Surfactant Instabilities on Thin Films
by
Angelica Aessopos
B.S., University of Nottingham (2003)
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Master of Science
MASSACHUSETS INS
OF TECHNOLOGY
JULN
2005
at the
LIBRARIES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2005
© Angelica Aessopos. All rights reserved.
The author hereby grants to Massachusetts Institute of Technology permission to
reproduce and
to distribute copies of this thesis document in whole or in part.
Signature of Author .............................
.....
Department of Mechanical Engineering
May 6, 2005
Certified by .................................
Anette Hosoi
Assistant Professor of Mechanical Engineering
Thesis Supervisor
A ccepted by....................................
BARKER
Lallit Anand
Professor of Mechanical Engineering
Chairman, Committee on Graduate Students
E
Surfactant Instabilities on Thin Films
by
Angelica Aessopos
Submitted to the Department of Mechanical Engineering
on May 6, 2005, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
The deposition of a surfactant drop over a thin liquid film may be accompanied by a fingering
instability. In this work, we present experimental results which identify the critical parameters
that govern the shape and extend of the fingering phenomena. It was found that the normalized
wavelength, A/t, scales with Marangoni number, Ma=Ayt/paD, to the -1 exponent for any
Marangoni higher than 4.3 - 107 . On the other end , for Marangoni < 4.3. 107 the normalized
wavelength scales with Ma to the -0.4 but becomes in addition a funcion of the Prandtl number,
Pr=v/D, which demonstrates the critical significance of bulk diffusion on the spreding behavior.
Finally, we present a numerical implementation of a mathematical model which is capable of
reproducing the experimentally observed trends.
Thesis Supervisor: Anette Hosoi
Title: Assistant Professor of Mechanical Engineering
2
Contents
1
INTRODUCTION
10
1.1
Problem Statement and Industrial Context
. . . . . . . . . . . . . . . . . . . .
10
1.2
Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.3
Research objectives and Thesis outline . . . . . . . . . . . . . . . . . . . . . . .
13
2 EXPERIMENTS
15
2.1
Experimental literature review
15
2.2
Experimental details .........
18
2.3
2.4
2.5
18
..........
2.2.1
Materials
2.2.2
Visualization Technique and Experimental procedure
21
24
Experimental Results and Analysis
2.3.1
Flow visualization pictures
24
2.3.2
Image Analysis . . . . . . .
24
Results and Discussion . . . . . . .
33
2.4.1
Results at steady state .
33
2.4.2
Source of instability
.
42
2.4.3
Transient region
. . . . . .
52
Solubility Effects . . . . . . . . . .
55
62
3 SIMULATIONS
3.1
Introduction . . . . . . . . . . . . .
62
3.2
Model Formulation . . . . . . . . .
64
3.2.1
Governing equations and scaling
3
. . . . . . . . . . . . . . . . . . . . . . . 65
3.2.2
3.3
Dimensionless momentum equations and boundary conditions:
. . . . . . 69
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 CONCLUSION AND FUTURE WORK
79
A Laplace Equation
81
B Wavelength versus thickness plots
83
C Dimensionless momentum equations.
85
D Boundary condition at the interface
87
E Bibliography
89
4
List of Figures
1-1
Fingering patterns at t=0.31 sec generated after the deposition of a 9 pl droplet
of 1.2 CMC SDS solution on a thin water film of approximate initial thickness 25
/um [taken (Ref. 26) with author's permition]. G shows the corresponding side
view .
2-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
a. Fingering patterns produced when a drop of 0.4 cmc (3.2 mM) SDS is deposited on a 25 pm water film (taken from Afsar-Siddiquid, Luckham, and Matar
[Ref. 261. Used with permition from the author). b. Line of food die spreading
on a thin honey film of approximate thickness 0.5 mm. . . . . . . . . . . . . . . . 16
2-2
Dynamic viscosity vs. the concentration of PEO in water. The solid line indincates a power law fit to the data. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-3
19
a: Photo of a drop (5% of PEO 300000 gr/mol in water) captured with a high
resolution camera, b: contour of drop obtained from the analysis of the photo
using ImageJ software
2-4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
a: Profiles obtained from numerical solution of Laplace equation, b: relation
between the numerical curve giving the best fit and the ratio of the drop's radius
over the capillary length. These curves are created in IgorPro by Mark Fermigier. 22
2-5
Schematic diagram of the experimental setup. . . . . . . . . . . . . . . . . . . . . 23
2-6
Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 5 percent PEO (by mass).
2-7
. . . . . . . 25
Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 6 percent PEO (by mass).
5
. . . . . . . 26
2-8
Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 7 percent PEO (in mass).
2-9
. . . . . . . 27
Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 8 percent PEO (by mass). . . . . . . . 28
2-10 Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 9 percent PEO (by mass). . . . . . . . 29
2-11 Time sequence of surfactant drolet spreading on viscous films of different thicknesses. The film is an aqueous solution of 10 percent PEO (by mass).
. . . . . . 30
2-12 Image analysis proceedure using matlab image processing toolbox. The axes in
the image desplayed by matlab (a) are given in pixels. . . . . . . . . . . . . . . . 31
2-13 Image analysis to convert pixels in mm.
. . . . . . . . . . . . . . . . . . . . . . . 33
2-14 The analysis was repeated for many radii between rmin and rma,
intervals.
in 10 pixels
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2-15 Plots of wavalength and frequency of the fingers for different radii of circle plotted. Each plot is for a different viscosity of the sublayer and shows data for all
the thicknesses of sublayer.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2-16 Plots of wavalength and frequency of fingers for different radii of circle plotted.
Each plot is for a different viscosity of the sublayer and shows data for all the
thicknesses of sublayer.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2-17 Plots showing how the length of the fingers (plot a) and the width of the fingers
(plot b) changes with the thickness of the sublayer. All the viscosities are ploted.
38
2-18 Fingering patterns produced after 0.5 s when a 9-IL drop of 1.2 cmc SDS is
deposited on (a) a 25pm water film and (b) a 100 /Lm water film (taken from
Afsar-Siddiqui et al [26]. Used with permition from the author.)
. . . . . . . . . 39
2-19 a) The average fingers wavelength against the thickness. b) The total area covered by the fingers against the thickness. . . . . . . . . . . . . . . . . . . . . . . . 40
2-20 3-dimensional view of how the area of the fingers changes with the viscosity and
the thickness of the sublayer. The data comes from graph 2-19. Three different
views of the plot are shown.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2-21 The total area covered by the core of the flower (up to rmin) against the thickness. 42
6
2-22 Wavelength of the instability measured at rmin vs.
time.
Data is given for
surfactant spreading on two different sublayers. . . . . . . . . . . . . . . . . . . . 45
2-23 Plot of wavelength at rmin against thickness for different viscosities.
. . . . . . . 46
2-24 a. Plot of A/t versus Prandtl for fixed Marangoni numbers. b. Plot of A/t versus
Marangoni for fixed Prandtl numbers.
2-25 Plot of log(A/t) versus log(Ma).
. . . . . . . . . . . . . . . . . . . . . . . . 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2-26 Two views of a 3-d plot of A/t against Ma and Pr. . . . . . . . . . . . . . . . . . 49
2-27 Experimenta data of Matar et al. [25] (taken with author's permition). . . . . . . 50
2-28 A/t vs Ma. This graph contains our experimental results and existing data [25].
.
51
2-29 Surfactant drop spreading on a 3mm sublayer of aqueous solution of 5% PEO.
The intensity of the line is drawn in the middle of each flower is plotted.
. . .
