force and a repulsive squeeze film effect used to
prevent the component from touching the gripper (see later, “Ultrasonic
levitation,” p.23).
In absence of this repulsive squeeze film, the high contact forces during
the picking phase can lead to the formation of cracks at the surface, as
illustrated in Fig. 3.5.
4. The magnetic gripping, and particularly the magnetic levitation in which
the force comes from a magnetic field generated by magnets (Fig. 3.1d).
Three different types can be used: permanent magnets, electromagnets,
or superconducting magnets. The use of electromagnetic levitation is limited to materials with high electrical conductivity and to low-temperature
applications [96, 128, 140]. Recently, magnetic levitation has also been
3.2 Presentation of Gripping Principles
17
Vacuum nozzle
Crushed circular
shape
Propagated crack
Fig. 3.5. The mechanical damage at contact (Courtesy of Assembléon.)
Dm
Lm
V
D
Lp
Dp
Fig. 3.6. The electrostatic handling (Reprinted with permission from [151].
Copyright 2003, American Institute of Physics.)
used by [27] to propose new force sensor for microassembly, with a stiffness of about 0.02 N m−1 .
5. The electrostatic gripper , including the electrostatic levitation [52, 53,
56, 58, 83, 171]. A first example of a micromanipulation task driven by
the electrostatic effect is presented in [151] and depicted in Fig. 3.6. The
authors proposed to use the electrostatic force in order to handle (=catch,
move, release) a spherical particle near a substrate plate. All the objects
(gripper, sphere, and substrate) are conductive.
The principle in this case is to use the adhesion force to perform the gripping task and to impose a detachment voltage for release. The adhesion is
modeled according to the JKR model [93, 94] and the electrostatic force
18
3 Handling Principles for Microassembly
Electrodes
Top view
Side view
D
Fig. 3.7. Electrodes of the electrostatic gripper (Courtesy of J. Hesselbach, TU
Braunschweig, [83].)
is computed by the boundary elements method. The effects of roughness,
the use of nonconductive materials, the electric discharge, and the tunneling current have not been taken into consideration. The results are given
as a function of the geometry ratio, and seem to be independent from the
actual size.
A second example, a micromanipulation driven by the electrostatic force
is presented in [83] and shown in Fig. 3.7 (data are given in Table 3.2).
The authors described the handling of insulating objects: the gripping,
the centering, the moving, and the releasing tasks are performed by controlling the electrostatic force. It turns out that the humidity rate greatly
influences the reliability of this kind of electrostatic gripper. Above 65%
RH3 , objects with a size >400 µm cannot be gripped, even at maximum
voltage (1, 200 V). As the adhesion influences the releasing task, the reliability of this kind of gripper becomes problematic.
As far as the electric levitation is concerned, different kinds of particles
can be manipulated such as, for example, conductive, semiconductors, and
dielectric materials. A distinction can be made between the following:
(a) Electrostatic. Static electrical fields can be used to levitate uncharged
small particles by induced polarization of the sample. This technique is suitable only for polar liquids and low temperature fields. At
high temperature the static charges are not maintained and gradually
degenerating over time. On the other hand relatively large particles
can be levitated [56, 92].
(b) Electrodynamic. The particles are charged and held stationary using
a combination of static (DC) and oscillating (AC) electric fields. This
technique is able to suspend and levitate small charged particle. The
disadvantages lie in the poor particle stability and in the limited particle size (up to 200 µm) [11].
3
RH = Relative humidity.
3.2 Presentation of Gripping Principles
19
Table 3.2. Properties of the electrostatic gripping [83]
Part
Substrate
Parameter
Profile
Roughness
Material
Handled components
Spheres
Material
Diameter
Cubic valve flap
Material
Characteristic size
Working conditions Voltage
Humidity rate
Electrode
Inner diameter
Outer diameter
Gripping force
(an estimation)
Value
Unit
Flat
“smooth”
Pyrex wafer
Glass
100–800
Insulator
80
300–1200
<65
120
560
50
10−6 m
10−6 m
V
%
10−6 m
10−6 m
µN
α
L
h
R
l
D
(a)
(b)
(c)
Fig. 3.8. The push–pull manipulation. (a) Geometrical data; (b) example of tip:
CS21 (Courtesy Mikromasch.); (c) example of tip: NSC12 (Courtesy Mikromasch.)
Finally, let us cite two additional devices that are currently being studied
to achieve feeding and centering of millimetric components: Fantoni and
Santochi [57] have developed a electrostatic feeder and Porta and Santochi
[144] are currently developing a centering device based on electrostatic
field gradient (components are attracted toward high gradient area). This
kind of devices already give satisfactory results for conductive parts but
still suffer from a lack of precision as far as nonconductive materials are
concerned.
6. The “push-pull” or adhesion based handling (Fig. 3.8) has already been
described for example in [152]. It consists in pushing small spheres with
the tip of an AFM4 (Atomic force microscope) located on a “plane” surface
(Fig. 3.8a). Adhesive forces can prevent the separation of the tip from the
sphere, leading to “adhesive pulling.” However, in some cases the release
task can be performed, but a detailed study of the phenomena acting at
4
For an introduction to the atomic forces microscopy, see [31]
20
3 Handling Principles for Microassembly
Table 3.3. Properties of the push–pull handling
Part
Substrate
Handled sphere
Cantilever
Tip
Working space
Parameter
Profile
Roughness
Material
Diameter
Material
Length (L)
Height (h)
Tilt angle (α)
Height (l)
Radius (R)
PI Range: X,Y ,Z
AFM Range: X and Y
AFM Range: Z
Value
Flat
Unknown
Glass
50
Polystyrene
350
2
30
10
10
15 × 15 × 15
45
4
Unit
10−6 m
10−6 m
10−6 m
◦
10−6 m
10−9 m
mm3
10−6 m
10−6 m
the interface sphere-tip helps to understand and control this manipulation:
Details of this manipulation are given in Table 3.3. The cantilevers are
commercialized on http://www.spmtips.com/, and two cantilevers have
been used: the CSC21 and the NSC12 (Fig. 3.8b and c).
Another team, the Laboratoire de Robotique de Paris (LRP), has studied
the adhesion based micromanipulation and has proposed related release
strategies (by rolling, scrapping, and imposing high dynamics). A pointer
to their work can be found in [76].
7. The capillary or surface tension based gripper , in which the surface tension forces can be used to get parts stuck to the gripper [12, 13, 26, 72,
108, 109, 112, 138, 153]. [13] used a low viscosity liquid such as ethanol
that evaporates without leaving particles on the part’s surface. Several
parameters intervene in the accuracy with which the part can be handled,
such as the gripper’s shape, the part’s shape, and the difference of size
between the gripper and the component. The so-called centering effect
occurs and causes an orientation of the component at the gripper’s shape.
The lifting force depends on the type of adhesive, the gripper’s surface,
the gripping distance, the adhesive volume, and the materials. The second
example [72] of capillary gripper was used to manipulate small Si plates
of 2 × 2 mm2 .
Authors of [26] suggest to vary the gripper curvature in order to vary the
capillary force, according to the well-known Israelachvili approximation
(18.2). Therefore, as illustrated in Fig. 3.9a, the extremity of the gripper
is made of a deformable membrane whose curvature increases under the
action of a liquid (i.e., this liquid is inside the gripper, and only used to
drive the shape of the membrane. It can be totally different from the liquid constituting the meniscus between the gripper and the component).
Nowadays, the main limitation of the method is its difficulty to be miniaturized. Recently, Pagano et al. [138] has proposed a new design (Fig. 3.9b)
3.2 Presentation of Gripping Principles
21
Actuation volume
First electrode
R
Zero curvature
Curvature = 2/R
(a)
First electrode
Actuation volume
Squeezed EAP
Non squeezed EAP
Second electrode
Second electrode
(b)
Fig. 3.9. Some examples of surface tension based gripper designs. (a) Variable
curvature driven by a liquid; (b) variable curvature driven by an actuated EAP
[138] (Copyrights CIRP.)
where the deformation of the membrane is obtained by using an electronic
EAP5 between two electrodes which can squeeze it to provoke a change of
the curvature of the gripper tip. This principle is still to be implemented.
Saito et al. [153] has proposed to use a gripper with a hemispherical concavity in order to increase the contact conformity between the gripper and
a spherical component (Fig. 3.10a). This allows to pick it up from a flat
plane. To achieve the release, the authors propose to increase the volume
of the liquid, which leads to a force decrease. This principle has still to be
implemented.
8. The ice or cryogenic gripper , also called ice gripper [131, 101, 114, 158]. A
gripper that was developed in the context of an Eureka project, including
the participation of the Swiss Centre for Electronics and Microtechnology
(CSEM), AP Technologies, and Sysmelec, is described in [131]. This
method is based on the adhesive properties of ice to pick up the microparts. The gripper developed by the CSEM6 first sprays a drop of
water onto the object and then it gets close to the object until it touches
it. As soon as there is a contact between the gripper and the object,
water freezes and the component is stuck to the gripper so that it can
be manipulated at will. It should be noted that the gripping strength of
ice is 20–100 times stronger than that obtained with vacuum grippers.
To release the object, the tip of the gripper is simply warmed up to the
phase change temperature of the liquid interface. A prototype was able
to handle components sized between 0.1 and 5 mm, with an accuracy of
1 µm and a rate of 1,000 cycles per hour. The advantages of the cryogenic
gripper are as follows: High adhesion forces, the surfaces of the object are
not damaged during the handling process, the handling process is almost
independent from materials properties, and short pickup and release times
[158]. An example of millimetric component hold by a cryogenic gripper
is given in Fig. 3.10b.
5
6
EAP = Electro Active Polymer.
Source: http://www.devicelink.com/emdm/archive/98/09/tech.html.
22
3 Handling Principles for Microassembly
Gripper
Component
Plane
(a)
(b)
Fig. 3.10. (a) Examples of surface tension gripper based on conformity to increase
the capillary force (Reprinted with permission from [153]. Copyright 2005, American
Institute of Physics.); (b) Example of cryogenic gripper (Courtesy of Defeng Lang,
TU Delft.)
Table 3.4. Examples of phase changes proposed by [114] as gripping principles
Intermediate
Water
Magnetic-rheological fluid
Thermoplastic polymer
Thermoset polymer
Process
Heating–Cooling
Electromagnetic field
Heating–Cooling
UV based principle
A new adaptation of this principle has recently been proposed in [121],
using a submerged gripper. In this case, the liquid environment ensures
a low adhesion between the gripper and the component and a rigid link
between them can be created or suppressed, thanks to the control of a
Peltier element.
Note that the cryogenic gripping principle has been generalized under the
terminology of phase changing gripping [114], whose several examples are
given in Table 3.4.
9. The aerodynamic levitation: according to the direction of the flow, two
different approaches can be used:
(a) Bernoulli levitation. The sample is held below the manipulator through
which air flows downward (Fig. 3.1i). Because of the high pressure
supply, air radially flows between the gripper and the component.
The velocity increase induces a dynamic pressure decrease (Bernoulli
effect), leading to an upwards attracting force on the component
[54, 72, 180].
(b) Air cushion levitation. The sample is held above the manipulator (Fig.
3.1j). Pressurized air flows upwards through several holes that are
drilled all over the gripper and leads to a repulsive levitation force
that counterbalances the weight of the component [66].
3.2 Presentation of Gripping Principles
23
Reflector
z
z
z
+
g
Heavy sphere
n λ2
Stable
Unstable
+
-
Pressure
Velocity
Piston sound source
Force
Fig. 3.11. Levitation of dense spheres in an acoustic standing wave [147]
(Copyrights CIRP.)
Recently, Nyhuis and Fiege [132] have proposed a feeding concept using aerodynamics orientation methods. According to the authors, these
latter fall into two categories: passive orientation methods, where incorrectly oriented parts are rejected by means of aerodynamic baffles, and
active orientation methods, where the workpiece is actively moved into
the desired position.
10. The ultrasonic levitation, divided into the following:
(a) Standing waves levitation. Small components can be levitated in the
pressure nodes of an acoustical standing wave between a vibrating
plate and a reflector (Fig. 3.11). Moreover, because of the pressure
distribution, a radial flow, whose velocity decreases toward the periphery, induces radial centering by Bernoulli effect reaching up to 30% of
the levitation force [14, 38, 40, 43, 68, 74, 86, 99, 148, 186, 188]. The
company Dantec Dynamics7 can be asked for information concerning
an ultrasonic levitator [44].
(b) Squeeze film levitation or near field levitation. The reflector of the
standing wave levitation is replaced by the levitated object (Fig. 3.1g).
Consequently, any weight can be levitated if the separation distance
between the object and the vibrating plate is small enough (see the
typical profile of force as a function of separation distance in Fig. 3.12).
This technique is often referred to as near field levitation according
to several authors [86, 78, 172, 183].
The combined use of vacuum attraction and squeeze film repulsion
has already been illustrated in Fig. 3.4.
7
http://www.dantecdynamics.com.
3 Handling Principles for Microassembly
Levitation force
24
Near-field levitation
Levitation distance
λ/2
λ/2
λ/2
Fig. 3.12. Levitation force as a function of levitation distance [147] (Copyrights
CIRP.)
Beam axis z
Light intensity
i
FD
a
o
FD
FRi
Beam radius r
B
f
FRo
b
(a)
Fig. 3.13. Optical forces acting on a sphere (Reprinted figure with permission from
[7]. Copyright (1970) by the American Physical Society.)
11. The optical or laser gripper [5, 7, 8, 10, 9, 122, 146, 164]. Because of both
beam reflection and refraction, the component undergoes an axial force
that always pushes it forward in the direction of the beam and a radial
gradient force that traps it in the center of the beam. This effect can be
understood as follows for a sphere with higher refractive index than that
of its surrounding medium (water, oil): As depicted in Fig. 3.13 the rays a,
B, and b undergo both reflection and refraction (also called deflection) at
input and output interfaces. Because light carries momentum, the changes
in ray directions cause (and are caused, i.e., the action/reaction principle
3.3 Classification of Gripping Principles
25
is applied) the forces FRi and FRo due to reflection and the forces FDi and
FDo due to deflection. These forces accelerate the sphere in the +z direction
of the beam. For the stronger ray a, FRi and FRo are balanced to first order
while FDi and FDo radially add in the −r direction, leading to a net inward
radial force (the beam axis z represented in Fig. 3.13 is the center of the
beam). For similar reasons, the weaker ray b leads to a radial outward
force of smaller magnitude. Therefore, the sphere is trapped in a stable
way in the center of the beam. Nevertheless, it is preferred [146] to use
this principle in a liquid environment because the condition on refractive
index is respected and because this medium damps the oscillations of
the component. This technique is restricted to very small transparent
dielectric samples. Note that this principle leads to a gripping force that
hardly reaches 1 nN [141], which represents the weight of a 33 µm edge
cube with a density of 2700 kg m−3 (i.e., aluminum component).
12. The gripping based on van der Waals forces has been proposed in [4, 59]
but no prototype was built. Actually, it seems that the adhesion based
gripper probably uses van der Waals forces, combined with surface tension
or electrostatic effects.
3.3 Classification of Gripping Principles
We classify the handling principles according to the way they tackle the surface
force disturbing problematics arising from the downscaling. We distinguish
four strategies as indicated in Fig. 3.14. The first two strategies consist in
downscaling the existing equipment and in proposing some solutions to reduce
or overcome the sticking aspects. The last two solutions are more microassembly oriented in the sense that they are based on strategies taking advantage
of downscaling (typically these principles will probably not be adapted to the
manipulation of conventional macroproducts) or avoiding any contact.
Microassembly
Feeding
Handling & Orientation
With contact
1 - Reduction
2 - Overcoming
Joining
Without contact
3 - Exploitation
4 - No surface forces
Fig. 3.14. Four strategies as far as surface forces are concerned
26
3 Handling Principles for Microassembly
1. Downscaling Approach I: How to Reduce the Surface Forces
Effects?
The first way to deal with these surface forces is a downscaling approach
that consists in performing assembly on a conventional way with downscaled grippers. To avoid or to reduce the sticking effects, the surface forces
effects are reduced by an adapted choice of the manipulation parameters.
To decrease van der Waals forces, it is suggested to
• Decrease surface pressure (=gripping force), to keep the contact area
as small as possible
• Prefer hard materials, to keep the contact area as small as possible
[58]
• Increase roughness [3, 58, 178, 192]
• Control roughness profile [107]
To decrease the surface tension effects, it is proposed to
• Reduce the number of contact points
• Use a microheater to evaporate the water bridge [3]
• Use anti-adhesive, hydrophobic coating [3]
• Handle in a liquid medium [170, 181]
• Work with a low humidity rate [58, 80, 192]
• Work under vacuum
To reduce the electrostatic force, it is proposed to
• Ground conductive materials [59]
• Use materials with a small contact potential difference [58]
• Use conductive materials that does not easily form insulating oxides
[58]
• Ionize the surrounding environment
2. Downscaling Approach II: How to Overcome the Surface Force
Effects?
This second downscaling approach summarizes several techniques proposed by authors confronted with sticking effects during the release task.
These propositions are far from being solutions, because they come from a
narrow point of view and cannot result from an integrated approach. For
example, it has been suggested to work in dry air environment to avoid the
formation of liquid bridges, but the breakdown voltage is then increased
leading to larger electrostatic forces. However, these suggestions represent practical solutions for those who are confronted to sticking problems.
General solutions (independently of the used gripping principle) suggest
the following:
• Use a large probe to pick up the micropart by adhesion and a smaller
probe to release it (a smaller probe reduces the contact area and consequently, the gravity becomes dominant again and the object will stay
in place when removing the small probe) [12, 190]
• Change the pressure by controlling the temperature [3]
3.3 Classification of Gripping Principles
27
• Use or, even better, control the adhesion between the handled component and the substrate
• Improve the previous solution by joining the component to the substrate at the right place (for example by gluing) [21], [12]
• Induce a relative motion between the component and the gripper by
stripping off the component on a sharp edge [190]
• Use vibrations to perform the release task [22]
• Inject gas: A small puff of gas pushes the object while removing the
gripper [12, 190]
• A similar approach to the vibration method is the dynamic approach
that consists in communicating to the gripper an acceleration bigger
than that the sticking force can communicate to the component [75,
149]
Gripping principle related solutions consist in the following:
• Using a positive pressure (vacuum gripper) [190]
• Modifying the relative orientation of the component and the gripper
[97] (when dealing with van der Waals forces, [59, 107])
• Destructing the gripping mechanism (for instance, with a gripper using
the surface tension force to pick an object, the object can be released
by heating the gripper and evaporating the adhesive liquid. Another
example is the ice microgripper described in [131])
• Several propositions that still have to be validated, related to the capillary force: see later
3. Microworld Driven Approach I: How to Take Advantage of the
Downscaling?
Although adhesion can be reduced or overcome in some cases, it is sometimes interesting to exploit the surface forces effects in order to perform
the handling task. This approach, made possible by the interesting ratio surface forces/component weight, is followed by several authors who
propose a lot of gripping principles related to microhandling:
• Surface tension effects
• van der Waals force
• Cryogenic gripping
• Laser gripping
• Bernoulli effect
• Handling in a fluid medium [187]
4. Microworld Driven Approach II: Handling Without Contact
Finally, these unavoidable difficulties lead to a fourth approach: If the
gap between the component and the gripper can always remain larger
than the cut-off lengths of the physical principles leading to adhesion,
the handling task can be performed without paying attention to the
surface forces (typically, this cut-off length is 10 nm for van der Waals
forces [89]). The advantages of this noncontact handling approach can
28
3 Handling Principles for Microassembly
be summarized as follows. Some advantages cited are not limited to the
handling of microcomponents.
• Surface forces can be completely neglected
• The friction effect is drastically reduced, which enables high resolution
and accuracy motion devices [96]
• Handling of tricky (fragile, freshly painted, sensitive, or micron-sized
structured surfaces) components is made possible because high local
contact pressure by direct mechanical contact is avoided [189]. Handling of nonrigid products is also possible due to the field of force
[54]
• Contamination from and of the end-effector (in food handling [54] or
in presence of lubricant [140]) can be totally avoided
• In materials science, measurements of some physical properties are
allowed avoiding undesired contamination from the container and eliminating wall-driven heterogeneous nucleation [63]
The literature review highlights five distinct levitation techniques:
(a) Magnetic levitation
(b) Electric levitation
(c) Optical levitation
(d) Aerodynamic levitation
(e) Ultrasonic acoustical levitation
Once the object is levitated, another principle can be used for the horizontal displacement of the component ([66] therefore uses the electric field)
or the whole levitation system can be moved. The control of the levitation
height should also give an additional degree of freedom when handling the
component.
Additional information on this topic can be found in [113, 177].
3.4 Comparison between Gripping Principles
To compare all these principles, a classification scheme is required. A modified
version of an already proposed scheme [169] takes the following aspects into
account:
•
•
•
•
•
•
•
•
•
Material type of the handled component (for example, dielectric properties,
porosity)
Surface properties
Specific grip force
Force control aspects
Remarks on possible limitations
Accuracy
Sensitivity to adhesion
Environment
Downscaling law
3.5 Conclusions
•
•
29
Applications field
Handled components
Let us emphasize the fact that the required information is not available in
all cases, leading to empty cells in Tables 3.5, 3.6 and 3.7. Note that all the
aspects related to the component are listed in the norm DIN 325638 (Production equipment for microsystems–System for classification of components for
microsystems).
Other classification schemes could be based on different criteria. For example, the actuation type can help to distinguish different miniaturized tweezers.
Moreover, a lot of grippers based on several technologies have been designed:
piezoelectric actuated [29] microgrippers, thermally actuated grippers [119],
SMA9 actuated grippers [17], piezoelectric bimorph gripper [2]. Other grippers
already integrate a force feedback: gripper with an integrated piezo-resistive
force sensor in the range of 1 µN [191], gripper with an attached strain gage
[97], AFM force measurement in the range of 1 nN.
3.5 Conclusions
In conclusion of this chapter on forces and gripping principles, we note that
not all presented principles can be used to handle components with a size in
the range between 10 µm and 1 mm. For example, the laser gripping principle
must be put aside because of the low force it develops (0.7 nN). The cryogenic
and the Bernoulli principles seem too “exotic” and their study does not help
to understand the underlying phenomena mentioned in the microassembly literature. As far as the other principles are concerned, we preferred to focus on
microworld driven gripping principles (van der Waals, electrostatic, surface
tension forces). Note that the vacuum gripper is an interesting alternative
to the downscaled mechanical gripper (it allows the handling of plane components, which is impossible with a two-finger gripper if the lateral faces of
the components are not high enough). Nevertheless, the surface tension based
gripping seems to be more promising (for example, in terms of downscaling
laws) as presented in Chap. 6. Moreover, the study of the capillary force is
a good way to enter the microworld because it is involved at many scales
from the nanoscale (capillary condensation) up to the submillimetric scale.
This will be discussed again in the chapter devoted to the capillary forces.
8
9
This norm takes the following parameters into account: geometrical shape, length,
width, height and mass of the microcomponent, the shape (for example, a hole)
and profile (for example, convex surface) of the surface available for handling, the
number of available faces available for handling [...], material characteristics and
sensitivity to environmental conditions (for example, UV radiation), mechanical
properties, roughness characteristics (Ra ), physical properties, clean room related
aspects.
Shape Memory Alloy.
Material limitation
None
Surface properties Force
0.1 µN [98],
1 mN [69],
2–5 N [141]
None
None
Condition on the
refractive index [7]
11 Standing
waves
12 Squ. film
13 Optical
Not limited
0.1–10 pN
150 µN [63]
5 Electrostatic Better control with Rough surface
Not limited
conductive materials is a drawback [167]
6 Push-Pull
7 Capillary
Hydrophilic,
oleophilic [182]
8 Cryogenic
None [167]
1 Nmm−2 [167]
9 Bernoulli
Not too compliant,
0.1–10 N
not too porous
[54]
10 Air cushion None
2 Form closure None
3 Vacuum
Limits on
Contact may
porosity [167]
cause damage [167]
4 Magnetic
Ferromagnetic [167]
Not limited
Principle
1 Friction
Table 3.5. Comparison between the gripping principles
Remarks
At least 2 accessible
surfaces [167]
Euler instabilities (plate
does not remain parallel
to the nozzle surface)
Horizontal instabilities
due to the absence of
centering effect [66]
Axial stabilization [16]
Orientation systems [15]
Centering effect [43]
Centering effect [79]
Only planar objects
Damage due to
radiation pressure[10]
Reduced ergonomy
due to the reflector
External pressurized
air supply is needed
External pressurized
air supply is needed
Open (closed) loop for
transfer (positioning)[140]
Disturbing forces
RH should be
due to triboelectrification <40–60%
Control
Important for
fragile parts [167]
30
3 Handling Principles for Microassembly
Accuracy in the
gripper part relation
Not simultaneous
contact of fingers
may introduce errors
[167]
8
9
10
11
12
13
Cryogenic
Bernoulli
Air cushion
Stand. waves
Squeeze film
Optical
6 Push-Pull
7 Capillary
5 Electrostatic
4 Magnetic
Environment
No
No
No
No
contact
contact
contact
contact
Not in vacuum
Not in vacuum
Not in vacuum
Not in vacuum
Should be in a liquid
Not in a liquid
Depends on contact area
Not in vacuum
and surface properties
Blow away is a release method [167]
With or without contact
Sensitivity
to adhesion
Triboelectrification
may be a problem
Adhesion is the working principle
Automatic centering [141] Adhesion is the working principle
20 µm [182]
2 Form closed
3 Vacuum
Relatively
inaccurate [167]
1 Friction
Principle
Table 3.6. Comparison between the gripping principles
L [141]
L2 [141]
(limit = max
magnetic induction)
L2 [141]
(limit = breakdown voltage)
L2 [141]
(limit = suction force)
L2 [141]
(limit = strength)
Downscaling law
3.5 Conclusions
31
Applications
13 Optical
12
10
11
8
9
Crystals manipulation [10]
Handling of components with only one
accessible surface
Ball placement for a watch ball bearing
Cryogenic
Clamp before machining
Bernoulli
Food handling [54]
Semiconductor devices handling [18, 123]
Air cushion Microparts transfer system [66]
Stand. waves Crystals manipulations [63]
Small part handling in a commercialised
device [160]
Squeeze film Wafers transfer [148]
7 Capillary
6 Push–Pull
2 Form closed
3 Vacuum
Typical in SMD components handling
4 Magnetic
Containerless crystal growth [128]
Wafer transfer [140]
5 Electrostatic Aligning, positioning and transferring
components [56]
Principle
1 Friction
Planar objects (Φ 200 mm wafers) [148]
Bakelite plate (90 mm × 65 mm2 , 8.6 g) [78]
Spheres (5 µm), bacteria, cells [5]
Silica particles (25 nm–10 µm) in water [8]
Crystals seeds (1–25 µm) [10]
Aluminum blocks (65–130 g), jelly blocks (35–175 g) [54]
Flat disk (R up to 150 mm and m up to 2 kg [180]
Millimeter sized polymer plate (PMMA) [66]
Φ 3 mm spherical crystals [63]
Microgears (wheel Φ 3 mm, shaft Φ 0.8 mm) [85]
0.3 mm and 0.5 mm diameter balls [112]
Water, ice, sugar, quartz [128]
8 in. wafer [140], 4.2 mm × 4.2 mm2 components [182]
Metallic cylinders (Φ 0.25 − 1 mm and 1–4 mm length) [56]
Glass spheres (Φ 100–800 µm) [83]
Not suited for sensitive IC components [141]
Typically micrometric sized components
like small spheres (polystyrene, glass)
2 × 2 mm2 Si components [70]
Handled object
2.7 µm polystyrene spheres, red-blood cells,
protozoa [98], Φ 1 mm glass balls [165]
Microbe [134]
Table 3.7. Comparison between the gripping principles (continued): examples of applications fields and handled components (SMD
states for Surface Mounted Device)
32
3 Handling Principles for Microassembly
3.5 Conclusions
33
Electrostatic forces are probably a good candidate to develop microhandling
tools, but we decided to put them aside because of the electromagnetic perturbations they induced on microcomponents, which often include electronics.
Additionally, since they are long range, electrostatic force are suspected to disturb the manipulation task in case of electrostatic actuation of a two-fingered
gripper. Therefore, despite its theoretically less favorable response time, thermal actuation is currently envisaged as an alternative to actuations using a
high electric field.
In what follows, a preliminary theoretical study has been led to assess the
role of capillary forces.
The first reason for a theoretical modeling is that even if these effects can
be measured with adapted force sensors such as AFM, they cannot be separated from other contributions (all forces contribute to the global measured
force amount). A model is therefore necessary to discriminate each contribution. The second justification is that we intended to investigate the influence
of a large number of parameters before setting up a experimental validation.
Consequently, these effects have first been assessed by simulation. The third
reason is that a simulation tool could help to better design microgrippers.
4
Conclusions
As it has been shown throughout this first part, many current researches focus
on grippers miniaturization. At scales larger than typically 1 mm, the reference
gripping technique for components (typically SMD1 components) is the vacuum gripping. This technique is also applied for submillimetric applications;
however, it seems to reach its limits2 .
Therefore, it turns out from this literature review that at the considered
scale (0.1–1 mm), effects due to both gravity (depending on the cube of the
characteristic size of the component) and vacuum suction (depending on the
section area of the gripper, i.e., on the second power of the characteristic size)
become overcome by capillary effects. Indeed, although the capillary forces are
often referred to as surface forces, they actually vary linearly as a function of
the characteristic size, as it will be shown in Sect. 6.6. Consequently, scaling
laws are more favorable to a capillary gripper than to a vacuum gripper.
Moreover, the miniaturization will reinforce this advantage.
Let us note that both capillary and vacuum grippers are well adapted to
the picking of flat components (with a low aspect ratio), which is not the case
of the two-fingered gripper, requiring opposite faces to grip the component.
Another advantage of the capillary gripper is the contact damping due to
the liquid film (see later Fig. 11.5). At the contrary, the vacuum gripper can
lead to cracks at the component surface, as indicated in Fig. 3.5.
A final argument concerning the good cycle time of the physical principle
will be given in Chap. 12, showing that the performances of a capillary gripper
can be assumed to be better than the typical picking time of the SMD assembly
machines [174].
1
2
A Surface Mount Device is a component mounted on the surface of a printed
circuit board.
However, this limit is not clearly stated, a few tens of millimeters can be suggested
as a lower limit.
Part II
Modeling and Simulation of Capillary Forces
5
Introduction
Beside van der Waals and electrostatic forces, surface tension effects are often
cited in the microassembly literature as being of utmost importance. The most
frequently cited effects are the influence on sticking during the release task
[58], the negative effect on the reliability of microswitches that can collapse by
capillary forces [126], and the positive effect of surface tension when chosen
as gripping principle [70, 100].
This second part will consequently focus on the description of the underlying parameters of capillary and the modeling of the surface tension effects
in the simulation of the handling task. As the problematics of surface tension effects is far from being easy, we will first introduce a few parameters
in Chap. 6 (those involved in the justification of the capillary as a suitable
gripping principle: surface tension, Young-Dupré equation, Laplace equation).
After a review of the ways to take these surface effects into account in handling
(Chap. 7), we will present the static simulation at constant volume (Chap. 8).
Two kinds of capillary force models, namely the energy based method and
the Laplace equation based method, are compared and their equivalence is
proven in Chap. 9. The application of these results to a watch ball bearing
is presented in Chap. 10. We will then expound the advantages but also the
limitations of this approach and introduce in Chap. 11 a second set of parameters and phenomena (static contact angle hysteresis, dynamic contact angle,
inertial effects of the meniscus, influence of surface roughness, and surface
heterogeneities (impurities)). A study of the limits of the proposed method is
presented in Chap. 12. Finally, models are developed to predict the separation distance between a gripper and a component that provokes their adhesion
and to predict the separation distance causing the rupture of the liquid bridge
linking them with each other (Chap. 13).
6
First Set of Parameters
6.1 Introduction
The reader will find in this chapter a brief introduction to the concepts of
surface tension, contact angle, and Laplace equation, which will be used to
model the liquid bridge and compute the capillary forces as it will be explained
in Chap. 7. Based on these concepts, some arguments will be given at the end
of the chapter to justify a priori the use of surface tension as a gripping
principle.
6.2 Surface Tension
A first representation of the three states of matter is to consider that a solid is
characterized by a volume and a shape and conversely, a gas does not have any
own volume or own shape. A liquid is somewhere in between, having its own
volume but no own shape. Usually, if a liquid is not contained, it spreads out.
However, when we look at soap bubbles or small water droplets, we observe
that they behave as if their surface was an elastic membrane, characterized
by a “surface tension” that acts against their deformations.
This surface tension is presented in a didactic way in [46]. The classical
explanation of this phenomenon is based on the fact that in a liquid, the
mutual attraction between the molecules overcomes the thermal agitation. All
molecules inside the liquid are equally attracted by their neighboring molecules, but the molecules located at the interface between the liquid and (for
example) a gas suffer from a so-called “attraction default” (see Fig. 6.1).
From a thermodynamic point of view, the energetic state of a molecule
near the surface is less favorable, leading to a global shape of the liquid that
minimizes the interface area (this explains why wet hair stick together). This
introduces the concept of surface energy (or surface tension), which has the
dimensions of an energy by surface unit (J m−2 ). The mechanical point of
view considers the surface tension as a tensile force by length unit (N m−1 ).
42
6 First Set of Parameters
Fig. 6.1. Illustration of the attraction default
The surface tension is denoted by γ and its numerical value depends on the
molecular interactions: In most oils, the molecular interaction is van der Waals
interaction, leading to quite low surface tensions (γ ≈ 20 mN m−1 ). As far as
water is concerned, because of the hydrogen bonding, the molecular attraction
is larger (γ ≈ 72 mN m−1 ). Typical values for conventional liquid range from 20
(silicone oil) to 72 mN m−1 (water at 20◦ C). For example, the following values
for ethanol (23 mN m−1 ), acetone (24 mN m−1 ), and glycerol (63 mN m−1 ) are
given in [46]. Surface tension is an important parameter in the perspective of a
downscaling of the assembly equipment, because the force it generates linearly
decreases with the size while the weight decreases more quickly. While surface
tension has been pointed out as being one of the disturbing effects in MEMS
(stiction problems [104, 125, 184]), other uses have been positively considered
[20, 81, 117, 55].
6.3 Young–Dupré Equation and Static Contact Angle
In the subsection 6.2 we saw that an interface between a vapor and a liquid
could be characterized by an interfacial tension, denoted by γ and expressed
as an energy by surface unit or as a force by length unit. Interfacial tensions
can also be defined at the interfaces between a liquid and a solid (γSL ) and
between a solid and a vapor (γSV ). Typical values of γSV are given in [133]:
Nylon (Polyamid) 6.6 (41.4 mN m−1 ), PE High density (30.3–35.1 mN m−1 ),
PE Low density (32.1–33.2 mN m−1 ), PET (40.9–42.4 mN m−1 ), PMMA
(44.9–45.8 mN m−1 ), PP (29.7), PTFE (20.0–21.8 mN m−1 ).
The surface tension γ will indifferently be denoted by γLV . When a droplet
is posed on a solid substrate (see Fig. 6.2), the liquid spreads out and we can
distinguish three phases (vapor, liquid, and solid) separated by three interfaces
that join one another at the triple line, also called contact line.
At this triple line, the liquid–vapor interface makes an angle θ with the
substrate. If the contact line is at equilibrium, θ is called the static contact
angle, which is linked to the interfacial tensions by the Young–Dupré equation
[1, 89]:
γLV cosθ + γSL = γSV .
(6.1)
6.4 Laplace Equation
g
43
LV
Contact line
liquid
vapor
θ
g
g
SL
SV
solid
Fig. 6.2. The Young–Dupré equation
∆A
gas
liquid
θ
solid
Fig. 6.3. Small displacement of the contact line
This equation can be written immediately by considering the balance of the
forces acting on the contact line. A second approach is based on the fact that
at the equilibrium the energy must be extremal and that any displacement of
the contact line (see Fig. 6.3) leads to an energy variation equal to zero:
∆G = ∆A(γSL − γSV ) + ∆AγLV cosθ
(6.2)
lim∆A→∞ ∆G
∆A = 0,
where ∆A and ∆G state for the variation of interface area and energy
during the considered displacement. We will first consider that this contact
angle is constant as it is determined from the interfacial tensions that depend
on materials in presence. This assumption will be used in the static simulation
with constant volume.
Measured values of the contact angles will be given in Sects. 17.9.5
and 19.6.
6.4 Laplace Equation
Because of the surface tension, there exists a pressure difference across the
interface between a liquid and a gas. In the case of a soap bubble for example,
the pressure inside the bubble is bigger to compensate the outside pressure and
to overcome the tension effect. In a more general case, the pressure difference
44
6 First Set of Parameters
dS
Pout
u
v+dv
v
u+du
R1
R2
dq1
Pin
dq2
Fig. 6.4. Surface element of an interface between a liquid and a gas
is linked to the curvature of the interface according to the Laplace equation
that will now be established. Let us assume a curved surface S at equilibrium
on which we draw a net of coordinate curves u and v that intersect one another
with an angle of 90◦ . Let us now consider a surface element dS limited by the
curves u, v, u + du, and v + dv (Fig. 6.4).
The different forces that act on this surface element are (we only consider
their components along the normal to the surface element, the positive direction being that of the external normal) the forces exerted by the internal and
external pressures on dS (pin dS and pout dS),
pin dS, pout dS
(6.3)
and the force exerted by the surface tension along the line v and v + dv
(see Fig. 6.5)
2γdv sin
dθ1
,
2
(6.4)
where
•
•
factor 2 represents the fact that the surface tension acts along v and v + dv
dv = R1 dθ2
A similar equation can be written as far as the surface tension along u and
u + du is concerned:
2γdu sin
dθ2
.
2
(6.5)
We can now write the force balance along the normal to the surface
element as
6.5 Effects of a Liquid Bridge on the Adhesion Between Two Solids
45
n
du
γ
γ
R1
dq1
Fig. 6.5. Detail of the surface tension acting along v and v + dv
2γdu sin
dθ1
dθ2
+ 2γdv sin
+ pout dS = pin dS.
2
2
(6.6)
Using the definition1 of dS = dudv, du = R2 dθ1 , dv = R1 dθ2 and using
the classical approximation for small angles (sin x ≈ x), 6.6 can be rewritten as
1
1
+
(6.7)
= pin − pout .
γ
R1
R2
As (1/R1 + 1/R2 ) represents the double of the mean surface curvature H
[47], 6.7 can be finally rewritten into the Laplace equation [1]:
2γH = pin − pout .
(6.8)
6.5 Effects of a Liquid Bridge on the Adhesion
Between Two Solids
Let us now consider two solids linked by a liquid bridge2 , also called meniscus
(Fig. 6.6). To link this to the general frame of micromanipulation, let us call
the upper solid the “tool” or the “gripper” (it will be used as a gripper) and
1
2
Outside this section, θ1 and θ2 will denote the contact angles between the liquid
and, respectively, the component and the gripper.
The presented configuration is axially symmetric, to introduce the capillary force
from a “mechanical” point of view, i.e., using concepts like pressure or tensions. In
a more general case, the configuration is not axially symmetric and an energetic
approach has to be implemented, see therefore Chap. 7.
46
6 First Set of Parameters
z
Tool
Gripper equationz2(r)
pout
qs
r2
h
Interface
q2
z
q1
pin
Object
ρ
r'
Liquid
bridge
r1
r
Substrate
Fig. 6.6. Effects of a liquid bridge linking two solid objects (Reprinted with permission from [108]. Copyright 2005 American Chemical Society.)
the lower one as the object (it will be used as micropart or microcomponent).
Since axial symmetry is assumed, it can be seen in Fig. 6.6 that the contact line between the meniscus and the object (the gripper) is a circle with
a radius r1 (r2 ). The pressure inside the meniscus is denoted by pin and that
outside the meniscus by pout . θ1 is the contact angle between the object and
the meniscus and θ2 is the angle between the gripper and the meniscus. z represents the separation distance (also called the gap) between the component
and the gripper. h is called the immersion height. At its neck, the principal
curvature radii are ρ (in a plane perpendicular to the z axis, i.e., parallel to
the component) and ρ (in the plane rz).
The object is submitted to the “Laplace” force, arising from the pressure difference pin − pout , and to the “tension” force, directly exerted by the
surface tension. In what follows, we will consider that these two forces constitute what we will call the capillary force.3 The “Laplace force” is due to the
Laplace pressure difference that acts over an area πr12 (see Fig. 6.7) and can
be attractive or repulsive according to the sign of the pressure difference, i.e.,
according to the sign of the mean curvature: A concave meniscus will lead to
an attractive force while a convex one will induce a repulsive force.
FL = 2γHπr12 .
3
(6.9)
Marmur [124] uses the terms “capillary” force for the term arising from the pressure difference and “interfacial tension force” for that exerted by the surface
tension.
6.6 A Priori Justification of a Capillary Gripper
47
p in
Object
r1
pout
Fig. 6.7. Origin of the Laplace force: attractive case (Reprinted with permission
from [108]. Copyright 2005 American Chemical Society.)
