Peeling Models for Polydimethylsiloxane (PDMS)
by
Chetak M. Reshamwala
S.B., Mechanical Engineering (2001)
Massachusetts Institute of Technology
SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE IN MECHANICAL ENGINEERING
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 2003
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUL 0 8 2003
©2003 Massachusetts Institute of Technology
All rights reserved'
LIBRARIES
...............................
Department of Mechanical Engineering
May 9, 2003
Signature of Author........
C ertified by .............................................................
........................
Sanjay E. Sarma
Associate Professor of Mechanical Engineering
Thesis Supervisor
A ccepted by ...........................................................................
Ain A. Sonin
Professor of Mechanical Engineering
Chairman, Department Committee on Graduate Students
Peeling Models for Polydimethylsiloxane (PDMS)
by
Chetak M. Reshamwala
Submitted to the Department of Mechanical Engineering
on May 9, 2003 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
The objective of this study was to determine and model the significant peeling characteristics of
the polymer polydimethylsiloxane (PDMS) from a treated silicon substrate. Because of its
mechanical, chemical, and material properties, PDMS is very significant to the field of bioMEMS
(biological Micro-Electro-Mechanical Systems). PDMS devices are microfabricated using soft
lithographic techniques. A polymer membrane is peeled off of a treated silicon substrate carefully
to ensure it does not tear for use in bioMEMS applications. Theoretical peeling models of PDMS
were created using both a classical beam bending theory, as well as by utilizing Griffith theory,
which is an energy approach to peeling.
A simple peel test experiment was conducted to verify the validity of the theoretical models. The
experimental analysis gave initial indications that the theoretical models can be used to predict the
behavior of PDMS during the peeling process. This information can aid in the development of
automated technology to peel PDMS membranes rather than manually as is done currently. The
models allow for a prediction of what will happen if a machine pulls on a PDMS membrane displacing its tip or edge by a given amount. The models determined that approximately 2 mN of
force is required at a PDMS beam tip adhered to a positive PMMA photoresist-coated silicon substrate to begin the peeling process for a 1.8 mm thick PDMS beam. As peeling continues, the
force at the beam tip needed to peel an entire 10 cm long beam decreases to approximately 1 mN.
Thesis Supervisor: Sanjay E. Sarma
Title: Associate Professor of Mechanical Engineering
Acknowledgements
I would like to thank Sanjay Sarma for his endless guidance and motivation during this challenging project. Sanjay acted as my advisor, mentor, and friend. I am especially grateful for his continual confidence and support throughout the process. He blessed me with his wise advice in
regards to my project, my future, and life in general. Working with Sanjay was a great opportunity and sincere delight for me.
Thanks also to my colleagues Taejung Kim and Sriram Krishnan for their support in assisting me
through the course of the project, especially in regards to the theory and providing feedback on
the proposed theoretical models. I learned countless important lessons from them about research.
Thanks to the School of Engineering and the Department of Mechanical Engineering for their
generosity and commitment to education. This research could not have been possible without
them.
Thanks to my friends at Surface Logix, Inc., namely Olivier Schueller, Bernardo Aumond, Amar
Kendale, and Aaron Raphel, for giving me the opportunity to understand and learn about the field
of bioMEMS. They helped me considerably with my background in the current state-of-the-art
technologies and applications for my project.
Rummington, Rumbleberry, and Rummybear: Thanks for making home a fun place to be. Thanks
for bringing balance. --Rummio
And thanks to my family - my parents, my brother, and his wife - for always supporting me with
their love, patience, and understanding. Thanks for imparting in me the importance of education.
Table of Contents
1 I Introduction ............................................................................................................................
1.1 A bout Polydim ethylsiloxane.......................................................................................
1.2 Properties of Polydim ethylsiloxane ...........................................................................
2
Statem ent of Problem .............................................................................................................
2.1 Motivation 1: Cell M otility A ssay .............................................................................
2.2 M otivation 2: Kinase Enzym e A ssay.........................................................................
2.3 M otivation 3: Soft Lithographic M icrocontact Printing .............................................
2.4 Statem ent of Problem ..................................................................................................
3
Theoretical A nalysis...............................................................................................................21
3.1 Model 1: Static Equilibrium .......................................................................................
3.2 M odel 2: Beam supported by infinite springs.................................................................25
3.3 Model 3: Beam supported by sem i-infinite springs ........................................................
3.4 Model 4: Energy Approach.........................................................................................
3.5 Theoretical Model Com parison ..................................................................................
11
11
13
15
15
16
18
18
21
34
42
47
4
Experimental A nalysis.......................................................................................................
4.1 Experim ental Setup ....................................................................................................
4.2 Experim ental Results ..................................................................................................
49
49
49
5
Conclusion ..............................................................................................................................
55
A ppendix A D erivatives of fl,f 2 ,f 3 , andf 4 ..........................................
.................... .... .. ........ . .
57
A ppendix B M odel 3 m atrix setup ...........................................................................................
A ppendix C M odel 4 equation solution....................................................................................
59
61
References .....................................................................................................................................
63
7
8
List of Figures and Tables
Figure 1.1:
Figure 1.2:
Figure 1.3:
Figure 1.4:
Figure 1.5:
Table 1.1:
Figure 2.1:
Figure 2.2:
Figure 2.3:
Figure 2.4:
Figure 2.5:
Figure 2.6:
Figure 2.7:
Figure 2.8:
Figure 2.9:
Figure 3.1:
Figure 3.2:
Figure 3.3:
Figure 3.4:
Figure 3.5:
Figure 3.6:
Figure 3.7:
Gradient formation in PDMS channels for chemotaxis studies............................11
Microfermentor sketch and photograph................................................................12
12
Soft lithography process steps .............................................................................
13
PD M S chemical structure ....................................................................................
14
184-PDMS static tensile stress-strain behavior ....................................................
14
Typical Properties of Sylgard@ 184-PDMS Silicone Elastomer ..........................
15
96-well PDMS device and thin PDMS membrane with microscopic holes ......
Thin PDMS membrane being peeled from substrate ................................................ 15
16
Z eiss A xiovert.......................................................................................................
16
Cells spreading from their prearranged pattern ........................................................
Multi-well PDMS device sealed to rigid substrate coated with gold....................16
17
PDMS device being peeled off of rigid substrate ..................................................
Molecular Dynamics Typhoon 9410....................................................................17
New peptide substrate identification....................................................................17
18
Soft lithographic microcontact printing ................................................................
21
Sign convention ....................................................................................................
21
Cross-sectional dimensions..................................................................................
22
Sketch of static equilibrium state ..............................................................................
Figure 3.8:
Figure 3.9:
Figure 3.10:
Figure 3.11:
Figure 3.12:
Figure 3.13:
Figure 3.14:
Figure 3.15:
Figure 3.16:
Figure 3.17:
Figure 3.18:
Figure 3.19:
Free body of entire beam ......................................................................................
Possible reaction force distribution at B ...............................................................
Sketch of beam supported by springs .......................................................................
Sketch of beam supported by springs after deformation ...................
Spring force diagram for infinite springs .................................................................
Displacement diagram for beam with infinite springs ....................
Load intensity diagram for beam with infinite springs ....................
Shear force diagram for beam with infinite springs..................................................33
Bending moment diagram for beam with infinite springs ..................
Spring force diagram for semi-infinite springs .........................................................
Sketch of beam supported by semi-infinite springs after deformation .........
PDMS beam broken up into two beam segments connected at x* ............
Arbitrary shape for reaction distribution r(x).......................................................22
23
Free body in region AB of beam ...........................................................................
23
Magnitude w per unit length for reaction distribution r(x) ...................................
Existence of upward reaction force RB at point B along beam............................24
9
24
25
25
26
26
31
32
33
34
35
35
35
Figure 3.20: Free body diagram of first beam segment.............................................................
Figure 3.21: Free body diagram of second beam segment............................................................37
Figure 3.22: Root-finding curves to obtain x*...........................................................................
39
Figure 3.23: Displacement diagram for beam with semi-infinite springs .................
40
Figure 3.24: Load intensity diagram for beam with semi-infinite springs ................
40
Figure 3.25: Shear force diagram for beam with semi-infinite springs..................41
42
Figure 3.26: Bending moment diagram for beam with semi-infinite springs ..............
Figure 3.27: Cross-section of PDMS beam separating from its substrate.................................43
45
Figure 3.28: Energy curves for beam peeling problem .................................................................
46
Figure 3.29: Separation point vs. beam tip displacement for beam peeling problem ........
48
Table 3.1: Theoretical model com parison.............................................................................
50
Figure 4.1: Theoretical curves for peel test experiments .......................................................
50
Table 4.1: Experimental data for peel test experiments........................................................
Figure 4.2: Experimental data plotted with theoretical curves for peel test experiments ..... 51
Figure 4.3: Experimental data plotted with more accurate theoretical curves ........................ 52
Figure 4.4: Beam displacement diagram for peel test experiments....................53
Figure 4.5: Beam shear force diagram for peel test experiments ................................................ 53
10
Chapter 1
Introduction
1.1
About Polydimethylsiloxane
Polydimethylsiloxane (PDMS) is a polymer that is well-suited for many applications, which is
why it is the world's most common silicone [1]. It is a commercially available type of silicone
rubber. PDMS is physically and chemically stable, as well as clean-room compatible. In addition, it has a low curing temperature and high flexibility.
PDMS is currently used as an adhesive in wafer bonding; that is to say, the mechanical interconnection layer between two silicon wafers. Also, PDMS is used for micromachined mechanical
and chemical sensors, as, for example, the spring material in accelerometers. Moreover, due to its
low curing temperature, PDMS is used in sensors with integrated electronics. All in all, its applications range from contact lenses and medical devices to elastomers, caulking, lubricating oils,
and heat resistant tiles [1].
Because of its mechanical, chemical, and material properties, PDMS is very significant to the
field of bioMEMS (biological Micro-Electro-Mechanical Systems), where the applications are
biological analysis, medical diagnosis, chemical analysis and synthesis, drug discovery, and drug
delivery. In general, the advantages of microsystems are that they are close to some biological
length scales, they require shorter processing times, less sample and reagents are required, they
are disposable, automation is possible, and parallel operations are possible to achieve higher
throughputs, to name a few. In the field of bioMEMS, the material requirements are different
from that for MEMS. The desired properties of bioMEMS materials are that they be biocompatible, chemically modifiable, easy to fabricate, economic, soft, and compliant. Since silicon does
not satisfy all these requirements, polymers like PDMS are good materials especially for biological applications.