53
2-30 Superposition of all the intensity profiles of the images till 1 minute after the
deposition of the drop, in 5 seconds intervals
. . . . . . . . . . . . . . . . . . . . 54
2-31 Radius of the spreading drop vs time. Drop spreads on two different sublayers:
(1) 5% PEO (in water), 0.5 mm thickness and (2) 8% PEO (in water), 7 mm
thickness. The spreading behavior highly agrees with the t 1 / 4 scaling suggested
by Grotberg and co-workers [16, 13].
. . . . . . . . . . . . . . . . . . . . . . . . 54
2-32 Plot of Viscosity against shear rate for the visco-elastic aqueous solution of 1%
Poly(ethylene oxide) of molecular mass 8 000 000 g/mol . . . . . . . . . . . . . . 56
2-33 Drop of 100% pure surfactant spreading on a newtonian (a) and a viscoelastic
(b) thin film. Lucopotedes are deposited on the thin film's surface, in order to
make its deformation more clear. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2-34 Drop of food die spreading on a thin viscoelastic sublayer. . . . . . . . . . . . . . 58
3-1
Schematic of the flow geometry, representing a drop laden with soluble surfactant
deposited on an uncontaminated thin liquid film, resting on a horizonal rigid
support (taken from [10]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3-2
Schematic representation of the surfactant drop spreading on the thin liquid film.
3-3
Approximation of the velocity profile.
3-4
Approximation f6r the concentration profile. . . . . . . . . . . . . . . . . . . . . . 72
65
. . . . . . . . . . . . . . . . . . . . . . . . 71
7
3-5
Time sequence of surfactant droplet spreading on a 2mm film. The film is an
aqueous solution of 8 percent PEO (in mass) with IL = 13 Pas. The colorbar
shows the surfactant concentration.
3-6
. . . . . . . . . . . . . . . . . . . . . . . . . 75
Time sequence of surfactant droplet spreading on a 2mm film. The film is an
aqueous solution of 8 percent PEO (in mass) with y = 13 Pas. The colorbar
shows the surfactant concentration.
3-7
. . . . . . . . . . . . . . . . . . . . . . . . . 76
Time sequence of surfactant droplet spreading on a 2mm film with viscosity
p = 0.2 Pas. The colorbar shows the surfactant concentration.
3-8
. . . . . . . . . . 77
Surfactant drop spreading on thin film. The picture is captured 20 min after the
drop is deposited. The film is an aqueous solution of 8 percent PEO (in mass)
with p = 13 Pas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3-9
Profile 0.01 second after the drop is deposited on a 2mm film of aqueous solution
with 8% PEO. The spreading is characterized by the propagation of a thickened
front at the leading edge together with an accompanying trailing thinned region.
78
B-1 Wavelength versus thickness plots. a: the wavelength is measurd at a radius
(rmax+rmin)/2, b. the wavelength is measured at rmax, c.
wavelength of the wavelengths measured at all the radii.
8
this is an average
. . . . . . . . . . . . . 84
List of Tables
2.1
Physical and viscous properties of the thin film fluid: distilled water mixed with
different concentrations in mass of PEO 300000 gr/mol.
. . . . . . . . . . . . . . 19
2.2
Physical properties of ethylene glycol.
2.3
Variables describing the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4
Physical and viscous properties of viscoelastic and newtonian PEO . . . . . . . . 55
2.5
Spreading behavior of different surfactants deposited on Newtonian and Non-
. . . . . . . . . . . . . . . . . . . . . . . . 19
Newtonian sublayers. All the surfactants are soluble in both sublayers. . . . . . . 56
2.6
Physical properties of liquids used in experiments . . . . . . . . . . . . . . . . . . 59
2.7
Spreading Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
9
Chapter 1
INTRODUCTION
1.1
Problem Statement and Industrial Context
The spreading of a surfactant droplet over a thin liquid film can lead to rapid and spontaneous
spreading. The physical explanation of such a phenomenon has been attributed to Marangoni
stresses'. These stresses tend to pull liquid and surfactant towards regions of higher surface
tension and thus a Marangoni flow is created. Experimental evidences over the last 15 years
report the presence of a hydrodynamic instability which produces inhomogeneous surface coverage. Figure (1-1) shows an example of such an experiment: Upon deposition of the drop on
the substrate, it appears that liquid in the thin film is swept away from the vicinity of the drop
causing thinning of the film near the drop edge while a thickened front travels away from the
drop. Behind this front, fingers develop (see e.g. [3, 13]). This spreading is neither a uniform
nor a stable process.
This previously described mode of instability has received considerable attention in the literature as the spreading of a surfactant on thin films is of immense industrial importance. Its
application ranges from industrial and biological to household processes: these include coating
flow technology, detergency, ink jet printing, film drainage in emulsions and foams, and drying
of semi-conductor wafers in microelectronics [9,6]. Surfactant is also naturally produced inside
the lungs of mammalian systems and plays a vital role in maintaining the lungs compliance by
1Marangoni stresses are shear stresses generated at free liquid surfaces due to gradients in surface tension.
10
I thin region
I
thickened front
Figure 1-1: Fingering patterns at t=0.31 sec generated after the deposition of a 9 /11 droplet of
1.2 CMC SDS solution on a thin water film of approximate initial thickness 25 Am [taken (Ref.
26) with author's permition]. G shows the corresponding side view.
reducing the surface tension of the liquid film, which coats the interior of pulmonary airways.
Insufficient supply of lung surfactant often occurs in premature infants and can give rise to serious respiratory difficulties, often with fatal consequences. An effective technique for treatment
is through inhalation of surfactant (see e.g. [13]).
Many recent medications and antibiotics
are delivered in this manner. It is readily understood that the effectiveness of all the above
processes is significantly improved when the spreading of the surfactant is rapid, uniform, stable
and continuous, and when the liquid layer is not ruptured until complete coverage is obtained.
There is a growing body of experimental and numerical investigations of this phenomenon,
yet an in-depth understanding is still out of reach. Additional study of the spreading process
of a surfactant on a thin, viscous layer of fluid is therefore of extreme importance.
1.2
Literature review
The spreading process of a surfactant droplet has been extensively discussed over the past fifteen years in both the theoretical and experimental literature. Experimental and theoretical
investigations on the spreading of an insoluble surfactant, conclude that the spreading induces
large deformations in the substrate [13-18].
A lubrication model has been developed to de-
11
scribe the spreading of a insoluble surfactant monolayer driven by Marangoni stresses, capillary
effects arising from the surface deformation, gravity, surface and bulk diffusion of surfactant,
intermolecular forces and non-Newtonian effects in the absence of perturbations [5, 7, 13-18].
These studies show that the spreading produces significant film thinning near the surfactant
deposition, while a thickened rim, approximately twice the height of the uncontaminated film,
propagates away from the drop.
This is indicated in Figure (1-1).
The thinning occurs to
balance the surface stress caused by the large surfactant concentration gradient in this region
[see e.g.
13].
It was observed experimentally that if the initial gradients in surface tension
are significantly large, the deformation of the sublayer might be severe enough to make van
der Waals forces operative and cause rupture in the thinnest part of the film, leaving a dry
ring in finite time [17].
In the presence of significant solubility effects these features become
accentuated [14]. Also, a thin substrate and slow diffusion induce larger deformations in the
film (see e.g. 10, 15, 16)
In addition to these deformations observed in the film, experimental studies have demonstrated the existence of another instability, that manifests itself via the formation of surfactant
coated fingers that appear at the backside of the surfactant leading edge. [19-28]. These fingering patterns resemble in shape to those arising in viscous fingering [20]. Typical fingers developed during surfactant spreading are shown in figure 1-1. These studies have shown that fingers
are produced almost instantaneously and undergo branching, coalescence and tip-splitting as
they spread. The onset time and radius of the fingering phenomenon as well as the width of
the fingers decrease with increasing initial film thickness and surfactant concentration.