γ
q1
f1
Object
a
gz
q1
g SL
f1
g SV
a
Fig. 6.8. Origin of the tension force and detail (Reprinted with permission from
[108]. Copyright 2005 American Chemical Society.)
The “tension force” implies the force directly exerted by the liquid on the
solid surface. As illustrated in Fig. 6.8, the surface tension γ acting along the
contact circle must be projected on the vertical direction, leading to
FT = 2πr1 γ sin(θ1 + φ1 ).
(6.10)
Therefore, the capillary force is given by
FC = FT + FL = 2πr1 γ sin(θ1 + φ1 ) + 2γHπr12 ,
(6.11)
φ1 denotes the slope of the component at the location of the contact line: it
will be considered equal to zero in the following.
6.6 A Priori Justification of a Capillary Gripper
To roughly estimate the order of magnitude of the capillary and tension forces,
let us assume that ρ ≈ r1 = 1 mm, ρ ≈ −10 µm and θ1 = 30◦ . Therefore, the
48
6 First Set of Parameters
meniscus has a mean curvature H = 12 (1/ρ +1/ρ) = −4.95×104 m−1 . If water
is used, the surface tension γ is equal to 72 × 10−3 N m−1 . Consequently, we
can expect that the gripping force Fgrip exerted by a water meniscus on a
microcomponent is about
Fgrip = FL + FT ≈ (−22.4 + 0.5) mN = −21.9 mN.
(6.12)
It is important to note that it is about the weight of a small cube with
a 9.4 mm edge and 2, 700 kg m−3 density (aluminum). The first temporary
conclusion is that the surface tension effects are large enough to pick up what
we defined as microcomponents.
A second interesting element is the downscaling behavior of the capillary
force. Most authors consider that the capillary force is a surface force. However, it must not be understood as being a force that depends on the second
power of a characteristic length. This remark has already been pointed out in
[19] and is well illustrated by the behavior of the capillary force that depends
on the first power of a characteristic length: indeed let us assume a configuration (i.e., given γ, θ1 , θ2 , geometries) leading to curvature radii ρ and ρ
and foot radius r1 (we assume φ1 = 0). Therefore, the force can be written as
follows:
1
1
+
(6.13)
FC = 2πr1 γ sin θ1 + γ
πr12 .
ρ
ρ
If the characteristic size is reduced 10 times,
10 10
FC
r1
r12
=
.
FC = 2π γ sin θ1 + γ
+
π
10
ρ
ρ
100
10
(6.14)
The downscaling of this gripping principle is consequently very promising and
at least more promising than the vacuum gripper whose force directly depends
on the suction area (that is r12 ).
A third advantage of the capillary gripper is that the squeeze film effect
avoids damage to the handled part. We will discuss it again in subsection 11.4
(this damage at contact is often cited as a main drawback of the mechanical
gripper). Moreover, like the vacuum gripper, the capillary gripper is particularly well suited to pick up plane objects or components with a very low
aspect ratio. These microparts do not offer enough space on their sides to be
gripped by a mechanical gripper. Finally, the study and the modeling of the
capillary as a gripping principle is a good way to enter microassembly. Capillary is involved on many scales, from the capillary condensation at nanotips
to the reliability of microsystems.
The use of capillary as a gripping principle is cited in [167] and has been
tested in [70]. Grutzeck [70] set up a manipulation station using a gripper tip
made of a silicon chip, with a small hole used to generate the gripping droplet.
The gripped object was a silicon chip with a weight of 0.219 mN. In this
application, the measured forces ranged from 11 to 21 mN with drop volumes
between 1.6 and 0.19 µL (separation distance = 12 µm). The parameters of
the object and the gripper are summarized below.
6.7 Conclusions
49
Table 6.1. Parameters of the manipulation proposed in [70]
Element
Gripper
Description
Value
Plane tip made of a silicon chip
4 × 4 mm2 ,
600 µm height
Φ400 µm
Droplet supply through a hole
Object
Plane tip made of a silicon chip
Weight
Surface properties of the chips
Roughness
Advancing contact angle (with water)
Receding contact angle (with water)
Advancing contact angle (with methanol)
Receding contact angle (with methanol)
600 µm height
0.219 mN
1.36 µm
71◦
50◦
14◦
13.5◦
6.7 Conclusions
In this section, we presented some fundamental parameters involved in the
capillary force (surface tension, contact angle, Laplace equation). According
to 6.11, this capillary force can be calculated if the geometry of the liquid
bridge is known but some approximations can be used in order to evaluate
the force more quickly. This will be discussed in the next chapter.
7
State of the Art on the Capillary Force Models
at Equilibrium
7.1 Introduction
We will now compare several methods to estimate the capillary force in several
configurations. Each method will be presented with one or two configurations
but not all configurations will be detailed for all methods. The results are
summarized at the end of this chapter. Most often the capillary forces are
approximated by several formulations that include the following assumptions:
(1) axial symmetry of the liquid bridge, (2) gravity effects on the meniscus
shape are neglected (in other words, the Bond number BO ≡ ∆ρgL2 /γ is
assumed to be vanishing,
which means meniscus with a size smaller than the
capillary length LC ≡ γ/∆ρg - see later (8.1)).
The main approaches are the following:
•
•
The energetic approach,1 consisting in deriving the interfacial energy W
with respect to the separation distance between the gripper and the object
Geometrical approximations of the meniscus shape
– “Arc”, “circle”, or “toroidal” approximation
– “Parabolic” approximation
7.2 Energetic Approach: Interaction Between Two
Parallel Plates
This method consists in the following:
•
1
Writing the interfacial energy W of the system as a function of the parameters defining the geometry of the system
To be exact, the energetic approach requires the accurate knowledge of the meniscus shape, which can be computed by energy minimization (see for example the
well known Surface Evolver software) or by numerically solving the Laplace equation in the axially symmetric case, see therefore Chap. 8).
52
7 State of the Art on the Capillary Force Models at Equilibrium
z
S
r2
q2
z
L
V
q1
S
r1
r0
Fig. 7.1. Example of the energetic method: case of two parallel plates
•
•
Deriving this energy with respect to one of the parameters (the separation
distance z is often used) to calculate the capillary force as a function of
this parameter
Estimating the derivative of the other parameters with respect to the
chosen parameter by assuming a mathematical relation (for example, the
conservation of the liquid volume)
This approach can be illustrated by the case of two parallel plates linked by
a meniscus, as represented in Fig. 7.1.
The system has three phases (S, solid; L, liquid; V, vapor) and three
interfaces (LV, liquid–vapor; SL, solid–liquid; SV, solid–vapor) leading to a
total energy equal to:
W = WSL + WSV + WLV = γSL SSL + γSV SSV + γΣ,
(7.1)
where
WSL = γSL1 πr12 + γSL2 πr22 ,
(7.2)
WSV = γSV1 (πr02 − πr12 ) + γSV2 (πr02 − πr22 ),
(7.3)
WLV = γΣ.
(7.4)
In these equations, r0 is an arbitrary constant radius, larger than the
maximum between r1 and r2 and γSLi (γSVi ) state for the interfacial energy
between solid i and the liquid (vapor). Σ states for the area of the liquid–vapor
interface (the lateral area of the meniscus). As we try to get the expression
of the force F acting on one of the plates along the vertical z as a function of
the separation distance z, (7.1) must be derived with respect to z:
7.2 Energetic Approach: Interaction Between Two Parallel Plates
F =−
53
dW
dz
dr1
dr2
dr1
− γSL2 2πr2
+ γSV1 2πr1
dz
dz
dz
dΣ
dr2
+ γSV2 2πr2
−γ
.
(7.5)
dz
dz
To calculate all the derivatives involved in this expression, additional
assumptions must be stated. The first (and not restrictive) assumption is
that the volume of the meniscus remains constant (we consequently do not
consider the evaporation of the liquid). Unfortunately, this assumption is not
sufficient and more restrictive assumptions must be added:
= −γSL1 2πr1
1. The separation distance z is small compared to the radius r1 and r2 ;
henceforth, we neglect the term depending on the lateral area Σ
2. The liquid volume can be approximated by
V ≈ πr12 z.
(7.6)
Consequently, the conservation of the volume leads to
dV
dr1
= 2πr1
+ πr12 = 0
dz
dz
(7.7)
dr1
r1
=− .
dz
2z
(7.8)
and
Moreover, let us make a third assumption:
3. The derivative of r2 with respect to z can be written in the same way,
leading to
r2
dr2
=− .
dz
2z
(7.9)
With (7.8),(7.9), and (6.1), (7.5) can now be rewritten as
F =−
πγ 2
(r cos θ1 + r22 cos θ2 )
z 1
(7.10)
or, in the case of two plates made of the same material
F =−
2πγ 2
r cos θ.
z
(7.11)
If we take the lateral area Σ of the meniscus into consideration, the previous assumptions must be replaced by these ones:
1. The meniscus shape can be approximated by a cylinder of radius r and
height z
54
7 State of the Art on the Capillary Force Models at Equilibrium
2. The liquid volume is exactly given by V = πr2 z
The lateral area Σ is given by
Σ = 2πrz.
(7.12)
So, instead of neglecting it, we can write the derivative of Σ toward z as
dΣ
= 2πr.
dz
(7.13)
Consequently, still considering the Young–Dupré equation (6.1), (7.11) can
now be rewritten as
F =−
2πγ 2
(r cos θ + rz).
z
(7.14)
Or, if we write this latter equation as a function of the liquid volume V ,
1
F =−
1
2γV cos θ 2γπ 2 V 2
−
.
1
z2
z2
(7.15)
Finally, a last expression similar to (7.15) can be found in [60]:
1
lim F = −
z→0
1
2γV cos θ 2γπ 2 V 2 sin θ
−
.
1
z2
z2
(7.16)
To be comparable with (7.14) and (7.11), this last equation is rewritten to
F =−
2πγ 2
(r cos θ + rz sin θ).
z
(7.17)
We now propose several plots of (7.14), (7.11), and (7.17):
1. Parallel plates, influence of the contact line radius r for a contact angle
equal to 15◦ (Figs. 7.2–7.5)
2. Parallel plates, influence of the contact angle for a contact line radius r
of 100 µm (Figs. 7.6–7.9).
It can be shown in these figures that the force given by (7.17) lies between
the results of (7.14) and (7.11). Moreover, these three approximations tend to
one another when z decreases. However, these equations are only approximations, and so it cannot be concluded that one would be more accurate than
the other.
7.3 Energetic Approach: Other Configurations
55
R=1e−006m, θ=15°
0
10
0.01m
0.001m
−5
Force [N]
10
0.0001m
−10
1e−005m
10
1e−006m
−15
10
10
−10
10
−8
−6
−4
10
10
Separation distance [m]
10
−2
Fig. 7.2. Comparison between (7.11), (7.14), and (7.17) for a contact line radius
π
, and a surface tension γ = 72 mN m−1 . Solid line:
r = 1 µm, a contact angle θ = 12
(7.11), dashed line: (7.14), and dash–dot line: (7.17)
R=1e−005m, θ=15°
0
10
0.01m
0.001m
−5
Force [N]
10
0.0001m
−10
1e−005m
10
1e−006m
−15
10
10
−10
10
−8
−6
−4
10
10
Separation distance [m]
10
−2
Fig. 7.3. Comparison between (7.11), (7.14), and (7.17) for a contact line radius
π
, and a surface tension γ = 72 mN m−1 . Solid line:
r = 10 µm, contact angle θ = 12
(7.11), dashed line: (7.14), and dash–dot line: (7.17)
7.3 Energetic Approach: Other Configurations
Another important interaction is that between a sphere and a plane (see
Fig. 7.10). It has been presented in [89] Chap. 15.6. (Details are also proposed
in Appendix A.1.2).
Let us consider a sphere with radius R, located at a distance z from a
surface. The liquid meniscus rises up to a height z + h (the immersion height
h is so that the filling angle φ is small). By considering a constant volume
of liquid, Israelachvili [89] calculated that the attractive force between the
sphere and the surface due to the liquid bridge is
F =−
4πRγ cos θ
1 + hz
(7.18)
56
7 State of the Art on the Capillary Force Models at Equilibrium
R=0.0001m, θ=15°
5
10
0
Force [N]
10
0.01m
0.001m
−5
10
0.0001m
−10
1e−005m
10
1e−006m
−15
10
10
−10
10
−8
−6
−4
10
10
Separation distance [m]
10
−2
Fig. 7.4. Comparison between (7.11), (7.14), and (7.17) for a contact line radius
π
and a surface tension γ = 72 mN m−1 . Solid
r = 100 µm, a contact angle θ = 12
line: (7.11), dashed line: (7.14), and dash–dot line: (7.17)
R=0.001m, θ=15°
5
10
0
Force [N]
10
0.01m
0.001m
−5
10
0.0001m
−10
1e−005m
10
1e−006m
−15
10
10
−10
10
−8
−6
−4
10
10
Separation distance [m]
10
−2
Fig. 7.5. Comparison between (7.11), (7.14), and (7.17) for a contact line radius
π
, and a surface tension γ = 72 mN m−1 . Solid line:
r = 1 mm, a contact angle θ = 12
(7.11), dashed line: (7.14), and dash–dot line: (7.17)
and that maximum attraction occurs at z = 0, where
F = −4πRγ cos θ.
(7.19)
More rigorous expressions valid for large φ and different contact angles on
each surface are given by [137].
Finally, similar developments lead to the force between two spheres with
different radii R1 and R2 and different material (θ1 = θ2 ) (please refer to
Appendix A.1.3 for further details):
F = −4πγR cos θ,
(7.20)
where 2 cos θ ≡ cos θ1 + cos θ2 and 1/R ≡ 1/R1 + 1/R2 , so that it can be
concluded that at contact and with small amounts of liquid (φ <<), the force
7.4 Geometrical Approach: Circle Approximation
57
R=0.0001m, θ=0°
5
10
0
Force [N]
10
0.01m
0.001m
−5
10
0.0001m
−10
1e−005m
10
1e−006m
−15
10
10
−10
10
−8
−6
−4
10
10
Separation distance [m]
10
−2
Fig. 7.6. Comparison between (7.11), (7.14), and (7.17) for a contact line radius
r = 100 µm, a contact angle θ = 0, and a surface tension γ = 72 mN m−1 . Solid line:
(7.11), dashed line: (7.14), and dash-dot line: (7.17)
R=0.0001m, θ=15°
5
10
0
Force [N]
10
0.01m
0.001m
−5
10
0.0001m
−10
1e−005m
10
1e−006m
−15
10
10
−10
10
−8
−6
−4
10
10
Separation distance [m]
10
−2
Fig. 7.7. Comparison between (7.11), (7.14), and (7.17) for a contact line radius
π
, and a surface tension γ = 72 mN m−1 . Solid
r = 100 µm, a contact angle θ = 12
line: (7.11), dashed line: (7.14), and dash-dot line: (7.17)
between two spheres with radii R1 and R2 is equal to that between a plane
and a sphere of radius R given by 1/R = 1/R1 + 1/R2 .
7.4 Geometrical Approach: Circle Approximation
Another widespread method in the literature to calculate the capillary force
is to approximate the meniscus by an arc (part of a circle). This method
requires the determination of five parameters (ro , zo , ρ, θmin , and θmax ) that
are represented in Fig. 7.11.
Several conditions can be expressed to determine these five parameters: the
contact angles on each plate (therefore θmin = θ1 and θmax = π − θ2 ), the separation distance z. Another condition can be the volume of liquid or the mean
58
7 State of the Art on the Capillary Force Models at Equilibrium
R=0.0001m, θ=30°
5
10
0
Force [N]
10
0.01m
0.001m
−5
10
0.0001m
−10
1e−005m
10
1e−006m
−15
10
10
−10
10
−8
−6
−4
10
10
Separation distance [m]
10
−2
Fig. 7.8. Comparison between (7.11), (7.14), and (7.17) for a contact line radius
r = 100 µm, a contact angle θ = π/6, and a surface tension γ = 72 mN m−1 . Solid
line: (7.11), dashed line: (7.14), and dash-dot line: (7.17)
R=0.0001m, θ=80°
0
10
0.01m
0.001m
−5
Force [N]
10
0.0001m
−10
1e−005m
10
1e−006m
−15
10
10
−10
10
−8
−6
−4
10
10
Separation distance [m]
10
−2
Fig. 7.9. Comparison between (7.11), (7.14), and (7.17) for a contact line radius
r = 100 µm, a contact angle θ = 80◦ , and a surface tension γ = 72 mN m−1 . Solid
line: (7.11), dashed line: (7.14), and dash-dot line: (7.17)
curvature of the meniscus. This last condition contains the largest approximation of this method: it assumes that the curvature will be constant but an axially symmetric meniscus with a circular profile cannot have a constant curvature. The circle approximation method consequently consists in the following:
•
•
•
Determining the circle parameters
Computing its mean curvature (at the neck)
Calculating the “Laplace” and “tension” components of the capillary force
Let us illustrate this method in the case of two parallel plates (see Fig. 7.11).
Because of the geometry of the problem, we can express the circle radius ρ as
a function of the separation distance z and the contact angles θ1 and θ2 :
7.4 Geometrical Approach: Circle Approximation
59
R
φ
h
z
Fig. 7.10. Interaction between a sphere and a plane [89]: z is the gap between the
sphere and the plane, h is the immersion height, Φ is the filling angle, and R is the
sphere radius
z
θ2
θmax
z
ρ'
θ1
ρ
(r0 ,z0)
θmin
r
r1
Fig. 7.11. Circle approximation of the meniscus: z is the gap between both plates,
θ1 and θ2 are the contact angles, ρ and ρ are the two curvature radii at the neck
ρ=
z
.
cos θ1 + cos θ2
(7.21)
The coordinate z0 of the circle’s center is given by
z0 = ρ cos θ1 =
z cos θ1
.
cos θ1 + cos θ2
(7.22)
The coordinate ro can be expressed as a function of ρ and ρ :
r0 = ρ + ρ .
(7.23)
We miss an additional condition in order to determine ρ : we will assume
a constant volume V that can be expressed as
V = πρ2 z.
(7.24)
60
7 State of the Art on the Capillary Force Models at Equilibrium
The arc is now completely determined by
θmin = θ1 ,
(7.25)
θmax = π − θ2 ,
z
,
ρ=
cos θ + cos θ2
1
V
,
ρ =
πz
r0 = ρ + ρ ,
(7.26)
(7.27)
(7.28)
(7.29)
and we can compute the mean curvature H of this meniscus:
2H =
1
1
− .
ρ
ρ
(7.30)
ρ is positive but ρ must be counted negative because it contributes to the
concavity of the meniscus. The difference of pressure across the interface is
given by the Laplace equation (6.8)
pin − pout = 2γH.
(7.31)
Both components of the capillary force can now be calculated as a function
of r1 , the radius of the circular contact line on the plate 1, which is given by
r1 = (ρ + ρ ) − ρ sin θ1 .
(7.32)
Henceforth, we deduce the “Laplace” force
FL = πr12 (pin − pout )
(7.33)
and the “tension” force
FT = 2πr1 γ sin θ1 ,
(7.34)
leading to a global capillary force equal to
F = πr12 γ2H + 2πr1 γ sin θ1
(7.35)
1
1
−
(7.36)
= πγr12
+ 2πγr1 sin θ1
ρ
ρ
cos θ1 + cos θ2
πz
−
= πγr12
(7.37)
+ 2πγr1 sin θ1 ,
V
z
where r1 = cos θ1 +z cos θ2 + V /πz − cos θ1 +z cos θ2 sin θ1 .
Several authors used this method among whom Stifter et al. [161] and
Marmur [124] apply the arc approximation to study the capillary force in
surface force apparatus.
7.6 Comparisons and Summary
61
7.5 Geometrical Approach: Parabolic Approximation
In some cases the arc approximation can lead to numerical difficulties. For
example, Pepin et al. [142] emphasizes the switch from a convex meniscus to
a concave meniscus: The center of the circle first tends to −∞ before diverging
to +∞. This difficulty can be avoided by using a parabolic approximation of
the meniscus:
r(z) = az 2 + bz + c.
(7.38)
The three unknown a, b, and c can be determined by imposing the contact
angles θ1 and θ2 and the last unknown can be determined if either the pressure
(i.e., the curvature: see also the Laplace equation) or the volume is imposed.
For example, in the case of flat components, the slope must match the contact
angle θ1 :
b=−
cos θ1
.
sin θ1
(7.39)
The two other unknown can be determined by using an iterative scheme:
1. First choose a starting radius r
2. Compute the gripper point corresponding to r (i.e., determine z(r)) and
the slope tan p of the gripper at this point
1
( tan(θ12 + p) − b)
3. The parabola must match the contact angle θ2 ⇒ a = 2z
4. The point (r, z) belongs to the parabola too: c = r − az 2 − bz
5. Determine the corresponding volume v(r)
6. Compare v(r) and the prescribed volume V
• If v(r) > V , decrease r
• Otherwise, increase r
With this geometrical shape, the switch from a convex to a concave meniscus
smoothly occurs when a passes through zero before changing its sign.
7.6 Comparisons and Summary
Comparisons between the different formulations of the energetic method have
already been proposed in the ad hoc section. In conclusion of this section devoted to the approximations, we propose the graphical comparison (Fig. 7.12)
of the meniscus shapes between the geometrical methods.
The impact on the pressure difference and the force is plotted in Fig. 7.13.
As the difference between the “arc” and the “parabolic” approximations must
be explained (the “better” approximation must be determined), we will turn
ourselves to numerical solutions: According to the Laplace equation, we will
compute the shape of a meniscus at equilibrium for a given volume. These
simulations are of the highest interest as far as the use of capillary force as
62
7 State of the Art on the Capillary Force Models at Equilibrium
−4
x 10
z [m]
6
4
2
0
2
4
6
r [m]
8
10
12
−4
x 10
Fig. 7.12. Comparison between the arc (solid line) and the parabolic (dashed line)
approximations (Reprinted with permission from [108]. Copyright 2005 American
Chemical Society).
500
8
0
Force [mN]
6
∆p
−500
−1000
−1500
4
2
−2000
−2500
0
200
400
Gap [µm]
(a)
600
800
0
0
200
400
Gap [µm]
600
800
(b)
Fig. 7.13. Comparison between the meniscus shape models: arc (Open square) and
parabola (Open triangle) models. (a) Pressure difference ∆p across the LV interface;
(b) Force (Both reprinted with permission from [108]. Copyright 2005 American
Chemical Society).
gripping principle is concerned. These simulations will consequently directly
be related to the problematics of microgripping and microassembly presented
in the state of the art of this work.
Table 7.1 summarizes several classical approximations found in the literature and gives the corresponding references and assumptions (see Fig. 6.6 and
Appendix A.1 for more details). The considered assumptions are as follows:
1.
2.
3.
4.
5.
Parallel plates
Spherical tip (radius R) near a plate
Arc approximation of the interface (where ρ = constant is the radius)
Energetic formulation
The radii r1 and r2 of the two circular contact lines are very small compared to R
7.6 Comparisons and Summary
63
Table 7.1. Summary of the capillary forces (sign “−” has been omitted) (Reprinted
with permission from [111], Copyrights 2006 Koninklijke Brill N.V.)
Ref.
[124]
[124]
Eq.7.10
Eq.7.14
[89]
[161]
[71]
[70]
[48]
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
Force
F = 4πRγ cos θ
(r1 /R)2
r1 /R
F = 4πRγ cos θ{ 12 [ z/R+1−(1−(r
2 1/2 ] − [ 4 cos θ ]}
1 /R) )
πγ
2
2
F = z (r1 cos θ1 + r2 cos θ2 )
F = 2πγ
r2 cos θ + 2πrγ
z
4πRγ cos θ
F = 1+(z/h)
F = πγρ’2 ( ρ1 − ρ1 )
F = 2πγρ + γρ πρ2
z
F = 2πγρ + γρ π 2 sin
θ
F = 2πr2 γ sin(θ2 + φ) + πr22 γ( ρ1 − r12 )
Assumptions
2,5,6,7,8,9
2,3,6,7,15
1,4,7,8,10,11
1,4,7,8,11,12
2,4,7,10,13,14
2,3,15
1,3,8,9
1,3,7,9
2,3
r1 = r2 = r (“symmetric case”)
The contact angles are equal: θ1 = θ2 = θ
The gap z is very small compared to the radius r of the contact line
The curvature of the interface in the horizontal plane is negligible
| ρ1 | << | ρ1 |
Contribution of the interfacial energy liquid–vapor is neglected
Constant volume V ≈ πr12 z
Interfacial area liquid–vapor S ≈ 2πr1 z
Constant volume V = Vcylinder − Vspherical cap , cf. Appendix A.1.2
Immersion height h is small (φ ≈ 0, cf. Fig. 7.10)
Interfacial tension force is neglected
To end this section, let us add a recently published model [145] giving an
analytical expression for the capillary force between two spheres with radii R1
and R2 as a function of the separation distance z:
Fsphere/sphere = −
2R cos θ
,
1 + z/(2h)
(7.40)
1 R2
where R is the equivalent radius given by R = R2R
, 2 cos θ = cos θ1 +
1 +R2
cos θ2 , z is the separation distance or gap, and h is the immersion height,
approximately given by [145]
h=
z
(−1 + 1 + 2V /(Rz 2 )),
2
where V is the volume of the liquid bridge.
(7.41)
8
Static Simulation at Constant Volume of Liquid
8.1 Introduction
This chapter details the computation of capillary forces based on the solving
of the Laplace equation (6.8) in order to compute the meniscus shape. The
equivalence between this method and the energy minimization approach will
be treated in the next chapter.
8.2 Description of the Problem
The problem set out in this section is the simulation of a handling task based
on the capillary force as gripping principle. The object is pulled toward the
gripper by the capillary force exerted by the liquid bridge. The simulation
objectives will consist in determining this force and predicting the ability to
perform the manipulation. As explained in the previous sections, the capillary
force can be determined from the geometry of the system (i.e., liquid bridge,
gripper, and component).
In Fig. 8.2, we call “gripper” or “tool” the upper solid, characterized by
a profile described by the equation z = z2 (r). The lower solid is the handled
component (“object”) whose profile is described by the equation z = z1 (r).
θ1 (θ2 ) is the contact angle between the liquid and the object (the gripper).
We consider only axially symmetric menisci. The “manipulation” task can be
split into three subtasks: the “gripping”(or “picking”) task, the “handling”
task, and the “release” task. The gripper can be conical, spherical, parabolic,
or cylindrical (other geometries can be added but have not been taken into
account yet). They are characterized by two parameters: the gripper width
and the gripper parameter p, defined for each geometry in Fig. 8.1. The cylindrical gripper presents a horizontal plane tip and is actually a conical gripper
with p = 0.
66
8 Static Simulation at Constant Volume of Liquid
Gripper width
p (radius)
Gripper width
Gripper width
p (aperture angle)
p (radius)
Fig. 8.1. Geometries of the gripper: conical, spherical, and parabolic gripper
z
Tool
Gripper equation z2(r)
pout
qs
r2
h
Interface
q2
z
q1
pin
Object
Substrate
ρ'
ρ
Liquid
bridge
r1
r
Fig. 8.2. Gripper, liquid bridge, and microcomponent (Reprinted with permission
from [108]. Copyright 2005 American Chemical Society.)
8.3 Assumptions
This simulation is based on several assumptions.
Gravity effects are neglected. Indeed, Charlaix [32] shows the existence of
a cut-off length LC between capillary and gravity effects, given by
γ
LC =
.
(8.1)
ρg
In the case of water (γ ≈ 72 × 10−3 J m−2 , ρ ≈ 103 kg m−3 , g ≈ 10 m s−2 ),
LC ≈ 2.5 mm. This capillary length can also be interpreted from the ratio
between the hydrostatic pressure (ρgL) and the Laplace pressure γ/L:
γ 1
L2
∆PLap
=
= C2 .
2
∆PHyd
ρg L
L
(8.2)
8.4 Equations and Numerical Simulation
67
This result shows that the gravity effects in the meniscus can be neglected
if its characteristic length L is smaller than typically 1 mm. Let us note that
this restriction applies to the meniscus and not to the solid parts (i.e., “gripper” and the handled part): to be more accurate, this restriction applies to
the meniscus height.
A second assumption is the static approximation: Liquid bridges are assumed to be at equilibrium. Moreover, the static contact angle hysteresis is
neglected (i.e., the advancing contact angle is equal to the receding contact
angle, see also p. 97). The validity of this assumption therefore depends on
the liquid (for example, the difference between the advancing and the receding
contact angles for methanol on Si chips is only 0.5◦ but can be quite larger, up
to 20◦ in the case of water on the same substrate [70]). In the case of a 1 µL
water droplet and a conical gripper (aperture angle of the cone is equal to
20◦ ) separated by a distance of 1 nm from a plane substrate, this assumption
can lead to important variations of the capillary force. In the case of a contact
angle θ2 = 40◦ , a contact angle θ1 of 50◦ (71◦ ) leads to a 922 µN (690 µN)
capillary force (own results). However, in the picking task, the capillary force
should be bigger than the weight. If we assume small vertical accelerations of
the gripper, this implies that the gripped object will stay in close contact with
the gripper and that the meniscus will be in a relative equilibrium position.
Nevertheless, errors can occur in the simulation of the release task because
the increasing separation distance between the object and the gripper implies
the motion of the contact line.
A third assumption is that not only the weight of the meniscus but also the
inertial forces acting on the meniscus can be neglected during the simulation.
This last assumption must be discussed in the case of a release task based
on an important vertical acceleration communicated to the gripper, but is
suitable in the simulation of the gripping task (see Chap. 12).
Finally, we consider smooth surfaces for both solids and the evaporation
of the meniscus is neglected (the volume of liquid is constant).
8.4 Equations and Numerical Simulation
As presented in Chap. 6, the capillary force can be determined from the geometry of the liquid bridge, using (6.9) and (6.10). Therefore, the problem will
consist in determining the shape of the meniscus from its differential formulation given by (6.8). This equation states that the mean curvature H of the
meniscus is determined by (or determines) the pressure difference across the
interface. If equilibrium is assumed and the liquid bridge height is smaller
than LC , the pressure inside the liquid bridge will be constant. Therefore, H
will be constant.
To link the general formulation of equation (6.8) to the parametric description of a surface, let Σ be a surface given by its vectorial equation:
Σ(u, v) ≡ OP = r(u, v),
(8.3)
68
8 Static Simulation at Constant Volume of Liquid
where O is a reference point and P a point of Σ, determined by the two
parameters u and v. Let us assume the following differential operator [47]:
∂r ∂r
. ,
∂u ∂u
∂r ∂r
. ,
F =
∂u ∂v
∂r ∂r
. ,
G=
∂v ∂v
2
∂ r
L=
.1n ,
∂u2
∂2r
M =
.1n ,
∂u∂v
∂2r
N =
.1n .
∂v 2
The mean curvature H is given by (see Appendix B.2)
1
EN + GL − 2F M
1 1
,
+
=
H=
2 R1
R2
2(EG − F 2 )
(8.4)
E=
(8.5)
(8.6)
(8.7)
(8.8)
(8.9)
(8.10)
where R1 and R2 represent the two principal curvature radii. For a zaxially symmetric meniscus, the surface equation is given by
Σ(θ, z) ≡ r(θ, z) = r(z)1̄r + z 1̄z .
(8.11)
Using (8.10) and (8.11) the mean curvature of an axially symmetric surface
is given by
2H =
r
1
−
,
(1 + r2 )3/2
r(1 + r2 )1/2
(8.12)
where r = ∂r/∂z and r = ∂ 2 r/∂z 2 . Since the orientation of 1̄n is not oneto-one, the sign of H must be determined. In Fig. 8.3, we see a spherical soap
z
r''<0
r>0
z(r)
r
Fig. 8.3. Case of a spherical bubble with pin > pout
8.4 Equations and Numerical Simulation
69
z
rP
fP
P
q2
zP
q1
r
Fig. 8.4. Boundary conditions depend on the geometry and materials
bubble in air with an internal pressure higher than the external one, allowing
to write
<0
r
∆p
1
pin − pout
−
≡
.
+
=
2 1/2
γ
γ
(1 + r2 )3/2
r (1 + r )
>0
>0
>0
(8.13)
>0
Let us put (8.13) as a system of two first-order differential equations:
dr
dz
du
1+u2
dz =
r
u=
−
∆p
γ (1
(8.14)
+ u2 )3/2 .
To solve these equations, ∆p must be known and boundary conditions have
to be set (Fig. 8.4): let us assume that we know the point P of the meniscus
in contact with the gripper. Therefore, zP and rP = r(zP ) are given by the
initial coordinates of P and the slope of the meniscus in P is given by
1
uP =
,
dr
|zP = { tan(φP +θ2 )
dz
0,
if (θ2 + φP ) =
if (θ2 + φP ) =
π
2
π
2
,
(8.15)
where the contact angles θ2 and φP depend on the gripper geometry.
In our problem (how to determine the meniscus for given contact angles θ1
and θ2 and liquid volume V ), only θ2 is known. Indeed, ∆p and the position of
P are a priori unknown. Ref. [48] already suggested to iterate on ∆p to adjust
θ1 to the prescribed value. The typical evolution of θ1 as a function of ∆p in our
problem in shown in Fig. 8.5: Increasing pressure difference (i.e., more negative
∆p) leads to a more curved meniscus, and consequently, to smaller θ1 . Details
will be put forward in Chap. 14. P is still unknown and the condition on V has
not yet been used. Therefore, a second iteration loop is used [110] to determine
P : An initial position of P is guessed in order to solve the first iteration loop
70
8 Static Simulation at Constant Volume of Liquid
100
90
80
70
θ1
60
50
40
30
20
10
0
−300
−250
−200
−150
dp
−100
−50
0
Fig. 8.5. Contact angle θ1 (◦ ) as a function of the pressure difference across the
interface dp (Pa)
x 105
z (m)
8
6
Gripper
Starting point i+1
Starting point i
Meniscus
Starting point 1
4
2
0
0.5
1
1.5
r (m)
Meniscus ij (starting point i, ∆ pj)
2
x 10−4
Fig. 8.6. The double iterative scheme for a spherical gripper (R = 0.1 mm), water,
θ1 = θ2 = 30◦ , V = 4.5 nL, z = 0. Meniscus ij is obtained with the ith starting point
and the jth pressure difference (Reprinted with permission from [108]. Copyright
2005 American Chemical Society.)
(i.e., determine a meniscus that would be correct as far as contact angles
are concerned), leading to a candidate whose volume is computed. If this
volume is smaller (larger) than the prescribed one, P is moved away (closer)
from the symmetry axis (this is achieved by dichotomous search). In Fig. 8.6,
P is successively defined by the following radii: 50, 75, 87.5, 93.8, 96.9, and
98.4 µm. This double iterative scheme is illustrated in Fig. 8.7: it is actually
an application of the so-called shooting method.
8.5 Discussion and Conclusions
71
Choose a meniscus starting point
Choose a pressure difference
Contact angle OK?
No
Volume OK?
No
Compute the capillary force
Fig. 8.7. View of the resolution (Reprinted with permission from [108]. Copyright
2005 American Chemical Society.)
θ1
θ2
γ
Simulation
Geometries
Force
V
z
Fig. 8.8. Inputs and outputs of the static simulation
8.5 Discussion and Conclusions
The simulation described in this section can be qualified as a static simulation
at constant volume. Its main characteristics are the following:
•
•
•
•
Actual geometries of the gripper and the object can be taken into account
Material properties are taken into consideration with the contact angles,
the surface tension
Liquid volume is a user parameter
Capillary force is computed
The main assumptions of the model are as follows:
•
•
•
•
•
Gravity is neglected toward the capillary force
Contact angle hysteresis is not taken into consideration
Dynamic contact angle is not taken into account
Materials are assumed to be smooth (no roughness) and without any impurities
Inertial forces acting on the liquid bridges are neglected, which is not valid
if the release task is based on acceleration of the gripper
72
8 Static Simulation at Constant Volume of Liquid
Not all these assumptions are restrictive, and we will consider additional models in Chap. 11 that could be used to extend the validity of this simulation to
more complex configurations involving surface roughness or surface impurities
(i.e., how to modify θ1 and θ2 in Fig. 8.8). Chapter 9 will establish on a formal way the equivalence between the energetic approach presented in Chap. 7
and the approach based on the sum of the “tension” term and the “Laplace”
term as illustrated in the current chapter. Obviously, the reader can also go
directly to Chap. 10, dealing with the application of these models to a watch
ball bearing case study.
9
Comparisons Between the Capillary Force
Models
9.1 Introduction
This chapter gives evidence of the equivalence between the energetic approach
and the direct formulation based on the Laplace and the tension terms:
F = FL + F T = −
dW
,
dz
(9.1)
where FL is given by (6.9), FT by (6.10) and W by (7.1). z is the separation
distance between both solids.
9.2 Qualitative Arguments
The energetic approach involves both Laplace and tension terms. Let us illustrate this in the case of two parallel plates (see Fig. 7.1 reprinted as Fig. 9.1)
separated by a distance z (for convenience, both contact angles have been
chosen to be equal to θ). Based on the arguments of Sect. 7.2, the force was
given by (7.15)
F =−
2γV cos θ 2γπ 1/2 V 1/2
−
.
1
z2
z2
(9.2)
Let us now consider the case θ = π/2 to compare the force derived from
the energy with that from the meniscus geometry. Indeed, in this case, the
approximation of the meniscus shape by a cylindrical volume is exact (stripped
lateral area in Fig. 9.1), and we can directly compute the mean curvature of
this cylinder. Consequently, we have to compare the force derived from the
energy:
πV
(9.3)
F = −γ
z
74
9 Comparisons Between the Capillary Force Models
z
S
z
r2
L
q2
V
q1
S
r1
r0
Fig. 9.1. Case of two parallel plates separated by a gap z
and the one established from the mean curvature, which is given by:
2H =
1
+ 0,
r
(9.4)
leading to a pressure difference
∆p = 2Hγ =
γ
r
(9.5)
and henceforth to a “Laplace” term of the force equal to
FL = πr2 ∆p = πγr
(9.6)
Note that this term is positive, i.e., repulsive, because the meniscus is convex,
leading to a positive pressure difference. The ‘tension’ term of the force FT
can be written as
FT = −2πrγ,
(9.7)
leading to a total capillary force equal to
F = FL + FT = −γπr.
(9.8)
Assuming a cylindrical shape for the meniscus (V = πr2 z as already stated
in (7.6)), the latter equation can be rewritten into
πV
.
(9.9)
F = FL + FT = −γ
z
Since (9.3) and (9.9) are equal, we conclude that the force derived from the
energy exactly represents both the terms of the capillary force (note well that
the expression F = 4πγR cos θ proposed at (15.35) of [89] has been derived
this way, consequently including both terms).
9.3 Analytical Arguments
75
z
R
Fig. 9.2. A sphere (radius R) and a plate separated by a gap z: both contact angles
are equal to θ = π/2
Another argument is geometric. Let us consider the case depicted in
Fig. 9.2 where both contact angle are equal to π/2. The idea behind the
following intuitive argumentation is to prove that the energetic method well
involves both terms. On the one hand, the case depicted in Fig. 9.2 can be
modeled, thanks to (7.18) where h is set equal to 0: F = 4πRγ cos θ. As indicated in Sect. A.1.3, this expression is based on the energetic approach. In the
proposed case, this equation leads to F = 0. On the other hand, the meniscus
is clearly convex, leading to a repulsive “Laplace” force which is here counted
strictly positive. Now, if we take the (always) attractive “tension,” we see
(qualitatively) that we could have a total force equal to zero. This argument
also conveys the idea that both approaches are equivalent.
9.3 Analytical Arguments
9.3.1 Definition of the Case Study
We propose to demonstrate the equivalence of the approaches on a prism–
plane configuration. The prism is defined by its length in the y direction,
L, and its angular aperture φ (see Fig. 9.3). Its location is defined by the
distance1 D between its apex A and the plane. Let us assume a volume of
liquid V wetting the plane with a contact angle θ1 and the prism with a contact
angle θ2 . Since the curvature of the meniscus in the direction y perpendicular
to 0xz is equal to zero, the Laplace equation (8.13) becomes
x
∆p
,
=
γ
(1 + x2 )3/2
(9.10)
where x = dx/dz.
1
For the sake of clarity, since z will be used as one of the coordinates, the gap is
noted D in this chapter.
76
9 Comparisons Between the Capillary Force Models
z
α
θ2
x2
φ
ρ
θ2
h
A
D
O
C
l
φ
ρ
θ1
zo
x
x1
xo
Fig. 9.3. Prism–plane configuration
Assuming a vanishing Bond number, the hydrostatic pressure inside the
meniscus is neglected by comparison to the Laplace pressure difference ∆p,
which is therefore constant in all the meniscus. Therefore, the right hand side
of (9.10) is constant and this equation can be integrated twice with respect
to z to find the relation x = x(z), with two integration constants and the
undefined pressure difference ∆p. A more straightforward derivation is based
on the fact that since one of the curvature radii is infinite and that the total
curvature 2H is a constant, the second curvature radius (1 + x2 )3/2 /x is
a constant: let us denote it ρ. Therefore, the meniscus profile is a curve with
constant curvature, i.e., a circle given by the equation
(x − x0 )2 + (z − z0 )2 = ρ2 ,
(9.11)
where x0 and z0 are the coordinates of the circle center. Once again, three
parameters are to be determined: x0 , z0 , and ρ. This can be done using
three boundary conditions: both contact angles θ1 and θ2 and the volume of
liquid V .