Figure 1.1: Gradient formation in PDMS channels for chemotaxis studies (Whitesides).
Figure 1.1 shows an example of a bioMEMS device. A gradient is formed in PDMS channels for
chemotaxis studies (courtesy of Whitesides, Harvard University). Chemotaxis is the process of
directed migration by cells in gradients of soluble molecules called chemoattractants. This process is important in the study of cancer and wound healing [2].
11
PDMS is also currently being used as a thin film in, for example, microfermentors. The
fabrication and assembly of microfermentors is done using soft lithographic techniques. The
microfermentor is created in a PDMS body with a PDMS aeration membrane used for oxygen
transport. Figure 1.2 shows both a sketch of a microfermentor and a top view photograph of a
microfermentor in use with a 5 gl volume of liquid in it (courtesy of A. Zanzotto, MIT).
5mm
PDMS aeration membrane
(1 00.Um)
-a-
PDMS body
(300 pm)
Volume - 5 pl
Glass slide
Figure 1.2: Microfermentor sketch and photograph (A. Zanzotto).
The fabrication method used to create PDMS devices is known as soft lithography. The process
of soft lithography is a set of non-photolithographic microfabrication techniques that makes possible the creation of complex microstructures in biocompatible materials. These microfabrication
techniques include microcontact printing (gCP), replica molding, and patterning using elastomeric membranes. The central component in soft lithography is PDMS, which is an elastomer
that can be textured on the micron scale. PDMS usually serves as the functional microstructure
itself, as, for example, a microfluidic network as shown in the previous two examples. The main
advantages of soft lithographic techniques over conventional microfabrication methods are the
diversity of materials that can be patterned, compatibility with biological systems, low cost, the
ability to create 3-D structures, and patterning of non-planar surfaces. Figure 1.3 shows a diagram of the process of soft lithography (adapted from Whitesides).
aacd
Prr-mlvatiw
I
lithoush
phoicol1;hgrnpli)?
photoreist
'si ic n wf
----------------
.
lithographic tciique
Si
Sfer
I
eLst PDI
dastomerie
replica nwding
n
alremv
dement
j
nicroconact printing
nexr-field phase shift lithroraphy ;ticroniolding
*1
lsoerf
~?Si
micrminsfer molding
"dry pattrning
in capil lriWs
.nicroreaclor
stems
Figure 1.3: Soft lithography process steps (adapted from Whitesides).
12
Currently, PDMS fabrication is a rapid prototyping technique. Relatively simple PDMS devices
can be fabricated within 36 hours, with a minimum feature size satisfactory to many microfluidics
applications of approximately 15 tm, for less than $100.
1.2
Properties of Polydimethylsiloxane
Compared to other polymers, polydimethylsiloxane has one of the lowest glass transition temperatures (Tg = -125'C). Because of its low glass transition temperature, PDMS has a unique flexibility with a shear elastic modulus, G, of approximately 250 kPa. For rubber elastic materials,
Young's modulus, E, is approximately equal to three times the shear elastic modulus, so Young's
modulus approximately equals 750 kPa. Moreover, PDMS has high gas permeability, high com0
0
pressibility, usability over a wide range of temperatures from -100 C to +100 C, low chemical
reactivity (except at extreme values of pH), and is essentially non-toxic in nature [3].
The PDMS elastomer is composed of two parts, the polymer and the curing agent. They are
mixed by a certain volume ratio and allowed to cure in open air. In this study, Dow Coming's
Sylgard@ 184-PDMS with a volume ratio of 10:1 was used. The density, p,of 10:1 184-PDMS is
920 kg/m 3 . Young's modulus, E, is 750 kPa, and the Poisson ratio, v, is 0.5.
Once the two components, the polymer and curing agent, have been mixed, they will begin to
react with each other. The mixture can solidify at room temperature within 24 to 48 hours. If the
mixture is cured at 65 0 C, the PDMS will cure in approximately four hours. And when dealing
with thin PDMS membranes on the order of 250 jtm or less in thickness, the membrane can be put
on a heat plate at 100'C and the PDMS will cure in a few minutes.
CH-
CH
CH 3
CH 3
i-
Si--
Si-CH
I
CH
3
I
CH 3
I
GH3
Figure 1.4: PDMS chemical structure.
Figure 1.4 shows a sketch of the chemical structure of PDMS. PDMS has alternating silicon and
oxygen elements on its backbone, with methyl groups hanging off of the silicon elements. PDMS
is inherently hydrophobic. However, it can be put through a plasma oxidation process to make it
hydrophilic. Upon treatment in an air or oxygen plasma, PDMS will seal to itself, glass, silicon,
silicon nitride, and some plastic materials. Contacting two hydrophilic PDMS surfaces will create
the formation of covalent bonds and an irreversible seal between the two bodies.
PDMS can vary in its mechanical properties in that Young's modulus, E, can be in the range of
0.1 to 1 MPa, depending on the type of PDMS used, the volume ratio of the polymer and curing
13
agent, and the curing conditions. Figure 1.5 shows the static tensile stress-strain behavior of
184-PDMS (adapted from [4]).
£44~OMS
a.
1.0.
4
po
Static Sirain (%)
Figure 1.5: 184-PDMS static tensile stress-strain behavior (adapted from [4]).
According to Dow Coming's official product data sheet, Table 1.1 summarizes the typical
properties for Sylgard@ 184-PDMS silicone elastomer.
Unit
Property
Value
As Supplied
mPa.s
Viscosity at 23 0 C (Base)
Mixing ratio by weight (Base:Curing Agent)
Viscosity at 23'C, immediately after mixing with Curing Agent mPa.s
hours
Pot life at 23 0 C
Physical properties, cured 4 hours at 65'C
Color
5500
10:1
4000
2
Clear
50
Durometer hardness, Shore A
MPa
%
kN/m
7.1
140
2.6
Volume coefficient of thermal expansion
1/K
9.6 x 10-4
Coefficient of thermal conductivity
Electrical properties, cured 4 hours at 65 0 C
Dielectric strength
W/(m.K)
0.17
kV/mm
21
Tensile strength
Elongation at break
Tear strength - die B
1.05
Specific gravity at 23'C
Permittivity at 100 Hz
Permittivity at 100 kHz
2.75
2.75
Dissipation factor at 100 Hz
Dissipation factor at 1 kHz
Volume resistivity
0.001
0.001
5 x 1015
Ohm.cm
600
Comparative tracking index (IEC 112)
Table 1.1: Typical Properties of Sylgard@ 184-PDMS Silicone Elastomer.
14
Chapter 2
Statement of Problem
2.1
Motivation 1: Cell Motility Assay
A cell motility assay (CMA) measures the adhesion and movement of cells from a prearranged
pattern on an engineered surface that has biologically-specific ligands on it. A ligand is an ion, a
molecule, or a molecular group that binds to another chemical entity to form a larger complex.
Cell motility plays a significant role in the development and growth of many diseases, especially
in the fields of oncology and immunology. Quantified cell spreading on biological surfaces is
used as a metric for cell activity.
In the first step of conducting a CMA experiment, each well of a 96-well PDMS device has a thin
membrane, also fabricated from PDMS, containing microscopic holes in it for patterning cells on
a biological surface. Each well will have a solution of cells in it so that, once settled, the cells will
be prearranged in a pattern determined by the microscopic holes in the thin PDMS membrane.
Figure 2.1 is a photograph of a 96-well device as well as a magnification of the thin PDMS
membrane with microscopic holes in it (courtesy of Surface Logix, Inc. (SLx)).
Figure 2.1: 96-well PDMS device and thin PDMS membrane with microscopic holes.
Once the cells have been patterned on the biological surface, the PDMS device is removed in
order to enable the cells to spread from the prearranged pattern. The removal process, or peeling
process, is accomplished extremely carefully to ensure that the membrane does not tear.
Figure 2.2 is a photograph of a thin PDMS membrane being removed from a biological substrate
during the peeling process (SLx).
Figure 2.2: Thin PDMS membrane being peeled from substrate.
15
Once the PDMS device is removed, the cells are then allowed to spread from the prearranged
pattern. The surface of the biological substrate, which has biologically-specific ligands on it that
aid or obstruct the cell motility depending on the particular experiment that is conducted, is then
analyzed using an optical scanner or microscope. Figure 2.3 shows an example of an optical
microscope in use analyzing a cell motility assay (SLx). In this case, the Zeiss Axiovert is shown.
Figure 2.3: Zeiss Axiovert.
Using an optical scanner or microscope, cell motility or spreading is observed. The cell spreading
can be quantified using computer algorithms. Figure 2.4 shows first cells in their prearranged
pattern and then their resultant cell spreading due to the biologically-specific ligands on the
substrate (SLx).
Figure 2.4: Cells spreading from their prearranged pattern.
2.2
Motivation 2: Kinase Enzyme Assay
A kinase is any of various enzymes that catalyzes the transfer of a phosphate group from a donor,
such as ADP or ATP, to an acceptor. Kinase activity in each well of a device is determined by the
amount of phosphorylation of peptides anchored at the surface of the substrate. Peptide substrates
for kinases are introduced on biologically inert surfaces on rigid base plates of multi-well PDMS
devices. Peptides can be patterned uniformly and discretely in wells or in microarrays within
each well. Figure 2.5 shows a PDMS device sealed to a rigid substrate coated with a molecular
layer of gold (SLx).
Figure 2.5: Multi-well PDMS device sealed to rigid substrate coated with gold.
16
Once the peptides are patterned, the soft polymeric PDMS well structure is peeled off the rigid
substrate, once again carefully to ensure the PDMS device does not tear (Figure 2.6, SLx). The
surface of the substrate is then washed and labeled with a phosphopeptide specific antibody.
Figure 2.6: PDMS device being peeled off of rigid substrate.