The fingering depends on the initial surfactant concentration and the thickness of the initial
sublayer, which strongly indicates that Marangoni stresses are in some way responsible for this
spreading behavior [20, 3, 29, 31]. This suggestion is further verified by the fact that the fingers
never appear on perfectly dry substrates and that the spreading velocities are very high [20,
21,31].
Several studies have tried to explain the physical mechanism responsible for this instability.
These studies primarily model an insoluble surfactant coated drop spreading on an initially
uniform thickness. Troian et al. first proposed a lubrication model, which exploited certain
mathematical similarities between the present instability and viscous fingering and showed
12
that the fingering instability is driven by the Marangoni effect [30]. In a series of numerical
studies, Matar and Troian [1-4] considered the evolution of imposed transverse disturbances
superimposed on a base state composed of a deposited surfactant monolayer upon uniform
thickness film. Their results suggest that in the absence of van der Waals forces, growth can
not be sustained and the spreading process is asymptotically stable. This was confirmed by
Fisher and Troian despite the use of an alternative method of perturbation growth [11]. This
is clearly not consistent with the experimental observations.
However, recent experimental studies, present results that the fingering instability persists in
regimes where significant van der Waals forces cannot be the mechanism. In particular, fingers
were observed in an experiment even though the surfactant perfectly wetted the underlying
substrate [23, 24, 22]. Moreover, experimental studies related the average finger wavelength to
the thickness of the sublayer to the power 2/3, a scaling which is consistent with a Marangonidriven rather than a van der Waals-driven fingering instability [25, 26].
Warner et al. examined the linear and nonlinear stability of a thick surfactant deposition
spreading over a film of much smaller thickness and found disturbance growth despite the
absence of intermolecular forces. This growth is amplified by increasing the initial thickness
ratio of deposition to the thin film and decreasing the magnitude of capillarity and surface
diffusion [9].
Recent experimental studies involving sparingly soluble and highly soluble surfactants,
showed that the surfactant solubility also affects the stability of the spreading and the shape of
the fingers [25,26]. It was shown that the fingers are more pronounced and more accentuated
for a highly soluble surfactant. These effects have been recently confirmed for the first time,
numerically by Warner et al [10, 9].
1.3
Research objectives and Thesis outline
The precise mechanism for the fingering instability still remains unclear. The main objective of
this work is to investigate experimentally and numerically the fingering instability and identify
the crucial parameters responsible for the fingers. The relative strength of the forces participating in the spreading process is to be assessed in order to explain the observed behavior.
13
This thesis is divided in two major parts. The first part deals with the experimental study
of the phenomenon. First a brief literature review on previous experimental studies on the phenomenon is presented (Chapter 2.1.) Chapter 2.2 describes the apparatus and materials used.
Chapter 2.3, presents the results obtained from the flow visualization technique. The onset of
the instability and the steady state are discussed in Chapter 2.4. Experiments were repeated
for a set of liquids of different physical properties and the crucial parameters responsible for
the instability have been identified in Chapter 2.5.
The second part of this thesis Chapter 3, focuses on the numerical work done. Chapter
3.1, presents a brief literature review. The model formulation is described in Chapter 3.2. This
analysis is currently restricted to two-dimensional analysis. Some results are shown in Chapter
3.3. Finally, a brief conclusion and future work are presented in Chapter 4.
14
Chapter 2
EXPERIMENTS
This chapter is devoted into the experimental study of the fingering instability. Flow, was visualized using dye and a digital camera and the resulting images were analyzed by computational
methods.
The effect of varying the viscosity and film thickness of the thin sublayer on the
spreading exponent and wavelength of the fingers is examined using a soluble surfactant on
aqueous films. A new scaling law that describes the wavelength of the instability is determined.
Experiments were repeated for a set of liquids of different physical properties and the crucial
parameters responsible for the instability have been identified.
2.1
Experimental literature review
Marmur and Lelah, were the first to report fingering patterns during the spreading of various
aqueous surfactant solutions on what they believed to be dry glass [19]. Spreading surfactant
droplets at concentrations above the critical micelle concentration (cmc) 1 were accompanied
by "fingers" of surfactant originating near the point of original deposition, which appeared to
branch as they developed. Since then, the phenomenon has been observed by several independent groups [20- 27]. In these experiments it was observed that upon deposition of the drop,
liquid is swept away from the sublayer, causing a thinning of the film near the drop edge while
'Surfactants in solution tend to form aggregates of colloidal dimensions which exist in equilibrium with the
molecules or ions from which they are formed. Such aggregates are termed micelles. There is a critical value of
surfactant concentration (cmc) below which no micelles are detected and above which, virtually all surfactant
molecules form micelles. Many properties of surfactant solutions, if plotted against the concentration, appear to
change at a different rate above and below the cme.
15
thickened corona
travels away from
the drop
thinning of the film
Figure 2-1: a. Fingering patterns produced when a drop of 0.4 cmc (3.2 mM) SDS is deposited
on a 25 pm water film (taken from Afsar-Siddiquid, Luckham, and Matar [Ref. 26]. Used with
permition from the author). b. Line of food die spreading on a thin honey film of approximate
thickness 0.5 mm.
a thickened corona travels away from the drop (see Figure (2-1)). The drop rapidly spreads
into the thinned region and begins propagating fingers behind the spreading front. The fingers
are produced nearly instantaneously and undergo branching, coalescence and tip-splitting as
they spread. Typical patterns which develop during the spreading of surfactant on a thin water
film are shown in Figure 2-1. Recent experiments, show a very detailed view of the fingers (see
[28]).
Troian et al. conducted experiments using aqueous AOT solutions (sparingly soluble surfactant) on a water film in a closed environment to control evaporation effects [20]. They observed
fingering both above and below the cmc.
The experiments were performed in an open-cell
geometry with no external pressure gradient forcing the movement of the interface. These features rule out the Saffman-Taylor instability. The shape of the fingers depends on the initial
surfactant concentration and the thickness of the initial sublayer, which strongly indicates that
Marangoni stresses are in some way responsible for this spreading behavior [20, 29]. This suggestion is further verified by the fact that the fingers never appear on perfectly dry substrates
and that the spreading velocities are very high.
16
Zhu et al, studied the spreading of aqueous mixtures of surfactants on a hydrophobic surface and found that the presence of water vapor is necessary for superspreading and fingering
instability to occur. They speculate that the water vapor provides a thin high tension film at
the leading edge of the spreading drop, and so spreading is driven by the Marangoni effect. It
was found that the radius of the spreading drop varies as the square root of time during the
initial spreading. This time dependence is consistent with, but not unique to Marangoni flow
[31].
Frank and Garoff reported seeing fingers when spreading ionic and nonionic surfactant
solutions on microscopically thin film in a vertical geometry [21]. Their experiments were carried
at ambient humidity. They confirmed that the presence of both surface tension gradients and
thin water films ahead of the advancing surfactant front are prerequisites for the instability to
occur.
Grotberg and co-workers performed experiments using an oleic acid monolayer on glycerol
films. They observed film thinning in the smaller film thicknesses whereas reverse flow was
observed in the thicker films where gravitational effects become significant.
radius was found to advance in time as t 1/ 4 [17].
The spreading
These results agreed with their theoretical
results [16,13].
Cazabat and co-worker studied the nonionic CnEm surfactants spreading in ethylene and
diethylene glycol over a range of relative humidity [22-24]. They present experimental results
proving that fingering instability exists at regimes where significant van der Waals forces cannot
be the mechanism.
In particular, fingers were observed in an experiment even though the
surfactant perfectly wetted the underlying substrate.
Recent experiment by Afsar-Siddiqui et al. involving sparingly soluble (AOT) and highly
soluble surfactant (SDS), showed that increasing the surfactant solubility results in more pronounced and more accentuated fingers with shorter onset radii and times [25, 26]. This was
attributed to the greater degree of thinning achieved during the spreading of the soluble surfactant.