9.3.2 Preliminary Computations
Let us express x0 , z0 and ρ as functions of known data (φ, D, θ1 , θ2 ) and
the immersion height h, which is still unknown at this step, but which will be
determined using the condition on the volume of liquid V . Note that x2 is an
intermediary variable and that x1 will be used later. For sake of convenience,
the notation α = θ2 + φ has been adopted in the following equations:
9.3 Analytical Arguments
77
h
,
tan φ
D+h
ρ=
,
cos θ1 + cos α
z0 = ρ cos θ1 ,
x0 = x2 − (z0 − D − h) tan α,
(9.14)
(9.15)
x1 = x0 − z0 tan θ1 .
(9.16)
Additional useful relations are the meniscus equation:
x = x0 − ρ2 − (z − z0 )2 ,
(9.17)
(9.12)
x2 =
(9.13)
the meniscus slope x :
x = −
z − z0
,
x − x0
(9.18)
and finally, the rewritten Laplace equation linking ∆p and ρ:
∆p =
γ
.
ρ
(9.19)
h is still to be determined using the volume of liquid V (see next step).
9.3.3 Determination of the Immersion Height h
The volume of liquid can be used to determine the value of the immersion
height h, starting from the following expression of V as illustrated in Fig. 9.4:
V = 2LA
= 2L[x0 (h + D) − AI − AII − AIII − AIV ],
(9.20)
(9.21)
z
x2
D
O
AII
AI
h
C
AIV
A
AIII
zo
x
x1
xo
Fig. 9.4. Determination of the immersion height from the volume of liquid
78
9 Comparisons Between the Capillary Force Models
where
x2 h
,
2
(x0 − x2 )(D + h − z0 )
,
AII =
2
z0 (x0 − x1 )
,
AIII =
2
ρ2 (π − α − θ1 )
AIV =
.
2
Therefore, the equation giving the volume V can be rewritten as
(9.22)
AI =
(9.23)
(9.24)
(9.25)
x2 h
2
2
ρ (π − α − θ1 ) (x0 − x2 )(D + h − z0 ) z0 (x0 − x1 )
−
−
−
2
2
2
V = 2L x0 (D + h) −
(9.26)
= L 2x2 D + x2 h
+ρ [sin α cos α + 2 sin α cos θ1 − π + α + θ1 − sin θ1 cos θ1 ]
2
= L h2
(9.27)
≡µ(cos θ1 +cos α)2
1
1
+ µ + 2hD
+ µ + µD2
tan φ
tan φ
(9.28)
This latter equation can be rewritten as a second-degree equation with respect
to the unknown h:
h2 + 2hD +
which leads to
h = −D ±
µD2 − V /L
= 0,
µ + tan1 φ
D2 −
D2 µ − V /L
.
µ + tan1 φ
(9.29)
(9.30)
The “−” solution makes no physical sense since the immersion height cannot
be negative. Consequently
D2 µ − V /L
h = −D + D2 −
(9.31)
µ + tan1 φ
and the variation of h with respect to a variation of the separation distance
D (it will be used in what follows) is given by
dh
D
1
= −1 +
.
dD
D + h 1 + µ tan φ
(9.32)
9.3 Analytical Arguments
79
9.3.4 Laplace Equation Based Formulation of the Capillary Force
As it has been previously explained, the capillary force can be written as
the sum of a term depending on the Laplace pressure difference ∆p and the
so-called tension term:
F = 2Lx1 ∆p + 2Lγ sin θ1
x1
= 2Lγ
+ sin θ1
ρ
x0
= 2Lγ
ρ
x2
D + h − z0
= 2Lγ
+
tan α
ρ
ρ
h cos θ1 + cos α
+ sin α .
= 2Lγ
D+h
tan φ
(9.33)
(9.34)
(9.35)
(9.36)
(9.37)
Using (9.31) the force can be expressed as a function of the volume of liquid
V , the separation distance D, and the angles of the problem: contact angles
θ1 and θ2 on the one hand and the prism angle φ on the other hand. Let us
remind that α = θ2 + φ.
9.3.5 Energetic Formulation of the Capillary Force
As previously explained, the energetic or thermodynamic approach is based on
the differentiation of the total surface energy W with respect to the separation
distance D:
W = γ(Σ − A1 cos θ1 − A2 cos θ2 ),
(9.38)
where
Σ = 2L = 2Lρ(π − α − θ1 ),
A1 = 2Lx1 ,
h
.
A2 = 2L
sin φ
Consequently, the reduced surface energy W/(2Lγ) can be written as
arbitrary constant)
W
h
cos θ2
= ρ(π − α − θ1 ) − cos θ1
+ ρ sin α − ρ sin θ1 − h
2Lγ
tan φ
sin φ
π − α − θ1 − sin α cos θ1 + sin θ1 cos θ1
= (D + h)
cos θ1 + cos α
(9.41)
(+ any
(9.42)
≡β
cos θ1
cos θ2
+
tan φ
sin φ
cos θ2
cos θ1
−
.
= Dβ + h β −
tan φ
sin φ
−h
(9.39)
(9.40)
(9.43)
(9.44)
80
9 Comparisons Between the Capillary Force Models
To compute the force from the energy, the latter equation has to be derived
with respect to D using (9.32) (β is constant with respect to D):
dW 1
D
1
cos θ2
cos θ1
= β + −1 +
−
β−
(9.45)
dD 2Lγ
D + h 1 + µ tan φ
tan φ
sin φ
cos θ2
cos θ1
+
=
tan φ
sin φ
D
1
cos θ2
cos θ1
+
−
β−
,
(9.46)
D + h 1 + µ tan φ
tan φ
sin φ
where µ, h, and β have been defined in (9.27), (9.31), and (9.43). All the other
parameters are given data. It should now be proved that (9.37) and (9.46) are
equivalent.
9.3.6 Equivalence of Both Formulations
Equation (9.46) can be rewritten as
dW
cos θ1
cos θ2
D
π − α − θ1 − sin α cos θ1 + sin θ1 cos θ1
=
+
+
2LγdD
tan φ
sin φ
D+h
(cos θ1 + cos α)(1 + µ tan φ)
cos θ1
cos θ2
1
+
−
(9.47)
1 + µ tan φ tan φ
sin φ
It is shown in Appendix C that the expression in brackets in (9.47) is equal
to −(cos θ1 + cos α)/ tan φ. Therefore, (9.47) can be rewritten into
dW 1
cos θ1
cos θ2
D
cos φ
=
+
−
(cos θ1 + cos α)
dD 2Lγ
tan φ
sin φ
D+h
sin φ
(cos θ1 cos φ + cos θ1 )(D + h) − D cos φ(cos θ1 + cos θ2 cos φ − sin θ2 sin φ)
.
=
(D + h) sin φ
(9.48)
To let appear the term sin α present in (9.37), let us add and substract sin α
simultaneously to the latter equation: after some (tedious) calculations and
using the relation α = θ2 + φ, the following expression can be obtained
dW 1
h cos φ(cos θ1 + cos θ2 cos φ − sin θ2 sin φ)
=
+ sin α
dD 2Lγ
(D + h) sin φ
h cos θ1 + cos α
=
+ sin α.
D+h
tan φ
As a conclusion, the latter equation leads to a force given by
h cos θ1 + cos α
dW
F =−
= −2Lγ
+ sin α .
dD
D+h
tan φ
(9.49)
(9.50)
The negative sign in front of 2L indicates that the force is attractive. Consequently, it is concluded that the force computation based on the Laplace
equation (9.37) and the expression obtained from the energy formulation
(9.50) are equal.
9.4 Conclusions
81
9.4 Conclusions
It is shown that both approaches are equivalent; it means the energetic
approach already involves the tension term and the Laplace term on an implicit way. Consequently, the energetic approach as proposed by Israelachvili
(see (7.18)) includes both terms, even if, for zero separation distance, the pressure term usually dominates the tension one. For axially symmetric configurations, the method based on the Laplace equation will be preferred because
it can be easily numerically solved.
10
Example 1: Application to the Modeling
of a Microgripper for Watch Bearings
10.1 Introduction
This chapter aims at applying the force models presented in the previous
chapters to the case study of a watch ball bearing.
10.2 Presentation of the Case Study
The proposed case study consists in the design of a capillary gripper to
assemble the balls of a watch bearing (Fig. 10.1a), described in [139]. It is
made of the following components:
•
•
•
•
•
A stainless steel (4C27A) inner ring1
A stainless steel (4C27A) outer ring2
An austenitic steel (AISI 301) cage3
Zirconium oxide balls, with diameters 300 and 500 µm
A stainless steel (4C27A) cone4
The assembly task can be sketched by the insertion of a ball in a hole, such
as displayed in Fig. 10.1b. One of the requirements is to avoid the conventional
tweezers and vacuum grippers, because of the scratches they provoke on the
balls (Fig. 3.5). Due to the very small weight of the balls (about 3.8 µN), the
surface tension based gripper is largely strong enough since it generates forces
up to 150 µN. The handling scheme is illustrated in Fig. 10.1b: The picking
force is provided by the capillary force and the releasing task is ensured by
laterally moving the gripper once the ball is in the hole. Since the gripper
uses capillary forces, a liquid has to be dispensed before each manipulation,
1
2
3
4
In
In
In
In
French:
French:
French:
French:
le noyau.
la bague.
la cage.
le cône.
84
10 Example 1: Application to the Modeling of a Microgripper
(a)
(b)
Fig. 10.1. (a) Ball bearing view (courtesy of MPS – http://www.mpsag.com/).
(b) Handling scheme of the capillary gripping for the insertion of a ball in a hole
(Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)
Ø0.1
Liquid channel
M 1.7
Reference surface
Gripper tip
20˚
Ø0.5
(a)
(b)
(c)
Fig. 10.2. Designed capillary gripper. (a) Schematic view: the so-called reference
surface is the surface which contacts the ball during the handling; (b) prototype (the
distance between the graduations is 1 mm); (c) tip detail (the conical surface inclined
to an angle 20◦ can be seen) (Reprinted with permission from [112]. Copyright 2006
Institute of Physics.)
but there is no need to eject this liquid (such as for example in ink-jet printing): it is sufficient to bring a bit of liquid in contact with the ball to pick up
(see further). Besides this dispensing functionality, the other functions of the
developed gripper tip can be summarized as follows: (1) to develop a picking
force larger than the weight of the object (W ≈ 3.8 µN); (2) to develop a picking force large enough to handle the component with reasonable accelerations
(manual handling); (3) to center the ball with respect to the gripper to ensure
its positioning; (4) to release the ball once it is inserted in the hole. The proposed gripper is shown in Fig. 10.2b and c. Prototypes have been machined
in stainless steel and some of them have been coated with a hydrophobic
silane-based coating.
Two solutions have been tried to supply the gripping liquid: (1) to drive
the pressure through the gripper channel (see Fig. 10.2a); (2) to dip the tip
in the liquid. Finally, the first solution has been discarded because of the
10.2 Presentation of the Case Study
G
Response time
F
Force
85
Viscosity
Liquid
Surface tension
Roughness
Material
Impurities
B
A
Contact angle
Solid
Channel
C
Geometry
Gripper geometry
Volume of liquid
D
Hanging volume
E
Actual volume
Fig. 10.3. Overview of the required models
instabilities of the generated droplet when its height approaches half the
diameter of the gripper. It has been taken advantage of the good repeatability of the volume transferred to the gripper by dipping it into the liquid
(and consequently, the internal channel has been suppressed).
The study of this surface tension gripper involves several models: the
network linking the parameters is shown in Fig. 10.3. The solid considered
in this problem is related to its wetting properties, determined at least by the
type of material, the surface roughness, and the surface impurities, i.e., the
chemical heterogeneousness of the surface. These parameters therefore define
the kind of solid through model A, which is unknown. This solid surface,
together with the surface tension of the liquid, determines the contact angle,
through model B. What we call model B covers for example the Cassie or
Wenzel models which take the surface roughness into account (see Sect. 11.3).
Unfortunately, these ones are not sufficient to quantitatively describe all the
information of contact angles: the hysteresis for example cannot be predicted.
Therefore, model B is also considered as unknown, and the contact angles will
be measured as inputs of the models F and G (see Sect. 19.6 for the contact
angles measurement).
The grippers are essentially defined by their so-called geometry (i.e.,
the diameter of their tip, which is equal to the ball diameter, the value of the
angle of their conical tip (20◦ in Fig. 10.2a)), and for some of the grippers
the presence of an internal channel initially intended for the liquid supply. The
characterization of these geometries (model C) is summarized in Sect. 19.2.
The proposed tip dipping method is labeled D in Fig. 10.3 because it
determines the volume of liquid hanging below the gripper tip. Unfortunately,
at the first contact between the gripper and the component, a part of this
86
10 Example 1: Application to the Modeling of a Microgripper
liquid is left on the component and therefore the hanging volume of liquid
is different from the actual volume of liquid involved in handling the task.
Unfortunately, the relation between both volumes (model E) is neither easy
to described nor easy to measure.
In a static description, the force developed by the gripper depends on
the surface tension, the contact angles, the gripper, geometry and the actual
volume of liquid. This relation is called model F and can be described either
analytically (see Sect. 10.3) or numerically (see Sect. 10.4). Note that both
models have been benchmarked (see Sect. 10.5) and that an experimental
validation is partially presented in Sect. 20.5.
10.3 Analytical Model Based on the Circle
Approximation
The theoretical capillary force developed by the gripper can be computed by
using the so-called circle approximation presented in Sect. 7.4 (the detailed
problem is shown in Fig. 10.4). Let us consider the situation depicted in
Fig. 10.4b: the meniscus wets the ball along the circle containing B. (The
D
d
Gripper bearing surface
J
F
E
θ1
I
θ2
x
H
α
B
R
R2
β
A
z
I
G
C
α
C
β
RB
O
B
θ
Rsinb
R1
R
Handled ball
Symmetry axis
(a)
β
A
(b)
Fig. 10.4. (a) The configuration to model, including a cylindrical approached meniscus shape; (b) detailed view of the meniscus and the contact angles (Reprinted with
permission from [112]. Copyright 2006 Institute of Physics.)
10.3 Analytical Model Based on the Circle Approximation
87
symmetry axis of Fig. 10.4b is perpendicular to this circle and contains its
center. The position of this circle, whose radius is equal to RB , is determined
by the filling angle β.)
The contact angle θ1 is the angle between the tangent to the meniscus on
the one hand and the tangent to the ball on the other hand: It is determined
by the wetting properties of the materials, i.e., by the triple {handling liquid,
material of the ball, surrounding environment}. On the gripper side (point C),
since the gripper and the ball can be made of different materials, the contact
angle θ2 can be different from θ1 . The circle approximation assumes that the
meniscus has a circular shape, centered in O and with a radius R2 . Note that
in Fig. 10.4a, D has been chosen two times the ball radius; indeed, the gripper
has been designed as large as possible (for manufacturing reasons and to get
a larger capillary force) without being larger than the ball to allow it to enter
inside the bearing. With these assumptions, the capillary force5 exerted by
the meniscus on both solids can be computed according to (6.11) and using
2H =
1
1
+
,
R1
R2
(10.1)
where R1 and R2 are the principal curvature radii of the meniscus. R2 can be
directly read in Fig. 10.4b and, according to its definition, R1 is given by
R1 =
Since θ =
ten as
R1 =
RB
.
cos θ
π
2
(10.2)
− θ1 − β and RB = R sin β, the curvature radius R1 can be writ-
R sin β
,
sin (θ1 + β)
(10.3)
where R is the ball radius, β is the filling angle (which is a function of the
dispensed volume of liquid as it will be discussed later), and θ1 is the contact
angle at the ball side.
R2 can be determined by writing the distance z (Fig. 10.4b) on two different ways:
z = R(1 − cos(β − α)) = R2 (cos θ2 + cos(θ1 + β − α)),
(10.4)
where α is given by the geometry of the gripper (in Fig. 10.2a, α = 20◦ ).
Consequently, the capillary force of (6.11) can be rewritten into
sin(β + θ1 ) cos θ2 + cos(θ1 + β − α)
+
sin β
1 − cos(β − α)
+2πR sin βγ sin(β + θ1 )
F = πR sin2 βγ
5
(10.5)
Since only equilibrium situation is considered, and since the meniscus weight can
be neglected – due to a vanishing Bond number’ – the forces exerted by the
ball and the gripper onto the meniscus balance. Therefore, the (reaction) forces
exerted by the meniscus on the ball and on the gripper are equal.
88
10 Example 1: Application to the Modeling of a Microgripper
This equation gives the force as a function of the handling liquid (γ), the
materials (θ1 , θ2 ), the size of the ball (radius R), the volume of liquid through
the filling angle β, and the gripper geometry (α). Let us note that the meniscus
is not defined for β = α, leading to nonphysical result for the force (the force
tends toward infinity, which is discussed again in Sect. 10.6).
The link between the dispensed volume of liquid V and the filling angle β
can be determined as follows (see also the result in Fig. 10.5): In Fig. 10.4a,
the dispensed volume of liquid fills the BCEF GH area between the ball and
the gripper, therefore
V = V1 − V 2 − V 3 ,
(10.6)
where V1 is the sum of the volumes of the cylinder BCF G and the cone
CEF , V2 is the volume of the spherical cap of the ball limited by the filling
angle β, and V3 is the volume EJF . These volumes are given by the following
equations:
π
πR3 sin β 2
(1 − cos(β − α)) + R3 sin β 3 tan α,
cos α
3
1
2πR3
3
V2 =
1 − cos β + cos3 β ,
3
2
2
3
d
V3 = π tan α.
24
V1 =
(10.7)
(10.8)
(10.9)
The results of this analytical model are given in Sect. 10.5, together with
those of the numerical model presented in Sect. 10.4.
60
Filling angle [˚]
50
40
30
20
10
0
0
0.5
1
Volume of liquid [m³]
1.5
2
x 10−12
Fig. 10.5. Filling angle β in degree as a function of the volume of liquid in m3
10.4 Numerical Model Based on the Laplace Equation
89
10.4 Numerical Model Based on the Laplace Equation
The method presented in Sect. 8.4 has been adapted to the current case study,
that is the two first-order differential equation systems (8.14) can be rewritten
into
Y = f (z, Y ),
where
(10.10)
r
Y =
,
r
f (z, Y ) =
1+Y (2,1)2
Y (1,1)
Y (2, 1)
∆p
− γ (1 + Y (2, 1)2 )3/2
.
As already mentioned, this equation can be solved only if ∆p is known (and
actually it is not in advance) and with the following initial conditions for
z0 = R cos β (see Fig. 10.6):
R sin β
.
(10.11)
Y0 =
tan β + θ1 − π2
Since ∆p is not known in advance, it will be guessed and iterated until the
computed meniscus converges to the angle θ2 on the gripper side.
The algorithm can be expressed as follows:
1. Choose β
2. Choose ∆p
z
z + r tanα - R cosα - R sinαtanα = 0
z
z + r tanα - R cosα- R sinα tanα = 0
θ1
90−b
θ
α
B
r = r(z)
α
B
z0
Event detection
z0
A
A
R
R
β
β
z - r cotanb = 0
r
r
r0
r0
Symmetry axis
Symmetry axis
(a)
Fig. 10.6. (a) Initial conditions: tan θ =
interval for the pressure difference
(b)
dr
dz
; (b) rough evaluation of the initial
z=0
90
10 Example 1: Application to the Modeling of a Microgripper
We propose to evaluate an approximated initial interval for ∆p, so that a
dichotomic search can be achieved (the nonanalytical function between θ2
and ∆p is monotonic). To do so, let us compute the distance L between
A and B in Fig. 10.6, given by
L2 = (rA − rB )2 + (zA − zB )2
2
R tan β
− rB
= R sin β −
cos(α)(1 + tan α tan β)
2
R
+ cos β −
.
cos α(1 + tan α tan β)
(10.12)
From this distance, we deduce the radii ρ1 and ρ2 of curvature of the
extremal approached menisci shown in Fig. 10.6b under the form of a
circle:
L
(10.13)
ρ1 = ,
2
L
ρ2 = − .
(10.14)
2
The corresponding pressure differences6 are therefore given by
4γ
,
(10.15)
∆pmax =
L
4γ
(10.16)
∆pmin = − .
L
The search for the right ∆p is done within this interval by dichotomic
search.
3. Integrate (10.10) with the initial conditions (10.11) and the value of ∆p
chosen in point 2. This integration has to be led until the computed meniscus crosses the profile of the gripper, i.e., while
z + r tan α − R cos α − R sin α tan α ≤ 0.
(10.17)
This “until” condition is achieved by the events detection embedded in
the ode suite of Matlab.
4. Compute the angle made by the meniscus and the gripper (see Fig. 10.7):
π
− φ + α,
(10.18)
2
dr where tan φ = dz
. The pressure difference is adapted by dichotomic
end
search, i.e.,
θ2 =
∆pmax
∆pmin
(i)
(i)
∆p(i+1)
6
= ∆p(i) ,
if θ2 is to small
= ∆p(i) ,
∆pmin
=
if θ2 is to large
(i)
+ ∆pmax
2
(i)
.
These values have been actually doubled in the simulation code.
(10.19)
10.4 Numerical Model Based on the Laplace Equation
91
z
φ
α
θ2
α
B
z0
A
R
β
r
r0
Symmetry axis
Fig. 10.7. Determination of the computed value of θ2
z
z
d/2
C
V4
V = V1+V2-V3-V 4
D
α
V2
α
V3
B
V1
A
E
R
R
β
β
r
Symmetry axis
r
Symmetry axis
(a)
(b)
Fig. 10.8. Determination of the volume of liquid V = V1 + V2 − V3 − V4 . If there is
no internal channel, V4 = 0 (the diameter d vanishes)
While the value if θ2 does not fall within an error interval [θ2 (1 −
error), θ2 (1 + error)], reiterate the previous steps from point 3.
5. Compute the volume of liquid corresponding to the value of β chosen in
point 1 (see Fig. 10.8):
V = V1 + V 2 − V 3 − V 4
(10.20)
92
10 Example 1: Application to the Modeling of a Microgripper
z
z
q1
p1
90−b
q1
θ
FT
α
C
B
FL
p0
α
FT(attractive)
p2
B
A
A
FL(repulsive)
β
θ
90−b
FL(attractive)
R
β
R
r
r0
Symmetry axis
r0
r
Symmetry axis
(a)
(b)
Fig. 10.9. (a) Closed gripper (no dispensing channel): The tension force is always
attractive while the Laplace force (as depicted) is attractive if the pressure inside
the liquid is smaller than the surrounding pressure. (b) Gripper with a dispensing
channel: If the tightness in C is considered, the Laplace force is repulsive between
the symmetry axis and C and becomes attractive between C and A, at least if
the curvature of the meniscus is negative (as shown). The tension force is always
attractive
with
V1 =
n−1
πY (1, i)2 (zi+1 − zi ),
(10.21)
i=1
π
Y (1, end)3 tan α,
3
1
2πR3
3
V3 =
1 − cos β + cos3 β ,
3
2
2
3
d
V4 = π tan α.
24
6. Compute the capillary force.
In the case of Fig. 10.9a, the force is given by
V2 =
FL = − π(R cos β)2 ∆p,
π
,
FT = 2πR cos βγ cos β + θ1 −
2
F = FL + FT .
(10.22)
(10.23)
(10.24)
(10.25)
(10.26)
(10.27)
The sign ‘−’ indicates that a negative pressure difference ∆p leads to an
attractive force. In the case of a gripper fed with a pressure p1 (case of an
internal channel), the Laplace term should be replaced by
FL = −π(R sin α)2 (p1 − p0 ) − πR2 (sin2 β − sin2 α)∆p,
(10.28)
10.5 Benchmark
93
where in this case ∆p = p2 −p0 . Let us note that an attractive force is counted
positive in these equations.
10.5 Benchmark
The first result will be a validation by benchmarking the numerical model
in the case of a cylindrical gripper (α = 0) for which the capillary force is
expressed by the Israelachvili approximation [89]:
F = 2πRγ(cos θ1 + cos θ2 ).
(10.29)
The force approximation is F ≈ 0.196 mN (see Fig. 10.10).
The influence of the gripper angle α is plotted in Fig. 10.11, from which it
can be seen that α influences the position of the force peak (a finer analysis
will be led in next section concerning the significance of the apparently infinite
peak).
The influence of the coating of the gripper (i.e., the influence of the contact
angle θ2 is shown in Fig. 10.12, from which it can be seen that the force is still
attractive with nonwetting coated grippers (i.e., θ2 > 90◦ ), which is interesting
for our application.
To conclude, it should also be observed that (1) it is almost impossible to
work with filling angles smaller than the gripper angle because the volume
to dispense is much too small and the sensitivity of the filling angle (and
therefore of the force) is much too large with respect to a small variation of
the dispensed volume of liquid; (2) the force is of the order of 0.1 mN (for a
gripper diameter of 0.5 mm).
2
x 10−4
← FIsraelachvili
1.9
Force [N]
1.8
1.7
1.6
1.5
1.4
1.3
0
10
20
30
40
50
60
Filling angle [°]
Fig. 10.10. Benchmarking of the proposed numerical model (solid line) by comparison with the analytical approximation of (10.29) (square). Let us note that this
approximation is valid for a vanishing filling angle. Value of the parameters: α = 0,
θ1 = θ2 = 30◦ , R = 250 µm (or D = 500 µm), d = 100 µm, γ = 72 × 10−3 N m−1 ,
error on θ2 < 1%
10 Example 1: Application to the Modeling of a Microgripper
4
5
6
4
4
3
2
2
0
1
0
10
20
30
40
Filling angle [°]
◦
0.01
3
0.008
2.5
2
0.006
1.5
0.004
1
0.5
0.002
0
−0.5
0
0
60
50
0.012
3.5
6
8
−2
x 10−12
−3
x 10
7
Force [N]
Volume of liquid [m³]
−12
Force [N]
x 10
10
Volume of liquid [m³]
94
10
20
30
40
Filling angle [°]
0
60
50
(b) α = 10◦
(a) α = 5
Fig. 10.11. Comparison between the numerical (solid lines) and the analytical
(dashed lines) models for different geometries. The increasing curves with circle
marks represent the volume of liquid (left-hand side y label) and the curves with
the peak represent the force (right-hand side y label). Value of the parameters:
θ1 = θ2 = 30◦ , R = 250 µm (or D = 500 µm), d = 100 µm, γ = 72 × 10−3 N m−1 ,
error on θ2 < 1%
0.012
0.008
2.5
2
0.006
1.5
0.004
1
0.5
0.002
0
−0.5
0
10
20
30
40
Filling angle [°]
50
0
60
Volume of liquid [m³]
0.01
3
Force [N]
Volume of liquid [m³]
3.5
3
x 10−3
1.4
x 10−12
1.2
2.5
1
2
1.5
0.8
1
0.6
0.5
0.4
0
0.2
−0.5
0
(a) θ2 = 30◦
10
20
30
40
50
Force [N]
x 10−12
4
0
60
Filling angle [°]
(b) θ2 = 110◦
Fig. 10.12. Comparison between the numerical (solid lines) and the analytical
(dashed lines) models for different coatings. The increasing curves with circle marks
represent the volume of liquid (left-hand side y label) and the curves with the peak
represent the force (right-hand side y label). Value of the parameters: α = 20,
θ1 = 30◦ , R = 250 µm (or D = 500 µm), d = 100 µm, γ = 72 × 10−3 Nm−1 , error on
θ2 < 1%
10.6 Pressure Difference Saturation
At first sight, the meniscus seems not to be defined for β = α: since the gap
between the gripper and the component leads to zero in this point, the pressure difference given by the Laplace equation (∆p = 2Hγ) leads to infinity
as shown in Fig. 10.13. In this figure, the force (circle marks) is maximum
for β = α = 20◦ (by the way, let us note the respective contributions of
the tension term (− − − − −) and laplace term (triangular marks, which
10.6 Pressure Difference Saturation
3
95
x 10−12
0.01
0.009
2.5
0.007
2
0.006
1.5
0.005
0.004
1
Force [N]
Volume of liquid [m³]
0.008
0.003
0.002
0.5
0.001
0
0
10
20
30
Filling angle [°]
40
50
0
60
Fig. 10.13. Force as a function of the filling angle β (D = 0.5 mm, α = 20◦ ,
θ1 = 8◦ , θ2 = 50◦ , γ = 72 mN m−1 ). Solid line: volume of liquid, ———–: tension
term, triangles: laplace term, circles: total capillary force, · · · · · · : limiting value due
to cavitation (Reprinted with permission from [112]. Copyright 2006 Institute of
Physics.)
3
x 10−12
x 10−3
Shift
3.5
3
2
2.5
2
1.5
1.5
Force [N]
Volume of liquid [m³]
2.5
1
1
0.5
0
0.5
0
10
20
30
Filling angle [°]
40
50
0
60
Fig. 10.14. Shift of the maximal force due to the pressure difference saturation
(D = 0.5 mm, α = 20◦ , θ1 = 8◦ , θ2 = 50◦ , γ = 72 mN m−1 ). Solid line: volume
of liquid, − − −−: tension term, dots: Laplace term, circles: total capillary force
(Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)
96
10 Example 1: Application to the Modeling of a Microgripper
are superposed here with the circle marks)). Nevertheless, the pressure difference cannot be unlimited, since the external pressure is assumed to be the
atmospheric pressure. Consequently, the pressure difference used in the force
calculation cannot be larger than 101,400 Pa, or even less if the cavitation of
the liquid is assumed to occur at the vapor pressure. For example, in the case
of water at 20◦ C, the vapor pressure is equal to 2,839 Pa. For isopropanol,
the vapor pressure at 20◦ C is equal to 44,029 Pa. Therefore, the Laplace force
cannot exceed a value FM = Area× (ambient pressure−vapor pressure). This
limiting value is plotted in Fig. 10.13 (· · · · · · ). This leads to a shift of the
maximal force as illustrated in Fig. 10.14.
10.7 Conclusions
This chapter has illustrated on a watch bearing related case study the arc circle
approximation and the numerical approach in the preceding chapters. As a
summary of this chapter, let us note that the developed force is about 100 µN
for a 500 µm diameter gripper, that this force is increasing with decreasing
volumes of liquid (however behind the typical force peak), that the force
still remains attractive in the case of moderate non wetting grippers (θ2 =
110◦ ). The experimental characterization of the involved parameters will be
described in Chap. 19 and gripping experiments will be lead in Chap. 20.
11
Second Set of Parameters
11.1 Introduction
This chapter presents more advanced aspects related to the capillary force
modeling. In particular, as we have seen in Fig. 10.3, the contact angle which
is used as an input of the model depend on several parameters such as for
example surface heterogeneities, surface impurities, and surface roughness.
Moreover, equilibrium state has been assumed in all the force models. The
limitations implied by this assumption will be considered in Chap. 12, but
the damping effect occurring when considering the dynamics of the problem
will be briefly considered in this chapter.
11.2 Surface Heterogeneities and Surface Impurities
Let us assume a heterogeneous surface containing two materials 1 and 2. A
fraction f1 of this surface is characterized by a surface energy leading to a
contact angle θ1 and the other part of the surface (fraction f2 = 1 − f1 ) leads
to the contact angle θ2 (Fig. 11.1a). The theoretical contact angle given by
the Young equation (6.1) is modified into an effective contact angle θC given
by the Cassie equation [1, 90]:
cos θC = f1 cos θ1 + f2 cos θ2 .
(11.1)
Another expression has been proposed by [90] but it seems that for the same
values of θ1 , θ2 , f1 , and f2 , it will always predict a smaller contact angle than
that obtained with (11.1)
(1 + cos θC )2 = f1 (1 + cos θ1)2 + f2 (1 + cos θ2 )2 .
(11.2)
98
11 Second Set of Parameters
qR
θΑ
Hysteresis
Hydrophilic region
leads to a receding angle
smaller than expected
Hydrophobic region
leads to an advancing angle
higher than expected
(a)
fi
qi
Model
qC
(b)
Fig. 11.1. Influence of surface impurities or surface heterogeneities: (a) Macroscopic
example of a smooth heterogeneous surface [133], illustrating the advancing contact
angle θA when the meniscus is about to move downward and the receding contact
angle θR when the meniscus is about to move upward. The contact angle hysteresis
is equal to θA − θR ; (b) Schematic model to modify contact angles
11.3 Surface Roughness
Let us assume a droplet placed on a rough substrate: Due to the roughness
asperities, the actual area is bigger than the apparent one. Let us now introduce δ the ratio of the actual interface area to the apparent one. The area of
the actual (i.e., rough) area of the solid–vapor (solid–liquid) interface is denoted by ASV (ASL ). The apparent surface is a projection of the rough surface
(see Fig. 11.2):
δ=
ASL
ASV
=
.
AApparent
AApparent
(11.3)
Using δ, (6.2) can now be rewritten into
∆G = δ ∆AApparent γSL − δ ∆AApparent γSV + ∆AApparent γ cos θ
. (11.4)
lim∆A→0 ∆G
∆A = 0
Combining (11.4) and the expression of the contact angle given by the
Young equation, the effective contact angle θrough can be expressed as a function of the surface ratio δ and the contact angle θsmooth made of the liquid on
a plane smooth substrate made of the same material:
cos θrough = δ cos θsmooth .
(11.5)
This approach was first proposed by Wenzel and more detailed information can
be found in [1] and [73]. Henceforth, (11.5) can feed the previous simulation
with contact angles corresponding to actual rough surfaces. That is important
if the simulation is used to design gripper tips that usually present roughness
profiles.
From (11.5), we see that angles lower than 90◦ are decreased by roughness,
while the angle increases if θ is larger than 90◦ . This means that the gripping
force of a capillary gripper could be increased by using rough gripper tips.
11.4 Static Contact Angle Hysteresis
γ
99
Rough surface
vapour
liquid
γSL θ
Apparent
surface
gSV
Projection lines
solid
(a)
(b)
δ
qsmooth
Model
qrough
(c)
Fig. 11.2. Influence of surface roughness: (a) Contact line on a rough substrate;
(b) Actual and apparent surfaces; (c) Model to modify contact angles
It must be noted that surface roughness can lead to condensing humid air
in small cavities of the surface and hence to an attractive force Lcp due to
liquid bridging [103]:
Al γ
,
(11.6)
Lcp =
rk
where Al is the surface area where meniscus formation occurs and rk is the
Kelvin radius given by the Kelvin equation [1]1 :
γv
rk =
,
(11.7)
RT log (p0 /p)
where v is the molar volume of the liquid, R is the perfect gas constant, T is
the absolute temperature, and p0 /p is the relative vapor pressure (= relative
humidity for water). Israelachvili [89] gives γv/RT =0.54 nm for water at 20◦ .
11.4 Static Contact Angle Hysteresis
When the contact line is about to move, one observes contact angle changing.
The receding angle is smaller than the static angle while the observed angle,
when moving forward, is larger than the static contact angle. A model has been
proposed by Zisman (see Adamson [1]) who observed that cos θA (advancing
angle) is usually a monotonic function of γ. Henceforth, he proposed the
following equation:
cos θA = a − bγ
(11.8)
Reference [73] cited [95] for a detailed study of the effect of roughness on
contact angle hysteresis. This hysteresis implies that even at equilibrium, the
contact angle value is not unique. A way to improve the static simulation can
be to use the receding contact angle during the picking task: indeed, at least
at the beginning of the picking task, the object is in contact with the gripper
and the meniscus spreads out over its maximum area. Consequently, during
the vertical translation of the gripper, the meniscus is supposed to retract
(see Fig. 11.3). Hence, the contact angle value should be that of the receding
angle. Nevertheless, experiments show that the actual angle is closer to the
advancing contact angle: This will be discussed in Sect. 18.3.
1
log = loge = ln = log10 .
100
11 Second Set of Parameters
(a) Advancing contact angle θA
(b) Receding contact angle θR
Fig. 11.3. Contact angle hysteresis
3
2.5
log θ
(180°)
2
1.5
1
0.5
−5
−4
−3
−2
−1
log Ca
0
1
2
Fig. 11.4. Dynamic contact angle θ as a function of the capillary number Ca =
µV
(µ=dynamic viscosity (Pa s), V =velocity of the contact line (m s−1 )) for several
γ
silicone oils flowing in a 1.955 mm diameter glass capillary. (Reprinted from [84],
with permission from Elsevier)
11.5 Dynamic Spreading
Unfortunately, the contact angle hysteresis cannot be put aside only by choosing the advancing or the receding angle, depending on the relative motion of
the object and the gripper. Indeed, the contact angle also depends on the
velocity of the contact line. This phenomenon is described in [84]. We illustrate it with their results in Fig. 11.4. According to a representation in [46],
the results are presented in the logarithmic scale, showing the law:
√
3
(11.9)
θ ≈ V,
where θ is the dynamic contact angle and V is the velocity of the contact line.
More information on the way to model the contact angle will be searched
in order to use it in the simulation. We see that the problem will be more
complex than just feeding the static simulation with different contact angles
as those suggested in sect. 11.4: The spreading of the liquid bridge is a dynamic
process. Reference [71] presents results of a force measure during the approach
of a tip with a 0.28 µL droplet hung on it: We can see the evolution of this
force in Fig. 11.5. An interesting observation is that during phase C (A, B,
and C are included in the approach phase of the tip toward the surface),
11.6 Conclusions
101
Force (mN)
20
10
0
6
8
10
12
14
16
18
20
22
Time (s)
Fig. 11.5. Force as a function of time: repulsive force briefly occurs during phase
C (A, B, C = approach phase) – Water droplet (volume=0.28 µL on a Si chip [72]):
This effect highly depends on the volume of liquid and is not observed with a 0.19 µL
volume. (Reprinted with permission from [72], Copyright 2002 Springer)
the capillary force becomes repulsive: Benefit can be taken from this positive
pressure in a capillary gripper because this damping effect prevents damage
at contact, which is one of the main drawbacks of a vacuum or mechanical
gripper.
11.6 Conclusions
If roughness and impurities can be taken into account by correcting the contact
angle with (11.1), (11.2), and (11.5), dynamic effects cannot be obtained by a
static simulation: It is one of the perspectives to develop a dynamic simulation.
Moreover, a companion topic linked to these simulations is the problematics of
capillary condensation. This problematics can be divided into two problems:
–
–
The capillary condensation at the tip of a surface force apparatus (SFA):
A modeling of the condensation menisci based on the arc approximation is
presented in [161]. Our static simulation could be adapted to this problem
easily because only one iteration loop is required on the starting position
of the liquid bridge in order to adjust the contact angle θ1 . Indeed, ∆p is
directly given by (11.7).
The capillary condensation due to roughness of the microsystems. This
problem is addressed in [176] and in [126]. These authors investigated this
aspect because of its implications on the reliability of microsystems but
we have not found any study of this subject in relation with micromanipulation or microassembly.
The problematics of capillary condensation has been pointed out as a
source of MEMS breakdown [104, 125, 179, 184] and has been applied to
microassembly by [36, 33].
12
Limits of the Static Simulation
12.1 Introduction
This chapter aims at validating the static simulation at constant volume in
the limits of an assembly case study. Basically, the static simulation cannot
afford to give information on the typical cycle time of an surface tension
based assembly task (in other words, since the time is not a parameter of the
static simulations, the latter cannot output any characteristic time). Therefore, some approximations will be tempted using dimensional analysis and
a 1D analytical model derived from the so-called Lucas–Washburn equation.
Moreover, high accelerations are usually applied in assembly machines (up to
10g): therefore, it should be checked whether this acceleration can deform the
equilibrium meniscus shape or not.
12.2 Performances of the Assembly Machines
We will denote by a the typical acceleration of an assembly machine, v its typical velocity, τpick and τplace the typical pick and place operation times. Values
are given in Table 12.1 (source: [174]). As far as acceleration is concerned
modern linear motors [61] reach up to 100 m s−2 .
12.3 Nondimensional Numbers and Buckingham
π Theorem
An adimensional analysis has been achieved, involving all the parameters
defined in Table 12.2. Knowing the SI dimensions of each dimensional
parameter, the dimensional matrix D can be built, whose element Dij is the
exponent of the jth dimension in the ith parameter:
104
12 Limits of the Static Simulation
Table 12.1. Typical values of an assembly machine
a
20 m s−2
v
1.5 m s−1
τpick
60 ms
τplace
60 ms
Table 12.2. List of the parameters involved in the simulation (Ci =coefficient)
Ci Parameter M L T
C1
µ
1 −1 −1
γ
1
0 −2
C2
C3
d
0
1 0
γSL
1
0 −2
C4
γSV
1
0 −2
C5
F
1
1 −2
C6
a
0
1 −2
C7
C8
v
0
1 −1
r
0
1 0
C9
ρ
1 −3 0
C10
C11
g
0
1 −2
Description
Dynamic viscosity of the used liquid
Surface tension
Characteristic size of the meniscus
Interfacial energy solid–liquid
Interfacial energy solid–vapor
Force exerted by the meniscus
Gripper acceleration
Gripper characteristic velocity
Gripper travelling range
Density
Gravity
⎛
1 −1
⎜1 0
⎜
⎜0 1
⎜
⎜1 0
⎜
⎜1 0
⎜
D=⎜
⎜1 1
⎜0 1
⎜
⎜0 1
⎜
⎜0 1
⎜
⎝1 −3
0 1
Value
1–50 mPa s
20–72 mN m−1
10−4 m
100 m s−2
0.1–0.3 m
1000 kg m−3
9.81 m s−2
⎞
−1
−2⎟
⎟
0⎟
⎟
−2⎟
⎟
−2⎟
⎟
−2⎟
⎟.