Since the substrate has been washed and labeled with a phosphopeptide specific antibody, the
substrate surface is then imaged using a fluorescent flatbed scanner, as for example the Molecular
Dynamics Typhoon 9410 shown in Figure 2.7 (SLx). The result is very sensitive detection of
kinase activity.
Figure 2.7: Molecular Dynamics Typhoon 9410.
Figure 2.8 shows the resultant image of a kinase enzyme assay after the substrate has been
rendered using a scanner (SLx). The resultant image is analyzed to identify new peptide
substrates for given kinase enzymes.
WA
WA
W
kAI
W
z
Na
N
Z
w
Z
Z
O
I
I
I
II
W'i
Z
Z
8eu
I n
n
L
Sre
ZAP70
EGFR
FGFR
Met
PDGFR
TEK
VEGFR
.. ~
58
S.
,
*,~
Pt
5
*
U
8
U
8
0
ML
Sw
SU
5%
Figure 2.8: New peptide substrate identification.
17
2.3
Motivation 3: Soft Lithographic Microcontact Printing
The fabrication method used to create PDMS devices is known as soft lithography. Soft
lithography includes a set of microfabrication processes, like micromolding, microcontact
printing ([tCP), and stenciling, for example. Unlike other conventional photolithographic
processes, soft lithography uses elastomeric polymers, like PDMS, that provide compatibility
with biological materials and flexibility in design. Soft lithography also enables the ability to
create a wide range of feature types and sizes, ranging from microscopic to macroscopic.
Moreover, soft lithography can be used to create microscopic features inside the macroscopic
structures of standard microtiter plates, like the microarrays in each well of a multi-well device
for cell motility assays. These types of devices and systems allow scientists to collect multiplexed
data, which was not available to them earlier, at the same time using established research tools
and technical expertise already in the field.
One significant technique of soft lithography is microcontact printing. Microcontact printing
involves inking a PDMS stamp (which was made using soft lithography), printing the ink onto a
gold-coated substrate, and then washing the surface with proteins or cells that will stick to the ink
to create a patterned biological surface (Figure 2.9, SLx).
SOFT LITHOGRAPHIC
MI CROC ON? ACT PRI NTINHG
10
IK
PRINT
IMONOLAYER
BIOCLOGICA LS
PR OTEI N.
CELLS
Figure 2.9: Soft lithographic microcontact printing.
2.4
Statement of Problem
For cell motility and kinase enzyme experiments, it is both interesting and important to
understand how PDMS membranes and devices are fabricated and removed, or peeled, from their
respective substrates. In both cases, the PDMS is peeled extremely carefully from the substrates
to ensure that it does not tear, and as a result, it is crucial to understand how to peel PDMS
membranes and devices without tearing them.
In microcontact printing applications, it is important to know and understand how a PDMS stamp
is fabricated. In the first step of PDMS stamp fabrication, the negative of the desired membrane
or stamp is created in a silicon wafer. This process is done using photoresist, ultraviolet light, and
etchants. Once the silicon wafer/photoresist master is created, PDMS is micromolded on top of
the silicon wafer using, for example, a spin-coating or spin-casting process, where the rotational
velocity of the wafer specifies the thickness of the PDMS membrane. PDMS membranes can also
18
be created using a vacuum or pumps to direct the flow of PDMS over a master. Once the PDMS
membrane has cured, it is peeled off very carefully from the silicon wafer, and used as the stamp.
If the silicon wafer has not been treated, the PDMS would create an irreversible seal with the
silicon, and the membrane would not be able to be peeled off the substrate. Covalent bonds
would exist between the wafer and membrane. To prevent this phenomenon from occurring, the
silicon wafer is treated. One way to treat the surface of the silicon wafer is to put it through a
fluorinated silanization process, where a molecular coating makes the substrate surface act like
teflon. As a result, there is no irreversible seal between the wafer and membrane. The dominant
forces between the silicon wafer and PDMS membrane are now adhesion forces and van der
Waals forces. In addition, there may exist electrostatic forces, or charge effects, with the PDMS
membrane that make it want to wrinkle or curl up and stick to itself during peeling.
The meaningful or significant question is what is the best way to peel a PDMS membrane or
stamp from a silicon wafer or rigid substrate without tearing it. In fact, the goal or objective is to
understand some of the peeling characteristics of PDMS that have to date not yet been
determined. The following theoretical analyses address some of the issues of adhesion and
peeling of PDMS devices. Both beam theory and energy criterion are utilized in the analyses.
Theoretical models were created to determine the PDMS device displacement from a substrate,
given external forces, adhesion forces, and the mechanical properties of PDMS.
An experimental analysis was then conducted to verify the usability and applicability of the
theoretical model. The experimental analysis involved a simple peel test of PDMS membranes
from a treated silicon substrate.
19
20
Chapter 3
Theoretical Analysis
3.1
Model 1: Static Equilibrium
The peeling process between a PDMS device and a silicon wafer is difficult to predict. In
particular, it is difficult to determine the "best" peeling process of a thin PDMS membrane in
order to ensure that the PDMS device does not tear. In order to create an accurate model of the
peeling process, it is first desired to examine the basic fundamentals of the process.
It was desired to determine the reactions with the silicon wafer, or rigid substrate, that the PDMS
device experienced. Understanding the reactions with the substrate will help in gaining insight
into the peeling phenomenon. In this first model, a simple, idealized, two-dimensional model of
the peeling process in static equilibrium was investigated.
Before the analysis was conducted, a few key beam bending assumptions were made. First, the
PDMS material was considered to be linear elastic, homogeneous, and isotropic. The sign
convention used to specify positive bending moments, Mb, shear forces, V, and net intensities of
loading, q, during beam bending is shown below in Figure 3.1.
+.V
Figure 3.1: Sign convention.
The cross-sectional dimensions of the beam were considered either constant or varying slowly
with position along the axis of the beam. In other words, as sketched in Figure 3.2, h/L << 1 and
b/L << 1. The xy-plane was considered to be the plane of symmetry. External forces were applied
parallel to the y-axis and lie in the xy-plane, and moments were applied parallel to the z-axis.
tx
Figure 3.2: Cross-sectional dimensions.
Now, consider a long, uniform beam of length L, weight w per unit length, and bending
modulus EI placed on a rigid, horizontal table such that a short segment BC of length a is lifted up
21
from the table by an upward vertical force F exerted at the tip of the beam C, as shown in
Figure 3.3. It is desired to determine the reactions with the table that the beam experiences, as
well as the value of the force F required to keep segment BC from touching the table.
Figure 3.3: Sketch of static equilibrium state.
The difficult part of this problem is indeed the determination of the reactions with the table. The
nature of the reaction between the table and the beam segment AB is not at all clear, as is indicated
in Figure 3.4 by the arbitrary shape shown for the reaction distribution r(x). It is possible,
however, to deduce the nature of this reaction distribution by considering the equilibrium,
geometric compatibility, and moment-curvature requirements for the beam segment AB.
F
rr
Figure 3.4: Arbitrary shape for reaction distribution r(x).
Begin by observing that, since the table is flat, the beam has zero curvature in the region AB. As a
result, from the moment-curvature relation,
2
Mb
dv
2
El'
dx
the bending moment must be zero throughout the segment of the beam from A to B. Therefore,
since the bending moment is constant, or zero, along the beam, from,
dMb +V =
0,
dx
22
the shear force, V, must also be zero in this region. Pursuing this reasoning one step further, from,
dV
+q = 0,
the net intensity of loading must be zero in the region AB because of the constant, or zero, shear
force. Therefore, for the free body of Figure 3.5,
Mb = V = 0,
r(x) = w.
r-G
Figure 3.5: Free body in region AB of beam.
In the free body of Figure 3.6, the reaction in the region between A and B of magnitude w per unit
length is shown.
Figure 3.6: Magnitude w per unit length for reaction distribution r(x).
In order to satisfy the requirement of moment equilibrium for this free body, a negative value for
the bending moment Mb should be obtained. However, a negative bending moment at this point
(a distance Ax to the right of B) is not compatible with the requirement that the beam has a
positive curvature in order to leave the surface of the substrate, since a positive curvature implies
a positive bending moment. A positive bending moment a distance Ax to the right of B requires
the existence of a significantly upward external force in this interval, and therefore, there must be
a concentratedupward reaction force at point B, as indicated on the following page in Figure 3.7.
23
)2"
A
/
Figure 3.7: Existence of upward reaction force RB at point B along beam.
It is important to note that the presence of RB is not in conflict with any of the previous arguments
which led to the uniformly distributed reaction in the region between A and B. Therefore, it was
determined that the reactions with the table were as shown in Figure 3.8.
F
C
A
T
T
Figure 3.8: Free body of entire beam.
By applying the equilibrium requirements to this free body, the magnitudes of the reaction
force RB and the applied force F were obtained. Summing all the moments about point B yields,
-wb - = 0>F
VMB = Fa-wa.- +wb12
2
2
=
-.
2
And summing the forces in the vertical y-direction yields,
ZFY = RB+F-wa+wb-wb = 0=>RB = wa-F =
a
In order to keep segment BC from touching the substrate, the forces F and RB split the weight of
segment BC evenly. The sudden appearance of a body force at B due to the table exerting a
concentrated upward reaction force on the beam is very interesting. The existence of this
phenomenon as a concentrated force only occurs in this ideal case. In the real-world situation, it
is more likely that the reaction force is a force distribution from the table supporting the beam, as,
for example, shown in Figure 3.9. On the other hand, depending on the method of peeling
conducted, for example, if the beam tip was peeled so that a negative bending moment was
24
applied at the beam tip (or the slope at C was flat), the concentrated reaction force distribution
may take on a different shape.
F
C
A-
Figure 3.9: Possible reaction force distribution at B.
Either way, it is necessary to create another model that will give more information into the peeling
process. It is essential to determine what the reaction force distribution from the table actually is
to ensure the beam satisfies its real boundary constraints and equilibrium conditions; that is to say,
what is the shape and magnitude of this force distribution given material properties and boundary
conditions? Answering this question will give greater insight into the peeling characteristics of
PDMS. The following model investigates this phenomenon more closely.
3.2
Model 2: Beam supported by infinite springs
In this second model, the same long, uniform beam as in the first model was investigated, except
in this case, the beam was supported by springs connecting the beam to the substrate, as shown
below in Figure 3.10.