They showed that the average finger wavelength scales as the initial film thickness
to the 2/3 power. This scaling is consistent with a Marangoni-driven rather than a van der
Waals-driven fingering instability, which would have given a thickness to the power 2 scaling.
Afsar-Siddiqui et al used surfactant concentrations below and above the cmc and investigated
17
the effect of varying surfactant concentration, initial film thickness and the effect of solubility
on the behavior of the spreading.
Clean surfactant deposited on a thin liquid support can also produce fingers.
He and
Ketterson spread a monolayer of insoluble ring shaped surfactant on a water film and observed
narrow branching fingers [27].
2.2
2.2.1
Experimental details.
Materials
The liquid substrates in our experiments are aqueous solutions of Poly(ethylene oxide) ,PEO,
of molecular mass of 300000 gr./mol.
Six liquids were created, each with a different mass
concentration of PEO in water. The concentration varied from 5% to 10%. The chains of this
PEO polymer are relatively short and do not create a viscoelastic network. The surface tension
of the liquids is almost independent of the nature and the concentration of the polymers; they all
lie around 60 mN/m. Varying the concentration of the solution changes only the viscosity of the
liquid significantly (see 2.1). The liquids' viscosities were measured with a rheometer situated
in the Hatsopoulos Microfluids Laboratory at a temperature of 20"C. All the viscosities are
constant for shear rates between 0.1 and 1000 s-1, hence for the time and length scales relevant
to our experiments, the mixture is Newtonian. Figure 2-2 shows the behavior of the dynamic
viscosity for different concentrations of PEO. The creation of the mixture is time consuming
because the polymers' chains are relatively large.
A day of continuous mixing is necessary
to create a completely homogeneous liquid. Non- homogeneous liquids may show completely
different responses to the spreading drop.
The surfactant drop is pure ethylene glycol colored with some blue methylene in order to
visualize the spreading. Methylene blue, is a very fine powder, that mixes will with ethylenalynicol. The physical properties of this solution are presented in table 2.2.
18
Molecular Mass (g/mol)
3 00 000
Newtonian
Percentage in Mass
5%
6%
7%
Density (g cm- 3 )
8%
9%
10%
13
8
15.9
6
0.98
Viscosity (Pa s)
Diffusion Coefficient .10-1 4 (m2 /s)
0.85
100
1.8
50
4.5
20
Surface Tension (mN/m)
6.8
10
60
Table 2.1: Physical and viscous properties of the thin film fluid: distilled water mixed with
different concentrations in mass of PEO 300000 gr/mol.
20
16
= 0.0009x4.2+
y12
F 2 = 0.9789
o
8-
0
0
2
6
4
concentration
10
8
Figure 2-2: Dynamic viscosity vs. the concentration of PEO in water. The solid line indincates
a power law fit to the data.
Surface tension (mN/m)
Density (gcrn-3 )
46
1.1
Viscosity at 250C (mPa s)
16.1
Table 2.2: Physical properties of ethylene glycol.
19
Diffusion coefficient measurements
The diffusion coefficient of the surfactant is defined as:
D=
KT
67rpr
1
==> D oc Ap
where K is the Boltzman constant, T is the absolute temperature, p is the viscosity and r is
the radius of the surfactant molecule. The coefficient of diffusion of small molecules in water is
on the order of D = 10-9 m 2 /s. However, the coefficient is inversely proportional to the liquid
viscosity, hence it is not certain what value is the relevant value for this problem. It could be
the diffusion of the surfactant into the viscous sublayer or the diffusion of the viscous sublayer
into the upper layer or most likely some combination of the two. Such complexities are ignored
in this analysis, and the problem is simplified to the diffusion of small surfactant molecules into
an aqueous solution (viscous sublayer). Thus, the diffusion coefficients are calculated as follows:
Dwater Awater
D
D
psublayer
~m
10-12
2/
Isublayer
where the viscosity of water is Awater
=
lcP = 10-3 Pa s. The diffusion coefficients are
calculated for the different viscosities of the sublayer are showed in Table 2.1.
Surface Tension Measurements
The surface tensions of the liquids were measured by the pendant drop method. The basic
premise of this method is that surface tension can be calculated from the shape of a drop as it
forms at the end of a capillary tip of known external radius. The opposing forces of gravity and
surface tension determine a droplet's shape. Thus, one can work backwards from a droplet's
shape and the known force of gravity to find its surface tension. This can be done by taking
pictures of drop shapes and fitting the Laplace equation of capillarity to their contours (this
equation is a second order non-linear equation and does not have an analytic solution, see
appendix A).
The shape of the drop was captured with a high resolution camera and the pictures were
20
b
Figure 2-3: a: Photo of a drop (5% of PEO 300000 gr/mol in water) captured with a high
resolution camera, b: contour of drop obtained from the analysis of the photo using ImageJ
software
analyzed with ImageJ software, in order to obtain the contour of the drop and save its x- and ycoordinates (see Fig 2-3). From this profile, the surface tension was obtained, using an IgorPro
routine created by Mark Fermigier (ESPCI, Paris). The drop's profile, was plotted in IgorPro
and was normalized so that the x- and y- radii of the drop equal 1. The apex of the drop was
placed at the origin of the coordinate system. This profile was superposed on top of profiles
obtained from the numerical solution of Laplace's equation (see Fig. 2-4 a). Then, by selecting
the best numerical profile, we find a value for the ratio (see Figure 2-4b):
drop radius
capillary length
R
L
Finally, the surface tension of the liquid can be obtained from the capillary length as follows:
L =
2.2.2
-pg
y=
L2pg
Visualization Technique and Experimental procedure
A specific volume of the sublayer fluid is deposited with a syringe on a glass petri dish of known
radius (two dishes were used: 5cm and 9 cm diameter). The volume deposited was determined
21
3.0-
2.5-
2.0-
1.5-
(R/L)
k
8
1.0-
rlll
7
6
0.5-
5
4
0.0 "
0.2
0.4
0.6
0.8
0
1.0
10
20
30
Curve number
giving best fit
a
b
Figure 2-4: a: Profiles obtained from numerical solution of Laplace equation, b: relation between
the numerical curve giving the best fit and the ratio of the drop's radius over the capillary length.
These curves are created in IgorPro by Mark Fermigier.
22
camera
syringe
Petri dish
syringe pump
Light box
Figure 2-5: Schematic diagram of the experimental setup.
from the thickness of the sublayer required. The glass plate was placed on an illuminated table
for clearer observation. A small ruler was placed on the table for scaling purposes. A bubble
level was used to check that the table was completely horizontal. Experiments show that even
a small slope of the dish, can dramatically affect the results, especially when low viscosities and
higher thicknesses are used. A 0.018 ml drop of colored surfactant is delicately deposited on
the still surface of a more viscous liquid with a 20 [L precision Hamilton syringe. In order to
minimize perturbations during the drop delivery, a syringe pump is used to deposit the drop.
The experiments are carried out in open environment with no external pressure gradient forcing
the movement. A high resolution camera (Canon EOS lOD Digital) is fixed from above to record
the images at a rate of 1 frame per 5 seconds. A schematic diagram of the experimental setup is
shown in Figure 2-5. The spreading was followed for approximately 1 minute after deposition.
Each spreading run was repeated 4-5 times to ensure reproducibility. After each experiment,
the petri dish was cleaned with distilled water.
The humidity of the environment was not controlled. When the sublayer was uniformly
deposited on the petri dish and it was ensured that its surface is fiat, the experiment was
performed immediately, so that the surface activity of the sublayer does not decrease. In fact,
if the sublayer rests in the dish more than 30 seconds, a water film accumulates at the top of
the film, which enormously alters the surface tension of the liquid and hence the behavior of
23
the surfactant drop. Also, all the solutions were used within 48 hours after being prepared to
avoid a decrease in surface activity [25]. The sublayer solutions were kept in sealed containers
and their exposure to air was minimal.