−2⎟
⎟
−1⎟
⎟
0⎟
⎟
0⎠
−2
These n = 11 dimensional parameters can be grouped to form m = n − k
adimensional numbers, where k states for the rank of the matrix D [162].
These adimensional numbers πi , (i = 1 : 10) are written as follows:
C4 C5 C6 C7 C8 C9 C10 C11
πi = µC1 γ C2 dC3 γSL
γSV F a v r ρ g ,
(12.1)
where the n = 11 coefficients Ci are the elements of a 11 × 1 vector C =
(C1 , C2 , ..., C11 )T , given by the solutions of the following equation [162]:
T
D C = 0.
(12.2)
3×11 11×1
It means that each vector C is a vector of the kernel of the matrix D, which
can be for example obtained using the Matlab command null(D,’r’). In this
case, this leads to 8 = 11 − 3 adimensional numbers (the rank of the matrix
D is equal to 3):
12.3 Nondimensional Numbers and Buckingham π Theorem
π1 =
π2 =
π3 =
π4 =
π5 =
π6 =
π7 =
π8 =
105
γSL
,
γ
γSV
,
γ
F
,
γd
µ2 da
,
γ2
µv
,
γ
r
,
d
γdρ
,
µ2
µ2 dg
.
γ2
These eight π terms can be recombined with one another, leading to eight
other formulations, among which:
(π5 .π7 ) → Re =
(π52 .π7 ) → We =
(π52 /π7 ) → Fr =
(π4 .π7 ) → Be =
(π7 .π8 ) → Bo =
=
(π2 − π1 ) → cos θ =
τ =
=
Inertial effects
ρdv
=
,
µ
Viscous effects
Inertial effects
v
,
=
Surface tension effects
γ/ρd
v
Intertial effects
√ =
,
Gravity effects
dg
Inertial effects
aρd2
=
,
γ
Surface effects
Gravity effects
dρg
=
γ/d
Surface tension effects
Hydrostatic pressure
,
Laplace pressure
γSV − γSL
= Young–Dupré equation
γ
µd/γ
µv
=
d/v
γ
Characteristic time of the droplet
.
Characteristic time of the manipulation
(12.3)
(12.4)
(12.5)
(12.6)
(12.7)
(12.8)
(12.9)
These results represent the classical adimensional numbers (Reynolds, Weber,
Froude and Bond number). Be is called “Bonding Effect Number” [19]. The
term µd/γ represents the characteristic droplet time, which is the time taken
by the droplet to move its interface along the characteristic distance d. By
comparison, the term vd represents a characteristic manipulation time at
106
12 Limits of the Static Simulation
release: indeed, if the object is being released, the meniscus is stretched over a
characteristic length d at the gripper velocity v. However, if the object is not
being released (that means that object and gripper remain in contact), the
manipulation time must be considered the one given in Table 12.2. Finally,
2
the term ρdµ has the dimensions of a time and gives an order of magnitude of
the damping time of the transitory effects occurring inside the droplet.
12.4 Another Approach: Use of a 1D Analytical Model
A first approximated description can be performed through a simplificated
1D geometry shown in Fig. 12.1. The capillary rise h of a liquid between
two infinite parallel plates separated by a distance d can be described by the
following equation:
12µhḣ
d(hḣdρ)
=−
+ 2γ cos θ − hdρg + po d,
dt
d
(12.10)
where h and ḣ are the position and the velocity of the liquid–gas interface, d
is the separation distance between the parallel plates, ρ is the liquid density,
µ its dynamic viscosity, γ its surface tension, g the gravity acceleration, and
po the pressure difference between points A and B. The left-hand side of this
equation states for the time derivative of the liquid momentum, i.e., the time
derivative of the mass×velocity product, where ḣ is the interface velocity and
hdρ is the liquid mass involved in the motion. This description is a bit similar
to the case in which a chain, initially at rest on a substrate, is pulled upward:
the mass of the system is increasing with time, and the pulling velocity can
vary. The first term on the right-hand side represents the viscous force, working against the motion: it can be estimated from a Hagen–Poiseuille stationary
flow. The velocity distribution is parabolic:
θ
θ
B
L
h
d
d
A
(a)
Fig. 12.1. 1D analytical model
(b)
h
12.4 Another Approach: Use of a 1D Analytical Model
v=−
1 dp
y(d − y),
2µ dx
107
(12.11)
where dp/dx is the (constant) pressure gradient along the motion axis, d is
the separation distance between both parallel plates, and y is the coordinate
perpendicular to the motion direction x (therefore 0 ≤ y ≤ d). Integrating
this velocity profile on a section perpendiular to the flow we can get the mass
flow:
d
ρv dy = · · · = −
Q=
ρd3 dp
.
12µ dx
(12.12)
0
Hence we can express the pressure gradient as a function of the mean velocity
v̄ = Q/(ρd):
−
dp
12µv̄
= 2 .
dx
d
(12.13)
Since the mean velocity is equal to h, the viscous force can be determined as
follows:
F =−
12µhḣ
dp
hd =
.
dx
d
(12.14)
The second term on the right-hand side is the driving surface tension term,
equal to the surface tension projected along the motion direction and acting along the perimeter of the triple line. In this case, this term (as the all
equation) is expressed as a force per unit length.
The third term is the weight of the volume of liquid and the last term
represents the force exerted by the pressure difference between A and B: It
can be considered as a driving or a resistance term depending on its sign.
Now that (12.10) is explained, we can divide it by ρd:
12µhḣ 2γ cos θ
po
d(hḣ)
=−
− gh + .
+
2
dt
ρd
ρd
ρ
(12.15)
In [156], Shoenfeld proposes the following variables substitution:
z = h2 → ż = 2hḣ
(12.16)
leading to replace (12.15) by:
√
1
6µż
2 cos θ po
z̈ = − 2 +
+
− g z.
2
ρd
ρd
ρ
(12.17)
The classical analytical approach consists in determining the equilibrium
position from (12.17) and in studying small oscillations around this equilibrium position. The equilibrium position is given by:
108
12 Limits of the Static Simulation
ho =
po
2γ cos θ
+ .
ρgd
ρg
(12.18)
Around this equilibrium position, we consider small oscillation of z:
z = zo + δz,
(12.19)
where zo = h2o .
Equation (12.17) becomes:
δz̈ +
√
gδz
4γ cos θ 2po
12µδ ż
+
− 2g zo ,
+√ =
2
ρd
zo
ρd
ρ
(12.20)
=0
which is similar to:
ẍ + 2λẋ + ω02 x = 0,
(12.21)
whose solutions are:
λx0 + v0
sinh(ωt)) if λ2 > ω02 ,
(12.22)
ω
λx0 + v0
sin(ωt)) if λ2 < ω02 ,
(12.23)
x(t) = exp(−λt)(x0 cos(ωt) +
ω
where ω = |λ2 − ω02 | and, in both cases, x(t = 0) = x0 and ẋ(t = 0) = v0 .
√
In our case, λ = 6µ/ρd, ω02 = g/ z0 , and the initial conditions are:
x(t) = exp(−λt)(x0 cosh(ωt) +
δz(t = 0) = δzo ,
δ ż(t = 0) = 0.
(12.24)
(12.25)
We conclude that the characteristic damping time is equal to (see also
Fig. 12.2)
τ=
1
ρd2
=
.
λ
6µ
(12.26)
We see that the expression ρd2 /µ and (12.26) differ from a factor 6.
12.5 Limitations of the Static Model
In order to set limitations to the proposed constant volume static simulation, characteristic times have been compared and the constant velocity or
accelerated motions have been analyzed.
–
For water (ρ = 1,000 kg m−3 , γ = 72 N m−1 , µ = 1 mPa s and d = 100 µm),
the characteristic times have been determined (Table 12.3.)
12.5 Limitations of the Static Model
109
x 10−4
4
dh [m]
2
0 Damping time
−2
−4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time [s]
x
10−3
h [m]
6
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time [s]
Fig. 12.2. µ = 0.001 Pa s, ρ = 1,000 kg m−3 , γ = 0.072 Nm−1 , θ = 60◦ , d = 0.001m,
τ = 0.16667 s
Table 12.3. Comparison between the different characteristic times (ms): The damping time given by (12.26) is six times smaller, i.e., 1.6 ms, but this is only an order
of magnitude
Time (ms) Water
µd/γ
0.001
ρd2 /µ
10
τpick or τplace 60
–
–
If the whole system including the component, the meniscus, and the gripper are moving at constant velocity, the static description at constant
volume is of course valid. However, it can be assumed that there are residual vortex, leading to velocities inside the liquid, and hence to pressure
gradient that could modify the capillary force computed by the static simulation. These transitory effects are damped after a characteristic time
ρd2 /µ(< τpick , τplace ).
In the case of accelerated motions, inertial effects must been taken into
account: the object will stick to the gripper only if the acceleration of the
gripper a is lower than the limit acceleration ã given by:
ã =
F
− g,
m + mliq
(12.27)
110
12 Limits of the Static Simulation
where F is the capillary force, m the mass of the component, and mliq
the mass of the liquid. The whole system is now in a motion with constant acceleration and the situation could be described at rest in a moving
coordinates system with constant acceleration. In this case, the interface
shape could be affected by this acceleration. In the same way as gravity
effects on the interface shape are neglected if the characteristic
length of
the meniscus is smaller than the capillary length Lc = γ/ρg, the effects
of the constant acceleration of the gripper are negligible if the characteristic
than the modified capillary length
size of the meniscus is smaller γ
γ
−2
Lc = (a+g)ρ . For a = 10 ms , Lc = (a+g)ρ
≈ 0.8 mm.
12.6 Conclusions
As a conclusion of this chapter, it can be stated that the characteristic time
of a surface tension based picking task could have typical cycle times down to
a few milliseconds. The second conclusion is that equilibrium is a reasonable
assumption, even in high throughput machines (a = 10g), as long as the
characteristic size of the meniscus is smaller than about 0.8 mm.
13
Approaching Contact Distance, Rupture
Criteria, and Volume Repartition After
Separation
13.1 Introduction
According to the volume of liquid and the shape of the gripper, the hanging
droplet will be transformed into a liquid bridge as soon as the separation
between the gripper and the component is smaller than zA , the approaching
contact distance. The first problem treated in this chapter is to compute this
distance according to the gripper geometry and the volume of liquid V .
The two other problems considered in this chapter are on the one hand
the determination of the separation distance (also called the gap) at which
the meniscus will break, and on the other hand, the residual volume of liquid
remaining on the component after this rupture.
13.2 Approaching Contact Distance
The general situation is depicted in Fig. 13.1: a droplet hangs below the
gripper, the bottom of it being at a distance zA from the apex of the gripper. If
gravity is neglected, this droplet has the shape of a spherical cap defined by its
radius R and the angle α (note that the volume of the gripper Vgripper located
below y must be subtracted from the cap volume, given in Appendix B.1).
First, the intersection point P between the meniscus and the gripper must
be found: P is defined by its coordinates (x, y) (the origin of this coordinates
frame is located at the gripper apex, O). To do so, the position of P is iterated
until the meniscus volume calculated corresponds to the prescribed volume V :
this is illustrated in Fig. 13.2.
Once x is chosen, y can be determined as a function of the gripper equation. Then, the mathematical relations used in Fig. 13.2 are given below (the
expression depending on α in (13.5) is detailed in Appendix B.1):
tan φ|(x,y) =
dy
|(x,y)
dx
(13.1)
112
13 Approaching and Rupture Distances
φ
θ
α
R
α
Vgripper
zA
V
P
φ
y
O
x
Fig. 13.1. Approaching contact distance—general case
x
y = y (x,gripper)
f = f (x,y)
α = α (x,y,q)
Vgripper =V (x,y,gripper)
R = R(α,x)
V = V(R,α) - Vgripper
No
V=Vprescribed
zA
Fig. 13.2. Approaching distance algorithm
α(x, y) = θ + φ(x, y)
x
R=
sin α
Vgripper = V (φ, gripper)
1
3
2π 3
R 1 − cos α + cos3 α − Vgripper
V =
3
2
2
(13.2)
(13.3)
(13.4)
(13.5)
where φ states for the gripper slope at P and θ is the contact angle . At the end
of each iteration, the computed volume corresponding to the chosen point P
13.3 Rupture Distance and Residual Volume of Liquid
113
is compared to the prescribed volume V : if it is smaller (larger), x is increased
(decreased). Finally, the approaching contact distance is determined by
zA = R(1 − cos α) − y.
(13.6)
Note that in the case of the conical gripper, φ is equal to the gripper parameter
p, and consequently, α = θ + p is constant. If moreover p = 0 (cylindrical
gripper), Vgripper ≡ 0 and R is directly given by (13.5) so that zA is computed
in only one iteration.
The approaching contact distance is one of the design elements of a capillary gripper: it gives partial information on the required range of the gripper.
Other elements are the receding distance at which the meniscus breaks into
two separated droplets.
13.3 Rupture Distance and Residual Volume of Liquid
The problem of the rupture criterion and the volume repartition can be
expressed as follows:
1. The expression of a rupture separation distance z ∗ at which the liquid
bridge will be replaced by two droplets adhering to the gripper on the one
hand and to the component on the other hand
2. The repartition of the meniscus volume V into the two residual volumes
V1 and V2 of the droplets after rupture (see Fig. 13.3)
These questions must be addressed both to design the gripper kinematics
(what range should the gripper move in order to break the liquid bridge?) and
to investigate the suitability of the gripping principle (what volume of liquid
still remains on the component after rupture? The answer to this question can
z
z
V2
V2
V1
V1
r
(a)
r
(b)
Fig. 13.3. The volume repartition. (a) Meniscus before break; (b) corresponding
droplets after break
114
13 Approaching and Rupture Distances
possibly lead to the computation of the evaporation time if the used liquid is
volatile enough – which is the case with water, but not with silicone oil).
In this chapter, the repartition of liquid will be studied from both an
energetic and a geometrical point of view: the energetic approach consists in
determining the repartition that extremes the interfacial energy of the system.
Unfortunately, it will be shown that this extremum is actually a maximum
and moreover the volume repartition does not agree with the experimental
results in the case of nonsymmetric configuration (not equal contact angles).
Therefore, a second approach called geometrical approach – already proposed
in [142] – will be followed and implemented.
Once the repartition after breaking is known, the breaking gap can be
evaluated by energetic considerations: it will be assumed that the bridge turns
itself toward two droplets when the total interfacial energy of the liquid bridge
equals that of the two droplets. According to Pepin et al. [142] the energy will
be evaluated only at the liquid–vapor interface. As it will be shown that this
criterion is inexact, we will propose to correct it by taking into account the
interfacial energy of all interfaces: liquid–vapor, solid–vapor, and solid–liquid.
13.4 Mathematical and Notation Preliminaries
1. The volume V and surface S of the portion of a sphere characterized
by a radius R and an angle θ are given by the following equations (see
Appendix B.1):
3
2πR3
a) Vi = 3 i (1 − 32 cos θi + cos2 θi )
b) Si = 2πRi2 (1 − cos θi )
2. The volume V is constant:
• for analytical developments, the meniscus is assumed to be of cylindrical shape V = πr2 z
• the volume of the two spherical caps after rupture is given by V = V1 +
V2 ⇒ dV2 = −dV1
3
1
3
3
3. Ai ≡ 2π
3 (1 − 2 cos θi + 2 cos θi ) (As Ai R represents the volume of a
spherical cap with radius R and limited by an angle θi , Ai ≥ 0)
4. αi ≡ 16 (2 − cos θi − cos2 θi ) ≥ 0
i
5. A
αi = 2π(1 − cos θi )
2 α1 3
6. K ≡ A
A1 ( α2 )
2
7. K = A
A1
8. 2 cos θ ≡ cos θ1R + cos θ2R
9. γSVi = γSLi + γ cos θi
10. θi = contact angle made by the ith droplet, usually the advancing contact
angle
13.5 Volume Repartition
115
13.5 Volume Repartition
Two models will be studied:
1. Energetic model
For small gaps the interfacial energy of the meniscus W is smaller than the
interfacial energy U of the two droplets configuration (basically because
the area of the meniscus is smaller than that of two droplets with equivalent volume). By increasing the gap, W can become larger than U , leading
to the rupture of the meniscus. Once the meniscus is broken, the liquid
volume is split into two spherical caps adhering to the component and
to the gripper. The underlying idea of this method consists in searching
for a repartition of the volumes V1 and V2 that (if it exists) extremes
the interfacial energy U (let us note U ∗ this extremum). Let us note
that the interfacial energy is the sum of the energy of the liquid–vapor
(LV), liquid–solid (LS), and solid–vapour (SV) interfaces. However, some
authors [142] claim to get valid results by considering only the energy
of the LV interface. Consequently, both assumptions will be considered
and the notations related to the second one will be added with the “LV”
subscript. The following results are shown in Appendix A.2 for a cylindrical gripper:
a) In the case of the LV interface,
V
(13.7)
V1 =
1 + K
KV
V2 =
(13.8)
1 + K
(13.9)
θ 1 > θ 2 ⇒ V 1 < V2
2/3
2
2
A1 α2
V
A2 α1
∗
= γ
+
.
(13.10)
ULV
α1
α2
α23 A1 + α13 A2
(13.11)
b) In the case of the whole interface,
V
(13.12)
V1 =
1+K
KV
V2 =
(13.13)
1+K
θ 1 > θ 2 ⇒ V 1 > V2
(13.14)
∗
2/3
1/3
(13.15)
U = 3γV (A1 + A2 ) .
We see from Figs. 13.4 and 13.5 that V1 /V = 0.5 for symmetric configurations (i.e., θ1 = θ2 ).
However, we can conclude from (13.14) that the proposed method
(at least in the case of the total interfacial energy model) cannot predict the volume repartition in a suitable way because experiments show
that the larger the contact angle the smaller the corresponding volume.
Maybe the trend derived from the LV model could fit the experimental
116
13 Approaching and Rupture Distances
V1/V
5
0.
99
82
0.64
0.5
49
91
973
0.699
64
6
0.749
55
0.7994
70
0.84937
0.94919
0.89928
80
0.5
60
928
493
7
0.7
994
6
0.7
49
55
0.6
99
0.
64
59
98
0.
64
2
97
3
30
1
99
54
3
0.
10
0.8
0.98928
0.4793
7
09.9746
4
0.955
69
0.964
64
97
20
99
54
0.
09
50
0.4
8
001
0.4
0.8
0.89
40
5
0.
1
0.949
19
θ2 [°]
50
009
0.45
7
502
0.3
036
0
.3
0
5
0.
9
00 8
5
0.44001
0.
5
27
50
504
2
0.30036 0.2 0054
98
0.2 5063
.3
0
9
5
0.1
0. 0.5
2
0.1007
098
50001
00..44500455643
2.10500072 0.050808
00.0.2
0.1
10
20
30
27
0.350
0.30036
45
0.250
4
0.2005
0.15063
0.10072
0.050808
0.050808
40
θ1 [°]
50
60
70
80
Fig. 13.4. Ratio V1 /V as a function of contact angles θ1 and θ2 after rupture of
the meniscus (cylindrical gripper) – Case of the LV interface
3
0.
25
0.
0.
35
V1/V
80
5
4
0.
0.2
5
0.1
0.
45
0.
0.1
0.0
5
70
5
0.6
0.7
25
0.
3
60
0.
2
0.
5
0.0
40
0.
25
30
.55
5 06
0. 50.
0..76
0
5
5
5
0.95
3..43.4 0.87
0.000 5 0.
.8
00.9
0.0
0.51
0.
15
10
0.
2
20
10
20
3 35
0. 00..40.45
5
5 .5
0. 00.6
5
6
0..7
0
5
0..78
0
5
0.8
0.9
30
5
0.7
0.8
5 0.55
0.6
5
0.6
0.7
5
0.7
0.8
0.
35
0. 4 5
0. 0.4
0.1
0.
15
θ2 [°]
50
5
0.5
0.6
0.85
0.9
.85
0
0.9
0.95
0.95
40
50
60
70
80
θ1 [°]
Fig. 13.5. Ratio V1 /V as a function of contact angles θ1 and θ2 after rupture of
the meniscus (cylindrical gripper) – Case of the total interface
observations, but unfortunately the total energy cannot be reduced to the
LV energy. Consequently, the suggested method is not suitable to predict
the volume repartition. Nevertheless, let us keep in mind the values of U ∗
∗
and ULV
: it will be later investigated if the comparison of the meniscus
∗
energy W (WLV ) with U ∗ (ULV
) provides a suitable rupture criterion, in a
13.6 Rupture Condition and Rupture Gap
117
z
V2
V1
∆z
zneck
ri
r
Fig. 13.6. Repartition of the volumes above and below the meniscus neck
symmetric configuration with θ1 = θ2 (in this case, the predicted volume
repartition corresponds to the trivial repartition V1 = V2 = V2 ). Otherwise, we will conclude that this method is never suitable.
2. Geometrical model
The geometrical model proposed in [142] assumes that the repartition of
the volume between the two droplets after rupture is equal to the distribution of the liquid above and below the meniscus neck right before rupture
occurs (Fig. 13.6):
V1 =
i=i
neck
πri2 ∆z
(13.16)
i=1
where the index ineck is defined by rineck = min(ri ). This method for
volume repartition is used in the Sect. 13.6 to determine V1 and V2 .
13.6 Rupture Condition and Rupture Gap
On the one hand, the interfacial energy W (z) of the meniscus increases as a
function of the gap; on the other hand, an interfacial energy U (V1 , V2 ) can
be associated to the volume repartition of the two droplets after rupture.
The idea in [142] consists in equaling both energies to get a rupture criterion
allowing to assess the rupture gap z ∗ :
W (z ∗ ) = U (V1 , V2 )
(13.17)
Two ways can be followed to evaluate the interfacial energy:
1. The LV interfacial energy approach proposed in [142] takes only the LV
interfacial energy WLV into account. WLV can be written as a function of
the surface tension γ and the lateral area of the meniscus Σ:
118
13 Approaching and Rupture Distances
WLV = γΣ
2
2π
z2
dr
=γ
dφ r 1 +
dz
dz
z1
0
2
n−1
ri+1 − ri
≈ γ2π∆z
ri 1 +
.
∆z
i=1
The developments of Appendix A.2 lead to
1/3
1/3
A1
A2
2/3
2/3
ULV (V1 , V2 ) = γ
V
+
V
α1 1
α2 2
(13.18)
(13.19)
(13.20)
(13.21)
The gap z ∗ for which the meniscus can break into two residual droplets
is given by the condition
WLV (z ∗ ) = ULV (V1 , V 2).
(13.22)
2. The total interfacial energy approach we propose consists in adapting the
criterion of [142] by considering the total interfacial energy W . By assuming that the sum of the liquid–vapor area ALV and the solid–vapor area
ASV is constant and by taking the Young–Dupré equation into consideration, W can be expressed as follows:
W = γΣ + γSL1 ASL1 + γSV1 ASV1 + γSL2 ASL2 + γSV2 ASV2
= γΣ + ASL1 γSL1 + (C1 − ASL1 )γSV1 + ASL2 γSL2
+(C2 − ASL2 )γSV2
= γΣ + ASL1 (γSL1 − γSV1 ) + ASL2 (γSV2 )
= γΣ − ASL1 γ cos θ1R − ASL2 γ cos θ2R .
(13.23)
(13.24)
(13.25)
(13.26)
Note that the arbitrary energy level has to be chosen so that C1 = C2 ≡ 0.
By considering flat components and axially symmetric grippers, W is
rewritten into
2
n−1
ri+1 − ri
ri 1 +
− πr12 γ cos θ1R
W = 2πγ∆z
∆z
i=1
+2π
j=j
m
rj
1 + rj2 ∆z
(13.27)
j=1
where the index jm is so that rjm = r2 , the radius of the intersection point
between the gripper and the meniscus. Note that rj = dr(z)
dz |j , where r(z)
is the gripper profile equation. Once again, developments of Appendix A.2
lead to
1/3
2/3
U (V1 , V2 ) = γ(3A1 V1
1/3
2/3
+ 3A2 V2
).
(13.28)
13.7 Analytical Benchmarks
119
The gap z ∗ for which the meniscus can break into two residual droplets is
given by the condition
W (z ∗ ) = U (V1 , V2 ).
(13.29)
13.7 Analytical Benchmarks
In (13.20) and (13.27), the energy is numerically evaluated. However, if we
assume a cylindrical shaped meniscus (Fig. 13.7), the lateral area Σ can be
expressed as
Σ = 2πrz
√
= 2 πzV .
(13.30)
Consequently, (13.20) and (13.27) can be expressed by
√ √ √
WLV ≈ 2 π V γ z
(13.31)
√ √ √
2V γ cos θR
W ≈ 2 π Vγ z−
z
cos θ
(13.32)
+cos θ
1R
2R
. By combining, respectively, (13.21) and (13.31)
where cos θR =
2
at the one hand and (13.28) and (13.32) at the other hand, the rupture criteria
13.22 and 13.29 can be written as follows:
√ √ ∗
=γ
2 π V γ zLV
1/3
1/3
A1
A
2/3
2/3
V
+ 2 V2
α1 1
α2
√ √ √
2V γ cos θR
= 3γV 2/3 (A1 + A2 )1/3 .
2 π V γ z∗ −
z∗
(13.33)
(13.34)
z
Approximation:
cylindrical shape
r
Fig. 13.7. The meniscus volume is approximated by a cylinder for the determination
of the lateral area Σ
120
13 Approaching and Rupture Distances
Let us now assume a symmetric case (θ1 = θ2 and the energetic approach
for volume repartition gives the trivial and suitable distribution V1 = V2 = V2 )
in order to further simplify these equations:
∗
=
zLV
V 1/3 −2/3 2/3 −2
4
A α
π
(13.35)
√ √ √
2V γ cos θrec
2 π V γ z∗ −
= 3γV 2/3 (2A)1/3 .
(13.36)
z∗
Finally, let us consider the trivial case θ1 = θ2 = π/2. This implies α = 1/3
and A = 2π/3, leading to a unique gap:
∗
z =
∗
zLV
=
V
π
1/3 81
4
1/3
.
(13.37)
Let us note that for θi = π/2, W and WLV tend to each other because this
contact angle corresponds to a neutral situation: the surface does not repel
and is not wetted by the liquid.
13.8 Summary of the Methods
Several approaches can be combined:
Useful information can be summarized as follows:
1. Case “0”
∗
z =
81V
4π
U ∗ = 3V 2/3
2. Case “1”
∗
ULV
=γ
1/3
(13.38)
4π
3
1/3
1/3
γ.
(13.39)
1/3
A1
A
2/3
2/3
V
+ 2 V2
α1 1
α2
(13.40)
V1 =
V
1 + K
(13.41)
V2 =
K V
1 + K
(13.42)
∗2
ULV
.
4πV γ 2
(13.43)
∗
zLV
=
3. Case “2”
U ∗ = 3γV 2/3 (A1 + A2 )1/3
(13.44)
13.8 Summary of the Methods
121
z
U
U*
q2
z
V2
V1 = 0
r
r
r
r*
(a)
(b)
r
Fig. 13.8. The algorithm. (a) Iterative search; (b) sketch of the droplets energy as
a function of the drop radius r
V
1+K
KV
V2 =
1+K
√
λ
∗
z | µ z∗ + ∗ = U ∗
z
V1 =
(13.45)
(13.46)
(13.47)
√ √
with λ = −2V γ cos θR and µ = 2 π V γ.
4. Cases “3 and 4”
The following algorithm has been used (Fig. 13.9a):
a) To determine the volume repartition V1 + V2 = V , search for a radius
r̄ so that the droplet hanging below the gripper that intersects the
gripper at r = r̄ has a volume V2 (r̄) = V . This means that the corresponding volume V1 for the droplet put on the component is equal to
zero, as illustrated in Fig. 13.8a
b) For r ∈ [0, r̄], compute V2 (r)
c) Deduce V1 = V − V2 (r) and hence U (V1 , V2 )
d) Search for U ∗ = max(U ) and the corresponding radius r∗ , as illustrated in Fig. 13.8b
e) Choose a starting gap z, as small as possible (to be sure that W < U )
f) Compute the corresponding meniscus and the associated energy W (z)
g) Compare W (z) > U ∗
• If the comparison is true, the rupture gap z ∗ is equal to z
• Otherwise, the gap must be increased and the steps [f–g] must be
iterated
5. Cases “5 and 6”
The following algorithm has been used (Fig. 13.9b):
a) Choose a starting gap z as small as possible (to be sure that W < U )
b) Compute the corresponding meniscus
122
13 Approaching and Rupture Distances
Search for r | V2(r) = V and V1=0
Choose a gap = z
Determine meniscus (z)
Choose a gap=z
Determine meniscus (z)
Compute zneck
Compute W(z)
Determine V1(z),V2(z),U(z) and W(z)
W>U*
W>U
No: increasegap
Determine r* corresponding to U*=max(U )
No: increasegap
For r C[0,r], compute V1(r), V2(r) and U(r)
Yes
Yes
Rupture gap z* = z
Rupture gap z*=z
(a)
(b)
Fig. 13.9. Algorithms used in the general case. (a) Volume repartition determined
with the energetic approach; (b) volume repartition determined with the geometrical
approach
Table 13.1. Summary of the methods
ID Volume repartition
Rupture criterion Geometrical configuration
V
2
0
Trivial: V1 = V2 =
1
2
3
4
5
6
Energetic approach
Energetic approach
Energetic approach
Energetic approach
Geometrical approach
Geometrical approach
LV
Total
LV
Total
LV
Total
Trivial: cylindrical gripper (p = 0)
and θ1 = θ2 = π2
Cylindrical gripper (p = 0)
Cylindrical gripper (p = 0)
General case
General case
General case
General case
c) Determine the corresponding energy W (z) and the position of the
meniscus neck and compute the volumes V1 below zneck and V2 above
zneck
d) Compute the energy U (z) associated to the liquid repartition V1 and
V2
e) Compare W (z) > U ∗
• If the comparison is true, the rupture gap z ∗ is equal to z
• Otherwise, the gap must be increased and the steps [b–e] must be
iterated
13.9 Comparison between the Methods
The different methods of Table 13.1 have been compared for cylindrical gripper, both for symmetric (θ1 = θ2 ) and nonsymmetric (θ1 = θ2 ) cases. The accuracy required on the volume in the computation of a meniscus was 1%
13.9 Comparison between the Methods
123
0.6
V2/V (squares)
0.5
0.4
0.3
0.2
0.1
0
20
30
40
50
60
70
80
90
100
110
θ1=θ2 [°]
Fig. 13.10. Normalized volumes V2 /V obtained by the different methods, between
which it can hardly be distinguished
gap3
Rupture gap and rupture neck [µm]
2000
gap1
gap5
1500
gap0
1000
z1
z5
gap4
gap6
z3
gap2
z4 ≈ z6
500
z2
0
20
30
40
50
60
70
80
90
100
110
θ1=θ2 [8]
Fig. 13.11. Gap and zneck
and the gripper was discretized with 1,000 points. The imposed volume was
V = 0.28 µL and the surface tension γ = 72 mN m−1 .
The results of the symmetric case are shown in Figs. 13.10 and 13.11.
Figure 13.10 shows the normalized volume V2 /V = 1/2 for all volume repartition methods (in the symmetric case). The rupture gap z ∗ and the rupture
neck height zneck are plotted in Fig. 13.11 (respectively represented by the lines
13 Approaching and Rupture Distances
Normalized volume V2/V for θ1 = 60° and V = 0.28µL
124
1
0.8
0.6
0.4
0.2
0
20
30
40
50
60
70
θ1 [°]
80
90
100
110
Fig. 13.12. Normalized volume V2 /V as a function of θ1 computed by the energetic
approach (stripped line) and by the geometrical approach (continuous line)
with circles and squares): it can be seen that all methods converge toward the
same result when the contact angles tend to π/2. For θ < π/2, the methods
based on a LV energy computation (dotted curves above the stripped line
corresponding to the trivial case 0) lead to unacceptable gaps that can even
reach up to 2 mm, which was never observed in the experiments. Moreover,
the trend does not seem to be physically acceptable: the gap cannot increase
by decreasing the angle, i.e., by increasing the wetting properties. The method
of case 2 (dotted-stripped line) is based on analytical developments assuming
that the meniscus is of cylindrical shape. Its results are consequently in very
good agreement with the numerical methods of cases 4 and 6 (continuous
lines) when θ → π/2 but are no longer correct for smaller contact angles. The
distinction between cases 4 and 6 requires a nonsymmetric simulation (see
results of Fig. 13.12). A nonsymmetric case has been studied for V = 0.28 µL
water, with θ2 = 60◦ . Figure 13.12 indicates that V2 (V1 ) is larger (smaller)
than 1/2 when θ1 < θ2 , when the energetic repartition method is used: this is
not physically correct because smaller θ corresponds to larger adhesion and
henceforth larger volumes.
13.10 Conclusions
We conclude this chapter by noting that
•
The repartition volume method can be based on the geometrical repartition method already proposed in [142]
13.10 Conclusions
•
125
The LV interfacial energy computation is not correct; henceforth, we propose to base the energy computations on all interfaces (LV, SV, and SV)
A final remark is that V1 is also the residual volume left on the component
after release. It should be noted from a surface tension micromanipulation
point of view that this residual volume of liquid might be a problem: in this
case, it is suggested to use a volatile liquid such as water or alcohols, whose
evaporation leads to the suppression of this residual volume of liquid. A comprehensive description of the evaporation phenomenon is beyond the scope of
this work.
Part III
Experimental Aspects
14
Example 2: Numerical Implementation
of the Proposed Models
14.1 Introduction
This chapter aims at presenting the numerical implementation of the proposed models. Therefore, the problem of computing the meniscus shape will
be addressed and coupled with a pseudodynamic simulation, in which the component is moved at each time step according to the capillary force acting on it
at this time step. This approach relies on the assumption stated in Chap. 12
that even in high acceleration assembly machines, the meniscus shape was
hardly modified by this acceleration, as long as the meniscus height would
remain smaller than the modified capillary length. Figure 14.1 details all the
inputs, related to the materials (liquid, material of the component...) and to
the user’s choices (volume of liquid, gripper...) while the outputs (meniscus
shape and pressure difference) are used to compute the capillary force and
move the component according to the Newton’s motion law.
14.2 Liquid Bridge Simulation for the Analysis
of a Meniscus
As explained in Chap. 8, the meniscus shape is obtained by a double iterative
scheme (Figs. 14.2 and 14.4). First, a so-called “starting point” is chosen along
the gripper profile: this point is supposed to be the first point of the meniscus
and moreover the slope of the gripper at this point and the contact angle are
known, providing initial conditions for the meniscus determination. For this
starting point, the pressure difference ∆p is adjusted so that the computed
meniscus respects the prescribed contact angle θ1 at the component side (see
the evolution of the contact angle θ1 as a function of ∆p for a given starting
point in Figs. 14.5 and 14.6 and the evolution of the contact angle for several
iterations of the starting point in Fig. 14.7). To do this, a starting pressure
difference is chosen (dpstart ) and decremented by the quantity dpinc until the
128
14 Example 2: Numerical Implementation of the Proposed Models
INTPUTS
OUTPUTS
Literature
models
Part and gripper material
Surrounding environment
q1,q2
Volume repartition
SIMULATION
TOOL
Liquid
γ
Volume of liquid
Gripper geometry
Energy
Shape of the interface
Rupture distance
Force exerted on the
component
Pressure difference
across the interface
Gap
Gripper kinematics
Newton's law
Component
kinematics
Fig. 14.1. Overview of the inputs and outputs of the implemented model
−5
x 10
Gripper
8
z (m)
6
Starting point i+1
Meniscus
Starting point i
Starting point 1
4
2
0
0.5
1
1.5
r (m)
2
x 10−4
Meniscus ij (starting point i, ∆ pj)
Fig. 14.2. The double iterative scheme for a spherical gripper (R = 0.1 mm), water,
θ1 = θ2 = 30◦ , V = 4.5 nL, z = 0. Meniscus ij is obtained with the ith starting point
and the jth pressure difference (Reprinted with permission from [108]. Copyright
2005 American Chemical Society)
θ
D pi−1
D pi
qup
θ
D p = f (θ)
q*
qlow
D pi+1
D p*
D plow
Dp
ε
D pup
Fig. 14.3. θ as a function of ∆p: the interval ∆pup − ∆plow is divided into n
subintervals
14.2 Liquid Bridge Simulation for the Analysis of a Meniscus
129
Choose a meniscus starting point
Choose a pressure difference
Contact angle OK?
No
No
Volume OK?
Compute the capillary force
Fig. 14.4. View of the resolution (Reprinted with permission from [108]. Copyright
2005 American Chemical Society)
100
90
80
70
θ1
60
50
40
30
20
10
0
−300
−250
−200
−150
−100
−50
0
dp
Fig. 14.5. First iterative scheme: ∆p (Pa) is adjusted to fit θ1 (at this stage,
the chosen volume has not been taken into account yet). Contact angle θ1 (◦ ) as a
function of the pressure difference ∆p (Pa) across the interface: typical evolution (it
is difficult to adjust ∆p for small angles)
contact angle θ is bounded by θlow and θup given by two pressure differences
∆plow and ∆pup (Fig. 14.3): the quantity dpinc is modified if the sensitivity
of θ is too small or too large.
In some cases, if the angle is too large, the starting pressure difference
(usually set around 10 Pa) is too small: it means that the pressure difference
130
14 Example 2: Numerical Implementation of the Proposed Models
x 10−5
15
Starting Point
z [m]
10
5
0
−101
1
1.3
−201
1.35
−301 −401 −501 −601 −701 −801
1.4
1.45
r [m]
1.5
x 10−3
Fig. 14.6. Evolution of the meniscus profile during the first iteration scheme (the
figures on right hand side of the curves indicate the pressure difference ∆p (Pa))
1.06
Normalized angle
1.04
1.02
1
0.98
0.96
0.94
1
2
3
4
Iteration #
5
6
7
Fig. 14.7. Evolution of the normalized contact angle θ1 (i.e., computed θ1 divided
by the prescribed one). The continuous lines state for the upper and lower imposed
acceptable limits (±5%)
is actually positive (i.e., the pressure inside the meniscus is larger than the external pressure), leading to a meniscus about to collapse (or even nonexisting).
Indeed, a positive pressure difference leads to a repulsive force that can be
compensated by the attractive tension force component. It should also be
emphasized that small contact angles can lead to instabilities because the
14.3 Evaluation of the Double Iterative Scheme
x 10−3
4
Normalized volume
2
131
r2 [m]
1.5
1
3
2
1
0
0.5
1
2
3
4
5
Iteration #
(a)
6
7
2
3
4
5
Iteration #
6
7
(b)
Fig. 14.8. Evolution of the starting point and the computed volume for a spherical
gripper (Φ = 8 mm), a gap z = 5 µm, water (γ = 72 × 10−3 N m−1 ), θ1 = θ2 = π3 ,
V = 100 nL. (a) Position of the iterated starting point (m); (b) Computed volume
at each iteration
pressure difference switches from a value where θ1 exists but is too large to
another value where θ1 does not exist, see Fig. 14.5.
Once a pressure difference has been found to respect the contact angle,
the volume associated to this meniscus is computed and compared with the
prescribed one: if it is smaller (larger), the starting point is chosen farther
from (nearer) the symmetry axis (see the typical evolution of the position r2
of the starting point in Fig. 14.8a and the corresponding computed volume
in Fig. 14.8b). The normalized volume presented in Fig. 14.8b is the computed volume at each iteration divided by the prescribed volume: it oscillates
until its value is inside the allowed error interval (represented by the two
continuous lines, corresponding to an acceptable error ±5%). The first iteration is not represented on this figure: the corresponding normalized volume is
about 70.
14.3 Evaluation of the Double Iterative Scheme
The influence of the number n of subintervals (see Fig. 14.3) in the determination of θ and ∆V
V on the accuracy of the meniscus position have been
investigated in the case of two parallel plates separated by a gap z = 50 µm.
The volume of liquid (here water, i.e. γ = 72 × 10−3 N m−1 ) has been fixed to
0.25 µL and the contact angles are both equal to π2 . In this case, the solution
to the problem is a cylindrical meniscus whose radius R can be determined
by expressing the volume:
V = πR2 z
(14.1)
132
14 Example 2: Numerical Implementation of the Proposed Models
hence R = 1.262 mm. This result can be used to calculate the mean curvature
H of the meniscus, since one curvature radius is given by R and the second
one tends to ∞:
2H =
1
1
1
+
=
= 792.4 m−1 .
R ∞
R
(14.2)
Consequently, by using (6.8),
∆p = 2Hγ = 57.07 Pa.
(14.3)
The results of this study are summarized in Tables 14.1 and 14.2 on the one
hand and Figs. 14.9 and 14.10 on the other hand.