1CD
Figure 3.10: Sketch of beam supported by springs.
Figure 3.10 shows a sketch of the beam model before an upward vertical force or moment is
applied to the tip of the beam. Figure 3.11 on the following page shows a sketch of the beam
model after an upward vertical force,f, and moment, m, has been applied to the tip of the beam,
and shows the possible resulting deformation.
25
Figure 3.11: Sketch of beam supported by springs after deformation.
There are a few assumptions that were made in this model in addition to the ones made in the first
model. For this analysis, the small-angle approximation for the curvature of the beam was
utilized, (0 < 4.70), even though in the real situation, the angle of bending, or peeling, may be
higher than 4.70. For the sake of simplifying the analysis, the PDMS was assumed to be
weightless. The springs supporting the beam and connected to the substrate were considered to be
infinitely linear in both positive and negative y-directions. The springs are called "infinite"
because they can stretch and compress infinitely in both positive and negative directions of y
without breaking. Figure 3.12 shows a qualitative graph of the spring force as a function of y.
Figure 3.12: Spring force diagram for infinite springs.
Now that the assumptions have been outlined, the analysis may begin. From Figure 3.11, the
springs apply a force per unit length as a function of position along the axis of the beam. This
force per unit length, or net intensity of loading, due to the springs was determined with the
following equation,
-'
qspring(x)
where k has units of force per length squared, or,
k
[F].
L2
26
Since the springs model the adhesion effects between the PDMS and the substrate, the substrate is
actually in contact with the PDMS. Therefore, let yo equal zero, thus shifting the x-axis so that it
is tangential to the bottom surface of the beam. Now,
qspring(x) = -ky(x).
Therefore, using a standard beam equation to characterize the beam properties, an ordinary
differential equation of the following form characterizing the beam was formulated,
4
4
EI
+ ky(x) = 0,
- _q(x) = 0 -> E I
dx4
d4
where I, the moment of inertia of the beam, is,
IW-.
-bh3
12
Dividing both sides of the equation by the bending modulus, EI, gives,
4
dy
k
dx 4'E
This ordinary differential equation (ODE) has a closed-form solution which was determined using
the analysis below. The characteristic equation of the ODE is,
r + EI
= 0.
ruri
It is desired to determine the roots of the equation above in order to determine the form of the
solution of the ODE. If both expressions in parentheses are set to zero, then the first equation in
parenthesis gives two roots of the equation,
_2
(r,2) = -
k
1
and the second equation in parenthesis gives two more roots of the equation,
(r 3 , 4 ) 2
3,4
27
k
El
Solving both these equations yields,
~
=
+
rl, 2, 3, 4
k
=
) 1/4
+Afi
j=+
All four roots are imaginary. Another way to write the imaginary number i using exponentials is
given below,
in
in
4
2
.
i= e
=
2
3in
3in
.
1+i
I= e
4
.
-
+ i
Substituting these expressions for i, and simplifying the resulting expression yields,
r23,
7+
k +
-'I
= +-
=+(1
--
=+-
i) .-
.
Finally, writing the four roots of the characteristic equation of the ODE in a form that is easy to
express gives,
rl, 2, 3, 4 = +[A+
Ai], where A
-
1/4
The four roots give a trigonometric form of the solution of the ODE. As a result, the expression
for the displacement of the beam, y(x), is given as,
.-.y(x) = eAx[ci cosAx + c2 sinAx] + e-Ax[c 3 cosAx + c 4 sinAx],
where cl, c 2 , c3 , and c4 are the constant coefficients in the expression. The constant coefficients
are functions of the material properties of the beam and the boundary constraints imposed on the
beam. With the closed-form solution characterizing the beam displacement, boundary conditions
must be used in order to determine the values of the constant coefficients.
The equation can be rewritten as,
y(x) = cieAxcosAx + c 2 eAxsinAx + c3 e-AxcosAx + c 4 e-AxsinAx.
28
Let,
f 1 (x) = eAxcosAx,
f 2 (x) = eAxsinAx,
f 3 (x) = e-AxcosAx,
f 4 (x) = e-AxsinAx,
then,
y(x) = cIf(x) + C2J 2 (X) + C3f 3 (X) + C4f 4 (X).
The standard beam equations relating the bending moments and shear forces in the beam to
derivatives of the displacement of the beam are given below,
y"xj
W
-
y"'(x) =-
Mb
El'
V
Four boundary equations are needed in order to solve for the constant coefficients. Two boundary
conditions are given for each free end of the beam. At x = 0, the displacement and slope of the
beam are both zero, however at x = L, the tip of the beam, there is both a force,f, and moment, rn,
at the tip. To simplify the analysis, rather than applying a force,f, to the tip, a displacement will
be applied to the tip, ytip. The force needed to apply that given displacement to the tip can be
determined from the shear force diagram of the beam or from the following equation,
f = - q(x)dx.
Also, rather than specifying a moment at the beam tip, the slope at the beam tip during peeling
will be zero, resulting in a negative applied moment at the tip. The value of this moment can be
determined from the bending moment diagram or from the following equation,
m = -fL-
q(x) -xdx.
As a result, the four boundary equations are given below.
y(O) = 0,
y'(0) = 0,
y(L) =y'jP
y'(L) = 0.
29
Using matrix algebra and setting up the matrices as below, we can solve for the constant
coefficients. Refer to Appendix A to see the expressions for the derivatives off,,f2,f3, andf 4 .
1 (0)
f 2 (0) f 3 (0) f 4 (O)
1 '(0)
f 2 '(0) f 3 () f 4 '(0) C2
C,
f 1 (L) f 2 (L) f 3 (L) f 4 (L)
C3
f 1 '(L) f 2 '(L) f 3 '(L) f4'(L) C4
0
[tip
0
_0
Using MATLAB, the values of the constant coefficients were determined in terms of k, EI, L, and
Yap. The values of the constant coefficients cl, c2 , c3 , and c4 are given below.
C1 =
2 =
sinhALcosAL + coshALsinAL
cosh2AL + 2(cosAL) 2 -3
- t'
sinhAL(sinAL - cosAL) - 2e-ALsinAL
cosh2AL+2(cosAL) 2 - 3
C3 = -C
1
,
C4 =-2c, - c2'
where again,
)k1/4
Using these values of the constant coefficients determined, graphs of the displacement, net
intensity of loading, shear force, and bending moment of the beam were created for certain given
mechanical properties and boundary conditions.
In the following figures, Young's modulus, E, was 0.75 MIPa, which is typical of a 10:1 polymer
to curing agent volume ratio of 184-PDMS. The beam was considered to be 4 mm wide, 1 mm
thick, and 10 cm long. The work of adhesion between PDMS and fluorinated silicon is
approximately 30 ergs/cm 2 , which is equal to 0.03 J/m 2 or 0.03 N/m [5]. It was unclear how to
apply the work of adhesion to determine a "pressure" constant k in units of force per length
squared for the elastic constant of the springs supporting the PDMS structure in the model . To
that end, it was assumed for the time being that k was 100 N/m2 , just so that qualitative graphs
could be created giving insight into the reaction force distribution. Later, after conducting an
experimental analysis, realistic values of k will be chosen to obtain both qualitative and
quantitative graphs of the peeling process.
The beam displacement was graphed for different applied tip displacements of the beam. The tip
displacements applied were 0 mm, 5 mm, 10 mm, and 20 mm. Figure 3.13 is a graph of the
30
displacement of the beam as a function of position along the axis of the beam for varying applied
displacements at the beam tip.
x10
Displacement Diagram for Beam with Infinite Springs
-3
--
4
Ip1 = 0 mm
y p1 2=5mm
y- =10 mm
--
ip4=20 mm
y
-
15- -
...............
10 F-.
5
...............
-~-__
___________
0
.......
_
J ..-.--------- J------.-.---...~ I -.....-.......-J..---~-.-------- I ___________ I
0.01
0.02
0.03
0.04
0.05
x (m)
__
___________
0.06
.......
..........
.............. ....... ........
........
...
a
...... .....
I
___________
0.07
I
___________
0.08
J
___________
0.09
0.1
Figure 3.13: Displacement diagram for beam with infinite springs.
Figure 3.13 shows that without the existence of the substrate, part of the beam in the approximate
region between x equals 5 cm to 7.5 cm would displace down below the x-axis. This is significant
because it shows that since there is a substrate, it must apply a reaction force in this region to
prevent the beam from displacing downward in the real case. In order to verify this phenomenon,
Figure 3.14 on the following page, a graph of the load intensity as a function of position along the
axis of the beam, was observed.
31
Load Intensity Diagram for Beam with Infinite Springs
0.5
1
z
= 0 mm
y
yp2= 5mm
yp3= 10 mm
y t 4 20 mm
-
0
0 01
0 02
-
0.03
0.04
0.05
x (M)
0.06
0.07
0.08
0.09
0.1
Figure 3.14: Load intensity diagram for beam with infinite springs.
Indeed this graph shows that in the region of x approximately between 5 cm and 7.5 cm the
springs are actually pushing up against the PDMS device, or that there is some type of reaction
from the table. After approximately 7.5 cm, the springs are pulling down on the PDMS device
trying to prevent it from peeling.
Figures 3.15 and 3.16 show the shear force and bending moment diagrams, respectively, for the
beam with infinite springs. From the shear force diagram, it is clear that there is some type of
reaction force distribution rather than a concentrated reaction force as indicated in the first model.
The reaction force distribution is in the region of x approximately between 6 cm and 8.5 cm. This
is interesting since the separation point of the PDMS structure from the substrate is at 7 .5 cm.
Also, the graph shows the shape of the distribution to be larger in magnitude closer to where the
beam leaves the substrate surface, and it slowly tapers in magnitude as it gets closer to x =0. The
magnitudes of the forces required to pull the beam tip were 0 mN, 5 mN, 10 mN, and 20 mN
respectively. The magnitudes of the reaction forces at their highest were approximately 0 mN,
0.34 mN, 0.67 mN, and 1.3 mN, at about x = 7.65 cm. This value of the beam position is
significant because it is the location of the separation point between the beam and the substrate
(Figure 3.13). Figure 3.16 shows that the bending moment is highest in the beam at
approximately x = 8.4 cm, which is approximately midway up the portion of the beam that is not
in contact with the substrate. The magnitudes of the bending moments at the tip are negative to
obey the zero-slope tip boundary condition. The zero-slope tip boundary condition also adds
bending energy to the beam to help it peel from the substrate surface.