2.3
2.3.1
Experimental Results and Analysis
Flow visualization pictures
The objective of this experiment is to study what is the effect of viscosity and thickness changes
of the sublayer on the fingering instability. The materials and experimental method presented
in section 2.2.1 and section 2.2.2 are used. Once the drop has been deposited on the thin film
the imaging collection was initiated. Images were taken in 5 seconds intervals for 1 minute.
The system reaches a steady state after the first 20 seconds, that is why only the first 20 second
are presented here.
Figures 2-6 to 2-11, present the time sequence of results obtained from the flow visualization
experiments for a number of viscosities and thicknesses of the sublayers.
2.3.2
Image Analysis
From the results obtained from the flow visualization technique (Figures 2-6 to 2-11) we obtain
qualitative data of the fingering instability. In an attempt to get more quantitative information,
such as finger width, length and wavelength, the results were analyzed using the matlab image
processing toolbox. A typical example of the procedure of the image analysis is presented in
Figure 2-12.
The results obtained are in the form of digital images. These colored images were read
into Matlab and were converted into intensity images. An intensity image is equivalent to a
"gray scale" image.
It represents each image as a matrix where every element has a value
corresponding to how bright/dark the pixel at the corresponding position should be colored.
There are two ways to represent the number that represents the brightness of the pixel: The
double class or uint8 class. The double class assigns a floating number between 0 and 1 to each
pixel. The value 0 corresponds to black and the value 1 corresponds to white. Since most of
the mathematical functions can only be applied to the double class, all images were converted
24
2
2
0
24
C.'-
0
0
2A
0
2e
0
Figure 2-6: Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 5 percent PEO (by mass).
25
Ilk
2
i
0
P0
li
0
2i
0
r1
o-4)
0
C.)
Figure 2-7: Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 6 percent PEO (by mass).
26
2
2Nk
0
0
C2
a)
E~
Figure 2-8: Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 7 percent PEG (in mass).
27
dab22aab
2
P
E
4EW
0
0--
o ~
o= Q
e'45g
Figure 2-9: Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 8 percent PEO (by mass).
28
II
1
l
2
220
0
22Q
4--
M
o ~
NZ
Figure 2-10: Time sequence of surfactant droplet spreading on viscous films of different thicknesses. The film is an aqueous solution of 9 percent PEO (by mass).
29
Sublayer: aqueous solution of 10% PEO
10%
1mm
3mm
5mm
6mm
7mm
5
sec
10
sec
20
sec
Figure 2-11: Time sequence of surfactant drolet spreading on viscous films of different thicknesses. The film is an aqueous solution of 10 percent PEO (by mass).
30
50-
U
IF,
Intensity along a circle of radius 130
a
I
0.85
100-
p
0.8
150-
0.75
200-
~0.7
250-
0.6
300-
0.55
~
b
0.5k
350-
0.450
a
40050
100
150
2010
250
300
0.25D.5
1
1.5
2
2.5
3
3.5
4
4.5
5
6
5.5
7
6.5
Anle for the center
350
Frequienc
o ,otr
radus 130
0.7
0
0
.5-
Aaverage
f
0.30.2
0
C
0
10
^5
L-
15
25
20
freguency Mliked)
30
35
40
Figure 2-12: Image analysis proceedure using matlab image processing toolbox. The axes in
the image desplayed by matlab (a) are given in pixels.
31
into intensity-double class images.
A circle of radius r was plotted on the image (see Fig.2-12 a). The radius r was given in
pixels. The axes in the image are also given in pixels. The center of the circle was located at
the center of the flower. For each image, the x and y coordinates of the center of the flower
(x and yc) were found by trial and error and a circle with a circumference of 2000 pixels was
plotted.
The perimeter of the circle drawn in Figure 2-12 a, intersects the fingers.
In order to
calculate the fingers' wavelength at this particular radius, the intensity of the circle's pixels was
found and plotted against the angle 0 [rad] (see Fig.2-12 b). This graph gives an estimate of
the fingers' wavelength from A = dO - r, where A is the wavelength of the instability.
A specific value for the average fingers' frequency [1/rad] was found by applying a Fast
Fourier Transform (FFT) to the intensities of the pixels (see Fig.2-12 c). FFT is used to find
the frequency components of a signal buried in a noisy angle domain signal. Using the FFT
plot, the ten most prominent fingers frequencies and respective wavelengths and dO were listed.
The wavelength [pixels], frequency [1/rad] and angle increment [rad] of the fingers are related
by: A = L and A = dO - r. The frequency and respective wavelength that best represented the
intensity plot was chosen. Most of the times, this frequency was not one of the 3 most important
detected from the FFT. The FFT often recognized some very small or very big frequencies as
the dominating ones. When choosing the correct frequency attention was given to the fact that
the corresponding wavelength (in pixels) fits approximately to any wavelength in Figure 2-12 a
and the corresponding dO fits to any dO in Figure 2-12 b.
All the length values in the analysis were given in pixels. A similar procedure was repeated
in order to convert the pixels in mm (see Fig. 2-13).
All the images included a ruler that
was placed on the illuminated table. A straight line was drawn on the ruler its intensity was
plotted. The FFT of the intensity vector, gave the maximum frequency [1/pixel] of the signal.
The average wavelength [pixels] was then found by:
Aaverage =
1
max
Therefore 1mm=Aaverage pixels.
32
I -I a
19
Reaon between pixels
FFlTo1 scading
m
0.35
1 mm=X pixels
0.6
0.55
nm
0.3-
0.25-
0.5
0.45
0.2-
A
0.5
0.15-
0.3
0.21
0.
.4
.
0.25 -b
0.100
100
10
1
0
250
0
0.05
0.1
0.15
0.2
02
0'3
0.3
A4
0.45
0.5
Figure 2-13: Image analysis to convert pixels in mm.
2.4
Results and Discussion
In this section we will present the results on the fingering instability obtained from the analysis
of the image data. The focus of this section is twofold: firstly we will analyze the fingering
instability at steady states and secondly we will study data obtained at early times and investigate the onset of the instability. This enables us to avoid dynamic effects contained in data
obtained in the transient region (see [23]).
2.4.1
Results at steady state
Experimental data is analyzed in order to quantify the shape of the fingers at steady state. We
will focus on the dependence of the area of the fingers and the wavelength of the instability on
the thickness and viscosity of the sublayer.
The images analyzed are captured 20 seconds after the deposition of the drop (see Tables
2-6-2-11). This time was the most representative of a steady state, because at later times the
drop stopped spreading, and was only slowly diffusing in the sublayer. As mentioned previously,
the data was analyzed by a Matlab program. The radii of the circles drawn on the spreading
drop, varied from rmin (radius to the average base of the fingers) to rmax (radius to the average
tip of the fingers) in 10 pixels intervals (see Fig. 2-14). rmin is the radius to the average base
33
50-
100-
150-
200-
250-
300-
400
50
100
150
200
250
300
350
Figure 2-14: The analysis was repeated for many radii between rmin and rmax, in 10 pixels
intervals.
of the fingers and rmax is the radius to the average tip of the fingers.
Plots 2-15 and 2-16 show the wavelength and frequency of the fingers for different radii of
circles drawn. Each plot corresponds to a different viscosity of sublayer and contains values
for all the thicknesses of sublayer. These plots clearly show that the wavelength of the fingers
changes with the radius. This is due to a combination of dynamic effects that come into play
during the spreading of the drop and conservation of mass. The frequency vs. radius plots show
the existent of a critical radius, where there is a negative jump in the frequency because some
fingers stopped expanding. This effect is more obvious at smaller viscosities and thicknesses.