In Table 14.2 and Fig. 14.10, the error on the radius Rneck is estimated as
follows (the volume of liquid is assumed to be proportional to the cube of the
characteristic size of the meniscus Rneck ):
3
αRneck
=V
⇒ Rneck =
(14.4)
V
α
13
(14.5)
∆Rneck
1 ∆V
=
Rneck
3 V
(14.6)
Table 14.1. Influence of the discretization n of the interval [∆plow − ∆pup ] with
dpinc = 1, dpstart = 100 Pa, ∆V /V = 1%
n
5
10
20
50
100
∆p (Pa)
56
55.56
57.05
57.22
57.17
error (%)
14.29
6.40
2.95
1.14
0.57
Computation time (s)
29
38
49
94
168
Table 14.2. Influence of the relative error ∆V /V with dpinc = 1, dpstart = 100,
n = 50
∆V /V (%)
10
5
3.5
2
1
0.5
0.1
Rneck (mm)
1.250
1.250
1.250
1.250
1.260
1.260
1.261
∆Rneck /Rneck (%)
3.33
1.67
1.17
0.67
0.33
0.17
0.03
Computation time (s)
22
21
22
23
97
96
128
14.4 Pseudodynamic Simulation
133
60
∆p = 57.07Pa
50
40
30
20
10
0
Computing time [s]
Pressure difference [Pa]
70
150
100
50
# points
# points
0
20
40
60
80
100
0
0
120
(a)
20
40
60
80
100
(b)
Fig. 14.9. Influence of the discretization n of the interval [∆plow − ∆pup ] with
dpinc = 1, dpstart = 100 Pa, ∆V /V = 1%.
150
R=1.26 mm
Computing time [s]
Rneck [mm]
1.5
1
0.5
0
0
2
4
6
8
10
100
50
0
Relative error on the volume [%]
(a)
0
2
4
6
8
10
Relative error on the volume [%]
(b)
Fig. 14.10. Influence of the relative error ∆V /V with dpinc = 1, dpstart = 100,
n = 50
14.4 Pseudodynamic Simulation
To exploit the results in a dynamic simulation (for example, to predict if
the omponent will stick to the gripper in an assembly machine or to design the
kinematics of the gripper if it is intended to use its acceleration to release the
component), we compute the position and velocity of the component at time
step i+1 by solving the motion equation (Newton’s equation); the acceleration
at time step i is deduced from the capillary force and the mass of the object.
The position and the velocity of the object at time step i are used as boundary
conditions. This procedure is summarized by the following equations: let us
start with Newton’s law.
m
d2 z
= mz̈ = F.
dt2
(14.7)
134
14 Example 2: Numerical Implementation of the Proposed Models
This second order equation is rewritten into a system of two first-order equations:
z
Y ≡
,
(14.8)
ż
Y (2, 1)
Ẏ ≡ f (t, Y ) =
.
(14.9)
F
m
Hence, the Runge-Kutta method [49] allows to compute Yi+1 from Yi and Fi :
1
Yi+1 = Yi + (K1i + 2K2i + 2K3i + K4i ),
(14.10)
6
where Yi (1, 1) is the position of the component at time step i and Yi (2, 1) is
its velocity. As the force Fi can be computed at time step i from the meniscus
geometry, the coefficients Kji can be determined by
K1i = ∆t.f (ti , Yi ),
∆t
K1i
, Yi +
),
K2i = ∆t.f (ti +
2
2
∆t
K2i
K3i = ∆t.f (ti +
, Yi +
),
2
2
K4i = ∆t.f (ti + ∆t, Yi + K3i ).
(14.11)
(14.12)
(14.13)
(14.14)
Figure 14.11 gives an overview of the complete algorithm. As far as the
force computing is concerned, the most accurate results are found with the
Time step i
Choose a meniscus starting point
Choose a pressure difference
No
Contact angle OK?
Volume OK?
No
Compute the capillary force
Compute object acceleration
xi
vi
Compute position and velocity at time step i+1
End simulation?
Stop
Fig. 14.11. Algorithm of the coupled problem
No
14.5 Conclusions
135
“Laplace” calculation of the meniscus (=based on the Laplace equation), but
for computing time reasons it is advised to use the “Arc” approximation (the
force obtained by the “Arc” model is about a few % (2–5%) lower than the
output by the Laplace model).
14.5 Conclusions
This chapter has put the emphasize on the numerical implementation of the
meniscus shape model based on the Laplace equation. From the knowledge
of this meniscus shape, the capillary force acting in the component can be
computed and the component moved according to the Newton’slaw. This implementation has been used to output the simulation results presented in
Chap. 18. To experimentally validate these results, an experimental test bed
has been set up, and this is explained in the third part of this book.
15
Conclusions of the Theoretical Study
of Capillary Forces
In this second part of the book, we have introduced the underlying parameters
(surface tension, advancing and receding contact angles) and models (Young–
Dupré, Laplace, Cassie, Wenzel) ruling capillary forces. We have summarized
approach and exact methods to compute these forces at equilibrium.
For general meniscus shapes, the use of an energy minimization software
such as Surface Evolver cannot be avoided, but we have shown how to solve the
Laplace equation to compute the meniscus shape in axially symmetric cases,
from which the force can be computed according to (6.11). This method has
been applied successfully to the case study of a watch ball bearing, illustrating the existence of a force optimum in the case of a conforming gripper
(i.e., a gripper whose geometry conforms with that of the component). The
equivalence between this approach and the interfacial energy differentiation
has been analytically proved in the case of a prism interacting with a plane.
From the microassembly point of view, the first conclusion is that the
capillary forces linearly depend on the characteristic size of the meniscus;
henceforth, they are of the utmost interest for miniaturization. Second, they
provide a suitable principle to pick up flat components whose only the top
surface is accessible. Additionally, it has been shown how surface impurities
or surface roughness theoretically affect the contact angle. Nevertheless, the
contact angle is considered as an input of the proposed models and will be
measured in the characterization stage (see the third part of this book). Moreover, experiment will help to chose between advancing and receding contact
angle. Since a useful release method in microassembly is to impose a high
acceleration to the gripper, the suitability of an equilibrium modeling has
been considered and it has been proved that it was an acceptable approach
as long as the characteristic size of the meniscus would be smaller than a
modified capillary length L̄c , given by
γ
L̄c =
(15.1)
(a + g)ρ
138
15 Conclusions of the Theoretical Study of Capillary Forces
Consequently, we have not investigated the case of a dynamic simulation.
Therefore, an exact estimation of characteristic times of this kind of picking
was impossible, but an estimation of the characteristic damping time of the
meniscus oscillations has been given on two different ways, on the first hand
using dimensional analysis and on the other hand solving the Lucas–Washburn
equation. An interesting damping phenomenon preventing high contact forces
to damage the component has been pointed out from a figure of the literature
(Fig. 11.5).
Finally, the conditions for the rupture of the meniscus have been studied
in the last chapter of this second part.
Third part is devoted to experimental aspects, including both characterization to get the model input and measurement of the outputs.
16
Introduction
The third part of this book concerns the experimental aspects, which essentially covers three points, i.e., characterization, validation, and results. The
characterization step consists in measuring the inputs of the models, that is,
the contact angles, the surface tension, and the volume of liquid. The validation stage is the experimental verification of the proposed models and simulations, mainly done by comparing two possible outputs, i.e., the capillary
force and the meniscus shape. Thirdly, we present in this part a collection of
results based on computations on the one hand and on experiments on the
other hand.
Therefore, this part falls into four chapters.
Chapter 17 details the requirements list and the set up of the designed test
bed, including the discussion of the possible sources of error. This test bed
includes two main functions: vision (contact angles, gap, meniscus shape...)
and force sensing. This chapter also summarizes the measurements of a first
characterization stage.
The first part of Chap. 18 contains the preliminary results that are used
to validate the simulation code. It includes two meniscus shape validations,
based on experiment and comparison with the analytical solution of the catenary curve (∆p = 0). The force output of the simulation is compared with
an analytical benchmark and with experiments. The second part of Chap. 18
presents a detailed study on the influence of each parameter on the capillary force: the gap, the gripper geometry, the surface tension, the materials
(through the contact angles). These results are compared with the simulation
outputs. Then, some release concepts are tested: the influence of the relative
orientation between the gripper and the component (the tilt angle breaks the
axial symmetry assumption of the numerical model), the impact of an auxiliary PTFE tip, the use of a dynamical release strategy. The rupture distance of
the meniscus is also experimentally studied in this chapter. The results of this
chapter will be discussed in Chap. 21, especially the role of each parameter
on the picking task and a quantified comparison of several release strategies,
142
16 Introduction
introducing the concept of adhesion ratio φ. These results are the first steps
toward design rules of a surface tension based gripper.
In the two last chapters, these concepts are then applied to the case
study of a watch ball bearing. Following an identical methodology, Chap. 19
concerns the characterization aspects (grippers geometry, contact angles, surface tension) while Chap. 20 includes the results obtained with the designed
microgripper. This chapter presents the studies of the picking task (with the
possible errors and some solutions) and the releasing task (including a study
on the reliability of this principle). The force model presented in Chap. 10 is
experimentally assessed. The force order of magnitude given by this model is
validated but a more accurate validation would require the exact determination of the involved volume of liquid. This is also exhaustively discussed in
Sect. 20.5.
The conclusions proposed in this part put forward some design rules for
surface tension based grippers.
17
Test Bed and Characterization
17.1 Introduction
The need for a test bed can be justified by several reasons: the main one
is probably the need for an experimental validation of the above described
simulation. Moreover, the inherent assumptions of the built model prevent us
from getting results from nonaxially symmetric configurations. A test bed can
then feed us with additional information such as, for example, the influence
of the gripper tilt on the capillary forces. A third reason is that the simulation has to be fed with input data such as the amount of liquid and the
contact angles: these inputs will be measured with this test bed. Moreover, it
is still not clear which contact angle (advancing one vs receding one) should
be used: the choice will be achieved by direct observation of the picking operation. Consequently, this chapter focuses on the requirement and the design
principles of a force measurement test bed, which can be sketched as shown in
Fig. 17.1. The information released in this chapter concerns the integration of
each function (force measurement, position sensing, liquid dispensing, vision),
the calibration and characterization of the test bed itself, and finally, the
characterization of the materials involved in the experiments and the related
contact angles.
17.2 Requirements
In Table 17.1, X axis refers to the optical axis of the camera (see Fig. 17.1)
and Z axis to the “vertical”1 axis, i.e., the symmetry axis of the gripper.
1
The term “vertical” usually refers to gravity, but here gravity is neglected so that
we use it to refer to the symmetry axis of the gripper.
144
17 Test Bed and Characterization
Positioning sensor
Upper solid
(gripper)
d2
Liquid dispensing
Z
(Compliance)
Meniscus
d1
Imaged scene
Force sensor
Y
X
(Deflection sensor)
Fig. 17.1. Principle of the force test bed: the variation of d1 is measured by a
noncontact displacement sensor. This variation, together with the stiffness of the
cantilevered beam, gives the amount of force applied onto this beam. This force
is exerted by a meniscus that has been dispensed between the beam, which acts
as a component, and the upper solid, which acts as a gripper. The position d2 of
the upper solid is also measured and the knowledge of both d1 and d2 gives the
gap between the upper solid and the beam. Finally, the stripped box indicates the
imaged scene.
Table 17.1. Requirements for the test bed
Category
Requirement
Force Measurement Measurement along the Z axis
Full scale: several milli-newton
Sensitivity: several micro-newton
Vision
Vision field: about 1 mm2
Range of focusing distance: several millimeter (X)
2 dof for the camera in the plane Y Z
Zoom ×40
Image acquisition and transfer to the work station
Backlight system with its power supply
Gripper interface
2 dof along X and Y , low accuracy
1 dof along Z axis, high resolution (≈ 1µm), range ≈ 5 mm
Interchangeable gripper interface
1 rotation dof (θx ) to tune the tilt of the gripper
(range ≈ 45◦ , accuracy better than 1◦ )
Components
Flat components, several materials
Drop delivery
Water and silicone oil delivery
Volume smaller than 1µL
Displacement
2 mm range along Z (to induce the bridge collapse)
measurements
1 µm accuracy
17.3 Test Bed Principles
145
17.3 Test Bed Principles
17.3.1 Force Measurement
According to Ref. [67], the sensing of small forces can be achieved by measuring the displacement caused by elastic deformation induced by the loading
of an elastic structure. Still according to Ref. [67], usual deformable sampling
bodies are presented in Fig. 17.2a. The most suitable body is the cantilevered
beam (Fig. 17.2b) because it can serve both as deformable body for the force
sensing and as component for the experiment. If the component material has
to be changed, another blade can be used or a small flat sample can be glued
on it.
Because of its small thickness, the blade extremity undergoes a deflection
δweight due to its own weight [150]:
δweight =
3 ρg L4
qL4
=
8EI
2 E h2
(17.1)
where L is the length of the beam (m), q is the distributed load per unit length
(N m−1 ), i.e., q = ρghb, where ρ is the density of the beam (kg m−3 ), g is the
well-known gravity constant (9.81 ms−2 ), h is the beam thickness (m), and b
is the beam width (m). E denotes the Young modulus of the material (Pa)
and I refers to the momentum of the beam section, i.e., I = bh3 /12 (m4 ).
Typical values are given in Tables 17.2 and 17.3.
Note that this deflection does not actually disturb the measurement because deflections are assumed to be small. Consequently, the linearity and
the superposition principle can be applied and the own deflection neglected.
F
δ
(a)
(b)
Fig. 17.2. Force sensing (a) deformable sampling bodies [67]; (b) chosen force
sensing principle
Table 17.2. Geometrical properties of the cantilevered blades
Symbol
L
h
b
Description
Length
Thickness
Width
Value
10–100
0.025–1
12
Units
10−3 m
10−3 m
10−3 m
146
17 Test Bed and Characterization
Table 17.3. Physical properties of the cantilevered blades
Symbol
E
ρ
Description
Young modulus
Density
Steel
210
7,800
Units
109 Pa
kg m−3
Actually, in (17.1), δweight cannot directly be reduced because it should imply
a loss of sensitivity as far as the force measure is concerned. Indeed, the
deflection δforce caused by an external load F (expressed in N) is given in
[150]:
δforce (L) =
4F L3
F L3
=
.
3EI
Ebh3
(17.2)
Consequently, it should be tried to maximize the following ratio M , describing
the relative importance due to force deflection compared to that due to weight
deflection:
M=
8F 1
δforce
.
=
δweight
3ρgb hL
(17.3)
If the density of the cantilever and its width are fixed, we have the following
two considerations for a given force:
1. The thickness h and the length L should be as small as possible in order
to reduce the relative importance of the weight
2. h should be as small as possible in order to maximize the force measurement sensitivity, but the length L should be maximized
We conclude that the thinnest cantilever should be chosen, and its length
adapted for minimal sensitivity. The deflection measuring system has been
implemented as follows: at equilibrium the capillary upward force F exerted
by the liquid bridge on a cantilevered beam is balanced by the elastic restoring
force due to the beam deflection δ (Fig. 17.2b). This deflection is directly measured by a noncontact displacement sensor (Keyence LC-2440 laser), whose
measuring range is equal to 3 mm and accuracy is guaranteed by the constructor to be 0.2 µm. The measured value can be read immediately on the
controller display or can be transmitted by a RS-232 connection in order to
achieve almost real time acquisition.
17.3.2 Drop Dispensing
One must distinguish between continuous flow and drop-on-demand generators. Basically, the main parameters governing the drop-on-demand delivering
are surface tension, viscosity, inner diameter of the ejection nozzle, ejection
pressure [19]. Several principles of drop-on-demand systems are described in
17.3 Test Bed Principles
147
[116]. The commercially available systems are compared in Table 17.4 (Microdrop and Gesim companies).
Drop-on-demand generators are quite expensive but allow to generate
smaller drops than the manual dispensing device. Nevertheless, it is not justified because the drop size range of the manual dispensing device (Eppendorf
company) is small enough to meet the assumptions of the model to be validated: a droplet diameter between 500 and 1,000 µm is smaller than the
capillary length (LC ≈ 2.5 mm). Moreover, the size that must be smaller
than the capillary length is actually the droplet height and not its diameter,
which is the case in near contact configurations. A drawback of both systems
(i.e., manual dispensing systems and automated drop-on-demand generators)
is that they are calibrated for water: the drop-on-demand generators can also
dispense liquids like silicone oil but, in this case, they require an additional
heating unit to decrease the liquid viscosity. For these experimental tests, a
manual dispensing system has been used. Since the liquid amount is an input
parameter of the simulation tool, it has to be known with accuracy. It is either
calibrated with the manual dispensing device (from 0.1 to 2.5 µL, with steps
equal to 2 nL) or it can be measured from the height h and the diameter D
of the droplet as depicted in Fig. 17.3. Indeed, the radius R and the cosine of
the angle cos θ can be expressed starting from
R(1 − cos θ) = h,
2R sin θ = D,
⇒2
(17.4)
(17.5)
h
sin θ + cos θ = 1,
D
(17.6)
Table 17.4. Comparison between commercial solutions for drop generation
Permissible liquid viscosity (mPa s)
Inner nozzle diameter µm
Dosing volume (nL)
Drop diameter (µm)
Drop range of flight (mm)
Price (e)
Actuation
Microdrop
10
30–70
0.048–0.144
45–65
20
6,300
Piezo
Gesim
5
?
0.1–1
58–124
1
4,470
Piezo
h
θ
D
Fig. 17.3. Volume measurement from a spherical cap picture
Eppendorf
water
100–2,500
576–1,684
Contact
200
Manual
148
17 Test Bed and Characterization
which leads to
D2 + 4h2
,
8h
D2 − 4h2
cos θ = 2
,
D + 4h2
hence the volume can be determined:
R=
V =
3
1
2πR3
(1 − cos θ + cos3 θ).
3
2
2
(17.7)
(17.8)
(17.9)
Let us note that this second method is not suitable for very small contact
angles, because the height h becomes too small to be measured accurately
from a number of pixels.
17.3.3 Vision
The used camera is the Keyence-CV050 CCD camera, mounted with a 50 mm
lens and a set of rings to tune the zoom. This device is coupled to a monitor
allowing to track the picture, to achieve several measurements,2 and to transfer
the acquired image to the work station. For imaging droplets, a backlight
illumination system is recommended.
17.4 CAD Model and Drawings
CAD drawings of the designed set up have been made with CATIA (V5)
and are presented in Fig. 17.4, where the following elements can be seen:
the beam is located just below the spherical tip stating for the gripper. Its
extremity is located in the view field of the camera and its cantilevered length
can be changed by modifying the casing to tune flexibility. The camera is
mounted on a plate that can be moved along the X axis and in the plane
Y Z with two manual stages. The gripper tip (here it is a spherical tip) can
be changed easily, thanks to the gripper interface. This one is mounted on
a manual stage allowing a relative motion along the Z axis with respect to
the upper displacement sensor. Both can be moved together in the XY plane,
thanks to a manual stage mounted on the top of a gantry. The role of the
latter is to carry a back light represented in Fig. 17.5 that is used to improve
the illumination conditions.
These drawings can be compared to the pictures of the set up shown in
Figs. 17.5, 17.6, and 17.7.
2
Contact angle measurement, gripper geometry, volume of liquid.
17.4 CAD Model and Drawings
Displacement
sensors
149
Z
Spherical
Tip
Z
270 mm
Gripper Interface
1D manual stages
Beam
XY
Gantry
Camera
X
Y
Casing
YZ
X
YZ
130 mm
2D manual stages
(a)
(b)
Fig. 17.4. Drawings of the experimental set up (a) left side view; (b) right side view
(Reprinted with permission from [108]. Copyright 2005 American Chemical Society)
Displacement sensor
Z axis
Gripper interface
XY
X axis
Spherical tip
Camera
Y axis
Back light
Fig. 17.5. Left side view (Reprinted with permission from [108]. Copyright 2005
American Chemical Society)
150
17 Test Bed and Characterization
40mm
(a)
(b)
Fig. 17.6. Experimental set up (a) front view; (b) right side view
(a)
(b)
Fig. 17.7. Experimental set up (a) side view; (b) detailed view
17.5 Characteristics of the Force Measurement Set Up
151
17.5 Characteristics of the Force Measurement Set Up
17.5.1 Typical Calibration
The typical way to calibrate the set up consists in measuring the deflection δ
due to a known force, such as, for example, the weight of a calibrated mass
m0 . Typical values are the following:
•
•
m0 = 301.6 mg ± 0.1
δ = 510 µm ± 20
The mass has been weighted with both a mechanical and an electronic balance,
with a 0.1 mg accuracy. The accuracy of the deflection measure lies more in
the fact that there is an error in the positioning of the calibrating mass on the
beam, leading to an error on the deflection that has been estimated at about
20 µm by estimating the extreme possible positions for the calibrating mass.
Consequently, the stiffness of the beam is given by
m0 g
= 5.80 N m−1 ,
(17.10)
k=
δ
with a relative error given by
∆k
∆m0
∆δ
20
∆δ
=
≈
≈
≈ 4%.
+
k
m0
δ
δ
580
(17.11)
This result can be improved by calibrating the system with a heavier mass, so
that δ is increased and that the error is decreased. Nevertheless, the deflection
cannot be increased above 2–3 mm so that the best accuracy could be
∆δ
20
∆k
≈
≈
≈ 1%.
k
δ
2000
(17.12)
17.5.2 Linearity
To assess the linearity of the set up, Fig. 17.8 presents the measured deflection
as a function of the applied force, i.e., the weight of reference masses (masses
mi and deflections δi are presented in Table 17.5).
If δi is supposed to be approached by δi ≈ ami + b where a and b are
obtained by the least square method:
N Σ(δi mi ) − Σδi Σmi
,
N Σm2i − (Σmi )2
Σδi Σm2i − Σ(δi mi )Σmi
,
b=
N Σm2i − (Σmi )2
a=
(17.13)
(17.14)
the linearity error is defined in [67] as
≡
max |ami + b − δi |
.
max δi
(17.15)
Since a = 1.7144 µm g−1 and b = 0.735 µm, the linearity error is about 0.33%.
152
17 Test Bed and Characterization
Measured Deflection [m]
1.5
x 10−3
1
0.5
0
0
0.002
0.004
0.006
0.008
0.01
Applied Force [N]
Fig. 17.8. Linearity of the force measurement set up
Table 17.5. Data for the linearity
mi (mass in g)
0 0.1225 0.2453 0.3643 0.4836 0.6049 0.7490 0.8608
δi (deflection in µm) 0 210
423
627
832
1034 1280 1481
17.5.3 Accuracy
As the measured force is given by
F = kδ,
(17.16)
the measurement error is given by
∆F
∆k ∆δ
=
+
.
F
k
δ
(17.17)
The relative error on the stiffness has been estimated at about 1–4% and the
absolute error on the deflection about 20 µm. If the stiffness of the beam is
adjusted (by changing its length for example) in order to get deflection about
600–2,000 µm, the measurement error of the force can be estimated at:
20
∆F
≈ 1 − 4% +
≈ 1 − 4% + 1 − 3.3% ≈ 2 − 7.3%.
F
600 − 2000
(17.18)
17.5.4 Influence of a Misalignment on the Force Measurement
It must be distinguished between the misalignment of the gripper and the
component on the one hand (Fig. 17.9) and the misalignment between the
gripper and the droplet on the other hand (Fig. 17.10).
1. Misalignment of the gripper and the component: when there is no alignment error between the component and the gripper (it is the case if their
17.5 Characteristics of the Force Measurement Set Up
z
z
O F
P
z'
ε
Spherical gripper
φ
Spherical component
Spherical component
P
Beam
Fmeas
153
F
O
Q
Spherical gripper
Beam
Fmeas
(a)
(b)
Fig. 17.9. Centering error between a spherical component and a spherical gripper.
(a) Without centering error; (b) with a centering error (a)
(b)
(c)
(d)
Fig. 17.10. Centering error between the delivered droplet and a spherical gripper
centers P and O are on the same vertical z axis, see Fig. 17.9a), the
measured force Fmeas (actually the restoring elastic force of the deflected
beam) is equal to the capillary force F between the component and the
gripper. Let us now assume that the gripper is not perfectly aligned with
the component and that there is a distance between the z axis of the
component and the z axis parallel to z and containing the center O of
the gripper (Fig. 17.9b): in this case Fmeas is equal to the projection of F :
Fmeas = F cos φ
= F 1 − sin2 φ
2
= F 1−
R1 + R2
2 1
≈ F 1−
2 R1 + R2
(17.19)
where φ can be read in Fig. 17.9b and R1 and R2 state for the component
and gripper radii. is assumed to be always smaller than 0.5 mm and
R1 + R2 ≈ 5 mm. Then the ratio Fmeas /F is about 99.5%. Consequently,
this error will be neglected.
2. Misalignment between the gripper and the droplet: this misalignment
(Fig. 17.10a) does not imply measurement error because once the liquid
bridge has linked the gripper and the component (Fig. 17.10b), there is
154
17 Test Bed and Characterization
a centering effect (Fig. 17.10c) of the liquid that makes the alignment
perfectly correct after the first rupture (Fig. 17.10d). Note that measurements have always been preceded by a blank trial.
17.6 Characteristics of the Contact Angles
Measurements
As the contact angles constitute inputs for the simulations, it was necessary to measure them according to the different solid–liquid combinations
(for example, steel–water, steel–silicone oil). The first idea was to assume
that a small droplet posed on a substrate would take a spherical cap shape
(see Fig. 17.11a), the contact angle θ can consequently be deduced from the
drop height h and the drop diameter D, measured on the screen of the CCD
camera controller. Using (17.6) and assuming t ≡ tan θ/2, sin θ = 2t/1 + t2
and cos θ = 1 − t2 /1 + t2 , we get
tan
θ
2h
=t=
.
2
D
(17.20)
Unfortunately, as already mentioned, this method does not hold when θ is
small because in this case it is difficult to determine h, accurately, and when
the hysteresis is large because it is difficult to know whether the angle made
by the spherical cap tends or not to the advancing or receding contact angles.
Consequently, the finally used method consists in transforming the droplet in
a meniscus by approaching the gripper near the blade on top of which the drop
is posed. Once the meniscus has been formed, the gripper is slightly moved
downwards (upwards) to force the contact line to move along the solids: in
this case we can be ensured to measure the advancing (receding) contact angle
as depicted in Fig. 17.11b. The contact angle is then determined by the angle
between the line tangent to the meniscus and the one tangent to the solid.
D
θ
h
θ
R
qR
qA
qA
(a)
(b)
qR
(c)
Fig. 17.11. Different measured contact angles. (a) Spherical cap – not well defined
angle; (b) advancing contact angle; (c) receding contact angle
17.8 Modus Operandi
155
The results presented in Table 17.9 show a dispersion of the angle measurements, which is quite moderate for silicone oil but which can be dramatically
high for water. Hopefully, the pertinent information seems to be cos θ more
than θ so that even with ∆θ ≈ 15◦ , the error on the cosine remains lower than
7%. Additional information as far as the contact angles are concerned can be
found in Sect. 18.3.
17.7 Surface Tension Measurement
If necessary, the surface tension can be measured, thanks to classical methods
described in [1], such as, for example, the Wilhelmy plate method or the
Du Nouÿ ring method. In this case, surface tension has not been measured
directly since the liquid which have been used in this part of the work are well
characterized ones, whose properties are given in Sect. 17.9.4.
17.8 Modus Operandi
1. The gripper and the beam are chosen and set up: a small flat component
made of whatever material that can be glued onto the beam, in order to
simulate the desired material. Note that this operation does not disturb
the force measurement as the weight of the additional component is quite
small.
2. The force measurement system is calibrated by measuring the deflection corresponding to a known force, i.e., the weight of a reference mass
(Fig. 17.12).
3. The gripper is moved downwards until the contact with the beam is
detected (Fig. 17.13).
(a)
(b)
Fig. 17.12. Test bench calibration. (a) Calibration mass; (b) corresponding
deflection
156
17 Test Bed and Characterization
(a)
(b)
Fig. 17.13. Initial configuration of the gripper and the beam. (a) Away from contact
(beam width = 12.7 mm); (b) at contact
(a)
(b)
Fig. 17.14. 0.5 µL water droplet. (a) Droplet, beam, and spherical tip gripper
(beam width = 12.7 mm); (b) detailed view
4. Displacement sensors are set to zero.
5. The gripper is moved upwards to free space between its tip and the beam.
A liquid droplet is then put on the beam right below the gripper tip
(Fig. 17.14a): its volume can be known either by using the calibrated
indication of the micro-pipetting device or by computing the volume from
the geometrical parameters of the spherical cap-shaped droplet shown in
Fig. 17.14b. A last solution consists in superposing the meniscus profile
and the computed shape for a given input volume. If the shapes correspond
with one another, then the volumes are equal.
6. The gripper is moved downwards until the droplet turns itself into a liquid
bridge. From this situation, the advancing (receding) contact angles are
obtained by slowly moving the gripper downwards (upwards) and measured directly on the screen of the camera monitor (Fig. 17.15). Results
are presented in Table 17.9.
17.8 Modus Operandi
(a)
157
(b)
Fig. 17.15. Steel component (St-004-02), water, spherical gripper (Φ = 13 mm).
(a) Advancing contact angle; (b) receding contact angle
Fig. 17.16. Deflection measurements (mm)
7. Force measurement: the separation distance can be tuned by moving the
gripper. The cantilever deflection sensor should now display a positive
value because the cantilevered blade is pulled upwards by the meniscus,
while the gripper sensor should display a negative value because the gripper is on a higher position than at the initialization time (see Fig. 17.16).
The sum of the values leads to the separation distance.
a) The force at contact can be measured from the maximum deflection
of the beam. Note that this maximum can be difficult to read because
this configuration is unstable: indeed, from the maximal deflection
situation, the beam quickly jumps downwards when the gripper height
increases. The reason therefore is that the capillary force becomes
lower than the elastic restoring force of the beam, and consequently,
there is a jump from this position to a lower one, corresponding to
another forces balance, given by a larger gap. According to the stiffness
of the beam, this second equilibrium position can exist or not (this
phenomenon is widely described in [31]).
158
17 Test Bed and Characterization
Fig. 17.17. Example of calibration with a ceramic slip gage (width = 1 mm)
b) Force–distance curve can be drawn by moving the gripper downwards
step by step until the physical contact of the gripper and the cantilever: during this phase, the deflection of the cantilever gives the
value of the pulling force exerted by the meniscus.
Note that in step 5, it can be necessary to calibrate the camera: this
is achieved by imaging a ceramic slip gage (width = 1 mm), as depicted in
Fig. 17.17.
We then conclude that the scale is about 330 ± 2 pixel mm−1 .
17.9 Characterization
17.9.1 Set of Available Grippers
First let us note that gripper here must be understood as the upper solid.
The case of an actual gripper will be considered in Chap. 19. As we wanted
to study the influence of the shape (both the gripper type and the gripper
parameter p), conical and spherical grippers have been considered.
•
The conical grippers have been turned in steel material:
– GC-St-0: cylindrical gripper p = 0◦ (polished, Ra = 0.23 ± 0.01 µm)
– GC-St-5: conical aperture angle p = 4.5◦ , Ra = 0.23 ± 0.01 µm
(Fig. 17.18a)
– GC-St-10: conical aperture angle p = 9.75◦ , Ra = 0.23 ± 0.01 µm
(Fig. 17.18b)
– GC-St-45: conical aperture angle p = 47.5◦ , Ra = 0.23 ± 0.01 µm
(Fig. 17.18c)
17.9 Characterization
(a)
(b)
159
(c)
Fig. 17.18. Geometry of the conical tips (a) GC-St-5; (b) GC-St-10; (c) GC-St-45
(a)
(b)
(c)
Fig. 17.19. Geometry of the spherical tips. (a) GS-St-3.2; (b) GS-St-7.9; (c) GSSt-13.0
•
•
The spherical grippers are made with steel ball bearings (the roughness
Ra has not been measured but values are given according to ISO3290:1998
found in [42], p. 595):
– GS-St-3.2: Φ3.2 mm (Ra < 0.010 µm) (Fig. 17.19a)
– GS-St-7.9: Φ7.9 mm (Ra < 0.010 µm) (Fig. 17.19b)
– GS-St-12.8: Φ12.8 mm (Ra < 0.014 µm)
– GS-St-13: Φ13.0 mm (Ra < 0.014 µm) (Fig. 17.19c)
Finally a foam gripper (2×2.5×2 mm3 ) has been tested in dynamic release
trials
17.9.2 Set of Available Components
The steel components have been cut in a steel blade with a thickness
102 ± 5 µm (accuracy referred by the supplier Precision Brand). The length
and the width have been roughly measured because of the shape errors of
the components, the mass with a 0.1 mg accuracy weighting balance. The
roughness has been measured with a Taylor Hobson device (±0.01 µm): two
values are given (when measured) parallel and perpendicular to the machining
direction of the blade (see Tables 17.6 and 17.7).
160
17 Test Bed and Characterization
Table 17.6. Steel components
Code Material Thickness (µm) Length (mm) Width (mm)
±5
St-1
Steel
102
3.00
3.30
St-2
Steel
102
5.50
5.60
St-3
Steel
102
7.55
8.30
St-4
Steel
102
12.70
15.00
Ra (µm) Mass (g)
±0.01
±0.0001
0.09–0.23 0.0072
0.09–0.23 0.0228
0.09–0.23 0.0469
0.09–0.23 0.1428
Table 17.7. Silicon components
Code Material Thickness (µm) Length (mm) Width (mm) Ra (µm) Mass (g)
±50
±0.01 ±0.0001
Si-1 Silicon
550
3.30
4.00
<0.03
0.0128
Si-2 Silicon
550
5.75
6.45
<0.03
0.0385
Si-3 Silicon
550
5.75
8.15
<0.03
0.0508
Si-4 Silicon
550
11.85
13.60
<0.03
0.1827
Si-5 Silicon
550
0.1550
Si-6 Silicon
550
0.1713
Si-7 Silicon
550
0.2850
Si-8 Silicon
550
0.2023
Table 17.8. Blades
Code
St-004-01
St-004-02
St-004-04
Si-004-01
Si-004-02
Material
Steel
Steel
Steel
Silicon
Silicon
Thickness (µm)
±5
102
102
102
102 + (550 ± 50)
102 + (550 ± 50)
Width (mm)
±0.13
12.70
12.70
12.70
12.70 (without object)
12.70 (without object)
Ra (µm)
±0.01
0.23
0.10–0.29
Not measured
<0.030
<0.030
The silicon components have been cleaved in a silicon wafer with a thickness 550 ± 50 µm.
17.9.3 Set of Available Blades
The basic blade consists in a 102 µm thick steel blade (St-004-XX). The
blades Si-004-XX consist in silicon flat components glued on the steel blade.
In this case the dimensions of the silicon components are indicated next to
the dimensions of the steel blade in Table 17.8. The width is given with the
supplier accuracy (±0.13 mm). Again two values for Ra are given parallel and
perpendicular to the machining directions.
17.9 Characterization
(a)
161
(b)
Fig. 17.20. Steel component (St-004-02), water, conical gripper (GC-St-10).
(a) Advancing angle; (b) receding angle
17.9.4 Available Liquids
Two liquids have been used:
1. Water (milli-Q purity grade), ρ = 1, 000 kg m−3 , γ = 72 × 10−3 N m−1 ,
µ25◦ = 1 × 10−3 Pa s
2. R47V50 silicone oil, ρ = 960 kg m−3 , γ = 20.8 × 10−3 N m−1 , µ25◦ =
48 × 10−3 Pa s
17.9.5 Contact Angles Characterization
The way the static contact angles have been determined has already been
briefly presented in Sect. 17.6. For each configuration (=a given liquid and
a given solid) the meniscus has been imaged five times for advancing and
receding contact angle (see for example Fig. 17.20). This was achieved with
an almost zero (a few µm s−1 ) velocity and so the measured angles correspond
to the advancing and receding static contact angles. For each image the angle
has been measured three or four times: the results are presented in Table 17.9.
In this table, liquid I was water and liquid II was silicone oil (R47V50).
The solids that have been tested are representative for all grippers and components intended to be used later: they have been chosen either because of
the constitutive material (steel, silicon) or because of their surface roughness.
Consequently, the GS-St-13.0 spherical gripper made of a steel ball (for ball
bearings) represents all spherical grippers. With the same idea, the GC-St-10
states for all conical grippers. The two tested components were steel (St-000402) and silicon (Si-004-01) blades. The measured angles have been averaged
(θA and θR ) and their standard deviation computed (σθA and σθR ). The relative errors on the cosines have been computed from measures that were inside
the intervals θ ± 2σ and θ ± 1.5σ. Finally, the hysteresis has been computed
as the difference between the mean advancing contact angle θA and the mean
receding contact angle θR .
162
17 Test Bed and Characterization
Table 17.9. Advancing and receding contact angles
Liquid
Solid
H2 0(I)
H2 0(I)
H2 0(I)
H2 0(I)
H2 0(I)
Oil(II)
Oil(II)
Oil(II)
Oil(II)
Oil(II)
St-004-02
Si-004-01
GS-St-13.0
GC-St-10
GC-St-00
St-004-02
Si-004-01
GS-St-13.0
GC-St-10
GC-St-00
θA
(◦ )
92
55
88
82
92
34
29
22
23
33
θR
(◦ )
56
33
37
25
39
16
16
12
13
14
σθA
(◦ )
1.9
6.0
8.1
12.9
2.4
7.1
9.5
5.6
6.0
7.2
cos θA
∆ cos θR ∆ cos θA ∆ cos θR
σθR ∆cos
Hysteresis
θA
cos θR
cos θA
cos θR
◦
( ) (1.5σθa ) (1.5σθr ) (2σθa ) (2σθr )
(◦ )
8.3
1.39
0.32
1.86
0.42
36
4.6
0.23
0.08
0.30
0.10
23
4.4
4.88
0.09
6.51
0.11
51
7.9
2.53
0.10
3.37
0.13
57
15.6 2.10
0.33
2.81
0.44
53
3.5
0.13
0.03
0.17
0.04
18
5.8
0.14
0.04
0.18
0.06
13
3.3
0.06
0.02
0.08
0.02
10
2.4
0.07
0.01
0.09
0.02
11
2.8
0.12
0.02
0.16
0.03
18
Additional tests have been led with PTFE. The receding contact angle with
R47V50 was [44, 46, 50]◦ → 47◦ , that one with water was [100, 99, 94]◦ → 96◦ .
17.10 Conclusions
This chapter presented the design of the test bench, its performances and
error sources, and the related modus operandi. Then the different grippers,
components, and blades were described and characterized. This equipment
has been used to measure the static contact angles and the contact angle
hysteresis, which will be used as inputs of the models in the next chapter.
18
Results
18.1 Introduction
Two groups of results are presented in this chapter. Section 18.2 summarizes
preliminary experiments, which have been led in order to validate the simulation tool: the corresponding results concern (1) the meniscus profile, (2)
the comparison between the simulation outputs and analytical results, and
(3) the comparison between the simulation outputs and experimental results.
Sections 18.3–18.12 collect both simulation and experimental results. The results obtained in relation with the watch bearing case study already presented
in Chap. 10 will be given in Chaps. 19 and 20.
18.2 Preliminary Results: Validation
of the Simulation Code
To validate the developed simulation code, we have compared its results as
far as both the meniscus shape and the capillary force are concerned. The
meniscus shape has been studied in two simplified cases and a comparison
has been led with experiment in a general case. Then, we have compared the
output force both with the approximations of the capillary force from the
literature and with experimental results.
18.2.1 Meniscus Profile
The first meniscus profile validation is the case of two parallel plates separated
by a distance b and for a difference of pressure equal to zero, leading to the
analytical equation of a catenary curve (see Appendix B.3):
r(z) = A cosh
z−B
A
(18.1)
164
18 Results
x 10−3
2
z [m]
1.5
1
Analytical solution
Numerical solution
0.5
0
0
0.5
1
1.5
2
2.5
r [m]
3
x 10−3
Fig. 18.1. Comparison between the numerical and analytical meniscus shape, with
∆p = 0, a = 3 mm, b = 2 mm, and θ2 = 60◦ (Reprinted with permission from [108].
Copyright 2005 American Chemical Society.)
where A = a sin θ and B = b − a sin θacosh(1/sin θ) (a, b, and θ are shown in
Fig. 18.1). For z = 0, the relative error between the numerical radius and the
analytical one is about 1.5%.
A second case has been tested, namely the case of a meniscus between
two parallel plates, with contact angles equal to 90◦ , leading to a cylindrical
meniscus (with radius R) whose principal curvature radii are R1 = ∞ and
R2 = R. This case has already been presented in Sect. 14.3: the graphical
output of the meniscus shape is presented in Fig. 18.2a.
The last verification operated on the meniscus shape is the comparison
between the output profile and the picture of the meniscus, as presented in
Fig. 18.2b.
This picture corresponds to a 12.7 mm diameter spherical steel gripper
catching a steel component (blade St-004-2) with 0.72 µL water. The gap in
this case is 265 µm. The dashed line states for the meniscus output by the
simulation tool.