32
Shear Force Diagram for Beam with Infinite Springs
0.025
=
-
mm
= 10mm
= 20 mm
y
0.02
0
-
- 4
0.015
-
-
...... ...
..........-.
............
...................-.
........-.
...... .......
0.01
..............
........
........
........
........
...
... .....
....
......
...
....
..
..
0.005
.... ..
...
...
. ....
...
...
......
...
....
...
...
...
.. ...
.....
...
...
...
...
..
....
..
..
...
...
...
....
...
...
...
...
0
-0.005
0
0.03
0.02
0.01
0.04
0.05
0.06
0.07
0.08
0.09
0.1
x (m)
Figure 3.15: Shear force diagram for beam with infinite springs.
4
x 10-'
Bending Moment Diagram for Beam with Infinite Springs
.....
....
2 ..........
.....
...
....
..
........
.....
-2
z
x
-14
-6
-8
~~
-10
-12
0
t 11p =
0
mm
Yp
yt Y23=5rmm
=10mm
ytlp3 = 10 mmn
= 20 mm
y
-
I
0.01
0.02
0.03
0.04
0.05
x (m)
0.06
0.07
0.08
0.09
0.1
Figure 3.16: Bending moment diagram for beam with infinite springs.
33
All in all, this second model showed that there was in fact a reaction force distribution rather than
a concentrated upward vertical force where the beam separates from the table. The graphs show
the magnitude and shape of this distribution for different given tip boundary conditions.
Despite this result, our model has some flaws and can be improved. One flaw is that as the tip
displacement is increased, the separation point between the PDMS beam and the substrate does
not move. The reason for this flaw is that an assumption was made that the springs do not break.
In fact, once the PDMS beam is separated from its substrate, the springs which model adhesion
have essentially broken and no longer apply a restoring force on the beam. The next model will
address the issue of using semi-infinite springs, or springs that will break after stretched a certain
given distance. This following model will give more accurate magnitudes of reaction force
distributions and moments in the beam during the peeling process. Moreover, it may give insight
into more realistic values for the pressure constant k and the applied forces at the tip of the beam.
For now, the chosen value of k and resulting tip forces are relatively realistic to a couple
significant digits, but the model is not as accurate as it could be. Ideally, after the next model is
formulated, an experiment could be run to determine realistic values of the tip forces, and as a
result, realistic values of k would be acquired to match the shape of the experimental beam. But
immediately, it is important to discuss another iteration in the creation of a more accurate model.
3.3
Model 3: Beam supported by semi-infinite springs
This third model is exactly the same problem as in the previous case, except for one thing; the
springs connecting the PDMS beam to its substrate are no longer infinitely linear in both
directions. Rather, the springs are termed semi-infinite, because they are still infinitely linear in
the negative y-direction, however, they are not infinitely linear in the positive y-direction. At a
certain critical value of the vertical displacement of each spring's end, the springs which model
adhesion will break. Figure 3.17 shows a qualitative graph of the spring force as a function of
vertical position. According to the graph, at a critical value of vertical displacement, Ycritical, the
springs break and no longer apply a force on the PDMS beam.
Figure 3.17: Spring force diagram for semi-infinite springs.
Again, the same long, uniform beam as in the first and second models was investigated. Before
deformation, the model is as shown in Figure 3.10. Figure 3.10 shows a sketch of the beam model
before an upward vertical force or moment is applied to the tip of the beam. Figure 3.18 shows a
sketch of the beam model after an upward vertical force,f, and moment, m, has been applied to
34
the tip of the beam, and shows the possible resulting deformation. Notice that some of the springs
are broken, because they have been stretched too far.
Figure 3.18: Sketch of beam supported by semi-infinite springs after deformation.
In order to conduct the analysis, the beam was broken up into two beam segments, as shown
below in Figure 3.19. The first beam segment has springs acting on it, represented by w(x), while
the second beam segment does not, since the springs have broken. Once a critical value of
vertical displacement, ycritical, is specified, it corresponds to a value of x = x*, a position along the
axis of the beam where the two beam segments meet (Figure 3.19).
Figure 3.19: PDMS beam broken up into two beam segments connected at x*.
The first beam segment is similar to the beam in the second model (Figure 3.20). The springs
apply a force per unit length as a function of position along the axis of the beam. At x*, where the
two beam segments were split, a shear force and bending moment appear in the beam, termed V1
and Mbl respectively, due to the split.
6
Le ( K)
Figure 3.20: Free body diagram of first beam segment.
Therefore, using a standard beam equation to characterize the beam properties, an ordinary
differential equation of the following form was formulated,
35
4
4
EI
-- q(x) = 0 -> EI
+ ky(x) = 0,
dx4
dx 4
where k is the pressure constant of the springs in units of N/m 2 . Dividing both sides of the
equation by the bending modulus, EI, gives,
4
dy
dx
k
0.
y=
A
This is the same ordinary differential equation as in the previous model, and it has a closed-form
solution. The expression for the displacement of the beam is given as,
y 1 (x) = c 1 eAxcosAx + c 2 eAxsinAx + c 3 e-AxcosAx + c 4 e-AxsinAx,
where,
Ak) 1/4,
and c1 , c2 , c3 , and c4 are the constant coefficients in the expression which will be determined by
the material properties of the beam and the boundary conditions imposed on the beam. With the
closed-form solution characterizing the beam displacement, boundary conditions must be used in
order to determine these values of the constant coefficients.
Four boundary equations are needed. Two boundary conditions are given for each free end of the
beam. At x = 0, there is zero displacement and zero slope, however at x = x*, the tip of the first
beam segment, there are shear forces and bending moments in the beam, V1 and Mbl, respectively.
ITHerefure, tle our uoundary equations are:
y1 (O) = 0,
Y1'(O) = 0,
I
-
Mbl
EI
V1
y1"(x*
EI
Unfortunately, there are four equations and six unknowns, c1 , c2 , c3 , c4 , V1 , and Mbl. In order to
determine the expressions for the remaining two variables, the second beam segment must be
analyzed.
36
Figure 3.21 shows a sketch of the free body diagram of the second beam segment. There are no
loading intensities acting on the beam, since the springs have broken, but there are forces and
moments on both free ends of the beam.
L
Figure 3.21: Free body diagram of second beam segment.
The ODE characterizing this beam segment is:
4
dy
EIy:.
-
4
=OEdy~
q(x) = 0->xEI
0.
Dividing both sides of the equation by the bending modulus, EI, gives,
4
d y
dx
4
-
0.
This ODE also has a closed-form solution, and the expression for the displacement of the second
beam segment is,
y 2 (X) =5
3 + c 6 x 2 +c 7 x +c 8
where c5 , c6 , c7 , and c8 are the constant coefficients of this expression. They are once again
functions of the material properties of the beam and the boundary constraints imposed on the
beam. Four boundary equations are needed in order to solve for the constant coefficients. Two
boundary conditions are given for each free end of the beam. At x = x*, there is a shear force and
bending moment acting on the beam end due to splitting the beam, and at x = L, the tip of the
second beam segment, there is a force, f, and bending moment, m, at the tip of the PDMS beam.
However, again for simplicity, it will be assumed that a displacement is applied at the tip, ytg, and
that the beam tip will have zero slope.
Therefore, the four boundary equations are:
-
Mb
-El
37
2
y X2 ) '
V
-
E I
y 2 (L) =yP,
y 2 '(L) = 0.
Now there are eight equations and twelve unknowns. The eight equations are the four boundary
conditions for the first beam segment and the four boundary conditions for the second beam
segment. The twelve unknowns are c1 , c2 , c3 , c4 , c 5 , c6 , c 7 , c 8 , Mbl, V1 , Mb2, and V2. Four more
equations are needed to characterize the beam.
The four desired equations can be created from the connection constraints for the two beam
segments. The connection constraints mandate that the position, slope, shear force, and moment
at the right end of the first beam segment and the left end of the second beam segment be the
same. Therefore, the four connection constraints are given below,
2 (x*) =
y 1 (x*)-y
0,
y'(x*)-y2'(x*)= 0,
y"(x*)- y2"(x*)= 0,
y'"t(x*) - y2'"(x*
= 0.
Essentially, the last two connection constraints say that Mbl = Mb2 and V1 = V2 . So now there are
12 equations and 12 unknowns. As in the second model, using matrix algebra and setting up the
matrices as shown in Appendix B, we can solve for the constant coefficients, and shear force and
bending moment at x = x*.
Using MATLAB, the values of the constant coefficients were determined in terms of k, El, L, ytip,
and x*. The functions were rather lengthy expressions of these variables. And rather than
showing what the constant coefficients equal, referring to Appendix B, and using software like
MTAr
lviI
i h
b
011C~ C-all UJLalWII L11%,
AD , -
1jtL)
fir
xrsin
--
FL-O
kj
O JLJI
t
rtcknnt
I
rnefft-'icint.
ILIL
Even though there are now expressions for the constant coefficients, they are all functions of x*,
which is an unknown. In order to be able to plot the beam displacement, x* must be determined.
Therefore, one more equation is needed; x* can be determined given YcriticalFrom the beam analysis of the first segment of the beam, it is known that,
yl(x*) = cieAx*cosAx* + c2 eAx*sinAx* + Ce-Ax*cosAx* + C4e-Ax*sinAx* = Ycritical,
or that,
ci eAx* cosAx* + c 2 e Ax* sinAx* + c 3 e-Ax* cosAx* + c 4 e-Ax* sinAx* - Ycritical = 0.
38
In order to determine the solution of the value of x*, one must find the root of.this equation.
Therefore, given material properties and boundary conditions, x* can be determined, and as a
result, the beam displacement (and other curves) can be plotted.