Figure 2-16 indicates that for higher viscosities, the frequency and wavelength of the fingers
remains fairly constant as the thickness increases.
There is fair degree of consistency in the wavelength of the instability for thicknesses larger
than or equal to 3mm. The behavior of the drop on the 0.5mm and 1mm thicknesses seems to
be different. This can be confirmed by observing the images obtained from the visualization
technique. It is clear that a pattern is observed in the spreading behavior of the drop after
the 3mm thickness: the rmin increases and the length of the fingers decreases. This has been
observed in previous experimental studies (see e.g. [23]). However, this pattern breaks down
for the two smallest thicknesses which may be dominated by different effects.
The length of the fingers was measured as:
34
5%
+0.5mm
0.004
18
-
5mm
0.003
E
16
7mm
-A-
- -9mm
-+--12mm
14
Ca
I
12
10
,0.002 -
C
a)
6
4
a)
C0.001 -
XX
8
2
00.007
0
0.5mm
-0.003
-
1 mm
-
4-'- 0.002
-
E
0.007
6%
12
7mm
10
- 9mm
-+-12mm
- ..
-
- -
> 0.0015
0.002 --
=
0.008
0.01
0
0.006
0.012
0.008
r [m]
-+-0.5mm
-31 mm
7%
-x- 5mm
0.0015
-
-
-
X
0.012
0.014
,12
'a
_10
+ 12mm
-15mm
x0
X
_I~iz 1'
0.001 0.007
0.01
14 -r
-*- 7mm
0.0025 ~ -+- 9mm
0.002
-
r [m]
3mm
-
(D
-
2
0.00
0.006
C
Cu
-
4
C.
0.0015-
_,)
-
6
C
E
--.-
8
C)
0.003
0.017
0.012
r [m]
3mm
-x- 5mm
x*
0.0025
0.017
0.012
r [m]
C6
(D
4)
I
0.011
0.009
0.013
r [m]
2
0.007
0.011
r [m]
0.015
Figure 2-15: Plots of wavalength and frequency of the fingers for different radii of circle plotted.
Each plot is for a different viscosity of the sublayer and shows data for all the thicknesses of
sublayer.
35
0.005 -
8%
-+-1mm
--- 3mm
5mm
7
-x-7mm
-
-0.004
E
0.003
-N-9mm
12mm
-+-25mm
6
35mm
-
t-+
x-
cu5
.
x
,
'-4
0
S3
'
52
0.002
1
0.001
0.007
0.005
0.009
0.011
0.013
0
0.005
0.007
r [m]
-1
-3
mm
mm
0.003
0.013
6
mm
-9 mm
+-12 mm
0.004
0.011
r [m]
9%
5 mm
0.009
"-7
5
0
-20 mm
-35 mm
"'I-
-
3
CU
2
0.002
-
0.001
-
0.0 05
1D
0.007
r [m]
0.009
0.011
0
0.005
0.007
0.009
0.01,
r [m]
Figure 2-16: Plots of wavalength and frequency of fingers for different radii of circle plotted.
Each plot is for a different viscosity of the sublayer and shows data for all the thicknesses of
sublayer.
36
1
max -
rmin
It must be noted that this formula takes into account only the longer fingers and neglects the
smaller ones. The length of the fingers is plotted against thickness in Fig. 2-17 a. A negative
trent is observed. For smaller thicknesses this slope is steeper, while the length of the fingers
that form on high viscosity sublayers remains fairly constant. It can be argued that the length
vs. thickness lines of the different viscosities cross at one point.
The average width of the fingers was measured manually, by approximating the shape of
one finger as a parallelogram. The width of the fingers is plotted against thickness for all the
viscosities in Fig. 2-17 b. The fingers are broad on thin sublayers and they become skinny
as the sublayer thickness increases.
This behavior is similar for all the viscosities. Smaller
viscosities produce thinner fingers.
The shape of the fingers has been studied experimentally by Afsar-Siddiqui et al.
[26].
They observe that with increasing film thickness, the fingers become shorter and straighter.
This agrees with our experimental results. However, they claim that there is an increase in
finger width with increasing film thickness, which is in contrast to our results. Figure 2-18 is
an example of their work and shows the difference in spreading patterns that arise at different
film thicknesses.
The shape of one finger is approximated as a parallelogram. Hence, the area of a finger is:
Area1 finger = length - width
The number of fingers in one flower is:
n = 2 .r
where
f
-f
=average frequency, i.e. the average value of the all frequencies obtained at the all the
radii. This frequency is plotted against the thickness in Plot 2-19a.
The area covered by all the fingers is:
37
0.008.
--
0.007.
-U0.006-
8%
9%
6%
0.005
-x-
0.004-
---
5%
7%
CO
0.003
-
0.002
0.001 0
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
a
thickness [m]
0.0017
8%
-U-9%
0.0015.
6%
S
0
0,
C
0.0013-
-x-5%
finI
0.0011
0
7%
0.0009
0.00070.00050
0.005
0.01
0.015
0.02
thickness [m]
0.025
0.03
0.035
b
Figure 2-17: Plots showing how the length of the fingers (plot a) and the width of the fingers
(plot b) changes with the thickness of the sublayer. All the viscosities are ploted.
38
Figure 2-18: Fingering patterns produced after 0.5 s when a 9-paL drop of 1.2 cmc SDS is
deposited on (a) a 251Lm water film and (b) a 100 pam water film (taken from Afsar-Siddiqui et
al [26]. Used with permition from the author.)
Areaau fingers
=
n
length width n
=
length width 2 - r f.
The total area covered by the fingers is plotted against the thickness of the sublayer in Plot
2-19. A three dimensional view of this plot is given in figure 2-20.
There is a decrease in
area covered by the fingers with increasing film thickness. This agrees with previous numerical
and experimental studies (see e.g. [20, 22, 4]). For high viscosities, the area covered by the
fingers remains fairly constant for different thicknesses. It is readily observed that the area vs.
thickness lines for all the viscosities cross at one point.
The area of the core of the spreading drop, is defined as:
Areacore = 7r - rin
Figure 2-21 is a plot of the Areacore against thickness. A positive trent is observed. With
increasing film thickness, the area covered by the fingers becomes smaller, while the area of the
core increases.
39
12
+4-8%
0
10
-4-9%
6%
8
ILO-
-x-5%
-*-7%
6
4
2
0
0.005
0.01
0.015
0.02
0.025
0.03
thickness [m]
0.035
a
0.0004
0.00035
r-"
04
C)
CD
a'
OC
-7%
0.0003
_U_
5%
0.00025 0.00025
6%
0.0002 0.0002
-v- 8%
0.00015
-
_
_
-
9%
0.0001
0.00005
0
0
0.005
0.01
0.015
0.02
thickness [m]
0.025
0.03
0.035
b
Figure 2-19: a) The average fingers wavelength against the thickness. b) The total area covered
by the fingers against the thickness.
40
x
10'
07
view b
15
5
0.015
0.01
0
0
viscsity [Pas]
0.005
thickness [m]
104
X10
-
43
3,5322.5
2.5-
-I-0.5
0.01
0
-
10
0.01
0
14
12
10
1
4
2
b
view b
view a
Figure 2-20: 3-dimensional view of how the area of the fingers changes with the viscosity and
the thickness of the sublayer. The data comes from graph 2-19. Three different views of the
plot are shown.
41
...........
0.0007
0.0006 -9/
0.0005
6%
x-5%
h 0.0004
0
00.0003--
---
-- X7
*0.00020.0001
0
0.01
0.02
thickness [m]
0.03
0.04
Figure 2-21: The total area covered by the core of the flower (up to rmin) against the thickness.