18.2.2 Comparison with the Analytical Expressions
Let us remind the analytical approximation of the capillary force between
a plane and a sphere (radius R), for a gap equal to zero and an equivalent
θ2
). As usual γ is the surface
contact angle θ (if θ1 = θ2 , cos θ ≡ cos θ1 +cos
2
tension of the liquid between the gripper and the component.
F = 4πRγ cos θ.
(18.2)
18.2 Preliminary Results: Validation of the Simulation Code
165
z [m]
x 10−4
6
5
4
3
2
1
0
−1
−2
−3
Measured interface
Liquid bridge
z=265µm
Simulation,V=0.72µL
0
0.2
0.4
0.6
0.8
r [m]
1
1.2
x 10−3
(a)
(b)
Fig. 18.2. Study of the meniscus shape (a) Simulation: meniscus shape with θ1 =
θ2 = 90◦ (V = 0.25 µL, z = 50 µm); (b) Comparison between the simulation and the
experimental meniscus shape (water, steel component, spherical steel gripper (R =
6.35 mm)) (Both reprinted with permission from [108]. Copyright 2005 American
Chemical Society)
V=0.1µl (water)
Analytical (water)
V=0.1µl (R47V50)
Analytical (R47V50)
V=0.5µl (water)
V=0.5µl (R47V50)
V=1µl (water)
V=1µl (R47V50)
0.012
0.01
Force [N]
0.008
0.006
0.004
0.002
0
10
20
30
40
50
Contact angles θ1 = θ2 [8]
60
70
Fig. 18.3. Comparison between the simulation results and the analytical
approximation F = 4πγR cos θ for a spherical gripper with diameter 13.0 mm, γ =
72 × 10−3 N m−1 (water) and γ = 20.8 × 10−3 N m−1 (R47V50). The results are presented for different volumes (0.1, 0.5, and 1 µL) and different contact angles simulating different materials; the simulation points tend to the analytical approximations
for water (solid line) and silicone oil (dashed line) (Reprinted with permission from
[108]. Copyright 2005 American Chemical Society)
166
18 Results
Figure 18.3 plots the force as a function of the equivalent contact angle θ for
a Φ 26 mm spherical gripper and for two liquids: the upper curve shows the
force for water and the lower one that for silicone oil (R47V50). It can be
seen on this picture that the results of the simulations tend to the analytical
approximations (solid and dashed lines).
18.2.3 Experimental Validation
Several experimental validations have been led: the tested configurations
are summarized in Table 18.1. Each experiment has been made 15 times,
and each time plotted in Figs. 18.4 and 18.5 by a dot. The “cross” signs
state for the boundaries of the error interval, centered on the mean value
of the 15 results. This error interval has been calculated as indicated in
Sect. 17.5.3:
∆k ∆δ
∆me g ∆δe
20 µm
10 µm
∆F
∆δ
=
+
=
+
≈0+
+
(18.3)
+
F
k
δ
me g
δe
δ
600 µm min(δi )
Table 18.1. Summary of the experimentally tested configurations: 1 refers to components and 2 refers to steel grippers, A refers to advancing and R refers to receding.
The angles values shown in this table have been separately measured (see Table 17.9)
# Gripper radius Component Liquid θ1R θ2R θ1A θ2A
γ
(mm)
(◦ ) (◦ ) (◦ ) (◦ ) (mN m−1 )
1
6.4
Steel
R47V50 16 12 34 22
20.8
2
6.4
Steel
R47V50 16 12 34 22
20.8
3
6.4
Steel
R47V50 16 12 34 22
20.8
4
1.6
Steel
R47V50 16 12 34 22
20.8
5
1.6
Steel
R47V50 16 12 34 22
20.8
6
1.6
Steel
R47V50 16 12 34 22
20.8
7
1.6
Silicon
R47V50 16 12 29 22
20.8
8
1.6
Silicon
R47V50 16 12 29 22
20.8
9
1.6
Silicon
R47V50 16 12 29 22
20.8
10
6.4
Silicon
R47V50 16 12 29 22
20.8
11
6.4
Silicon
R47V50 16 12 29 22
20.8
12
6.4
Silicon
R47V50 16 12 29 22
20.8
13
1.0
Silicon
R47V50 16 12 29 22
20.8
14
1.0
Silicon
R47V50 16 12 29 22
20.8
15
1.0
Silicon
R47V50 16 12 29 22
20.8
16
6.4
Silicon
Water 33 37 55 88
72.0
17
6.4
Silicon
Water 33 37 55 88
72.0
18
6.4
Silicon
Water 33 37 55 88
72.0
19
6.4
Silicon
Water 33 37 55 88
72.0
20
1.6
Silicon
Water 33 37 55 88
72.0
21
1.6
Silicon
Water 33 37 55 88
72.0
22
1.6
Silicon
Water 33 37 55 88
72.0
23
1.6
Silicon
Water 33 37 55 88
72.0
V
(µL)
0.1
0.2
0.5
0.1
0.2
0.5
0.1
0.2
0.5
0.1
0.2
0.5
0.1
0.2
0.5
0.1
0.2
0.5
1.0
0.1
0.2
0.5
1.0
18.2 Preliminary Results: Validation of the Simulation Code
1.8
167
x 10−3
1.6
1.4
Force [N]
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
Experiment number (R47V50)
14
16
Fig. 18.4. Comparisons between experiments, analytical approximation, and simulations (with silicone oil): closed circle, experimental points; plus, limits of the error
intervals; open circle, analytical approximations with θA and θR ; open down triangle,
simulation with θA ; open triangle, simulation with θR
5
x 10−3
Force [N]
4
3
2
1
0
16
17
18
19
20
21
22
Experiment number (Water)
23
24
Fig. 18.5. Comparisons between experiments, analytical approximation, and simulations (with water): close circle, experimental points; plus, limits of the error
intervals; open circle, analytical approximations with θA and θR ; open down triangle, simulation with θA ; open triangle, simulation with θR
168
18 Results
where k is the stiffness of the blade, δe is the calibration deflection (i.e., the
measured deflection of the beam when a known mass me is put on the beam), δ
is the beam deflection due to the capillary force. The relative error on the mass
me is supposed to be smaller than the other error sources and consequently
set to zero. The typical error on the stiffness is due to the positioning error of
the calibration mass (typically about 20 µm). As the calibration deflection is
typically 600 µm, this leads to a relative error on the stiffness of about 3.3%.
Finally, the deflection error is estimated at about 10 µm and is divided by the
smaller measured deflection δi . For each simulation, the following information
is plotted just right to the experimental points:
•
•
•
The analytical approximation is plotted for advancing and receding contact angles (◦) – the upper circle corresponds to the receding contact angle
(larger force)
The simulation result with the advancing contact angles ( )
The simulation result with the receding contact angles ( )
It can be seen on these pictures that the experimental results can be predicted by simulation for the experiments led with silicone oil, but in the case
of water, the simulated force with a receding contact angle is much too large:
the experimental value lies between the forces corresponding to the advancing
and the receding contact angles in simulations [20–23] but is a little smaller in
experiments [16–19]. The general trend is that the actual force rather corresponds to advancing contact angles (this is discussed again in Sect. 18.3). The
influence of the volume (that cannot be predicted by the analytical approximation: the underlying cylindrical approximation of the meniscus makes the
force independent of the volume of the liquid bridge) is difficult to determine
from our results: if there is an influence1 , it is smaller than our experimental and simulation accuracies. What can be observed is the influence of the
gripper radius: the larger it is the larger is the force as indicated by the comparison between experiments [1–3] and [4–6], experiments [10–12] and [7–9;
13–15] and, finally, experiments [16–19] and [20–23]. The following sections are
now devoted to the exploitation of both the simulation and the experimental
tools.
18.3 Advancing vs Receding Contact Angle
For liquid–solid combinations with small contact angles and low hysteresis,
the capillary force does not depend too much on the angle as explained in the
case of the sphere–plane interaction by the following equation:
cos θR
FR
=
.
FA
cos θA
1
Bhushan [24] cites Mazzone et al. [127]: “it can be shown that the force decreases
with the volume”.
18.3 Advancing vs Receding Contact Angle
169
In the case of silicone oil on a silicon component, θA = 29◦ and θR = 16◦ ,
leading to FR /FA = 1.10. However, for solid–liquid combinations with larger
angles and hysteresis such as, for example, water on silicon, θadv = 55◦ and
θrec = 33◦ can lead to FR /FA = 1.46. The choice of the right angle must consequently be addressed carefully as illustrated by the following experiment,
involving a spherical gripper GS-St-12.7 and a silicon component. The liquid
is water and the used volumes [0.1, 0.2, 0.5, 1] µL (see experiments [16-19] of
Fig. 18.5): the experimental results lead to F ≈ 1.34 mN ± 0.08. Let us now
consider the contact angles (A states for “Advancing angle assumption” and
R for “Receding angle assumption”):
Table 18.2 indicates that the angle to be taken into consideration seems
to be the advancing one. At first sight the upward motion of the gripper during the picking step can be thought to lead to a receding motion
of the liquid and consequently to a receding angle (Fig. 18.6a). However,
come into close contact with the component, the approaching motion of
the gripper pushes the liquid outwards, with an advancing contact angle
(Fig. 18.6b). At the beginning of the upward motion of the gripper, the
angle is still the advancing one because the gripper (and consequently the
liquid too) has not yet begun to move (Fig. 18.6c). Only when the gap
has begun to increase the contact angles move from advancing to receding ones (Fig. 18.6d). This assumption has been validated by direct observation of the scene with the CCD camera. θ1 and θ2 have been measured
15 times in picking situation, leading to the following (averaged) results:
θ1 ≈ 57◦ ± 16 and θ2 ≈ 83◦ ± 6 (angles to be compared with the 55◦ and
88◦ of Table 18.2).
Table 18.2. Measured contact angles of the steel–water and the silicon–water combinations
Solid–Liquid combination
Silicon–Water
Steel–Water
cos θ = 12 (cos θ1 + cos θ2 )
F = 4πγR cos θ
A
θ1adv = 55◦
θ2adv = 88◦
0.304
1.75 mN
R
θ1rec = 33◦
θ1rec = 37◦
0.819
4.71 mN
gap
(a)
(b)
Fig. 18.6. Advancing vs receding contact angles
(c)
(d)
170
18 Results
18.4 Influence of the Gap
18.4.1 Force–Distance Curve
This section can be seen as an additional validation of the simulation or
already be interpreted as a first knowledge toward a gripper based in the
surface tension effects: the force–distance curve. This curve (Fig. 18.7) plots
the capillary force exerted by a 7.9 mmΦ spherical gripper (GS-St-7.9) on a
silicon component (Si-004-01). The force is exerted by a 0.5 µL silicone oil
droplet (R47V50). This curve has been measured with an almost zero velocity (equilibrium curve). As far as the simulation is concerned, receding contact
angles have been input since the meniscus is stretched by moving the gripper
upwards. The correspondence between simulation and experiment can be seen
on this picture, although the rupture distance predicted by the simulation is
a little smaller than the measured one: this is discussed again in Sect. 18.11
devoted to the study of the separation distance. This result indicates that the
simulation tool can predict the capillary force with separation distances different from zero: this comes as a complement to the previous validations made
by comparing the simulation with the analytical approximations, which was
valid only at contact. From the point of view of a capillary gripper design,
this curves means that the sticking effect due to the capillary force can be
reduced or suppressed by increasing the distance between the gripper and the
component. This separation could be achieved either by dynamical effects (see
also Sect. 18.10) or by pushing the component away from the gripper with
1
x 10−3
Simulation
Experiments
Force [N]
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
Gap [m]
0.8
1
x 10−3
Fig. 18.7. Force–Distance curve for V = 0.5 µL R47V50, Si-component and GS-St7.9 (Both reprinted with permission from [108]. Copyright 2005 American Chemical
Society.)
18.5 Influence of the Gripper Geometry
171
Ratio Force / Total
1
0.8
0.6
Laplace
Tension
Total
0.4
0.2
0
0
1
2
3
4
Gap [m]
5
6
7
x 10−4
Fig. 18.8. Respective contribution of the “tension” and “Laplace” terms in the total
amount of the force—Ratios without dimensions (Both reprinted with permission
from [108]. Copyright 2005 American Chemical Society.)
a less adhesive auxiliary tool or tip. The curve plotted in Fig. 18.7 helps to
design the gripper by indicating the required range to separate the component
from the gripper and/or the residual force.
18.4.2 Tension Force vs. Laplace Force
The simulation results of Fig. 18.7 also allow to calculate and compare the
importance of the interfacial tension force and the Laplace term of the capillary force (see Fig. 18.8). The results presented in this figure justify some
approximations found in the literature, neglecting the “tension” term for small
gaps. Nevertheless, this assumption is no longer valid for larger gaps and the
tension term even becomes dominant.
18.5 Influence of the Gripper Geometry
The influence of the gripper geometry involves the study of the influence of the
radius in the case of a spherical gripper or the angular aperture p in the case
of a conical gripper. The results of Sects. 18.2.2 and 18.2.3 indicate that the
force is proportional to the gripper radius R as mentioned by the Israelachvili
[89] approximation:
F = 4πγR cos θ
(18.4)
172
18 Results
FRef = 0.87mN
F0 [N]
10−3.1
Ref
log10F0−log10Fref = 0.34 (log10V0 −log10 Vref)
10−3.3
−9
VRef = 10 m
10−3.5
10−11
10−10
V, Volume of liquid [m3]
3
10−9
Fig. 18.9. Influence of the angular aperture p for conical grippers: comparison
between simulation (solid lines) and experiments led with 0.1 µL (open circle), 0.2 µL
(open square), and 0.5 µL (open triangle) silicone oil (R47V50) with conical gripper
with p = 5◦ (GC-St-5), p = 10◦ (GC-St-10), and p = 45◦ (GC-St-45) (Both reprinted
with permission from [108]. Copyright 2005 American Chemical Society.)
In this case the volume of liquid plays no (or a minor) role. The situation is
different in the case of a conical gripper as indicated in Fig. 18.9, where it
can be seen that the larger the volume of liquid, the larger the capillary force.
This influence is nevertheless less significant than that of the angular aperture:
when the conical gripper diverges from the cylindrical tip (p = 0), the force
dramatically decreases. This result, however, indicates that a geometry tip
modification could be used to successfully decrease the force and hence achieve
the release task of a component caught by a capillary gripper. Figure 18.9
quantifies the influence of this geometry change.
18.6 Influence of the Surface Tension
The influence of surface tension has been investigated by simulation, for conical (GC-St-5) and spherical (Φ26 mm) grippers. For both grippers, the gap
was equal to zero and the gripper contact angle was set to θ2 = 20◦ . To achieve
this, it must be assumed that the gripper material changes when the liquid
(i.e., surface tension) changes. The volume of liquid was set to V = 0.5 µL.
Results are presented in Figs. 18.10 and 18.11.
One can see the linear influence of the surface tension on the capillary
force. This result can be explained by (6.11)
FC = FT + FL = 2πr1 γ sin(θ1 + φ1 ) + 2γHπr12 .
(18.5)
18.6 Influence of the Surface Tension
6
x 10−3
θ1= 208
5
Force [N]
173
θ1= 308
θ1= 408
θ1= 508
θ1= 608
θ1= 108
4
θ1= 708
3
2
1
0
10
20
30
40
γ [mN−1]
50
60
70
Fig. 18.10. Influence of surface tension for different contact angles θ1 – Conical tip
(p = 5◦ ), Vol = 0.5 µL, Gap = 0 µm, θ2 = 20◦
0.012
θ1= 108
Force [N]
0.01
θ1= 208
θ1= 308
θ1= 408
θ1= 508
θ1= 608
θ1= 708
0.008
0.006
0.004
0.002
0
10
20
30
40
γ [mNm−1]
50
60
70
Fig. 18.11. Influence of surface tension for different contact angles θ1 – Spherical
tip (R = 13 mm), Vol = 0.5 µL, Gap = 0 µm, θ2 = 20◦
In this equation, θ1 is imposed by the couple component material/liquid and
φ1 = 0 for flat components. r1 and H are related to the meniscus geometry.
Since this geometry must respect (8.13)
−
1
pin − pout
r
,
+
=
γ
(1 + r2 )3/2
(1 + r 2)1/2
(18.6)
it means that for identical boundary conditions (i.e., identical contact angles,
gap, and volume), different surface tensions γ1 and γ2 will give the same geometry with two different pressure differences ∆p1 and ∆p2 on the condition that
∆p1
∆p2
=
.
γ1
γ2
(18.7)
174
18 Results
x 10−5
7
6
z [m]
5
4
γ = 72e−3 Nm−1
3
γ = 10e−3 Nm−1
2
1
0
−1
1.18
1.2
1.22
1.24
1.26
1.28
r [m]
1.3
x 10−3
Fig. 18.12. Absence of influence of γ on the meniscus shape
Table 18.3. Comparison between two menisci got with two different surface tensions
∆p (Pa)
Rneck (mm)
∆p
γ
γ = 72 × 10−3 N m−1
−2451.51
1.236
34.049
γ = 10 × 10−3 N m−1
−340.327
1.235
34.033
This is illustrated in Fig. 18.12 in the case of a cylindrical gripper with a gap
z = 50 µm and contact angles θ1 = θ2 = 30◦ .
The comparison between these two menisci (Table 18.3) illustrates the fact
that the same geometries lead to the same r1 and H, and consequently, the
force is proportional to the surface tension.
18.7 Influence of the Contact Angle θ1
The influence of the contact angle θ1 has been studied by simulation for a
conical (GC-St-5, p = 5◦ ) and spherical (R = 13 mm) gripper with z = 0
and V = 0.5 µL. In these simulations, θ2 = 20◦ . The decrease of the force
as long as θ1 increases can be seen in Figs. 18.13 and 18.14. This behavior
can be approximated by a cosine function in the case of the spherical gripper
(Fig. 18.14).
18.8 Influence of the Relative Orientation
The influence of the relative orientation of the gripper with respect to the
component could not be simulated due to the axially symmetry assumption.
Consequently, this influence has been studied experimentally by tilting the
18.8 Influence of the Relative Orientation
6
x 10−3
5
Force [N]
175
γ = 70mNm−1
γ = 60mNm−1
4
3
2
γ = 50mNm−1
γ = 40mNm−1
γ = 30mNm−1
−1
γ = 20mNm
1
0
10
γ = 10mNm−1
20
30
40
θ1 [8]
50
60
70
Fig. 18.13. Influence of θ1 for different surface tensions γ—Conical tip (p = 5◦ ),
Vol = 0.5 µL, Gap = 0 µm, θ2 = 20◦
0.012
0.01
Force [N]
0.008
γ = 70mNm−1
γ = 60mNm−1
0.006
γ = 50mNm−1
γ = 40mNm−1
0.004
γ = 30mNm−1
γ = 20mNm−1
0.002
γ = 10mNm−1
0
10
20
30
40
θ1 [8]
50
60
70
Fig. 18.14. Influence of θ1 for different surface tensions γ – Spherical tip (R =
13 mm), Vol = 0.5 µL, Gap = 0 µm, θ2 = 20◦
gripper with respect to the component, thanks to a manual rotational stage
with 0.04◦ accuracy. The results are presented in Fig. 18.15 for a steel cylindrical gripper, a steel component, and a 0.1 µL R47V50 droplet. It can be
seen that the relative orientation plays a major role on the force: this can
dramatically influence the performances of a capillary gripper due to some
unavoidable machining error or misalignment. On the contrary, this effect can
be used to force the release of the component after it has been positioned.
Therefore, the gripper should be designed to allow a rotation degree of freedom. A rotation range of 10◦ can already decrease the force by a factor 5.
Finally, let us emphasize the fact that these results are quite similar to those
176
18 Results
7
x 10−3
6
Force [N]
5
4
3
2
1
0
0
5
10
15
20
Tilt angle [8]
25
30
35
Fig. 18.15. Influence of the relative orientation of a steel cylindrical gripper (except
for a tilt angle of 2◦ , the experimental points are within the interval error)
Q
N
F
A
F
Q
N
B
W
C
α
W
α
W
α
(a)
Fig. 18.16. Tilt release
presented for van der Waals forces in [107]. To experiment this strategy, the
component was picked (Fig. 18.16a) and moved to its final location. Then the
substrate was tilted with an angle α ≈ 6◦ and the component pushed against
the substrate (Q in Fig. 18.16b): initially, Q is applied in A but due to the
moment of the force N , the component begins to rotate around C and Q
moves from A to B. Therefore, the meniscus moves to B too and the capillary
force F is reduced because the gripper is no longer parallel to the component.
If F becomes smaller than the weight W of the component, the releasing operation can be proceeded. The tilt angle α has to be chosen according to the
component weight and to the desired force reduction (Fig. 18.15).
18.9 Auxiliary PTFE Tip
Among several release strategies, the use of an auxiliary thin tip has already
been envisaged [191]. Quantitative assessment of this strategy has been led
with a PTFE conical tip with an angular aperture p = 70◦ . The results are
18.10 Dynamical Release
177
x 10−4
1.2
Force [N]
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
5
6
7
Experiment number
◦
Fig. 18.17. Force of a p = 70 PTFE conical tip on a silicon component (the
experimental points are within the interval error)
presented in Fig. 18.172 . The first three tests have been led with silicone oil on
a silicon component (V = [0.1, 0.2, 0.5] µL) and the last four ones with water
on a silicon component (V = [0.1, 0.2, 0.5, 1] µL). The results show that the
forces are between 42 and 65 µN for silicone oil and between 50 and 90 µN for
water, i.e., of at least two orders of magnitude lower than the forces exerted by
the spherical grippers. Consequently, the substitution of materials and shape
could be very successful to achieve the release task.
18.10 Dynamical Release
18.10.1 Simulation Results
As already mentioned in [149], the release task in handling with adhesion
force can be achieved dynamically, by giving the gripper an upward acceleration a large enough to detach the component. The previously described
static simulation has been used to compute the capillary force exerted on
2
The error intervals are different because two different blades have been used to
avoid mixing water and silicone oil. Therefore, the cantilever length was different,
leading to different stiffnesses: 0.755 N m−1 for experiments [1–3] and 3.561 N m−1
for experiments [40–7]. Since a larger stiffness induces lower deflections, the relative errors as far as deflections are concerned are larger in experiments [4–7].
178
18 Results
(a)
(b)
(c)
(d)
Fig. 18.18. Manipulation by adhesion: (a) approach; (b) picking; (c) depose;
(d) release
the component at each time step and hence deduce its kinematics. The
canonical task is described in Fig. 18.18: the gripping phase is based on the
adhesion force (b) and the release (d) is achieved by giving to the gripper an
acceleration.
Several simulations have been led with a conical gripper (p = 0.1), equal
contact angles θ1 = θ2 = 30◦ and 0.28 µL water. The component was given
a mass between 25 and 200 mg: consequently, we will see an evolution from
perfect picking for the lighter mass3 to an impossible picking (i.e., a perfect
release) for the heaviest mass. To do this, the kinematics of the gripper (freely
determined by the user) was supposed to be a trapezoidal distribution of
velocity: first an increase ramp limited by a user defined acceleration a until
the velocity reaches the prescribed value. Then the velocity is kept constant
until the gripper is decelerated (deceleration = −a) to reach a zero velocity at
the end of the predefined time interval. An example is given in Fig. 18.19 where
it can be seen that the time interval has been set to 10 ms, the acceleration to
a = 100 m s−2 , and the velocity to v = 0.1 m s−1 (the position of the gripper is
deduced from this information by assuming an initial position equal to zero:
we see a parabolic trend between t = 0 ms and t = 1 ms, corresponding to a
linear velocity, and a linear evolution corresponding to the constant value of
the velocity between t = 1 ms and t = 9 ms. Between t = 9 ms and t = 10 ms,
the position z of the gripper shows a parabolic behavior again). The simulation
has been launched for three different values of the component mass: 25, 50,
and 200 mg. Each time, the kinematics of the gripper (defined by the user)
has been superposed to that of the component (computed), the gap between
the component and the gripper, and the corresponding value of the capillary
force have been plotted. Finally, the normal reaction N of the substrate below
the component is also represented (=contact force). The case of a perfect
picking is illustrated for m = 25 mg in Fig. 18.19, where the position, velocity,
and acceleration of the component are equal to those of the gripper. This is
confirmed in Fig. 18.20 by a gap and a contact force N equal to zero: this
means that the component sticks to the gripper all the time; the value of the
3
Here the term “light” must be related to the acceleration a. For lower acceleration,
a heavier mass will also be considered as light enough.
18.10 Dynamical Release
z [mm]
1
179
Gripper
Object
0.5
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
Time [ms]
7
8
9
10
vz [ms−1]
0.1
0.05
0
vz [ms−2]
100
0
−100
Fig. 18.19. m = 25 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1):
positions, velocities, and accelerations
Gap[mm]
1
0
Contact force [mN]
Capillary Force [mN]
−1
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
5
4
3
2
0.4
0.2
0
Time [ms]
Fig. 18.20. m = 25 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper
(p = 0.1): gap, capillary, and contact forces
180
18 Results
z [mm]
1
Gripper
Object
0.5
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
vz [ms−1]
0.1
0
− 0.1
az [ms−2]
100
0
−100
Time [ms]
Fig. 18.21. m = 200 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper
(p = 0.1): positions, velocities, and accelerations
capillary force F = 3.70 mN is well larger than the weight of the component
W = 0.25 mN.
The opposite case is the perfect release presented in Fig. 18.21 for m =
200 mg: the component begins to move upwards (at time t = 2.5 ms, its
velocity is equal to zero and its position is 10 µm above the substrate) before
falling and reaching the substrate at time t = 4.2 ms (this time is confirmed
by the apparition of a contact force N at t = 4.2 ms. Note that N is equal
to the weight W = 1.96 mN of the component). The small upward motion is
unavoidable since the component is initially in contact with the gripper, leading to an initial force F = 3.7 mN larger than the weight. The only way to
avoid this is to have a capillary force at contact smaller than the weight, but
in this case, the picking task cannot be performed. However, the gap increases
very quickly and the corresponding capillary force becomes smaller than the
weight at time t = 1.1 ms: this can be seen by observing that at this time
the force F is equal to the component weight W (Fig. 18.22) or by observing
that the acceleration of the component becomes equal to zero at this time
(Fig. 18.21). A last remark on Fig. 18.22 is the switch in capillary force at
time t = 3.4 ms from F = 0.41 mN to zero: this change is forced once the
capillary force becomes smaller than a predefined threshold (typically 10% of
the force computed at contact).
Between these two extreme situations, an evolution can be observed from
the perfect pick to the perfect release in Fig. 18.23. The latter figure is particularly interesting because the component begins to be released: indeed
18.10 Dynamical Release
181
Gap[mm]
1
0.5
Contact force [mN]
Capillary Force [mN]
0
0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
4
2
0
0
2
1
0
0
Time [ms]
Fig. 18.22. m = 200 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper
(p = 0.1): gap, capillary, and contact forces
z [mm]
1
Gripper
Object
0.5
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
vz [ms−1]
0.2
0.1
0
az [ms−2]
100
0
−100
Time [ms]
Fig. 18.23. m = 50 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1):
positions, velocities, and accelerations
182
18 Results
Gap[mm]
0.04
0.02
Contact force [mN]
Capillary Force [mN]
0
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
4
3
2
1
0.5
0
Time [ms]
Fig. 18.24. m = 50 mg, V = 0.28 µL water, θ1 = θ2 = 30◦ , conical gripper (p = 0.1):
gap, capillary, and contact forces
Fig. 18.23 indicates that until time t = 1 ms its acceleration is positive (the
component moves upwards and leaves the substrate as indicated by N = 0
in Fig. 18.24) but smaller than that of the gripper. It can also be seen that
its velocity is smaller than that of the gripper until t = 2 ms. Nevertheless, in
time t = 1 ms, the acceleration of the gripper is set to zero, so that between
t = 1 ms and t = 2 ms, the velocity of the component can reach up to that
of the gripper: the gap is consequently maximum at t = 2 ms as indicated
in 18.24. Between t = 2 ms and t = 3.5 ms, the velocity of the component
is larger than that of the gripper: the gap decreases and the capillary force
increases. After t = 3.5 ms, the component perfectly follows the motion of the
gripper. This figure indicates that the component would be finally released if
the acceleration duration of the gripper was larger, but at a height z at least
higher than 30 µm (this is the position of the component at time t = 1 ms).
18.10.2 Experimental Results
For the experimental testing of the dynamic release, a specific workbench has
been set up in collaboration with Maxime Frennet [61]. As shown in Sect. 18.9,
if the gripper acceleration is stopped too early, a component that has begun
to detach can stick to the gripper again due to its acquired upward velocity (Fig. 18.24). Consequently, the actuator has been programed in order to
first displace downwards to give the component a downwards velocity before
18.10 Dynamical Release
183
ε
F
ε
W
(a)
F
W
(b)
(c)
Fig. 18.25. Sources of errors in the tests for dynamic release. (a) Misalignment ;
(b) Droplet positioning error; (c) Axial symmetry error
beginning its upward acceleration : the full acceleration lies in the range between 108 and 123 m s−2 and is applied during 10 ms. Then the control mode
of the actuator must be switched from the “acceleration” mode to the “position” mode and the actuator decelerates before stabilizing about the position
at the switch time. Several gripper tips have been tried on this actuator (conical and spherical grippers in steel, foam plane gripper); the tests led with
the conical and spherical grippers did not give satisfaction, not because they
could not show the ability to catch and release components but well because
the actual forces seemed to be lower than the predicted ones. This could
be explained by misalignments or axial asymmetry as shown in Fig. 18.25:
they usually induce the rotation of the component along the gripper, leading
to unexpected configurations. The components used for these tests were the
components St-i and Si-i defined in Sect. 17.9.2. To increase the repeatability
of the handling task, a foam gripper has been tried: it consisted of a small
foam parallelepiped (2 × 2.5 × 2 mm3 , Length–Width–Thickness) glued at the
end of a support screwed in the movable axis of the actuator. As this kind of
gripper has not been modeled, the force has been measured separately on a
silicon component (Si-004-1) with 1 µL R47V50. The gripper has been dipped
once to a 1 µL silicone oil droplet. Then, after each measure, the component
was dried and the gripper applied again on the component with a force of
about 0.86 mN before measuring the capillary force linking the foam gripper
and the silicon component. This experiment has been led 30 times (results
are presented in Fig. 18.26). A second experiment consisted in measuring the
force seven times by dispensing a new 1 µL droplet before each new trial. The
mean value of these seven trials (F = 0.277 mN) has been used to roughly
predict if a component would stick or not to the gripper tip during a trial of
dynamic release. A range has consequently been defined for the masses:
F
a+g
F
= .
g
mmin =
(18.8)
mmax
(18.9)
184
18 Results
4
x 10−4
3.5
Available Force [N]
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
Number of pick operations
25
30
Fig. 18.26. Capabilities of a foam gripper
1.5
Lower Boundary of the Working Domain, m=2.5mg
Upper Boundary of the Working Domain, m=28.2mg
1
Shift between prediction and experiments
Picked but not released
Picked and Released
Not Picked
0.5
0
0
50
100
Mass [mg]
150
200
Fig. 18.27. Results of the dynamical handling
If a component has a mass smaller (larger) than mmin (mmax ), it cannot be
released (picked up). With a = 100 m s−2 , this led to mmin = 2.5 mg and
mmax = 28.2 mg (represented by the solid lines in Fig. 18.27). Then, experimental tests have been led with different masses: 12.8, 15.1, 17.1, 46.1, 54.1,
and 183.1 mg:
•
•
•
Components with m = 12.8 and 15.1 mg could not be released
Components with m = 17.1 and 46.1 mg could be picked up and released
Components with m = 54.1 and 183.1 mg could be picked up
18.11 Approaching Contact and Rupture Distances
185
The predicted results are not perfectly correct (there is a shift of about
23 mg with the experimental results), but the trend is indeed obtained experimentally: it can be distinguished between three intervals, indicating that a
handling window exists either by adapting the mass for a given gripper acceleration or by adapting the acceleration for a given component. Nevertheless,
this release method leads to a poor positioning accuracy. Indeed, in successful
release trials, the component could touch the substrate at a different position
from that where it was deposed by the gripper. This positioning error can
reach several tenths of millimeter, which is unavoidable. The situation was
hardly improved by increasing the downwards impulsion. The conclusion of
these tests are the following:
•
•
•
There is actually a handling window in dynamical manipulation
The positioning accuracy is not good (several tenths of millimeter), at least
for millimetric components
Consequently, this method cannot be used without a dramatical improvement of the accuracy (in particular, the alignment must be achieved very
carefully (Fig. 18.25). Maybe at lower scale, higher accelerations can be
reached with piezoelectric actuators, but it should be checked if the relative positioning error (i.e., error divided by component size) is actually
decreased
18.11 Approaching Contact and Rupture Distances
To investigate the approaching and rupture distances, the following protocol
(Fig. 18.28) has been applied to a spherical steel gripper with radius R =
1.6 mm and a silicon component, with volumes V = [0.1, 0.2, 0.5, 1] µL of water
and silicone oil:
1. The amount of liquid is dispensed directly on the component (Fig. 18.28a)
2. The gripper is moved downwards until contact with the droplet is detected:
this gives the initial approach distance (Fig. 18.28b)
Rupture
distance(1)
Initial approach
distance
(a)
(b)
(c)
Rupture
distance(2)
(d)
Fig. 18.28. Rupture protocol: (a) liquid dispensing; (b) initial contact; (c) rupture;
(d) second approach
18 Results
1200
1200
1000
1000
Distance [µm]
Distance [µm]
186
800
600
400
200
0
800
600
400
200
0
0.2 0.4 0.6 0.8
Volume of water [µL]
(a)
1
0
0
0.2
0.4
0.6
0.8
1
Volume of R47V50 [µL]
(b)
Fig. 18.29. Approaching and rupture distances. Experiments: initial approach
(plus), second approach (open triangle), first rupture (open square), second rupture
(asterisk ). Simulations: initial approach (dotted line), second approach (dot dashed
line), rupture (dashed line). As a comparison, the solid line states for z = V 1/3 .
(a) Water; (b) R47V50
3. The liquid bridge is stretched until rupture: this is called the rupture
distance 1 (Fig. 18.28c)
4. After the first rupture, the volume of liquid is now distributed over both
the gripper and the component: the approaching distance has now changed
into the approaching distance 2 (Fig. 18.28d)
5. The gripper is moved upwards and the rupture gap is measured for the
second time
The results presented in Fig. 18.29 show that the rupture gap is slightly
smaller at the second bridge collapse. The difference is between 6.5% and 9.9%
for water, 2.0% and 5.7% for silicone oil.
Simulations give the right trend and the order of magnitude although
they are not so accurate. The reason therefore should be further investigated.
Nevertheless, the simulation provides an order of magnitude for gripper
design: the approach and rupture distances supply information on the required
displacement range of the gripper for picking (approaching distance) and
release (rupture distance). If the gripper could be “moved away” from the
component, release would occur by meniscus collapse. This could be achieved
by using an auxiliary tip with hydro (oleo)phobic properties.
18.12 Shear Force
The shear force exerted by the meniscus has been tested experimentally (see
Fig. 18.30) because this test is out of the scope of the axial symmetry assumption. Trials have been led with water, a silicon component, and a steel spherical gripper R = 6.35 mm. The gap was about z = 200 µm. In comparison, the
18.13 Conclusions
187
x 10−4
Shear force [N]
2.5
2
1.5
1
0.5
0
0
1
2
3
4
Volume of water [m3]
5
x 10−10
Fig. 18.30. Shear force (experimental points and their averages)
forces generated in the z direction in the same conditions are, respectively,
F = 0.35 × 10−3 N, F = 0.58 × 10−3 N, and F = 1.00 × 10−3 N. The norm
of the shear force to overcome is consequently reduced to 28.7%, 24.1%, and
20.0% of the force along z. If the gripper is moved perpendicularly to the z
direction, the contact force must also be taken into account. In some cases,
the viscous drag is not neglected [25]. Here the measure only concerns the
shear capillary force because there was no contact between the gripper and
the component (gap z = 200 µm) and the shear velocity was almost zero.
18.13 Conclusions
This chapter has shown that the models proposed in Part II were in good
agreement with experiments and analytical benchmarks when available. The
results presented in this chapter have illustrated the influence of each parameter on the capillary force. These results will be summarized in Chap. 21,
together with the results of the case study. It will allow to sketch some design
rules for a surface tension gripper.
19
Example 3: Application to the Watch Bearing
Case Study: Characterization
19.1 Introduction
This chapter describes the characterization of the required parameters to be
used in the force model of Chap. 10. It focuses on the available grippers, the
different components, the kinds of liquids used in the micromanipulation and
their properties (surface tension, viscosity, and density). The liquid dispensing
method is also addressed and, finally, the measurement of the contact angles
and the contact angle hysteresis are described. Based on a Anova analysis, the
importance of the coating on the repeatability of the contact angles is shown
(the influence is here clearly negative to an unadapted choice of the coating).
19.2 Available Grippers
According to the specifications given in Chap. 10, twelve grippers have been
manufactured in stainless steel (sketches are given in Fig. 19.2). These grippers vary from one another concerning the presence/absence of an internal
channel initially foreseen for liquid supply (it will be shown later that it is
unnecessary), the diameter (0.3 mm or 0.5 mm), the presence or absence of
a hydrophobic coating, and the shape of the extremal tip (with or without
the conical concavity which is preferable to ensure centering (see Fig. 19.1)).
A summary of their properties is given in Table 19.1.
The geometrical characteristics of these 12 grippers have been measured
using the control points defined in Fig. 19.2.
The measured values for the 0.5 mm (0.3 mm) diameter grippers are presented in Table 19.2 (19.3).
190
19 Example 3: Application to the Watch Bearing Case Study
(a)
(b)
Fig. 19.1. Pictures of two grippers. (a) Gripper E with a cylindrical tip without
concavity, an internal Φ0.1 mm channel and no coating (the two marks below the
gripper are separated from 1 mm); (b) Gripper B with a concavity, an internal
Φ0.1 mm channel and an hydrophobic coating
Table 19.1. Properties of the grippers
A
B
C
D
E
F
G
H
I
J
K
L
Diameter (mm)
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.3
0.3
0.3
0.3
Channel
Yes
Yes
No
No
Yes
Yes
No
No
No
No
No
No
Concavity
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
Hydrophobic coating
Yes
Yes
Yes
Yes
No
No
No
No
Yes
Yes
No
No
αg
αd
y
t
w
z
x
Fig. 19.2. Measured points
19.4 Liquid Properties
191
Table 19.2. Characterization of the 500 µm diameter grippers
A
B
C
D
E
F
G
H
Average
Std
w (µm) tmin (µm) tmax (µm) x(µm) y(µm) z(µm)
494.8
907.1
917.0
448.6 511.3
864.3
537.7
864.3
870.9
537.7 527.8
834.6
491.5
1091.9
1101.8
494.8 498.1 1082.0
491.5
950.0
956.6
484.9 498.1
923.6
504.7
851.1
834.6
494.8 501.4
646.5
527.8
903.8
907.1
508.0 481.6
897.2
494.8
1095.2
1105.1
491.5 494.8 1098.5
511.3
966.5
983.0
501.4 501.4
963.2
506.8
953.7
959.5
495.2 501.8
913.7
17.6
94.5
100.0
24.8
13.4
144.1
αl (◦ ) αr (◦ )
135.5 136.0
134.5 135.4
140.0 139.7
138.9 140.5
134.9 138.9
135.0 135.6
137.7 138.9
139.4 140.1
137.0 13801
2.3
2.1
Table 19.3. Characterization of the 300 µm diameter grippers
w (µm) tmin (µm) tmax (µm) x(µm) y(µm)
I
1052.3
1078.7
300.2 296.9
J
316.7
1062.2
1065.5
316.7 303.5
K
313.4
1042.4
1055.6
303.5 293.6
L
283.7
1016.0
1025.9
296.9 287.0
Average 304.6
1043.2
1056.4
304.3 295.2
Std
18.2
19.9
22.4
8.7
6.9
z(µm)
992.9
1016.0
1022.6
963.2
998.7
26.80
αl (◦ )
136.8
139.6
140.1
139.1
138.9
1.5
αr (◦ )
142.8
140.4
141.5
140.5
141.3
1.1
Table 19.4. Surface tension
Liquid
Isopropanol
Water
L23
Surface tension (N m−1 )
23.2 × 10−3
73.0 × 10−3
33.5 × 10−3
19.3 Available Components
The components handled in this application are the Zirconium balls of the
chosen ball bearing. Two kinds of diameters are available, 0.3 mm and 0.5 mm,
to be picked by grippers with a corresponding diameter.
19.4 Liquid Properties
The properties that play a role in the presented models are the surface tension
γ (capillary force models) on the one hand and the dynamic viscosity µ and
the density ρ (characteristic times) on the other hand. The surface tensions
have been measured by the Du Nouÿ ring method [1] applied, thanks to a
Sigma 703 sensor: they are presented in Table 19.4. The densities have been
measured from the volume and the mass. Results are given in Table 19.5.
192
19 Example 3: Application to the Watch Bearing Case Study
Table 19.5. Density
Liquid
Isopropanol
Water
L23
Density (kg m−3 )
768
1000
906
Source
Own experiment
Classical value
Own experiment
Table 19.6. Dynamic viscosity
Liquid
Isopropanol
Water
L23
Dynamic viscosity at 20◦ C(Pa s)
2.27 × 10−3
1 × 10−3
21.7 × 10−3
Source
[62]
Classical value
MPS (24 cs) and density
Finally, the values of the dynamic viscosity, given by the suppliers, are
given in Table 19.6.