In the following figures, as in the previous model, Young's modulus, E, was 0.75 MPa. The beam
was considered to be 4 mm wide, 1 mm thick, and 10 cm long. It was assumed that k was
100 N/m 2 . The displacements applied at the tip of the beam were 0 mm, 5 mm, 10 mm, and
20 mm. Also, the critical y-value at which the springs break was chosen to be 1 mm.
Figure 3.22 is a graph of the root-finding curves used to determine the value of x* for each beam
condition. As expected, for no displacement of the beam tip, there is no root of the curve since the
springs have not yet broken. However, for the other curves there is one single root each. The
value of x*, the x-position at which the springs break, is 7.2 cm, 5.56 cm, and 3.3 cm, for beam tip
displacements of 5 mm, 10 mm, and 20 mm, respectively.
20
X10-
Root(s) of curves are x*
--
yt
-- y,
1
=50 mm
= 5 MM
ytip 3=10 mm
15 ..=
p4
........
..................
0~1.......
0
20 mm..
001
002
003
0.04
005
006
0.07
008
0.09
0.1
x (M)
Figure 3.22: Root-finding curves to obtain x*.
Figure 3.23 on the next page is a graph of the displacement of the beam as a function of position
along the axis of the beam for varying applied displacements at the beam tip. The attractive
feature about this figure is that is shows that as the beam tip displacement increases, the
separation point between the PDMS beam and the substrate moves to the left. This phenomenon
was impossible to predict in the previous model. As a result, it is clear that this model is more
accurate than the second model from a qualitative standpoint. Figure 3.24 on the next page is a
load intensity diagram of the beam as a function of position along the axis of the beam. This
figure shows that in all three cases with an applied tip displacement, the springs break at the same
load intensity of -0.098 N/m. Each curve has the same shape, except that each is shifted
39
depending on the separation point for its beam condition. Beyond x*, there is zero loading
intensity from the springs, which explains why the curves flatten out to zero.
Displacement Diagram for Beam with Semi-Infinite Springs
x10-.3
20
-
~
15 - --
-
-
10
Yi 1 = 0=mm
tip2 =5m
Yt1pi =10mm
4=20 mm
-
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
x (m)
Figure 3.23: Displacement diagram for beam with semi-infinite springs.
Load Intensity Diagram for Beam with Semi-infinite Springs
0.02
tipIp2i
......
----- ---... -
tip4
.- 0.04
...... .......
...
-.. ..
-0.06 --....... -
-0.08
n
.1 0
..
..
-.. ...
0.02
0.03
..-..
............. ......
-..
....
..
...
-..
. ..-..
. .........
...... .....
. ...........
....
-..
- ....---..
0.01
... ...
.....
... -.
..............
.
-..
...
.-..
.. ........~....
............
..
..........
.
1
5 mm
10 mm
20 mm
0.04
0.05
0.06
0.07
0.08
0.09
....-..
0.1
x (m)
Figure 3.24: Load intensity diagram for beam with semi-infinite springs.
40 '
Figure 3.25 shows the shear force diagram for the beam with semi-infinite springs. There are a
few interesting points to make from these curves. First of all, the shape of the curve shows that
there is in fact a force distribution rather than a concentrated vertical force near the separation
point in this more realistic model. The reaction forces are highest in magnitude right at the
separation point. More of the magnitude of the forces from the bending of the beam is in
equilibrium (or is "absorbed") by the springs further to the left of the separation point.
10,
Shear Force Diagram for Beam with Semi-Infinite Springs
-
2
m
. -..
-. p2 =...
0m
= 10 mm
y
yt1p4 20 mm
0
0.01
0.02
0.03
0.04
0.05
x (m)
0.06
0.07
0.08
0.09
0.1
Figure 3.25: Shear force diagram for beam with semi-infinite springs.
Of particular interest are the magnitudes of the shear forces to the right of the separation point. In
all the curves with tip displacements, the curves flatten out. The values of the forces applied at
the tip were 0.26 mN, 0.18 mN, and 0.13 mN, respectively for beam tip displacements of 5 mm,
10 mm, and 20 mm. It may seem counter-intuitive at first that for a higher beam tip displacement,
less force is required. In fact, that is not what the curves indicate. The curves only show a
particular instant in time. They show that as the separation point moves to the left, less force is
required to keep it moving to the left. Or in other words, a lot of force is needed at first to separate
the two bodies, and less and less force is needed as the "crack" propagates. This phenomenon
makes sense because the energy of adhesion between the two bodies is transformed into bending
energy in the beam which helps to peel the PDMS from the substrate rather than helping it to
adhere to the substrate as it was before.
Figure 3.26 on the following page shows the bending moment diagram for the beam with semiinfinite springs. Basically, these curves show that the bending moment is highest just before the
springs break. This observation leads to the thought that where the springs break, or where the
adhesive bonds break, is the point in the beam of highest vulnerability to tear. The curves have
similar shapes. As soon as these adhesive bonds break, the bending moment drops significantly
41
in each case. At the beam tips, there is a negative bending moment to obey the zero-slope
boundary condition. The zero-slope boundary condition helps to add bending energy to the beam,
thereby aiding it in the peeling process. Compared to the second model, this model shows that the
bending moment is zero at the location midway up the portion of the beam that is not in contact
with the substrate.
Bending Moment Diagram for Beam with Semi-Infinite Springs
X 10,
4 -
0E
z
---
-3
1.__
4;~~~~~
_i_
0 mm
t.....
_
J.L
_
.....
y Up2 = 5 mm
10 mmn......
-5Yia=
y=20 mm
0
0.01
0.02
0.03
..
.....
_.....
..
..... .....
...
.. ...
........ ...
....... .....
y 1=
-4
0.04
.....
0.05
x (m)
0.06
0.07
0. 8
~...........
....
0.09
0.1
Figure 3.26: Bending moment diagram for beam with semi-infinite springs.
All in all, it is clear that this model is much more accurate than the second model. The
displacement, load intensity, shear force, and bending moment diagrams are more realistic given
different tip displacements. Of course, the model could still be improved by considering springs
that do not compress at all, which is similar to the real phenomenon. The magnitudes of the
curves seem qualitatively close to real observations, but to verify that, some experiments will be
run to correlate the data. One more theoretical approach was attempted to model the peeling
process: an energy approach, which we describe below.
3.4
Model 4: Energy Approach
This section- discusses the peeling of an elastic PDMS beam from an adhesive substrate surface
using Griffith theory, or an energy approach rather than beam bending theory. Griffith theory
states that crack extension, or beam separation from a substrate surface, occurs when the energy
available for crack growth, or separation, is sufficient to overcome the resistance of the material to
separation from a surface due to surface energies, like in this case, adhesion energies.
42
In order to use Griffith theory, it is necessary to determine the stored elastic energy in the PDMS
beam, as well as the adhesion energy stored to hold the beam in contact with the substrate. To see
a sketch of the beam configuration, refer to Figure 3.27. The beam peeling configuration is
different than that observed during beam bending theory; that is to say, the peeling happens from
the left rather than the right. The reason for this change is that it simplifies the analysis
considerably. However, it is important to note that even if the analysis had been conducted with
peeling occurring from the right, that the same solution would obviously have been obtained.
I-W
-
Figure 3.27: Cross-section of PDMS beam separating from its substrate.
Figure 3.27 shows a cross section of a PDMS cantilever beam of length L, width b, thickness h,
height ytip, and Young's modulus E. For simplicity, the height of the beam tip, ytip, rather than the
force applied at the beam tip was given. The PDMS beam is adhering to the silicon substrate a
distance a = (L - s) from its tip.
The stored elastic energy of the beam is in the beam segment 0 x s. The energy in this region
induces a restoring force that helps to peel the PDMS beam from the silicon substrate. On the
other hand, the energy of adhesion is stored in the beam segment s x L. The energy in this
beam segment induces another force that holds the beam in contact with the substrate.
The equilibrium peel distance, or equilibrium separation position s*, is determined by the balance
of these two energies. At equilibrium, s* minimizes the total energy of the system, which
includes both bending and adhesion energies.
Since there are no external forces acting on the PDMS beam for 0
solution of,
4
Eld y
dx4
0, where I =
12
I is the moment of inertia of the beam with respect to the z-axis.
43
x
s, its deflection y(x) is the
The equation was solved using the following boundary conditions,
y'(O) = y'(s) = 0,
y(O) =Y
y(s) = 0 .
Refer to Appendix C to see how the solution was determined. The equation has the solution,
- 3
2
y(x) = Yap 1 +
.
Now that the expression for the beam position is known in the region 0 x 5 s, the bending
energy stored in the beam can be determined. The bending energy stored in a beam segment is
given by:
C
EJenn
i
=
b
2J
2 EI
M2
Ebe~dng
=
22
EIdy2
dx ,
2
El
2
=
El
d
d
2x ( dx 2
dx.
For the beam segment 0 5 x 5 s, and using information in Appendix C,
EI
bending
s
6 yi
2y
-
0
2
144y 2.
(144Y 2._
EI
2S 0
ss
F36yv
is]
F48y,
0
2
s2
36 2
+
Evaluating this integral yields,
7 f F144V2
Ebending= 2
S6
F144v? -2.s
3 o
S 5 2 10
31
S4
36y$,,
72y2,
3
S3
+
3
Therefore,
6Elyp
s3
Ebending
Now the energy due to the PDMS beam adhesion to the silicon substrate must be determined. The
interfacial adhesion energy stored in the PDMS beam segment s 5 x 5 L is the surface energy per
unit area of the bond ys, times the area of contact,
44
Eadhesion
-Ysb(L - s).
The parameter ys has units of J/m 2 or N/m, and is also known as the work of adhesion. The sign
of Eadhesion is negative because it is a binding energy. The total energy of the system is therefore
the sum of the elastic plus surface energies [6],
6EIy 2
Etotai = Ebending +Eadhesion =
s
-
-
s).
Energy Curves for Beam Peeling Problem
Eota
Ebending
-adhesion
0
Etotalmin
s=L
s*
Figure 3.28: Energy curves for beam peeling problem.
Figure 3.28 shows a typical curve of Etotal(s). This curve has a single minimum corresponding to
the equilibrium point, or separation point, s*. This variable is found by first setting dEtotal/ds = 0,
18EIyp
dEtotai
ds
t
-
se
+yb=0,
and then solving for s,
4= 18EIyt2,
ysb
18Ey2p bW
ysb
12
45
3Eh 3 y 2
2ys
to obtain s*,
1/4
3Eh3 2
2ys
The energy curve has a single equilibrium point if s* < L and no equilibrium point if s* > L.