2.4.2
Source of instability
The focus of this chapter is the origin of the fingering instability. In what follows, we assign
wavelength as the critical parameter quantifying the extent of fingering phenomena. A dimensional analysis is carried out in order to find a scaling law for the wavelength of the instability.
Experimental data is provided to demonstrate the validity of this law. As stated previously,the
fingers are produced nearly instantaneously when the drop comes into contact with the sublayer. Thus we concentrate our effort at time scales close to zero where the instability takes
place. This enable us to avoid dynamic effects contained in data obtained at later times.
Dimensional Analysis
A dimensional analysis is performed with the aim to reduce the dimension of the problem to
a smaller set of parameters [33]. As mentioned previously, the wavelength of the fingers is the
critical parameter characterizing the instability. Experiments show that the fingers wavelength
(A) depends upon the following dimensional variables: viscosity (p), thickness (t) and density
(p) of the sublayer, diffusion coefficient (D) and surface tension difference between the surfactant
and the sublayer (Ay). Table 2.3, lists the dimensions of each variable.
This problem contains six variables described by three dimensions. Therefore, the Buckingham Pi theorem guarantees that there will be exactly three independent dimensionless groups.
42
variable
dimensions
A
A
t
{L}
{4}
{L}
p
D
{}
AY
{}
Table 2.3: Variables describing the problem
Combining the variables gives the three pi groups:
Hi
12
13
=
D = Marangoni number
p D
/L = - V
Dp
D
=
Prandtl number
=A
t
Hence:
t -=f (11, l12)
HI is the Marangoni number that reflects the surface tension force, 112 is the Prandtl number
that reflects the ratio between molecular diffusion and momentum diffusion in the vertical
direction. Note that we have not included time in the analysis because we study the onset
of the fingering instability. At later times the wavelength is a function of more parameters
(including the radius and time) and the analysis is much more complicated. Figures 2-15 and
2-16 testify towards this notion.
Results at t -+
0
Experimentally, it is difficult to obtain data at such early times and small inaccuracies in the
time of capturing the first picture can cause huge errors. The initial behavior in the first second
is obviously subject to inertial effects [24]. At the deposition of the drop (t -+ 0) the fingers
spread with a very high velocity and only high speed photography can give accurate data. To
circumvent these difficulties, we studied the images obtained from the flow visualization and
drew the following conclusion: the wavelength measured at r -- rmin remains constant with
time. Graph 2-22, provides further evidence supporting this remark. The wavelength measured
43
at rmin is plotted against time for a drop spreading on two different sublayers. Time zero in
this plot is the time that the first picture was captured, which differs from the time the drop
was deposited on the sublayer. It is clear that the wavelength reaches a constant value in a few
seconds. Thus, the wavelength measurement at rmin is the most representative wavelength at
the onset of the instability. Graph 2-23, shows how this wavelength changes with thickness and
viscosity of sublayer. The data is fairly consistent.
The wavelength of the instability vs. thickness was plotted for the spreading drop captured
at steady state (20 seconds after the deposition of the drop). It has been mentioned previously
that the wavelength of the instability is smaller at the base and larger at the tips of the fingers.
Different values of the wavelength were plotted: the average of the wavelengths at all the radii,
the wavelength at a radius
T
"in"rmax,
and the wavelength at rmax (see Appendix B). The data
shown in these plots is fairly random controlled by other, more complicated effects. This is
another reason that the wavelength at rmin will be used.
Scaling law
Using the fingers' wavelengths at rmin (from section 2.4.2) and the values of the liquids' properties (from section 2.2.1), we calculated the three Pi groups (A/t, Pr, Ma) for all the viscosities
and thicknesses. Since p and Ay are constant in these experiments, constant viscosity signifies
constant Prandtl number and constant thickness is constant Marangoni number. Figure 2-24a
shows how A/t changes with Pr number for given Ma numbers. Figure 2-24b, is a plot of A/t
against Marangoni for fixed Prandtl numbers. We plot this data on a logarithmic scale and
perform a linear regression to retrieve a power law. The results are plotted in Figure 2-25. A
power law of the form:
y = 6 - 106x-O. 9 35
(2.1)
fits the data with R 2 =0.9587. All the experimental data is concentrated about a single line,
hence whatever the Prandtl number, the normalized wavelength (A/t) scales with the Marangoni
number to the -0.935
~ -1
power. Figure 2-24a, indicates that the wavelength over thickness
ratio is independent of the Prandtl number for Marangoni number < 4.2-107 . This region is
separated in the graph with a thick dashed line. These results can be summarized as follows:
44
1.60E-03
1.20E-03
S8.00E-04
5%, 0.5mm
*8%,7mm
.o E-4+
4.OOE-04
0.00E+
0
5
15
10
20
25
30
time [Sec]
Figure 2-22: Wavelength of the instability measured at rmin vs. time. Data is given for surfactant spreading on two different sublayers.
45
Wavelength at smallest radius
0.003
5%
0.0025
a6%
0.002
-+
9%
0.0015
0.001
0.0005
0
0.005
0.01
thickness
0.015
0.02
[m]
Figure 2-23: Plot of wavelength at rmin against thickness for different viscosities.
A
1
-
=
f(Ma) oc Ma, Ma < 4.3 - 10 7
M
-
=
f (Ma, Pr), Ma > 4.3 107
t
t
(2.2)
It has been stated previously that the behavior of the spreading drop is consistent for thicknesses bigger or equal to 3mm and that the pattern observed breaks down for smaller thicknesses
which are probably dominated by different effects (see section 2.4.1). It is interesting to note
that the critical Marangoni number, 4.3 - 10 7 corresponds to the 3mm thickness.
Therefore
Equation 2.2 verifies our previous observation and indicates that for low Ma the wavelength
is a function of both the Marangoni and the Prandtl number. Hence bulk diffusion plays a
significant role in the spreading behavior.
Comparison with existing data
It is hard to compare our results with previous results due to lack of experimental data of
the wavelength of the instability. Matar et al. are the only group that give such data. They
studied the spreading of a highly soluble and sparingly soluble surfactant solution across a thin
46
....
......
.
2.5
+
2
x
X/t
Ma=7.OOE+OE
Ma=1.40E+0i
--
1.5
-
Ma=4.20E+02
Ma=7.00E+02
Ma=9.80E+02
Ma=1.26E+OE
Ma=1.68E+OE
Ma=2.10E+OE
Ma=3.50E+OE
Ma=4.90E+OE
1
0.5
Independent
0
0.OOE+00
4.OOE+10
8.OOE+10
1.60E +11
1.20E+11
2.OOE+1 1
a
Pr
32.5
2 . --
-
Pi2=7.37E+0.8
-u--Pi2=3.31 E+0.9
2
Pi2=2.07E+1 0
--
1.5
Pi2=4.72E+10
Pi2=1.72E+1 1
11
0.5
-
0.00E+00
1.OOE+08
3.00E+08
2.00E+08
Ma
4.OOE+08
5.00E+08
b
Figure 2-24: a. Plot of A/t versus Prandtl for fixed Marangoni numbers. b. Plot of A/t versus
Marangoni for fixed Prandtl numbers.
47
10 -_--
-
-
- -
-
-
-
--
Ff=0.9587
0
0.1
0.01
1.OOE+06
1.OOE+07
1.OOE+08
1.OOE+09
log (Ma)
Figure 2-25: Plot of log(A/t) versus log(Ma).
water film [25, 26].
They investigated the role of solubility and surfactant concentration by
conducting experiments on thin films of the order of pim. By using image analysis software on
developed profiles they measured the width of the fingers and made the approximation that the
width is equal to the wavelength of instability. Figure 2-27, shows variation in finger width vs.
the thickness of the film for different surfactant concentrations.