19.5 Liquid Dispensing
It has first been envisaged to supply the liquid through the internal channel,
but it turned out that this method would lead to a complicated pressure
control solution, an instability of the hanging droplet and a configuration much
more difficult to model (in the force model presented in Chap. 10, the pressure
inside the meniscus is given by the external pressure and the pressure drop
across the interface, which is given by the surface tension and the curvature.
If the internal pressure is imposed, the volume of liquid has to be determined
in order to get a meniscus whose shape fits the contact angle as well as this –
now imposed – pressure difference).
The second solution would have been to extend the manual dispensing
solution applied in the first experiments of Chap. 18, but the main problem
with water and isopropanol is the evaporation with the time. Let us note
that it is not really a problem from an application point of view, because it
is interesting to eliminate the residual traces of liquid on the component. It
is, however, a drawback for gripper characterization because the evaporation
process can be faster than the typical experiment duration. For example, we
have experimented the following evaporation times for liquid droplets posed
on a flat substrate in the ambient conditions (see Fig. 19.3, 19.4 and 19.5):
1. 1 µL isopropanol evaporates in 15 s
2. 1 µL water evaporates in 10 min
3. Oil like R47V50 silicone oil or L23 synthetic oil does not evaporate or less
than 1%
Finally, the principle of liquid feeding which has been retained is to dip
the gripper tip to a liquid tank. A small liquid droplet hangs at the bottom
of the gripper, as illustrated in Fig. 19.6. One can expect the droplet height
19.5 Liquid Dispensing
(a)
(b)
(c)
(d)
193
(e)
Fig. 19.3. Evaporation of a water droplet (1). (a) t = 0 s; (b) t = 60 s ; (c) t = 120 s;
(d) t = 190 s; (e) t = 250 s
(a)
(b)
(c)
(d)
(e)
Fig. 19.4. Evaporation of a water droplet (2). (a) t = 310 s; (b) t = 370 s; (c)
t = 0430 s; (d) t = 500 s; (e) t = 560 s
20
V [nL]
15
10
5
0
0
100
200
300
400
Time [s]
500
600
700
Fig. 19.5. Water evaporation
d
h
Fig. 19.6. Tip dipping d is the gripper diameter and h is the height of the hanging
droplet
194
19 Example 3: Application to the Watch Bearing Case Study
Table 19.7. Anova of the hanging height model
Average H̄
A1
A2
Residual
Total Data
SS
35.29
0.02
0.02
0.13
35.46
DF
1
1
1
356
359
MS
35.29
0.022
00.07
0.00037
-
F
95000
59
44
1
-
p
< 10−9
< 10−9
< 10−9
-
h to depend on the following parameters: the liquid surface tension γ, the
gravity g, the liquid density ρ, the gripper diameter D, the contact angle θ.
Nondimensional formulated, it means that the ratio height/diameter can be
expressed as follows:
ρgd2
h
= f (θ,
) = f (θ, Bo ).
d
γ
(19.1)
Therefore, the problem is much easier to represent graphically: there are two
nondimensional input parameters and one nondimensional output parameter
H = hd .
Using the design of experiment theory, the linear model without interaction
has been tested:
H = H̄ + A1 θ + A2 Bo + ,
(19.2)
where the coefficient H̄, A1 , and A2 have been calculated by least squares,
and the corresponding Anova is shown in Table 19.7.
Using the genuine replications of the experiments (each point of the
experimental space has been replicated 10 times), the residual sum of squares
(0.13) has been split into the pure error (0.05) and the lack of fit of the model
(0.08). It is also possible to calculate a reliability interval for the coefficient
of the model as follows:
1. An estimation of the variance of the model is calculated:
ŝ =
(y − ŷ) (y − ŷ)
n−p
(19.3)
2. The impact of this estimated variance is amplified by the dispersion matrix:
Vb = (X X)−1 ŝ
(19.4)
3. Finally,
SEb =
diag(Vb )
(19.5)
19.6 Contact Angles
195
These calculations lead to the following reliability intervals given in
Table 19.8.
The conclusion to be drawn is that except the main effect given by H̄,
all the other parameters play no role in the volume of liquid obtained by tip
dipping. In other words, whatever the material and the liquid, the hanging
droplet has a height determined by the gripper diameter.
19.6 Contact Angles
To measure the contact angle hysteresis, the gripper was moved downwards
(Fig. 19.7a) to measure the advancing contact angle and upwards (Fig. 19.7b)
to measure the receding contact angle. Both angles were measured by moving
the gripper very slowly (a few microns per second, in a quasi static configuration). All combinations of liquids and of grippers have been considered [112],
each angle being measured 10 times. The average and the standard deviations
of each configuration are plotted in Figs. 19.8a and b and 19.9a and b. In
these figures, grippers A, B, C, D, I, J are coated and grippers E, F, G, H, K, L
are not.
From the detailed Anova analysis led in [112], it can be concluded that
the influence of the applied coating is not significant. Indeed, the main effect
Table 19.8. Error intervals
Average H̄
A1
A2
Value
0.31501
0.00191
−0.01298
Std
0.00126
0.00227
0.00195
θrec
θadv
(a)
(b)
Fig. 19.7. Illustration of the contact angle hysteresis (case of a stainless steel
Φ0.5 mm gripper and water) (Reprinted with permission from [112]. Copyright 2006
Institute of Physics.)
196
19 Example 3: Application to the Watch Bearing Case Study
100
Isopropanol
L23
Water
80
60
40
20
0
Std advancing angles
Mean advancing angles
100
Isopropanol
L23
Water
80
60
40
20
0
1 2 3 4 5 6 7 8 9 10 11 12
Grippers
A B C D I
(a)
J E F G H K L
Grippers
(b)
Fig. 19.8. Advancing contact angles at latest 4 days after coating. (a) Average values; (b) standard deviations of each ten of trials – Coated grippers: A, B, C, D, I, J;
noncoated grippers: E, F, G, H, K, L (Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)
100
Isopropanol
L23
Water
80
60
40
20
0
A B C D I J E F G H K L
Grippers
(a)
Std receding angles
Mean receding angles
100
Isopropanol
L23
Water
80
60
40
20
0
A B C D I J E F G H K L
Grippers
(b)
Fig. 19.9. Receding contact angles at latest 4 days after coating. (a) Average values;
(b) standard deviations of each ten of trials – Coated grippers: A, B, C, D, I, J; non
coated grippers: E, F, G, H, K, L (Reprinted with permission from [112]. Copyright
2006 Institute of Physics.)
of the coating is of the same order of magnitude – but a bit smaller – than
the main effect of the liquids on the advancing contact angle (4 days after
coating). It becomes totally negligible in the case of the receding contact
angle.
This data analysis leads to the conclusion that the coating of the grippers is not useful in our application since the main effect of the coating
is clearly negligible in comparison with the main effect of the liquid. More
physically, this coating has been optimized to be vaporized on a silicon substrate. This coating is just a monolayer, which can be damaged or pulled
off at some locations, leading to a quite heterogeneous surface. This physical
19.6 Contact Angles
197
suggestion is reinforced by larger standard deviations in the presence of coatings. The conclusion of this analysis is that in case of applying any coating,
the wear resistance should be addressed carefully since the repeated contact
between the gripper and the consecutive components induce damage onto
this gripper coating, leading to a lack of efficiency and an increase of the
dispersion.
20
Example 4: Application to the Watch Bearing
Case Study: Results
20.1 Introduction
In Chap. 10, we have briefly described the design of a surface tension based
gripper to be used in the placement of 0.3- or 0.5-mm diameter balls of a watch
ball bearing. Chapter 19 has described the characterization of this gripper
and the contact angles made by liquids that could be used (isopropanaol,
water, and oil). This chapter aims at detailing the results of the typical
micromanipulation task, illustrated in Figs. 20.1 and 20.2. The 0.5-mm diameter ball “floats” by surface tension on a spherical cap-shaped water drop
posed on a steel substrate (Fig. 20.1a), the gripper (ripper D, see Table 19.1
for details) is aligned above the ball (Fig. 20.1b) and then moved downwards
until contact with the ball. The result of the pick operation can be seen in
Fig. 20.1c. Then the gripper is moved toward the ball bearing (Fig. 20.1d)
and positioned above the cavity in which the ball is to be placed (Fig. 20.2a).
The ball is positioned in its final location (Fig. 20.2b) and the release takes
place by moving the gripper radially outwards (Fig. 20.2c and d).
20.2 Picking
20.2.1 Introduction
The picking task cannot be separated from the feeding solution. Indeed, small
components must first be separated and positioned before picking. Nevertheless, this task has to be done whatever the picking principle is, and some
solutions already exist: feeding trays, tapes, etc. As it is not the primary
point of this book, it has been considered that at worst the position of the
balls could be determined by vision. First some typical picking errors are presented in Sect. 20.3, before proposing solutions (Sect. 20.2.3) and tracks for
automated control (Sect. 20.2.4).
200
20 Example 4: Application to the Watch Bearing Case Study: Results
(a)
(b)
(c)
(d)
Fig. 20.1. Picking sequence: (a) the ball is initially placed on a small droplet
(bottom right); (b) alignment of the gripper; (c) ball picking; (d) displacement of
the gripper toward the ball bearing (Reprinted with permission from [112]. Copyright
2006 Institute of Physics.)
(a)
(b)
(c)
(d)
Fig. 20.2. (a) Positioning; (b) Placing of the ball the cage of the bearing; (c) and
(d) Release of the component by radial outwards motion of the gripper (Reprinted
with permission from [112]. Copyright 2006 Institute of Physics.)
(a)
(b)
(c)
(d)
Fig. 20.3. Picking errors: (a) centering error; (b-c) balls get stuck after water evaporation; (d) surface forces curiosity (Φ0.5 mm Zr O2 balls, conical grippers) (Reprinted
with permission from [112]. Copyright 2006 Institute of Physics.)
20.2.2 Errors
Figure 20.3 illustrates some typical picking errors. The first error (not shown
in the figure) is that when no ball is picked up by the capillary gripper. It
can happen when the gripper is not aligned with respect to the ball. There
is a small centering effect due to the conical geometry of the gripper, but we
experimented that this centering effect was limited by the friction. Usually,
by adding a bit of liquid and by positioning the gripper properly this could
be corrected.
Then second error type is a lack of centering, illustrated in Fig. 20.3a. In
this case, the ball must be released and the picking operation must be tried
again. To release the ball, it is sufficient to dip it to a liquid tank: since there
20.2 Picking
201
Fig. 20.4. Top view of the hexagonal network of the 10 balls: The gray ball is picked
by the gripper (Reprinted with permission from [112]. Copyright 2006 Institute of
Physics.)
is no more liquid–gas interface between the ball and the gripper, there is no
more capillary force and the weight of the ball pulls it downwards.
When working with water, other complications can occur, as illustrated in
Figs. 20.3b and c, showing the case of several balls sticking with each other,
probably due to the capillary forces caused by residual traces of liquid.1 In
some cases, these sticking balls form a regular hexagonal shaped alignment,
as shown in Fig. 20.4.
Finally, Fig. 20.3d illustrates the sensitivity of the micromanipulation task
to its environment. The ball seems to be levitated but the underlying reason
is the presence on a small dust linking the ball and the gripper, whose effect
dominates the weight of the ball. The phenomenon was very stable (a few
minutes) , henceforth discarding effects such as electrostatic levitation.
20.2.3 Solutions
Different solutions can be applied to these problems. The first category of
solutions aims at guaranteeing the picking of only one ball at a time. This
can be best performed by posing the balls at the interface of a liquid, such as
depicted in Fig. 20.5. These pictures illustrate the ability to pick one ball at a
time from an initial alignment of four balls (Fig. 20.5a). The coated concave
conical gripper is positioned above the second ball from left (Fig. 20.5b),
moved downwards in contact with this ball (Fig. 20.5c). Figure 20.5d shows
the pick operation of this ball. The pick operation is then improved with what
can be called a kind of “liquid feeding.” The efficiency of this strategy clearly
depends on the volume of liquid (which, for water, means dependent on time
because of the evaporation), on the kind of liquid, and on the number of balls
in the neighborhood of the picked ball. Indeed, in the case of regular balls
network with the L23 oil, the picking efficiency was not clearly improved by
1
It has been shown the large evaporation rate in Fig. 19.3 but when the meniscus
is trapped between two solid surfaces close to one another, it turns out a large
reduction of the evaporation.
202
20 Example 4: Application to the Watch Bearing Case Study: Results
(a)
(b)
(c)
(d)
Fig. 20.5. Coated conical tip (B): picking of sorted balls from a liquid (Reprinted
with permission from [112]. Copyright 2006 Institute of Physics.)
(a)
(b)
(c)
(d)
Fig. 20.6. Coated conical tip (B): picking of non sorted balls from a liquid
(Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)
the presence of a liquid film. The reason for this probably lies in the fact that
the contact angle between the ball (in zirconium) and the oil is much smaller
than the one got with water. Therefore, the capillary force between balls can
be stronger, the sum of the forces between the balls being larger than the
force between the gripper and the picked ball.
The pictures shown in Fig. 20.6 illustrate the “pick from wet bulk” strategy: a series of 10 balls float by surface tension on a spherical cap-shaped
water droplet put on a steel substrate. In Fig. 20.6, the gripper is dipped to
the balls, and picks up only one ball (Figs. 20.6b and 20.6).
The second category of solutions consists in releasing all the picked balls,
either when there is a centering error or when more than one ball has been
picked. A way to release all the balls is to dip the gripper to a liquid tank:
since there is no capillary force in the liquid, the weight of the balls pull them
downwards.
20.2.4 Automated Control
The picking task can be controlled by vision, using an image analysis based on
the recognition of the gripper blob (i.e., contour). The analysis routine starts
from the black and white images of the grippers, such as shown in Fig. 20.7.
The blob analysis is lead from the cropped area located just below the
gripper tip, such as indicated in Fig. 20.8.
Then, from the coordinates of the contour points of each blob, the area,
perimeter, and compacity of the blob are calculated. Let us consider that a
blob is defined by its coordinates:
20.2 Picking
(a)
(b)
(c)
203
(d)
Fig. 20.7. Black and white pictures of four picking results. (a) Correct picking; (b)
no ball is picked; (c) and (d) too many balls have been picked
(a)
(b)
(c)
(d)
Fig. 20.8. Corresponding blob analysis of the four picking results of Fig. 20.7
xi ,
1 ≤ i ≤ n,
(20.1)
yi ,
1 ≤ i ≤ n.
(20.2)
Then, the perimeter is given by
P =
ds =
n−1
dsi =
i=1
n−1
(xi+1 − xi )2 + (yi+1 − yi )2 .
(20.3)
i=1
The area calculation is based on the Stokes theorem:
¯ × F̄ ) · dS̄ = F̄ · dS̄.
(∇
(20.4)
Since the rotational of F̄ can be written in the cartesian coordinates as
¯ × F̄ = ( ∂Fy − ∂Fz )1̄x + ( ∂Fx − ∂Fz )1̄y + ( ∂Fy − ∂Fx )1̄z
∇
∂z
∂y
∂z
∂x
∂x
∂y
(20.5)
and
dS̄ = dS 1̄z ,
(20.6)
∂F
x
if we choose Fx ≡ 0 and Fy ≡ x, we will have ∂xy − ∂F
∂y = 1 and
¯
(∇ × F̄ ) · dS̄ =
dS = S = (Fx dx + Fy dy) = x dy.
(20.7)
Therefore, the area can be approximated by
A=
x dy =
n−1
i=1
xi (yi+1 − yi ).
(20.8)
204
20 Example 4: Application to the Watch Bearing Case Study: Results
4
0.8
2
0.6
1.5
0.4
1
0.2
0.5
0
0
x 10
1000
800
600
400
1
2
3
4
(a)
200
1
2
3
0
4
(b)
1
2
3
4
(c)
Fig. 20.9. (a) Compacity (without dimension); (b) Area (pixel2 ); (c) Perimeter
(pixel). Note that 1 = 1 ball, 2 = 0 ball, 3 = 3 horizontal balls, and 4 = 3 vertical
balls)
(a)
(b)
(c)
(d)
Fig. 20.10. (a) Successful placement: the ball remains in position; (b) Placement
error: the ball is released, but its equilibrium position is higher than the gripper; (c)
Placement error: the ball is not released and sticks to the gripper; (d) Placement
error: the gripper touches the cage (Reprinted with permission from [112]. Copyright
2006 Institute of Physics.)
Concerning the compacity, let us define it as the following ratio:
C = 4π
A
,
P2
(20.9)
so that the compacity of a disk is equal to 1. According to the results shown
in Fig. 20.9, the best strategy seems to use the compacity as an indicator,
since it is the largest in the case of a successful picking (1 ball) on the one
hand and because it is size independent on the other hand (there is no need
to calibrate the camera). These results suggest that the picking task can be
considered as successful when the compacity is larger than 0.7.
20.3 Placing
Several release strategies have been studied in the literature (a state of the art
can be found in [109]). The chosen strategy described below is inspired by the
so-called scraping release. Figure 20.10 illustrates several placement errors.
The placement reliability has been studied as a function of the ratio
between the height h between the gripper and the bearing on the one hand
20.4 Compliance Effect
205
Gripper L
Gripper D
Gripper H
Gripper J
Reliability as a function of the normalized gap
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0
0.1
0.2
0.3
0.4
0.5
1
0.5
0
1
Succeeded
Failed
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Fig. 20.11. Reliability of the placement as a function of the nondimensional gap
between the gripper and the bearing – Gripper J: 0.3 mm, coated; gripper
H: 0.5 mm, noncoated; gripper D: 0.5 mm, coated; gripper L: 0.3 mm, noncoated
(Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Fig. 20.12. Compliance of the surface tension gripping
and the gripper diameter d on the other hand. The use of this nondimensional
ratio allows the comparison between the Φ0.3 mm and the Φ0.5 mm grippers.
In Fig. 20.11, the reliability is calculated as the number of successes divided
by 10 (each placement has been replicated 10 times). It is represented as a
function of the ratio h/d. For gaps smaller than 0.2 times the ball diameter,
the placement is always successful.
20.4 Compliance Effect
The liquid meniscus provides a lateral compliance, which could be used to
balance positioning inaccuracies. This effect must be very dependent on the
volume of liquid. Since this latter could not be measured yet, this influence is
still to be studied. The volume of liquid might be measured with fluorescein or
its influence studied by simulation, using the Surface Evolver freeware since
the configuration is not axially symmetric (by definition).
AQ:
Citation of
Fig. 20.12
is missing.
206
20 Example 4: Application to the Watch Bearing Case Study: Results
(a)
(b)
(c)
Fig. 20.13. Parameters to be studied experimentally. (a) The prestressing ξ applied
onto the ball before picking; (b) the waiting time τ during which the prestressing is
applied; (c) the picking speed v
20.5 Force Measurement
20.5.1 Introduction
The force measurement is necessary to validate the force models described in
Chap. 10 and to study experimentally the influence of nonmodeled parameters
such as the prestressing ξ applied onto the ball before picking, the waiting time
τ during which this prestressing is applied, and the picking speed v. These
parameters are illustrated in Fig. 20.13.
20.5.2 Modification of the Force Measurement Test Bed
This first contact discards the main part of liquid so that the volume of liquid
becomes smaller than the conical cavity of the grippers (henceforth, it cannot
be seen with the camera). Then, the gripper is applied onto the ball n times
without refilling it and the capillary force between the gripper and the ball
is measured. After each rupture of the meniscus (i.e., after each contact), a
bit of liquid is left on the ball, so that the volume of liquid involved in the
manipulation is decreasing (however, it cannot be measured).
20.5.3 Comparison Between Models and Experiments
The first presented result is the comparison between models of Sect. 10.3
and Sect. 10.4 and the experimental measured force. From Fig. 10.14, two
regimes can be distinguished: for β < α (here α = 20◦ ), the force is increasing
with β while the volume of liquid remains quite constant. In the case β > α,
the volume of liquid quickly increases with the filling angle while the force
is decreasing. The first regime is not accessible at all, due to the dramatic
sensitivity of the force to the volume of liquid. The second regime encounters
severe limitations because the gripper is not optically transparent, hence the
volume of liquid cannot be accurately measured (this is still work in progress).
20.5 Force Measurement
207
x 10−4
Simulation result (H)
Force (N)
1.5
Gripper H
Simulation result (J)
1
Gripper J
Gripper B
0.5
Gripper F
Weight of a 0.5mm diameter ZrO2 ball
0
0
20
40
60
80
100
120
Experiment number
Fig. 20.14. Force measurements with the oil L23: The behavior of grippers J (•)
and H (2) is to be compared with simulation in the case of maximal filling angle β.
These simulation results only provide a comparison of the order of magnitude since
the volume of liquid is not accurately known (gripper H: 0.5 mm without channel,
noncoated; gripper J: 0.3 mm without channel, coated; gripper B (◦): 0.5 mm with
channel, coated; gripper F (∆): 0.5 mm with channel, noncoated). For grippers B
and F , the presence of an internal channel seems to reduce and stabilize the force
(Reprinted with permission from [112]. Copyright 2006 Institute of Physics.)
Therefore, to assess at least the order of magnitude of the force, the simulation
has been run until the liquid overflows, i.e., when β ≈ 60◦ . The underlying
justification is the assumption that if the contact between the gripper and the
ball is repeated without refilling the gripper, the amount of liquid involved in
the micromanipulation is assumed to decrease, hence the force increases. If the
reasoning is valid, the simulated force value should be a kind of limit toward
the experimental force should tend to. This trend is observed in Fig. 20.14
for grippers H and J, which have no internal channel. For grippers B and F
(with an internal channel), the simulation cannot be run because the effects of
the channel have not been modeled. Nevertheless, the presence of this channel
seems to decrease the force (the force generated by larger grippers B and F is
even smaller than the one generated by the smaller gripper J).
The conclusions at this stage are that the order of magnitude predicted by
the simulation is good but a finer analysis should be done with a transparent
gripper in order to measure both the volume of liquid and the force at the
same time. Deeper investigations could also be done on the role played by the
channel; however, its presence increases the complexity of manufacturing and
might be unnecessary for application purpose. Finally, since the force limit
does not seem to have been reached in Fig. 20.14, a second set of experiments
has been done with a linear motor instead of a manual stage to ensure the
gripper motion.
208
20 Example 4: Application to the Watch Bearing Case Study: Results
20.5.4 Ongoing Experimental Study
As already stated in the introduction, the influence of the following parameters
on the capillary force FC and the reliability of the process are to be studied: the
volume of liquid V , the prestressing ξ applied by the gripper on the component
before the picking step, the duration τ of this prestressing (also called delay
in what follows), the picking velocity v of the gripper, and the number of
repetitions n of the picking task. The following procedure has been applied:
1. Adding liquid (L23 oil, surface tension = 33.5 mN/m, dynamic viscosity =
21.7 mPa s, density = 906 kg/m3 , at 20◦ C) on the gripper
2. Moving the gripper on the ball and applying a prestressing of ξ mN
3. Waiting τ ms
4. Moving the gripper up at constant velocity v
In this procedure, attention should be paid to the fact that the picking task is
repeated n times (steps 2–4 are repeated n times) on the same ball, introducing artifact as far as the volume of liquid is concerned. Indeed, before the first
contact between the gripper and the ball, the volume of liquid is only on the
gripper side and it is larger than the available volume that can be filled between the gripper and the ball. Consequently, during the first picking (n = 1),
some liquid flow from the ball to the beam, leading to the decrease of the
actual volume of liquid involved in the manipulation. After some repetitions,
however, the system reaches a steady state since there is no leakage of liquid
any more (the beam becomes saturated and a nonvolatile liquid has been used
– silicone oil L23). For each set of input parameters (ξ, τ , v), the picking force
FC is measured n times during n repetitions (without refilling the system with
liquid). A typical evolution of FC is shown in Fig. 20.15 (n = 10,000).
In a first step, the curve grows up, reaches a maximum, and begins to
decrease. This is explained in [112] by the fact that by varying the volume
of liquid (cf. the experimental procedure which is described above), the force
increases, reaches its maximum, and decreases. After that, the capillary force
should stabilize: the liquid should stop flooding over the ball and the volume
would remain constant. However, the experiments have not been achieved as
much as 10,000 times.
In the following, attention will only be paid to the increasing part of the
curve (most of the experiments contain only 1,000 repetitions) and the experimental data will be fitted with the empirical curve FC = A − B e−Cn by the
least squares method (line in the lower part of Fig. 20.15). Consequently, it
will be focused on the influence of the liquid feeding and of the effects of ξ, τ ,
and v on the values of A, B, and C. The coefficient A represents the maximum
intensity of the capillary force, A − B is the capillary force developed with
initial volume of liquid and C describes how the capillary force evolves with
the samples n.
20.6 Conclusions
209
Capillary Force [mN]
0.1
0.05
0
0
2000
4000
6000
8000
10000
300
400
500
n
0.12
0.1
A
A–B
0.08
0
100
200
n
Fig. 20.15. Measure of capillary forces in an experiment of n = 10,000 repetitions,
with a prestressing ξ = 224 µN, a delay τ = 500 ms, and a picking speed v =
0.2 mm/s. The bottom subfigure shows the first 500 repetitions that have been
fitted with a curve FC = A − B e−Cn by the least squares method. Note that the
lengths A and A − B must be taken from the origin. For sake of clarity C has not
been represented (Reprinted with permission from [173], Copyright IWMF 2006.)
20.6 Conclusions
Following conclusions can be drawn from the above mentioned results.
When the picking error consists in an error centering or in picking up
more than one ball, the gripper can be dipped to liquid in order to release the
balls. Then, the picking task has to be performed again. When no ball can
be picked up, we suggest to tune the volume of liquid, but when the gripper
is centered above a ball without neighbors, this latter can always be picked
up. We proposed a vision control resting on a blob analysis and a compacity
analysis in order to automatically detect the error type.
The placing errors have been studied as a function of the gap between the
gripper and the bearing. When this gap is smaller than 1/5 of the gripper
diameter, the placing task is always successful.
Finally, the force results show that the generated capillary force (of the
order of 100 µN) is much larger than the weight (of the order of 4 µN).
Concerning the liquid supply of the gripper, it can be definitively concluded that the internal channel through which a pressure is applied is not
a good solution. Indeed, the manufacturing is more complicated, the liquid
feeding is subject to instabilities, and moreover, the generated force is smaller.
Therefore, the tip dipping strategy seems to be the good one.
Nevertheless, the validation of the proposed model has only been achieved
as far as the order of magnitude of the force is concerned. A more definitive
210
20 Example 4: Application to the Watch Bearing Case Study: Results
validation will be done as soon as the volume of liquid involved in the micromanipulation will be measured.
The repetitions of up to 10,000 picking operations without refilling the
gripper seems to be promising, but additional work will be done in a configuration which might be more realistic, i.e., where the ball would be dried
between two picking in order to better control the volume of liquid forming
the meniscus.
21
Conclusions
21.1 Introduction
To conclude this part, we propose to summarize the obtained results in terms
of picking on the one hand and of releasing on the other hand. Additionally,
some rules will be proposed to be applied to the design of a surface tension
based gripper.
21.2 Picking Operations
In terms of picking task, and especially in terms of picking force, the influence
of the following parameters can be pointed out:
1. Influence of contact angles (Figs. 18.13 and 18.14): contact angle θ1
depends on the combination component material-gripping liquid while θ2
depends on the combination gripping liquid-gripper material. Therefore,
if the component is imposed, the gripping liquid must be chosen to cause
small θ1 , in order to increase the force (it has been shown that the force
decreases as a function of θ1 , a little more slowly than the cosine function). Note that for a given component material, a low energy liquid (i.e.,
with small γ) provokes smaller θ1 but is less efficient as the force is also
proportional to γ. Therefore a compromise must be found.
Additionally, it must be noted that in the case of the conforming gripper
used in the watch bearing case study, the force is attractive even for nonwetting coating, as indicated in Fig. 10.12.
2. Influence of volume of liquid (Fig. 18.9): for spherical grippers, the volume
of liquid plays almost no role. Consequently the dispensing accuracy or
evaporation, for example, are not parameters of the utmost importance
here. On a different way, the force can remain quite steady if the amount
of liquid can be kept constant, which is more or less the case with a
foam gripper (Fig. 18.26). Nevertheless, for conical grippers, the larger
212
21 Conclusions
the volume, the larger the force. At the contrary, in the case of the conforming gripper of the watch bearing case study, the force increases with
decrease in volume of liquid (Fig. 10.14), which is of the utmost interest in a high throughput application. Consequently, no general rule can
be drawn concerning the influence of the volume of liquid: it depends
on the conformation between the component and the gripper geometries.
Moreover, the amount of liquid also determines the approaching contact
distance and the rupture distance.
3. Influence of the gripper’s geometry: cylindrical grippers lead to the largest
forces. As far as spherical grippers are concerned, the force is proportional
to their radius as illustrated by the Israelachvili approximation [89]:
F ≈ 4πRγ cos θ.
(21.1)
For conical grippers (Fig. 18.9) with silicone oil R47V501 , the developed
force FR47V50 is approximatively given by
log10
FR47V50
p
4
≈ − log10 ,
Fo
3
po
(21.2)
where po = 10◦ , p is the angular aperture (in ◦ ) of the considered conical
gripper, and Fo (in N) is given by
log10
Fo
V
≈ 0.33 log10 −9 ,
−3
0.87 × 10 N
10 L
(21.3)
where V is the volume of silicone oil (in L). The constants 0.87 × 10−3 N,
10−9 L, 0.33, and − 43 have been read in Fig. 18.9. Note that for other
liquids (i.e., other surface tensions γ), the force can be assumed to be
proportional to γ, and therefore
F
FR47V 50
=
γ
γR47V 50
.
(21.4)
This decrease of the force as a function of the angular aperture has to be
taken into account when considering the manufacturing accuracy of the
cone.
4. Influence of surface tension (Figs. 18.10 and 18.11): the force developed
by a liquid bridge is proportional to surface tension. Nevertheless, liquids
with a low surface tension better wet solids, leading to smaller contact
angles; therefore, a compromise has to be found.
5. Influence of the gap (Fig. 18.7): the force decreases with the separation
distance so that the maximal force is observed when gripper and component touch each other.
6. As an order of magnitude, forces up to several milli-Newton can be reached
easily (1 mN corresponds to a mass m = 100 mg). The rule of thumb is
1 mN for a 1 mm sized component.
1
γ = 20.9 mN m−1 .
21.3 Releasing Strategies
213
We have seen in Sect. 20.2 that in case of picking more than one component
at a time, the gripper could be cleaned by dipping it to a liquid tank. Of
course, a better solution would be to develop a specific feeder according to
the application. This will be considered in the perspectives of this work, at
least as far as the watch bearing is concerned.
21.3 Releasing Strategies
In a micro driven design, care should also be taken of the possibilities to release
the component. In macromanipulation, indeed, the picking force Fmax is replaced by Fmin = 0 at release while in the case of a surface tension based gripper (or any adhesive principle based gripper), there is always a residual adhesion force Fmin > 0 that can prevent the component from being released if Fmin
is larger than its weight W . Consequently, an adhesion or surface tension based
gripper can only be designed for a restricted mass range [mmin , mmax ] of components as illustrated in Table 21.1. To compare the different release strategies
in a quantified way, the following “adhesion” ratio has been introduced:
φ=
Fmin
,
Fmax
(21.5)
φ ranges from 0 (no residual adhesion, the releasing task is not a problem) to
1 (residual adhesion force is as large as the picking one, handling cannot be
performed). Let us now summarize the different release strategies (sorted by
decreasing φ):
1. Reduce the volume of the liquid bridge (Fig. 18.9): this strategy cannot be
used with spherical grippers since the developed force is almost independent from the amount of liquid but can be achieved for conical grippers.
Figure 18.9 indicates that if the volume is reduced 10 times, the force is
reduced by 1.541 (log F − log Fo = 0.432):
φ ≈ 0.649.
(21.6)
2. Tilt the gripper with respect to the component (Fig. 18.15): if the orientation of the gripper is changed from 0 to 5◦ , the ratio is
φ≈
2.4 mN
≈ 0.436
5.5 mN
(21.7)
Table 21.1. Mass range
Mass position
m > mmax
mmax > m > mmin
mmin > m
Description
Component cannot be picked up
Component can be picked up and released
Component can be picked up but cannot be released
214
21 Conclusions
and, if the gripper is tilted to 10◦
φ≈
1.2 mN
≈ 0.218.
5.5 mN
(21.8)
3. Move the gripper in the shear direction: in this case, if the component is
blocked laterally, the gripper can be moved in the xy plane (i.e., perpendicular to the axial symmetry axis z) and the force to generate depends
on the friction between the gripper and the component:
Fshear = fo F,
(21.9)
where fo is the static friction coefficient (“static” because the gripper
is moved from rest). In this case, the ratio of the forces to develop is
simply given by fo . This release is sometimes called scrapping release
method [50]. This method has been successfully implemented in the watch
bearing case study. Note that in order to achieve a reliable release of a
spherical component, the gap between the latter and the summit of the
cavity should not be larger than a fifth of the ball diameter (Fig. 20.11).
4. Increase the gap (Fig. 18.7): by increasing the gap, the force can be
reduced efficiently. In the example of Fig. 18.7, it can be seen that a
gap z = 500 µm leads to
φ≈
0.10 mN
≈ 0.105.
0.95 mN
(21.10)
5. Change the gripper geometry (Fig. 18.9): if the angular aperture was
increased from 0◦ to 5◦ , the force reduction would lead to
φ≈
2 mN
≈ 0.100.
20 mN
(21.11)
6. Use dynamical effects (Sect. 18.10): let us assume that the picking task of
a mass m is achieved with a zero velocity and a force F . Therefore, the
limit for m is Fg (g = 9.81 m s−1 ). If now an acceleration a is imposed to
the gripper, the heaviest acceptable mass is m = m g/a + g. Therefore,
the force seems to have been reduced by g/a + g. If a ≈ 100 m s−2 , the
ratio becomes
g
≈ 0.089.
(21.12)
φ=
a+g
Note that this method has to be envisaged carefully because of its low
positioning accuracy.
7. Use an auxiliary releasing tip (Fig. 18.17): for example, by substituting an
auxiliary PTFE tip to a steel cylindrical gripper generating forces about
5 mN, the force can be reduced to about 60 µm, leading to
φ≈
0.060 mN
≈ 0.012.
5 mN
(21.13)
21.4 Design Aspects
215
8. Control the contact angle (for example, using electrowetting): if the contact angle was controlled and set to π2 , the residual force could be avoided
as it essentially depends on cos θ. In this case
φ ≈ 0.
(21.14)
Several strategies can be combined, for example, a gripper tilting achieved
with a dynamical release: in this case the total efficiency of the release strategy
can be assessed by
Φ=
n
!
φi .
(21.15)
i=1
21.4 Design Aspects
In a given application, the component is imposed, that is its material and
its geometry cannot be changed. The freedom of the designer concerns the
handling liquid (surface tension, volatility, and dynamic viscosity), the gripper
(material and coating, geometry), and the releasing strategy.
•
•
•
•
Surface tension: The larger the surface tension, the larger the capillary
force. Nevertheless, a liquid with a lower surface tension will better wet
the component, leading to an opposite effect. This issue must be carefully
addressed.
Volatility: To achieve experimental measurements, we recommend to use
a nonvolatile liquid, such as, for example, the silicone oil we have used
in this work. Since it does not evaporate, the volume of liquid remains
constant during the experiment. Nevertheless, this can be a drawback from
the application point of view, since a nonvolatile liquid will leave residual
traces on the component. If the assembly task can be achieved very quickly
(less than 1 sec to fix the order of magnitude), volatile liquids such as
alcohols are suitable: they do not leave traces.
Contact Angle Hysteresis: To achieve experimental measurements, it is of
the utmost importance to know whether the advancing or the receding
contact angle should be used in the models. Moreover, the larger the hysteresis, the larger the variability of the results. Consequently, water is not
recommended. At the contrary, silicone oils or alcohols seem to be better;
however, it is recommended to first measure the contact angles with the
liquid one proposes to use.
Dynamic viscosity: The proposed force models rely on an equilibrium
assumption. Therefore, the viscosity does not play any role in these
models (i.e., the viscosity does not change the amount of force at equilibrium). Nevertheless, in high throughput machines, the transient effects
become dominant. We have proposed an approximation of the characteristic damping time τ by solving the Lucas-Washburn equation:
216
21 Conclusions
τ=
•
•
•
ρd2
1
=
.
λ
6µ
(21.16)
Implementing a dynamical simulation is one of the perspectives of this
work.
Gripper Material: A gripper with high energy (metals for example) leads
to smaller contact angles θ2 than a gripper with low energy (polymers
for example). Nevertheless, the results of Fig. 10.12 indicate that a nonwetting gripper can also lead to an attractive capillary force. The choice
of the material can also be ruled by manufacturing aspects.
Gripper Surface: According to the Wenzel model (11.5), the roughness
amplifies the wetting behavior: angles lower than 90◦ are decreased by
roughness, while the angle increases if it is larger than 90◦ . This means that
the gripping force of a capillary gripper could be increased by using rough
gripper tips. The surface impurities increase the contact angle hysteresis
(Fig. 11.1). Finally, we have observed that the presence of a coating could
be a drawback if it not mechanically resistant to wear (the gripper is
subject to many contacts). Of course, a wear resistant coating permits to
change the contact angle θ2 .
Gripper Geometry: The geometry of the gripper is of the utmost importance. Actually, the important parameter to keep in mind is that near
contact, the “pressure” or “Laplace” term of the capillary force dominates
the “tension” one (see Fig. 18.8). To maximize the capillary force, the
gripper should maximize the pressure difference pin − pout and also the
area on which this pressure difference is acting. The first objective can be
done with small θ1 and θ2 but also by keeping distance between the gripper and the component as small as possible, i.e., designing a conforming
geometry. This also respects the second condition since, for a given volume of liquid, a conforming gripper leads to a larger contact circle radius,
henceforth to a larger area. This principle is qualitatively discussed from
Fig. 21.1. We see on this picture that for the (hypothetic) depicted gripper, the distance between the gripper and the component is smaller for
a volume of liquid corresponding to a meniscus wetting in B than in A.
Hence, the pressure difference ∆pB is larger than the pressure difference
∆pA . Moreover, since the meniscus radius rB is larger than the meniscus
radius rA , the area over which this pressure difference acts is much larger,
leading to a larger force (FB > FA ). If the volume of liquid is increased
a bit more, the meniscus will wet the component at a distance rC > rB
from the symmetry axis. Therefore, the area is increased, but the pressure difference is reduced, since the distance between the component and
the gripper increases. Both effects fight one another, and qualitatively, the
force can increase or decrease, which is shown in the bottom right sketch
of Fig. 21.1. We also see on this picture the qualitative evolution of the
volume of liquid. It is clear that it will be difficult to master the volume
of liquid between VA and VB , since the sensitivity of the volume is very
21.4 Design Aspects
217
gripper
r
A
B
C
component
V
F
rA
rB
rC
r
rA
rB
rC
r
Fig. 21.1. Design rules
(a)
(b)
Fig. 21.2. (a) Example of a truncated conical gripper, which allows a large pressure
difference together with a large acting area; (b) Example of the gripper proposed
by Schmid et al. [155] (Courtesy of D. Schmid, EPFL/STI/IPR/LPM.)
low. At the contrary, there is a dramatic increase of the volume of liquid
as soon as r > rC . These principles can be applied to optimize the design
of a gripper intend to pick flat components. We see in Fig. 21.2a that
giving the gripper the shape of a truncated cone, we increase the “acting”
area2 by keeping a small distance between the gripper and the component,
2
The area on which the pressure difference acts.
218
•
21 Conclusions
leading to a large pressure difference. We let the reader compare this configuration with the spherical or the conical gripper presented in a previous
chapter. This has been applied in [155] to pick up square molybdenum dies
(100 × 100 × 20 µm3 ), simulating optical components. An example of the
used gripper is given in Fig. 21.2b.
Scaling Law: According to (6.14), the capillary force linearly depends
on the system size. Therefore, this principle is of the utmost interest in
miniaturization.
Part IV
General Conclusions and Perspectives
22
Conclusions and Perspectives
22.1 Conclusions
The goal defined at the very beginning of this work was to study the role of
surface forces in microassembly, and particularly, to assess the influence of
capillary forces in one of the assembly tasks, i.e., handling.
One of the first issues to be addressed was to grasp the framework of
microassembly. This has been led in Part I: it is shown that the classical
approaches to assemble components can be renewed by using different assembly models (such as, for example, internal or stochastic assembly) or different
handling principles. From a literature review on gripping principles, it turns
out that one of these handling principles, namely the surface tension based
handling, can lead to the set up of a capillary gripper prototype.