Therefore, the beam is adhered to the substrate if s* < L, and it is free or completely peeled if
s* > L.
Figure 3.29 is a curve of the separation point s* versus the beam tip displacement yt, given real
variables. Young's modulus, E, was 0.75 MPa. The beam was considered to be 1 mm thick and
10 cm long. Since the work of adhesion, ys, between PDMS and fluorinated silicon was
approximately 30 ergs/cm 2 , or 0.03 N/m [5], this value was used in the analysis. The separation
point was plotted for values of the beam tip displacement between zero and the length of the
beam.
Separation Point vs. Beam Tip Displacement for Beam Peeling Problem
14
r. ...... ..
.......
........
........
... ...
.
......
............
..
.........
..
.... .
....
..... ..
12
10
........
....
........ .... .....
.....
. .......
.....
..........
..
.....
....
7
....
.
8
.
.....
/.
.....
..
..
..
/..
6
.
.....
.......
...
...
...
Z
.-
.
/..
......................... ........
... ...... .............
.............. .. .
4
2
0
... .
. ......
1
2
...... ...
3
4
5
6
-
.. ..
_..
7
8
9
10
yfip (cm)
Figure 3.29: Separation point vs. beam tip displacement for beam peeling problem.
As can be observed from the graph, at approximately 5.1 cm for the beam tip displacement, the
separation point approaches and exceeds a length of 10 cm, which is the length of the beam.
Therefore, at this point, the entire beam has peeled off of the silicon substrate. It is interesting to
see how closely the data from a simple experimental peel test may coincide with this theoretical
data, since this theoretical model is idealized.
46
Overall, this energy model is far simpler and easier to conduct then the beam bending analyses. It
may also be more accurate. An advantage of this model is that it does not depend on spring
models for the adhesion between the beam and the silicon substrate. Therefore, values of the
pressure constant k do not need to be estimated or determined. Also, the width of the beam does
not come into play in the final solution of the energy approach. However, the energy approach
does depend on the work of adhesion, which is a variable that is not well-known for PDMS and
silicon, since the value can vary so much for different types of PDMS as well as different surface
chemistries on silicon substrates. The next step is to conduct an experimental analysis and see
how well the data correlates with these theoretical analyses.
3.5
Theoretical Model Comparison
Four theoretical models were created to understand and predict the peeling process of PDMS
from silicon. Each model has its own strengths and weaknesses. Table 3.1 on the following page
summarizes the strengths and weaknesses, or advantages and disadvantages, of each theoretical
model taking into account different considerations.
47
Model 1:
Static Equilibrium
Adv.
Disadv.
Gives information regarding existence of concentrated reaction force from substrate
Simple, idealized model; does not give information regarding shape and magnitude
of reaction force distribution
Unable to create relevant graphs to characterize beam, as can be done in other
models
Ignores adhesion between PDMS beam and substrate
Model 2:
Beam supported
by infinite springs
Adv.
Gives information regarding existence of reaction force distribution; shape and
magnitude determined given material, geometry, and boundary conditions
Disadv.
Introduces new variable; pressure constant k to help model adhesion, but k is an
unknown parameter, and must be estimated or chosen arbitrarily
Model flawed in that separation point does not move as beam tip displacement
increased; impossible to predict separation point
Shear force diagram not accurate; model claims more force is needed as peeling
continues, whereas in practice less force is needed as separation propagates
Model 3:
Beam supported
by semi-infinite
springs
Adv.
Best beam bending model; spring models for adhesion close to realistic
observations
As beam tip displacement increases, separation point moves as desired
Beam characteristics modeled resemble realistic observations of peeling
Disadv.
Introduces new variable; y-position of spring fracture ycritical; arbitrarily chosen
values of k and Ycritical used to match realistic observations
Root-finding step of x* necessary before model can display predictions of beam
characteristics
Model 4:
Adv.
Simple model, easy to conduct
Energv Annroach
No introduction of new variables (i.e., k, Ycriticai)
Model dependent on few variables; solution of separation point depends only on
Young's modulus, beam thickness, beam tip displacement, and work of adhesion
Work of adhesion parameter can be determined experimentally, and is used in field
for characterization of adhesion of different surfaces
Disadv.
Work of adhesion not well known for all types of treated silicon substrate
chemistries, but can be determined experimentally
Table 3.1: Theoretical model comparison.
48
Chapter 4
Experimental Analysis
4.1
Experimental Setup
A simple peel test experiment was conducted in order to obtain data to deny or verify the
theoretical models created. If the peel test did indeed verify the theoretical models, then insight
could be gained into values of the work of adhesion between a treated silicon wafer and PDMS, as
well as the pressure constant k used in the beam bending analysis.
Ideally, a fluorinated silicon wafer would have been used for the peel tests. Unfortunately, a
fluorinated silanization process was unavailable. Instead, a molecular layer of positive PMMA
photoresist was spun onto the silicon wafers used in the experiment. This molecular coating
prevented the PDMS from creating an irreversible bond with the silicon wafer, thereby making it
possible to peel the polymer.
Once the silicon wafer had been treated with a photoresist layer, a thin PDMS layer was molded
onto the silicon wafer and cured at 85*C for one hour. Upon taking the PDMS out of the oven, the
polymer was then cut into strips for peeling, approximately 6 mm wide. The PDMS layer was
approximately 1.8 mm thick, and each PDMS strip was about 9.5 cm long.
To begin the peeling process, a small clip was attached to the PDMS beam tip as a handle to
displace the tip of the PDMS beam. The separation point between the treated silicon wafer and
the PDMS was observed as a function of vertical displacement of the beam tip. Rulers and careful
human-eye observation were used to measure the data. The data was recorded and plotted against
the theoretical beam bending and energy models. The following section explains the results.
4.2
Experimental Results
Figure 4.1 on the following page shows the theoretical curve from the energy approach using
values of Young's modulus E of 0.75 MPa, the thickness h of 1.8 mm, and the work of adhesion y,
of 0.03 N/m. Once this curve was created, the beam bending theory with semi-infinite springs
was analyzed using values of the length L of 9.5 cm and the width b of 6 mm. The values of the
pressure constant k and the critical spring displacement were varied until they matched with the
energy theory. Using this approach, values of the pressure constant k of 1400 N/m 2 and the
critical spring displacement Ycritical of 500 pm gave the best similarity. Once the theoretical
curves had been determined, the peel tests were conducted.
49
Separation Point vs. Beam Tip Displacement (h = 1.8 mm)
14
12
10
-G-
8
U,
6
-Energy Theory
Beam Bending Theory
..
........---
-..
..
........
4
2
S
05
1.5
1
2
2.5
yi (cm)
3.5
3
4
4.5
5
Figure 4.1: Theoretical curves for peel test experiments.
Table 4.1 shows the data from four separate peel tests. The separation point was observed as a
function of the beam tip displacement. In the third peel test, the entire strip peeled at a beam tip
displacement of 2.5 cm, which is why there is no data point for a beam tip displacement of 3.0 cm.
Ytip (cm)
s*
1
(cm)
s* 2 (cm)
s*
3
(cm)
s* 4 (cm)
0.0
0.0
0.0
0.0
0.0
1.0
4.8
5.0
5.2
5.8
1.5
6.0
6.0
6.5
7.4
2.0
7.5
7.5
8.3
7.9
2.5
8.0
8.2
9.5
8.8
3.0
8.5
9.4
-
9.5
Table 4.1: Experimental data for peel test experiments.
Figure 4.2 shows the experimental data from Table 4.1 plotted with the theoretical beam bending
and energy curves from Figure 4.1.
50
Separation Point vs. Beam Tip Displacement (h
=
1.8 mm)
14
12-
10
4
0
-
8
.. ...............................................
.. ...
i(0
6 ---
..-
-Energy
Theory
negyThor
Beam Bending Theory
O Experinment 1
. --.
-.-
4
--
x
Expenment 2
o
Experiment 3
-
Experiment 4
M
...............
...................................... ....
....................'.
..
0
0.5
1
1.5
2.5
2
y
3
3.5
4
4.5
5
(cm)
Figure 4.2: Experimental data plotted with theoretical curves for peel test experiments.
The data points show that the experimental curves follow the same shape as the theoretical curves.
In other words, it appears that the peeling experiments verify that the separation point is a
1/4th-order curve of the beam tip displacement. However, it seems as though the theoretical
models overestimate the experimental data. All the experimental data points fall under the
theoretical curves. There are a few reasons why this is possible. First of all, the theoretical
models are idealizations, and many assumptions were made to create them. Putting the
assumptions aside though, one clear error could be in the value used for the work of adhesion. A
value of the work of adhesion between PDMS and fluorinated silicon was used. However, in the
experiments, fluorinated silicon was not used, rather silicon coated with positive PMIA
photoresist was used. Unfortunately, values of the work of adhesion between PDMS and
photoresist coated silicon could not be found. However, if the work of adhesion is higher in
value, for example, 0.06 N/m, then the experimental data closely resembles the theoretical data as
shown in Figure 4.3 on the following page. In order to match the energy curve in this case, the
2
beam bending theoretical curve used values of the pressure constant k of 2800 N/m and the
critical spring displacement Ycritical of 500 tm.
51
Separation Point vs. Beam Tip Displacement (h = 1.8 mm)
4 A
12
10
8
'
6. .
-
-.
.-
o
63
x
o
A
4
.0.... ........ ................. .. ... .
..
Energy Theory
Beam Bending Theory.
-
Experiment 1
Experiment 2
Experiment 3
Experiment 4
-
..... ...... .........
-
.....
2
0.5
1
1.5
2
2.5
ytlp (cm)
3
3.5
4
4.5
Figure 4.3: Experimental data plotted with more accurate theoretical curves.
One objective of the peel test experiments was to determine the work of adhesion experimentally
between PDMS and a treated silicon wafer. It seems like that in this case, a value of the work of
adhesion of 0.06 N/m is a good estimate for PDMS and positive PMMA photoresist-coated
silicon.