Among the advantages of such a gripper, let us cite a large force (several
milli-Newton) compared with the weight of small components, a favorable
downscaling law (i.e., better than that of the vacuum gripper since the force
only decreases as a linear function of the characteristic size), the ability to pick
up components that present only one accessible surface, a picking leading to
lower contact force than those exerted by a vacuum gripper. However, several
points must be assessed and mastered: is this kind of grippers suitable for use
in high acceleration assembly machines? In this case, how can the release task
be achieved? What is the residual volume of liquid left on the component after
it has been released? What are the design parameters of such a gripper?
To answer these issues, the modeling of capillary forces in Part II has been
adapted to the case study of a gripping task and a numerical code has been
written in order to simulate the working of the capillary handling according to
the following parameters: surface tension and volume of the used liquid (i.e.,
water and silicone oil), shape, material and size of the gripper (several conical
and spherical grippers made of steel or PTFE), material of the component
(steel, silicon), and kinematics of the gripper (traveling range and acceleration). These models output the meniscus shape, the capillary force it exerts,
and the critical distance at which the meniscus will break. Two examples are
222
22 Conclusions and Perspectives
developed: first, these models are applied to the case study of a watch ball
bearing with 0.3 and 0.5 mm diameter zirconium balls, and second, the numerical implementation of the double iterative scheme presented in Chap. 8
is exhaustively detailed.
Part III is devoted to the experimental aspects. A test bed has been set
up (chapter 17) in order to validate the simulation code and also to get
information being outside the axially symmetric assumptions framework of
the simulation (for example, to study the influence of tilting the gripper with
respect to the component). From these simulations and experiments, it turns
out that capillary forces are large enough to be used as a gripping principle
in microassembly (even in automated assembly machines with high accelerations): they can actually pick up millimetric components with accelerations
up to 100 ms−2 (10 G). The release capability has been formalized through
a so-called “adhesion” ratio φ that has been quantified for several release
strategies in Chap. 21: most promising ones probably consist in tilting the
gripper, moving the gripper laterally, increasing the gap, changing the gripper geometry, using an auxiliary releasing tip, or controlling the contact angle.
Dynamical release has been rejected because of its poor positioning accuracy.
As the presence of a residual volume of liquid on the component after releasing
is unavoidable, an additional model has been adapted in order to evaluate it.
Moreover, it has been shown that this residual volume could disappear in a
few seconds due to evaporation (at least for water at ambient temperature).
A list of design parameters has been established and the influence of each of
them has been quantitatively assessed. It has been shown that the developed
capillary force was proportional to surface tension γ, that the influence of the
volume of liquid would depend on the kind of gripper geometry (the force
does not depend on it in the case of a sphere–plane interaction, increases with
increasing volume in the case of the cone–plane interaction, decreases with
increasing volume in the case of the conforming watch bearing related gripper); the influence of the gripper size and that of the contact angles have been
described by mathematical laws (Chap. 21). This third part ends with design
suggestions.
An originality of this work lies in the classification of gripping principles
according to the way they tackle surface forces issues: it led to the proposal to
use capillary force as a gripping principle (even if it had already been partially
and independently suggested by two German researchers [13] and [72]). The
main contributions of this work are the transfer and the adaptation of scientific
knowledge from the chemists and physicists community to the microassembly
community. The main results of this book are the quantified comparisons of
picking strategies on the one hand and release strategies on the other hand.
22.2 Perspectives
223
22.2 Perspectives
The perspectives of this work are manyfold.
First, in terms of modeling, two topics emerge from this work: the capillary condensation and the dynamical simulation. The capillary condensation
problem is the companion topic of surface tension gripping. According to the
Kelvin equation (11.7), the humidity of the surrounding environment condensates in some conditions, leading to the formation of a liquid bridge at
the nanometric scale, and henceforth to adhesive capillary forces, which are
assumed to be responsible for MEMS breakdown and for a part of the pull-off
in atomic force microscopy. An introduction to this problem can be found in
[34]. The dynamical simulation of the meniscus evolution would allow to better
understand the transient phase and will probably of importance to model the
surface tension based gripping in high throughput assembly machines, or at
least as soon as the cycle time of the assembly process will become of the order
of magnitude of the characteristic damping time (about 10 ms for water).
Second, in terms of materials, a challenging topic is the active control
of the wetting properties. This would allow to switch on and off a surface
tension based gripper, making the adhesion ratio very small. This idea is
already approached with electrowetting [129] (i.e., control of the apparent
contact between a conductive substrate and a conductive droplet by applying
a voltage between the substrate and the droplet. To avoid the short-circuit,
the substrate is coated with a nonwetting and nonconductive coating) and
opto-electrowetting [37] (the impedance variation acting in the electrowetting
is controlled optically).
Third, in terms of technology, the surface tension based gripper should be
coupled with a dedicated powerful feeder, able to separate, orientate, and
position the components. This cannot be developed independently of the
application, so that the next step of this work will be the application of these
principles to an industrial case study as, for example, the assembly of SMD
components. An additional investigation track is the coupling between an
adhesive principle (surface tension effect) and a repulsive principle (ultrasonic
vibrations) in order to achieve a controllable noncontact gripping.
Part V
Appendices
A
Modeling Complements
This appendix aims at detailing the mathematical developments required
to calculate the analytical approximations of the capillary forces, based on
energetic approach (Chap. 7), and to establish the expression of the interfacial energies (L–V and total energies) used in Chap. 13 to study the rupture
conditions and the related volume repartition. It also illustrates the relation
between the capillary force and the interfacial energy, to illustrate the correctness of the interfacial energy expression we used in this appendix.
A.1 Analytical Approximations of the Capillary Forces
A.1.1 Preliminary
1. Definitions
1
2π
3
3
A(φ) ≡
1 − cos φ + cos φ ,
3
2
2
dA
= π sin3 φ.
dφ
2. Properties
φ2
φ4
+
+ O(φ6 ),
2
24
φ4
cos2 φ = 1 − φ2 +
+ O(φ6 ),
3
3
7
cos3 φ = 1 − φ2 + φ4 + O(φ6 ),
2
8
φ3
sin φ = φ −
+ O(φ5 ),
6
φ4
sin2 φ = φ2 −
+ O(φ6 ),
3
cos φ = 1 −
228
A Modeling Complements
sin3 φ = φ3 + O(φ5 ),
π
A(φ) = φ4 + O(φ6 ),
4
dA
= πφ3 + O(φ5 ),
dφ
1 − cos φ ≈
φ2
sin φ2
≈
.
2
2
A.1.2 Between a Sphere and a Plane
The force between a sphere and a plane is developed in [89]. The used notations
are defined in Fig. A.1.
In this figure, φ0 and r0 are arbitrary constants. Their exact value does
not play any role because the force will be calculated from the differentiation
of the interfacial energy W with respect to the gap z between the sphere and
the plane [89]:
F =−
dW
.
dz
(A.1)
Let us write the interfacial energy of the system:
W (z) = ASL γSL + ASV γSV + Σγ
= γSL πr2 + γSV π(r02 − r2 ) + γ2πr z + R(1 − cos φ)
(A.2)
2
2
+γSL 2πR (1 − cos φ) + γSV 2πR (1 − cos φ0 ) − (1 − cos φ) .
Since φ is assumed to be small, W can be rewritten as:
W (z) = πr2 (γSL − γSV ) + γ2πrz + γπrR sin2 φ + γSV πr02
+πR2 sin2 φ(γSL − γSV ) + γSV πR2 sin2 φ0
and, by considering the Young–Dupré equation (γ cos θ = −γSL + γSV ):
R
φ
φ0
h
z
r
r0
Fig. A.1. Studied configuration
A.1 Analytical Approximations of the Capillary Forces
229
W = −2πR2 sin2 φγ cos θ+γSV πr02 +γ2πrz +γπR2 sin3 φ+γSV πR2 sin2 φ0 .
(A.3)
Let us now consider the derivative of W :
dW
dφ
dφ
= −4πR2 sin φ cos φγ cos θ
+ γ2πR sin φ + γ2πzR cos φ
dz
dz
dz
dφ
2
2
+3γπR sin φ cos φ
dz
or, by assuming sin φ ≈ φ and cos φ ≈ 1:
dW
dφ
dφ
dφ
= −4πR2 φγ cos θ
+ γ2πRφ + γ2πRz
+ 3γπR2 φ2 .
dz
dz
dz
dz
(A.4)
(A.5)
The value of dφ/dz must be evaluated in (A.5). Therefore, the meniscus
volume is assumed to be constant, leading to dV /dz = 0. Moreover, the
meniscus will be assumed to be cylindrically shaped so that the volume is the
difference between the external liquid cylinder and the volume of the spherical
cap inside the external cylinder:
2πR3 3
cos3 φ 1 − cos φ +
.
V = πr2 z + R(1 − cos φ) −
3
2
2
(A.6)
Once again the assumption of small φ is made, leading to the following
approximation:
3
cos3 φ 2πR3 πR3 4
1 − cos φ +
= A(φ)R3 ≈
φ .
3
2
2
4
(A.7)
The final expression for V is now given by:
πr2 R
πR3 4
sin2 φ −
φ
2
4
πR3
πR3 4
sin4 φ −
φ
= πR2 sin2 φz +
2
4
so that:
dV
dφ
dφ
= 2πR2 z sin φ cos φ
+ πR2 sin φ + 2πR3 sin3 φ cos φ
dz
dz
dz
3 3 dφ
−πR φ
dz
dφ
dφ
2
= 2πR zφ
+ πR2 φ2 + πR3 φ3
dz
dz
=0
−πR2 φ2
dφ
=
⇒
2
dz
2πR φz + πR3 φ3
−1
= 2z
.
φ + Rφ
V = πr2 z +
(A.8)
(A.9)
(A.10)
(A.11)
230
A Modeling Complements
The total capillary force is then given by substituting this latter result into
(A.5):
F =−
4πR2 φγ cos θ
γ2πRz
3γπR2 φ2
− γ2πRφ + 2z
+ 2z
.
2z
φ + Rφ
φ + Rφ
φ + Rφ
Since h = R(1 − cos φ) ≈
R
2
sin2 φ ≈
(A.12)
R 2
2φ :
F =−
4πRγ cos θ
γ2πRz
3γπRφ
− γ2πRφ + 2z
+ 2z
2z
+
1
+
Rφ
2
Rφ
φ
Rφ2 + 1
=−
3γπRφ
γ2πRz
4πRγ cos θ
+ z
− γ2πRφ + 2z
.
z
h +1
h +1
φ + Rφ
The last three terms of (A.13) represent the contribution of the “LV” interface
to the total interfacial energy. Let us assess their relative importance with
respect to the first term. Their sum is given by:
πRγφ(Rφ2 − 2z)
.
Rφ2 + 2z
(A.13)
The ratio of the the first term to the sum of the last three ones is equal to:
4πRγ cos θ
z
h +1
πRγφ(Rφ2 −2z)
Rφ2 +2z
=
4 cos θh
.
φ(h − z)
(A.14)
If z = 0, this ratio tends toward infinity if φ tends to zero. Since φ cannot be
exactly equal to zero, the last three terms can be neglected with the (now)
classical assumption of small φ (φ ≈ sin φ). This leads to the well-known
approximation [89]:
Fmax = −4πRγ cos θ.
(A.15)
If z = 0 but by neglecting the contribution of lateral area to W , the total
capillary force can be rewritten as follows:
F =−
4πRγ cos θ
.
z
h +1
(A.16)
A.1.3 Between Two Spheres
Let us assume two spheres S1 and S2 characterized by their radius R1 and R2
such as depicted in Fig. A.2.
A.1 Analytical Approximations of the Capillary Forces
φ2
231
R2
SV2
SL2
h2
r
LV
z
h1
SL1
SV1
φ1
R1
Fig. A.2. Notations
Preliminaries
≡ R1 sin φ1 ≡ R2 sin φ2 ,
dφ1
dφ2
= R1 cos φ1
= R2 cos φ2
,
dz
dz
≈ R 1 φ1 ≈ R 2 φ2 ,
dφ1
dφ2
≈ R1
= R2
,
dz
dz
dφi
,
φi ≡
dz
dr
≈ R1 φ1 ≈ R2 φ2 .
r ≡
dz
r
dr
dz
r
dr
dz
Expression of the Interfacial Energy
The total interfacial energy W can be expressed as follows:
W = γ2πr(z + R1 (1 − cos φ1 ) + R2 (1 − cos φ2 )) + γSL2 2πR22 (1 − cos φ2 )
+γSV2 2πR22 (C − (1 − cos φ2 )) + γSL1 2πR12 (1 − cos φ1 )
+γSV2 2πR12 (C − (1 − cos φ1 )),
(A.17)
where C and C are arbitrary constants. With the assumptions that:
232
A Modeling Complements
C = C = 0,
φ → 0 ⇒ sin2 φ ≈ 2(1 − cos φ),
r = R1 sin φ1 = R2 sin φ2 ,
γSVi = γSLi + γ cos θi .
W can be rewritten into:
W = πrγ(2z + R1 sin2 φ1 + R2 sin2 φ2 ) + πγSL2 R22 sin2 φ2
−γSV2 πR22 sin2 φ2 + πγSL1 R12 sin2 φ1 − γSV1 πR12 sin2 φ1
= 2πrγz + πγR12 sin3 φ1 + πγR22 sin3 φ2 − πR22 sin2 φ2 γ cos θ2
−πR12 sin2 φ1 γ cos θ1
⇒
(A.18)
dW
= 2πγr + 2πγzr + 3πγR12 sin2 φ1 cos φ1 φ1
dz
+3πγR22 sin2 φ2 cos φ2 φ2 − 2πγR22 sin φ2 cos φ2 φ2 cos θ2
−2πγR12 sin φ1 cos φ1 φ1 cos θ1 .
(A.19)
As it can still be assumed that φ → 0, sin φ ≈ φ and cos φ ≈ 1, leading to:
dW
= 2πγr + 2πγzr + 3πγR12 φ21 φ1 + 3πγR22 φ22 φ2 − 2πγR22 φ2 φ2 cos θ2
dz
−2πγR12 φ1 φ1 cos θ1
(A.20)
(A.21)
It can be emphasized that r ≈ R1 φ1 ≈ R2 φ2 and that consequently,
φ2 = φ1 (R1 /R2 ). φ1 must still be determined, so the conservation of liquid
volume is assumed (dV /dz = 0). The volume of liquid can be approached by
the volume of the cylinder of radius r and height z + h1 + h2 decreased by
the volumes of two spherical of radii R1 and R2 , and limited by the angles φ1
and φ2 :
V = πr2 (z + R1 (1 − cos φ1 ) + R2 (1 − cos φ2 )) − A1 R13 − A2 R23
leading to:
dV
= 2πrr (z + R1 (1 − cos φ1 ) + R2 (1 − cos φ2 ))
dz
+πr2 (1 + R1 φ1 φ1 + R2 φ2 φ2 ) − πφ31 φ1 R13 − πφ32 φ2 R23
≡0
r2
2rz + rR1 φ21 + rR2 φ22 + r2 φ1 + r2 φ2 − φ31 R12 − R22 φ32
r
=−
2z + rφ1 + rφ2
r
=−
.
(A.22)
2z + r2 ( R11 + R12 )
⇔ R1 φ1 = −
A.2 Volume Repartition by the Energetic Approach
233
Equation (A.21) can now be rewritten into:
dW
πγr
=
dz
2z + r2 ( R11 +
1 (2z
R2 )
− R1 φ21 − R2 φ22 + 2r(cos θ1 + cos θ2 )).
At contact (z = 0), the assumption of small φi leads to:
dW
πγr
= 2 1
dz
r ( R1 +
1 2r(cos θ1
R2 )
+ cos θ2 ).
Let us note 2 cos θ ≡ cos θ1 + cos θ2 and 1/R ≡ 1/R1 + 1/R2 so that the last
equation finally leads to:
F ≡−
dW
= −4πγR cos θ.
dz
(A.23)
It can then be concluded that at contact and with small amounts of liquid
(φi ≈ sin φi ), the force between two spheres with radii R1 and R2 is equal to
that between a plane and a sphere of radius R given by 1/R = 1/R1 + 1/R2 .
A.2 Volume Repartition by the Energetic Approach
Let us assume an “analytical” configuration, i.e., a cylindrical gripper parallel
to the component.
A.2.1 Assumptions, Notations, and Mathematical Preliminaries
1. The volume V is constant
– The meniscus is assumed to be of cylindrical shape V = πr2 z
– The volume of both spherical caps after rupture is given by: V =
V1 + V2 ⇒ dV2 = −dV1
2. Ai ≡ (2π/3)(1 − (3/2) cos θi + (1/2) cos3 θi ) (As Ai R3 represents the
volume of a spherical cap with radius R and limited by an angle θi , Ai ≥ 0,
and dAi /dθi > 0)
3. αi ≡ (1/6)(2 − cos θi − cos2 θi ) ≥ 0
4. Ai /αi = 2π(1 − cos θi )
5. K ≡ (A2 /A1 )(α1 /α2 )3
6. K =
A2
A1
7. 2 cos θ ≡ cos θ1R + cos θ2R
8. γSVi = γSLi + γ cos θi
9. θi = contact angle made by the ith droplet, usually the advancing contact
angle
234
A Modeling Complements
10. λi ≡
Ai
α3i
θi
= 72π (2−cos1−cos
θi −cos2 θi )2
⎡
⎢
dλi
sin θi (2 − cos θi − cos2 θi )2
⎢
sign
= sign ⎢72π
dθi
⎣
(2 − cos θi − cos2 θi )4
>0
×
⎤
−2(1 − cos θi )(2 − cos θi − cos2 θi )(sin θi + sin 2θi ) ⎥
⎥
⎥
2
4
⎦
(2 − cos θi − cos θi )
>0
= sign(cos θi + 3 cos θi − 4)
= −1
2
(A.24)
11. The angles involved in W (Fig. A.3a) are receding ones, those involved in
U (Fig. A.3b) are between the receding and the advancing angles.
A.2.2 L–V Interfacial Energy
The energy of the LV interface WLV is given by:
WLV = 2πγrz
√ √ √
= 2γ π V z.
The energy of the two droplets is given by:
ULV = 2πγ[R12 (1 − cos θ1 ) + R22 (1 − cos θ2 )]
A1 2 A2 2
=γ
R +
R
α1 1 α2 2
1/3
1/3
A1
A2
2/3
2/3
=γ
V
+
V
since Vi = Ai Ri3 ,
α1 1
α2 2
R2
q2
z
z
q1
R1
r
(a)
Fig. A.3. Notations
(b)
A.2 Volume Repartition by the Energetic Approach
dULV
=0
dV1
1/3
1/3
2 A1
2 A2
−1/3
−1/3
=γ
V1
−
V
3 α1
3 α2 2
3
V2
A2 α1
⇒
=
≡ K .
V1
A1 α2
235
We conclude that the repartition of the volumes that extremes the interfacial
energy is given by:
V
,
(A.25)
1 + K
KV
V2 =
.
(A.26)
1 + K
∗
of ULV corresponding to this volumes repartition is given by:
The value ULV
⎤
⎡
α21 A2 2/3
1/3
1/3
(
)
2
1
A
A
A
α2
1
∗
⎦
ULV
= γ⎣ 1
+ 2
α1 (1 + A2 α313 )2/3
α2 (1 + A2 α313 )2/3
V1 =
=γ
A1 α2
2
A1 α22
A2 α1
+
α1
α2
A1 α2
V
α23 A1 + α13 A2
2/3
∗
Let us show that ULV
is a maximum by evaluating the second derivative of
ULV with respect to V1 :
1/3
1/3
d2 ULV
2γ A1
A2
−4/3
−4/3
=−
V
+
V
dV12
9
α1 1
α2 2
>0
<0
∗
d
is negative because Ai , αi , and Vi are positive. Consequently, ULV
is a maximum.
As a concluding remark of this subsection, it must be noted that if θ1 > θ2 ,
λ1 < λ2 , leading to V1 < V2 . Indeed, dλ/dθ < 0 ⇒ λ1 < λ2 , and V2 /V1 =
K = λ2 /λ1 .
2
ULV /dV12
A.2.3 Total Interfacial Energy
W = 2πγrz + πr2 γSL1 + πr2 γSL2 − πr2 γSV1 − πr2 γSV2
= 2πγrz − πr2 γ(cos θ1 + cos θ2 )
= 2πγrz − πr2 γ cos θ
√ √ √
2V γ cos θ
= 2γ π V z −
,
z
236
A Modeling Complements
(
U = γ 2π R12 (1 − cos θ1 ) + 2π R22 (1 − cos θ2 )
)
−πR12 sin2 θ1 cos θ1 − πR22 sin2 θ2 cos θ2
= 3γA1 R12 + 3γA2 R22
1/3 2/3
1/3 2/3
= γ 3A1 V1 + 3A2 V2
,
(A.27)
dU
=0
dV1
1/3 −1/3
1/3 −1/3
,
= γ 2A1 V1
− 2A2 V2
⇒
V2
A2
=
≡ K.
V1
A1
It can be pointed out that this last expression can be rewritten as:
V2
V1
=
⇔ R1 = R2 .
A2
A1
The volumes V1 and V2 are given by:
V
,
1+K
KV
V2 =
.
1+K
The value U ∗ of U corresponding to this volumes repartition is given by:
U ∗ = 3γ A1 R12 + A2 R22 .
V1 =
As R13 = R23 =
V1
A1
=
V2
A2
=
V
A1 +A2 ,
U ∗ can be written as follows:
U ∗ = 3γV 2/3 (A1 + A2 )1/3 .
Let us show that U ∗ is a maximum by evaluating the second derivative of U
with respect to V1 :
d2 U
2γ 1/3 −4/3
1/3 −4/3
= − (A1 V1
+ A2 V2
),
dV12
3 >0
<0
d
is negative because Ai and Vi are positive. Consequently, U ∗ is a
maximum.
As a concluding remark of this subsection, it must be noted that if θ1 > θ2 ,
A1 > A2 , leading to V1 > V2 .
2
U /dV12
B
Geometry Complements
This appendix covers several elements. First we remind the lateral area and
the volume of a spherical cap (= portion of a sphere): This result is used
in a widespread manner throughout this work. Then, elements of differential
geometry are given in order to calculate the mean curvature of an axially
symmetric surface (this result is used to get (8.13)). This appendix also gives
the equations of the catenary curve, used in the validation of the simulation
code (Sect. 18.2.1).
B.1 Area and Volume of a Spherical Cap
Let us consider a spherical cap ABC, defined by its radius R and the limiting
angle α such as depicted in Fig. B.1.
The lateral area and the volume of this spherical cap can be calculated as
follows:
2π
α
dϕ R2 sin θ dθ
S(α, R) =
0
0
= 2πR2 (1 − cos α)
(B.1)
and the volume V is given by:
2π
V (α, R) =
α
dϕ
0
R
dθ
0
r2 sin θ dr
r(θ)
1
2πR
3
3
=
1 − cos α + cos α ,
3
2
2
3
α
where r(θ) = R cos
cos θ .
(B.2)
238
B Geometry Complements
B
P2
P1
C
A
α
θ
r(q)
R
O
Fig. B.1. Definition of the spherical cap
B.2 Differential Geometry of Surfaces
B.2.1 Mean Curvature of a Surface
Let S be a surface given by its vectorial equation:
S(u, v) ≡ OP = r(u, v)
and let us define the following differential operator [47]:
⎧
∂r ∂r
E = ∂u
. ∂u
⎪
⎪
⎪
∂r
∂r
⎪
⎪
F
=
.
⎪
∂u ∂v
⎪
⎪
∂r ∂r
⎨
G = ∂v
. ∂v
,
∂2r
⎪
L = ∂u2 .1n
⎪
⎪
⎪
⎪
∂2r
⎪
M = ∂u∂v
.1n
⎪
⎪
⎩
∂2r
N = ∂v2 .1n
(B.3)
(B.4)
where 1̄n is the normal vector defined by:
1̄n =
∂ r̄
∂u
∂ r̄
|| ∂u
×
×
∂ ū
∂v
.
∂ ū
∂v ||
(B.5)
Note that the sign of this latter result does not make geometrical sense: The
permutation between u and v changes the orientation of 1̄n . According to [47],
the curvature Kn of a normal section of this surface is given by:
B.2 Differential Geometry of Surfaces
Kn =
1
L du2 + 2M du dv + N dv 2
=
Rn
E du2 + 2F du dv + G dv 2
or with λ ≡
Kn =
239
(B.6)
du
dv :
1
Lλ2 + 2M λ + N
.
=
Rn
Eλ2 + 2F λ + G
(B.7)
This equation can be rewritten as follows:
λ2 (LR − E) + 2λ(M R − F ) + N R − G = 0.
(B.8)
It means that if they exist, there are usually two directions λ1,2 for a given
curvature R:
F − M R ± (M R − F )2 − (LR − E)(N R − G)
.
(B.9)
λ1,2 =
LR − E
If the directions λ1 and λ2 are equal, they indicate a principal direction. In
this case, the corresponding curvature radius R is a principal curvature radius,
given by:
(M R − F )2 − (LR − E)(N R − G) = 0
(B.10)
that can be rewritten into:
1
1
(EG − F 2 ) + (2M F − EN − GL) + (LN − M 2 ) = 0.
R2
R
(B.11)
This equation has two solutions R1 and R2 whose sum is given by:
1
1
EN + GL − 2M F
+
=
R1
R2
EG − F 2
and the mean curvature is given by:
1 1
1
1 EN + GL − 2M F
H≡
+
.
=
2 R1
R2
2
EG − F 2
(B.12)
(B.13)
Since the sign of 1̄n is not yet defined, the sign of H is not determined.
B.2.2 Mean Curvature of an Axially Symmetric Surface
If the surface S is axially symmetric, its equation is given by:
S(θ, z) ≡ r(θ, z) = r(z)1̄r + z 1̄z
and leads to1 :
1
() =
d()
.
dz
(B.14)
240
B Geometry Complements
⎧
⎪
⎨E =
F =
⎪
⎩
G=
∂r ∂r
∂θ . ∂θ
∂r ∂r
∂θ . ∂z
∂r ∂r
∂z . ∂z
= r2
.
=0
2
=1+r
(B.15)
According to (B.5):
−r 1̄z + 1̄r
1̄n = √
1 + r2
that, combined with (B.4), leads to:
⎧
r
⎪
⎨ L = − √1+r2
M =0
.
⎪
⎩ N = √ r
1+r 2
(B.16)
(B.17)
The mean curvature of an axially symmetric surface is consequently written as:
H=
1
r
1
−
2 (1 + r2 )3/2
r(1 + r2 )1/2
(B.18)
B.3 Catenary Curve
In the case of ∆p = 0, (8.13) can be rewritten as:
dr 2
)
1 + ( dz
d2 r
.
=
dz 2
r
The solution to this equation looks like:
z−B
r(z) = A cosh
,
C
z−B
A
dr
= sinh
,
dz
C
C
z−B
A
d2 r
= 2 cosh
,
dz 2
C
C
(B.19)
(B.20)
(B.21)
(B.22)
and leads to the condition A = C. A and B can be determined with the
following boundary conditions:
r(b) = a,
1
,
r (b) =
tan θ
where a, b, and θ are represented in Fig. B.2.
(B.23)
(B.24)
B.3 Catenary Curve
241
x 10−3
2
z [m]
1.5
1
Analytical solution
Numerical solution
0.5
0
0
0.5
1
1.5
r [m]
2
2.5
3
x 10−3
Fig. B.2. Boundary conditions: a, b and θ
These equations are transformed into:
b−B
a = A cosh
A
b−B
1
= sinh
tan θ
A
and (B.25) can be written as:
b−B
a = A 1 + sinh2
A
1
= A 1+
tan2 θ
A
=±
sin θ
⇒ A = ±a sin θ.
(B.25)
(B.26)
(B.27)
(B.28)
(B.29)
(B.30)
By using this result in (B.25):
1
B = b ∓ a sin θa cosh
sin θ
(B.31)
because a cosh is a pair function. As a is a positive radius and 0 < θ < π:
A = a sin θ,
B = b − a sin θ a cosh
1
sin θ
(B.32)
.
(B.33)
C
Comparison Between Both Approaches
This appendix shows that the expression under brackets in (9.47), hereafter
noted B in (C.1), is equal to −(cos θ1 + cos α)/ tan φ, which leads to (9.48).
This step is used in the demonstration of equivalence of the Laplace equation
based method and the energy minimization method used to compute capillary
forces.
1+
1
sin α cos α+2 sin α cos θ1 −π+α+θ1 −sin θ1 cos θ1
(cos θ1 +cos α)2
tan φ
.f
=
≡n
=−n1
=−n2
2
. (π − α − θ ) sin φ − sin α cos θ sin φ + sin θ cos θ sin φ −(cos θ cos φ + cos θ )(cos θ + cos α) /
1
1
1
1
1
2
1
.
...
cos φ(cos θ1 + cos α)2 + (sin α cos α + 2 sin α cos θ1 −π + α + θ1 − sin θ1 cos θ1 ) sin φ
≡n1
cos φ(cos θ1 + cos α)2
.f
cos φ(cos θ1 + cos α)2 + (sin α cos α + 2 sin α cos θ1 − π + α + θ1 − sin θ1 cos θ1 ) sin φ
cos φ(cos θ1 + cos α)
=
...
sin φ
B=
Using (9.27), µ can be replaced, leading to:
≡f
The expression under brackets (let us note it B) can be reduced to a common denominator:
1
cos θ1
cos θ2
π − α − θ1 − sin α cos θ1 + sin θ1 cos θ1
−
+
B=
(cos θ1 + cos α)(1 + µ tan φ)
1 + µ tan φ tan φ
sin φ
(π − α − θ1 ) sin φ − sin α cos θ1 sin φ + sin θ1 cos θ1 sin φ − (cos θ1 cos φ + cos θ2 )(cos θ1 + cos α)
1
=
.
1 + µ tan φ
(cos α + cos θ1 ) sin φ
≡B
cos θ1
cos θ2
D . π − α − θ1 − sin α cos θ1 + sin θ1 cos θ1
1
cos θ1
cos θ2 /
dW
=
+
+
−
(
+
) .
2Lγ dD
tan φ
sin φ
D+h
(cos θ1 + cos α)(1 + µ tan φ)
1 + µ tan φ tan φ
sin φ
(C.2)
(C.1)
244
C Comparison Between Both Approaches
−(n1 +n2 )
=−1
(C.3)
Adding and subtracting sin α, the latter equation can be written as follows:
dW
cos θ1
cos θ2
D cos φ(cos θ1 + cos α)
=
+
−
− sin α + sin α
2Lγ dD
tan φ
sin φ
D+h
sin φ
(cos θ1 cos φ + cos θ2 )(D + h) − D cos φ(cos θ1 + cos α) − sin α sin φ(D + h)
=
+ sin α
(D + h) sin φ
= ...
h cos φ(cos θ1 + cos θ2 cos φ − sin θ2 sin φ)
=
+ sin α
(D + h) sin φ
h cos θ1 + cos α
= sin α +
.
D+h
tan φ
and the surface energy derivative given by (9.47) can be rewritten into:
dW
cos θ1
cos θ2
D cos φ(cos θ1 + cos α)
=
+
−
.
2Lγ dD
tan φ
sin φ
D+h
sin φ
(C.6)
(C.5)
Now, by replacing α by α = θ2 + φ and simplifying the latter expression, it can be shown that the expression under brackets
in (C.3) is equal to −1. Consequently, B is given by:
cos φ(cos θ1 + cos α)
(C.4)
B=−
sin φ
. (π − α − θ + sin θ cos θ ) sin φ − sin α cos θ sin φ − (cos θ cos φ + cos θ )(cos θ + cos α) /
1
1
1
1
1
2
1
...
.
(−π + α + θ1 − sin θ1 cos θ1 ) sin φ + cos φ(cos θ1 + cos α)2 + (sin α cos α + 2 sin α cos θ1 ) sin φ
n1 +n2
Let us rearrange the bracket of the latter equation in order to put n1 and n2 forward:
cos φ(cos θ1 + cos α)
...
B=
sin φ
C Comparison Between Both Approaches
245
246
C Comparison Between Both Approaches
The latter equation is equivalent to (9.37), which demonstrates the equivalence between the Laplace equation based and the energetic force formulations.
D
Symbols
Table D.1. Greek symbols
Symbol
α
β
δ
δe
∆p
φ
γ
γLV
γSL
γSV
µ
ρ
ρ
ρ
Σ
θ
θA
θR
θ1
θ2
Description
Angular beam deflection
Filling angle
Beam deflection
Beam deflection at calibration
Pressure difference across the LV interface = pin − pout
Misalignment
Residual adhesion ratio
Surface tension of a liquid
Energy of the interface Liquid–Vapor
Energy of the interface Solid–Liquid
Energy of the interface Solid–Vapor
Dynamic viscosity
Principal curvature radius of the meniscus (usually < 0)
Principal curvature radius of the meniscus (usually > 0)
Density
Lateral area of the liquid bridge
Contact angle
Advancing contact angle
Receding contact angle
Contact angle at the component side
Contact angle at the gripper side
Unit
deg
deg
m
m
Pa
m, deg
−
N m−1
J m−2
J m−2
J m−2
Pa s
m
m
kg m−3
m2
rad
rad
rad
rad
rad
248
D Symbols
Table D.2. Latin symbols
Symbol
a
b
Bo
C
Ca
d
D
dpinc
dpstart
E
F
FL
FT
g
G
k
K
h
H
I
L
LC
m0
M
N
p
p1 , p2
pin
pout
r1
r2
rneck
R
S
U
ULV
v
V
Description
Acceleration
Beam width
Bond number
Capacity of the SL interface in electrowetting
Capillary number (nondimensional)
Characteristic size of the meniscus in nondimensional analysis
Diameter of a droplet posed on the substrate
gap in Chap. 9
Pressure increment in θ1 search
Initial pressure difference in θ1 search
Young’s modulus
Differential operator to compute H
Differential operator to compute H
Laplace term of the capillary force
Interfacial tension term of the capillary force
Earth gravity
Differential operator to compute H
Stiffness of the cantilevered beam
Interaction constant depending on materials (VDW)
Beam thickness
Height of a droplet posed on the substrate
Immersion height
Capillary rise
Mean curvature
Inertia moment of the beam section
Length of the cantilevered beam
Differential operator to compute H
Capillary length
Standardized mass(es) used for calibration
Differential operator to compute H
Differential operator to compute H
Gripper parameter (radius or angular aperture)
Permanent dipoles (VDW)
Pressure in the liquid phase
Pressure in the vapor phase
Radius of the liquid bridge at the component side
Radius of the liquid bridge at the gripper side
Radius of the liquid bridge at the neck height
Radius of a droplet posed on the substrate
Radius of spherical grippers
Molar gas constant (8.314)
Area of a spherical cap
Total interfacial energy of the droplets (S–V, L–V, S–L)
Partial interfacial energy of the droplets (L–V)
Molar volume
Volume of liquid, usually between 0.1 and 1 µL
Unit
m s−2
m
−
F
−
m
m
Pa
Pa
GPa
N
N
9.81 m s−2
N m−1
J m6
m
m
m
m
m−1
m4
m
m
kg
m or deg
Pa
Pa
m
m
m
m
m
J K−1 mol−1
m2
J
J
m3 mol−1
m3
D Symbols
Table D.3. Latin symbols
Symbol Description
W
Component weight
Total interfacial energy of the meniscus (S–V, L–V, S–L)
Partial interfacial energy of the meniscus (L–V)
WLV
z
Separation distance between the gripper and the object
Approaching contact distance before the droplet
zA
hung to the gripper contacts the object
Distance at which the meniscus breaks
z∗
and is replaced by two droplets
Table D.4. Abbreviations
Abb.
AFM
CSEM
DIN
DMT
EAP
EDM
JKR
LAB
LIGA
LPM
LRP
MEL
MEMS
MOC
RH
SEM
SFA
SMA
SMD
VDW
Description
Atomic force microscope
Swiss Centre for Electronics and Microtechnology
German norms
Derjagin–Muller–Toporov
Electroactive polymer
Electrodischarge machining
Johnson–Kendall–Roberts model
Automation Laboratory of Besançon
Lithographie-Galvanoformung-Abformung
Microtechnology production laboratory (EPFL)
Robotics laboratory of Paris (Paris VI)
Mechanical engineering laboratory
Micro Electro Mechanical System
Microrobot on chip
Relative humidity
Scanning electron microscope
Surface force apparatus
Shape memory alloy
Surface mount device
van der Waals
Unit
N
J
J
m
m
m
249
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Index
Accuracy, 152
Adhesion, XIV, 42
Advancing contact angle, 161
AFM, XI, 19, 20, 29, 33
Anova, 189, 194
Approximations
Arc, Circle, Toroidal, 51
Parabolic, 51
Assembly
Combined part manufacturing and
assembly, 6
Design for microassembly, 4
Microassembly literature, 9
Parallel assembly, 8
Performances of the assembly
machines, 103
Product external/internal assembly
functions, 6
Requirements, 3
Self-assembly, 7
Stochastic assembly, 7
Taxonomy, 4
Assumptions of the static simulation,
65
Bernoulli, 22
Bond number, 51, 76, 105
Boundary conditions
Equilibrium, 69
Buckingham theorem, 103
CAD model, 148
Calibration, 151
Capillary
Gripper, 20, 27, 47
Length, 51, 66
Capillary force, 45
Between a sphere and a plane, 55
Between two parallel plates, 52
Between two spheres, 56
Laplace term, 46
Surface tension term, 46
Case study, 83, 127, 189, 199
Casimir, 11
Cassie, 97, 137
Catenary curve, 164, 240
CCD, 148
Circle approximation
See Approximations, 51
Components, 159
Condensation, 101
Contact
Distance of, 185
Line, 42, 43, 100
Contact angle, 43, 46, 112, 168, 174
Advancing contact angle, 161, 168
Dynamic contact angle, 100
Hysteresis, 99
Measurement, 154
Receding contact angle, 161, 168
Coulomb, 10, 11
Cryogenic gripper, 21, 27
Curvature
See Mean curvature, 45
Curvature radius, 68
262
Index
Damping
Effect, 101
Time, 106, 108
Debye, 11
DFµA, 4
Dimensional
Analysis, 103, 194
Matrix, 103
DIN, 29
Dispensing device, 146
DMT, 12
Drop generation, 146
Du Nouÿ, 155, 191
Dynamical release, 177
Push-pull, 19
Ultrasonic levitation, 23
Vacuum, 15
GS-St-#, 158
EAP, electroactive polymer, 21
EDM, 5
Energetic
Capillary force approximation, 51
Energy, 51
Interfacial energy W , 51, 52
JKR, 12, 17
Forces
Analytical approximations for
capillary forces, 227
Capillary force, 45, 47
Classification, 10
Coulomb, 10
Force-distance curve, 170
Laplace force, 46
Measurement, 145
Shear, 186
Tension force, 46, 171
Froude, 105
Gap, 46, 170
GC-St-#, 158
Gripper
Aerodynamic levitation, 22
Air cushion levitation, 22
Available grippers, 158
Capillary, 20, 27, 47
Cryogenic, 21, 27
Electrostatic, 17
Form closure, 15
Friction based, 13
Magnetic, 16
Optical, 24
Parameter, 113, 171
Hagen-Poiseuille, 106
Heterogeneities, 97
Hysteresis, 99
Impurities, 97
Interfacial energy, 52
Interfacial tension force
See Forces, 46
Israelachvili, 20, 55, 81, 93, 99, 171, 212
Keesom, 11
Kelvin
Equation, 99
Radius, 99
Lack of fit, 194
Laplace, 137
Equation, 43, 45, 61, 65
Force, 46, 171
Lifshitz, 10, 11
LIGA, 13
Limits, 103, 108
Linearity, 151
Error, 151
London, 10
Lucas-Washburn, 103, 138
Mean curvature, 45, 238
MEL, 5
MEMS, 11, 42, 101
Meniscus profile, 163
Microworld, IX
Mikromasch, 20
Misalignment, 152
Non dimensional numbers, 103
Numerical simulation, 65
Orientation (Relative), 174
Parabolic approximation, 51, 61
Picking strategies, 211
Polder, 11
Index
Protocol
See Test bed, 155
PTFE, 176
R47V50, 161
Receding contact angle, 161
Relative humidity, 18
Release
Dynamical, 177
Releasing strategies, 213
Repartition, 233
Requirements
Microassembly, 3
Test bed, 143
Reynolds, 105
Roughness
Influence on wetting properties, 98
Rupture, 111, 185
Rupture gap, 117
SEM, 15
SFA, 101
Shape memory alloy, 29
Shear force, 186
Shooting method, 70
Si-#, 159
Simulation
Algorithms, 127
Assumptions, 65
Limits of, 103
Numerical, 65
Static, 65
Validation of, 163
Spreading, 100
263
St-#, 159
Stiction, 42
Surface
Energy, 51
Impurities, 97
Mount Device, 35
Tension, 41, 172
Symbols, 247
Tension force, 171
Test bed, 143
Accuracy, 152
Available components, 159
Available grippers, 158
Calibration, 151
Linearity, 151
Misalignment, 152
Modus operandi, 155
Tilt, 174
Toroidal approximation
See Approximations, 51
Triple line
See Contact line, 42
van der Waals, 10, 11, 25, 27, 42, 176
Vision, 148
Volume repartition, 111, 115, 233
Weber, 105
Wenzel, 137
Wilhelmy, 155
Young-Dupré, 42, 137
Zisman model, 99