Using the beam bending model conditions in Figure 4.3, graphs were created for the beam
displacement and beam shear force as a function of position along the axis of the beam to describe
the peel tests experiments. Figure 4.4 and Figure 4.5 are the beam displacement diagram and
beam shear force diagram respectively.
Figure 4.5 shows that approximately 2 mN of force is required at the beam tip to begin the peeling
process. As peeling continues, the force at the beam tip needed decreases to approximately 1 mN.
The experimental analysis concluded that the models can be used to predict the behavior of
PDMS during the peeling process, at the very least to an order of magnitude. These experiments
were quite simple. A much more rigorous experiment should be run to determine the reliability of
this experimental data.
52
x 10-3
U3
Displacement Diagram for Beam with Semi-Infinite Springs
7n K
UPI
--
-
ii=5 mm
ytIp2 =10 mm
y
=10mm
yt p3 =15 mm
20 mm........
1P
15 .....-
.
.....
.....
....
.....
..
..
.....
...
.....
....
.....
...
10
~
-2
5
0
.........
0.01
0
..... .....
.i
0.04
0.03
0.02
... ..... ...
0.05
............ ... ................I... .
0.06
0.09
0.08
0.07
...
0.1
x (m)
Figure 4.4: Beam displacement diagram for peel test experiments.
2
Shear Force Diagram for Beam with Semi-Infinite Springs
X 10-3
1.5
-
0.5
-
----------
-.
ytp1 =5 rm
-..
--
0
-
- -
mm
t~ip3=15m
-
-
t2=10
-
-
yt4= 20 mm
-1.5
...
...
.....
...
...
...
...
....
...
..
...
...
...
...
...
...
....
...
...
...
....
:...
...
....
.. ...
...
.....
...
....
...
...
.....
...
...
.....
........
-1.5
-.....-
-..
........
.......
............
.............
-2
-2.5
-3
)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
x (m)
Figure 4.5: Beam shear force diagram for peel test experiments.
53
0.1
54
Chapter 5
Conclusion
The objective of this study was to determine and model some of the significant peeling characteristics of the polymer polydimethylsiloxane (PDMS) from a treated silicon substrate. Theoretical
peeling models of PDMS were created using both a classical beam bending theory, as well as utilizing Griffith theory, or an energy approach to peeling. Finally, a simple peel test experiment was
conducted to correlate the experimental data to the theoretical data. The experimental analysis
concluded that the theoretical models could be used to predict the behavior of PDMS during the
peeling process.
This study helped gain insight into PDMS peeling problems. The main result of this study was
the ability to predict the separation point between a PDMS body and a rigid substrate during the
peeling process. This information can aid in the development of automated technology to peel
PDMS membranes rather than using hands as is done currently. The models allow for a prediction of what will happen if a machine pulls on a PDMS membrane displacing its tip or edge by a
given amount.
The models determined that approximately 2 mN of force is required at a PDMS beam tip adhered
to a positive PMMA photoresist-coated silicon substrate to begin the peeling process for a 1.8 mm
thick PDMS beam. As peeling continues, the force at the beam tip needed to peel an entire 10 cm
long beam decreases to approximately 1 mN.
Regarding recommendations for future work, it is necessary to create more accurate and reliable
theoretical models. First of all, it may be beneficial to conduct an analysis removing the smallangle approximation from beam bending theory.
However, more importantly, in order to create more accurate models, one recommendation is to
create a model to determine the PDMS beam tip displacement trajectory to achieve a minimum
applied force on a PDMS body over a given time interval. This model would help in the development of a machine to conduct peeling.
Overall, this study mainly dealt with beam models. It is desirable to examine plate or membrane
models as well. One recommendation is to conduct a finite element analysis (FEA) of the peeling
process. Since the beam cases are relatively understood, exploring a plate or membrane analysis
using FEA would be very advantageous. Once the peeling process has been investigated using
FEA, obstructions in the peeling process could be added to the simulation. For example, when
peeling a membrane with holes in it, there are silicon or photoresist posts which obstruct the
peeling process. It is curious to know if and how these posts affect the peeling process. Finally,
again using FEA, one can determine the peeling front curve in a plate or membrane analysis,
rather than just the separation point as in the beam models. Finally, determining the membrane
corner or membrane edge displacement trajectory to have a minimum applied force or stress
across the PDMS membrane is crucial to understanding the best way to peel a thin PDMS
membrane.
55
All in all, most of these recommendation ask to examine the PDMS device in time-independent
states, or static states of equilibrium. However, it is also very important to examine the dynamics
of the peeling process. For example, what should the speed of the peeling be? Or what are the
dynamics of slow peeling of thin PDMS membranes from rigid substrates?
Finally, the data in this study was presented in a dimensional manner. However, it would be
advantageous to conduct a dimensional analysis using Buckingham Pi theorem to determine the
relationships of non-dimensional variables in the peeling process.
56
Appendix A
Derivatives off 1 ,f 2 ,f 3 , andf
4
The functions fi (x), f 2(x), f 3(x), andf 4 (x) are:
f1 (x) = eAxcosAx,
f 2 (x) = eAx sinAx,
f 3 (x) = e-AxcosAx,
f 4 (x) = e-AxsinAx.
The first derivatives of the functions are:
f 1 '(x) = AeAxcosAx -AeAxsinAx,
f 2 '(x) = AeAxsinAx + AeAxcosAx,
f 3 '(x) = -Ae-AxcosAx -Ae-AxsinAx,
f 4 '(x) = -Ae-AxsinAx+ Ae-AxcosAx.
57
58
Appendix B
Model 3 Matrix Setup
In Model 3,
y 1 (x) = cieAxcosAx + c2 eAxsinAx + c 3 e-AxcosAx + c 4 e-AxsinAx,
and,
2
6X
y 2 (x) = c 5 x3 +c
+c 7 x+ c 8 .
Let,
f1 (x) = eAxcosAx,
f 2 (x) = eAxsinAx,
f 3 (x) = eAxcosAx,
f 4 (x) = e-Ax sinAx,
f 5 (x) = X,
f6(X) = X2,
f7(x)=X
f8(x)= 1.
The 8 boundary conditions and 4 connection constraints are:
yi(O) = 0,
Y1'(O) = 0,
Mbl
EI
0,
y +
y 2"(x*)
+
-
EI
Y2 (L) = ytip
59
0,
y 2 '(L) = 0,
y 1(x*) -y
0,
2 (x*) =
Y1x*) - y2'(x*) = 0,
yI"(x*) -y
=
2 "(X*)
0,
y 1 I"(x*) - y 2'"(X*) = 0.
In order to solve for c1 , c2 , c3 , c4 , c5 , c6, c7 , c8 , Mb1, V 1, Mb2, and V2, matrix algebra was used,
and the matrices were set up as shown below.
fA(O)
f 2 (0)
f 3(0)
f 4 (O)
0
0
0
fA'(0)
f 2 '(0)
f 3'(0)
f 4 '(0)
0
0
fi"?(X*) f 2"(X*) f 3 I'(X*) f 4"(x*)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1 0
El
0
01
0
0
0
0
fi'"l(x*) f2III(X*) f3'I(X*)f4'"I(x*)
0
0
0
E
0
0
0
C2
C3
C4
f5 "(x*)
f6 I(x*)
f7 tf(x*)
f8"(x*)
0
0 -E
0
0
0
0
El
C5
C6
0
0
0
0
0
0
0
0
0
0
0
0
0
f 5 (L)
f 6 (L)
f 7 (L)
f 8(L)
0
0
0
0
C8
0
0
0
0
f 5 '(L)
f 61 (L)
f 7'(L)
f 8 (L)
0
0
0
0
Mbl
0
0
0
0
0
0
0
0
V1
ytip
0
0
0
0
Mb 2
-0
0
0
0
0
- V2
0
0
0
0
f1 (x*) f 2 (X*)
fi 1(X*)
f3(x*)
f 2 Yx*) f3(X*)
f5"(x*) f 6"'(x*) f 7 "(X*)
f8'"(x*)
f4 (X*) -f5(X*)
-f6 (x*) -f7(x*)
f 41 (X*)
-6
~f/fi*X*)
-f 8(x*)
f7'(X*) -f8'(x*)
f 1I"(x*) f 2 "'(x*) f 3 '"(x*) f 4 '"(x*) -f 5 "'(x*) -f 6 tf(X*) -f 7 l(X*) -f 8'"(x*)
LJ I
0
1
C1
V-/j2
V
ij
j
\:-Ij
4
\:-
/
j
)
V
/
jo
\7
60
/
.'
/ I'
Z'
O
C7
Appendix C
Model 4 Equation Solution
The deflection of the PDMS beam y(x) is the solution of,
4
0, where I =
dy
2
12
dx4
I is the moment of inertia of the beam with respect to the z-axis.
The equation was solved using the following boundary conditions,
y'(O) = y'(s) = 0,
y(O) =Yp,
y(s) = 0 .
Dividing both sides of the equation by EI yields,
4
d y -0
dx 4
Through the following series of integration, the equation has the solution,
3
dd3
y +
C, = 0,
dx
2
dy
dx2+cIx+c
dy
Yx-
2
0,
==0,
x2 C2x+ C3 = 0,
Y~2 xc
3
Y(X) + C'X
6
+ C3x+C4 =
+C2
0,
where c1 , c2 , c 3 , and c4 , are the constants of integration. Since y(O) = y
Yap + C4 = 0 => c 4 = -ytip
61
Since y'(0) = 0,
C3 = 0.
Since y'(s) = 0,
c1 s + c 2s = 0 => c 2 = -CI
.
Since y(s) = 0,
3 +
2
=0 ->cs
-y-2
-Y tP
)2 -C
C
+
0
> -Cj 123 - Ytip
Therefore,
6ytip
C2
-
2sy
-
S2
and,
( 12Yi
y(x) + (
s3
)
3 + 6yti x 2
+ (-yp) = 0.
+s2
-
Simplifying this equation yields,
-1
Y(X) + Yap - 2 ( XS)
=
and solving for y(x), the solution is,
y(x) = YP
2 Q- 3].
+
62
0,
0
-
=c
=
12yi
_
S3_
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63