They further supported their experimental data with a scaling analysis. Gravitational forces
and van der Waals interactions are not significant for the film thicknesses used. A balance
between Marangoni and capillary forces, gives: h 2Py ~
-h3 hyyy, where h denotes the local film
thickness, F the surfactant surface concentration, o- the surface tension and y the transient
coordinate. By assuming small variations in the local concentration gradient and that y ~ A
and h ~ t, where A is the finger's wavelength and t is the thickness of the sublayer, then
A ~ t 2 / 3 . The results in Figure 2-27, show good agreement with the 2/3 scaling law, suggesting
that Marangoni forces dominate the spreading.
In order to compare the data provided by Matar et al. with our results Matar's data is
used to calculate and plot the three Pi groups (A/t, Pr, Ma).
Approximate values for the
wavelength and thickness are obtained from Figure 2-27. The surfactant is sodium di-2-ethylhexyl sulfosuccinate mixed with water. Values of the surfactant's surface tension at different
concentrations can be obtained from published data [35]. The sublayer is water, with surface
48
5.
3
2
0
2
55
44
x 10
3
0.5
visc/(rho*D)
2
0
0
x 10
1
dgama*t/(visc*D)
4 .5 - -.....
3 .5 --
-.
....
S2.5 -...
0
0.5-
44
visc/(rho*D)
x Id,
daama*W/%vsc*D)
Figure 2-26: Two views of a 3-d plot of A/t against Ma and Pr.
49
0.5
0.30.14
S0.2
01
-Z
01
0
20
40
60
80
100
120
Film thickneas (pm)
Figure 5. Variation in finger width with variation in surfactant
concentration and water film thickness. (*) 1.2 and (0) 1.6
cmc. Each value is the average of 30 measurements over 3
runs. The solid lines are a power law fit to the data. The
exponents are (*) 0.59 and (U) 0.72. The regression coefficients
are (*) 0.96 and (U) 0.99.
Figure 2-27: Experimenta data of Matar et al. [25] (taken with author's permition).
tension 72 mN/m. These results and our experimental data presented previously are plotted in
Figure 2-28.
The importance of diffusion (Prandtl number) on thinner films observed in Fig. 2-24b, has
been mentioned by Matar et al [25]. They compare the Marangoni stresses to bulk diffusive
forces and find that in thin films, although Marangoni convection dominates, bulk diffusive
transport is fairly significant.
Our scaling law shows that faster diffusion of the surfactant in the sublayer (higher D)
creates higher wavelength of instability i.e. more unstable spreading; while no diffusion (D=0)
creates zero wavelength i.e. uniform spreading.
[25, 26] and Jensen et al. numerically [14].
Afsar et al. confirmed this experimentally
Jensen et al. suggest that in the case of soluble
surfactant, the surfactant will desorb from the surface to the bulk until both the bulk and the
surface concentrations are in local equilibrium. If the sorption is rapid then an advancing pulse
of fluid develops. If desorption is slow, initially the surfactant will spread as in the insoluble
case. When the desorption occurs, spreading rates will reduce and once the surface and bulk
concentration are in equilibrium, the pulse of fluid develops. Therefore, surface deformations
in the soluble case are more severe than in the insoluble case [14].
50
These observations are in
9
Ma=4.2E+07
Pi2=7.37E+0.8
-7 Pi2=3.31 E+0.9
Pi2=2.07E+10
-x- Pi2=4.72E+10
--Pi2=1.72E+1 1
e Pi2=1000, Matar]
-+-
6
5
3
2
0
0.OOE+00
10
1.OOE+08
2.OOE+08
Ma
3.OOE+08
4.OOE+08
1 .OOE+08
1 .OOE+09
.'Ma=4.2E+07
y =821.37X1
0.16
0.1
r1
0.0
1 .OOE+05
1 .OOE+06
1 .OOE+07
log (Ma)
Figure 2-28: A/t vs Ma. This graph contains our experimental results and existing data [25].
51
accordance with our scaling law.
2.4.3
Transient region
Profile of the sublayer
This analysis is performed in order to study the diffusion of the surfactant and the deformation
induced in the sublayer. Both are more accentuated for small viscosities and thicknesses bigger
or equal to 3mm. The Matlab routine mentioned previously is used. 100 straight lines were
drawn in the middle of an image (sublayer is 5% concentration and 3mm thick) and its intensity
was plotted (Fig. 2-29 shows just one line). This was repeated for all images captured every
5 seconds till 1 minute after the deposition of the drop. Figure 2-30 shows the average value
of intensity profiles. This figure confirms that upon deposition of the drop on the substrate,
the liquid in the thin film is swept away from the vicinity of the drop causing thinning of the
film near the drop edge while a thickened front travels away from the drop (see e.g. [3],[13]).
Thus at short times the centre of the drop plays the role of a reservoir [23]. It is observed
that 1 minute after the deposition of the drop the surfactant is still not in equilibrium in the
z-direction. However, the fingers have stopped spreading and are slowly diffusing.
Spreading behavior
From the flow visualization data we can study the dependence of the shape of the drop on time.
Figure 2-31 shows how the radius of the spreading drop varies with time for a drop spreading on
two sublayers of different thickness and viscosity. Grotberg and co-workers predicted that the
radius of the surfactant drop, driven by Marangoni with negligible gravitational and diffusion
effects advances in time as t1 / 4 [16, 13]. Other experimental studies show that the drop radius
scales like t 1/ 2 or t1/ 4 [15, 23, 31].
Our experimental values in Figure 2-31, are largely in
agreement with the t1/ 4 scaling, thus suggesting that marangoni convection is the main driving
force for the spreading at all film thicknesses. This highly agrees with the experimental results
of Matar et al. [25, 26]. In their work, they suggest that the t 1 / 4 scaling is is valid for all
thicknesses and surfactant concentration.
52
5 sec
0.8
0.0
5 sec0.
0207
0.3
00am
i
7
25 sec
00~
110
10
130
0:
0.8/
0.7
I0.
j
0.547
70
800
900
*
.
100
100
1200
1300
Figure 2-29: Surfactant drop spreading on a 3mm sublayer of aqueous solution of 5% PEO.
The intensity of the line is drawn in the middle of each flower is plotted.
53
0.9
0.80.7
0.6
0.5
time
0.3 -I
0.2
0.1
0
I
600
700
800
900
1300
1200
1100
1000
1400
Figure 2-30: Superposition of all the intensity profiles of the images till 1 minute after the
deposition of the drop, in 5 seconds intervals
0.025
Y=
* 7%, 0.5mm thickness
E
R2
* 8%, 7mm thickness
0.02
0.0066xP.2788
c 0.015
CL
0.932
3 172
80.0042e.
2=
0.005
0.9418
0
0
10
20
30
40
50
60
70
time (sec)
Figure 2-31: Radius of the spreading drop vs time. Drop spreads on two different sublayers:
(1) 5% PEO (in water), 0.5 mm thickness and (2) 8% PEO (in water), 7 mm thickness. The
spreading behavior highly agrees with the t1/ 4 scaling suggested by Grotberg and co-workers
[16, 13].
54
Molecular Mass (g/mol)
8 000 000
Solution
Percentage in Mass
viscoelastic
1%
Density (gcm- 3 )
Viscosity (Pa s)
0.98
see Fig. 2-32
0.98
4.5
60
60
Surface Tension (mN/m)
3 00 000
Newtonian
7%
Table 2.4: Physical and viscous properties of viscoelastic and newtonian PEO
2.5
Solubility Effects
This work was done in collaboration with Jose Bico and Mark Fermigier in ESPCI, Paris.
The objective is to identify the crucial parameters responsible for the formation of fingers.
To achieve this goal, the experiment was repeated with a numerous set of liquids of different
physical properties. the focus was on the effects of solubility, viscoelasticity of the sublayer and
surface tension difference between the two liquids. Quantitative data, such as the onset time of
the instability, the spreading area and the effect of the thickness of the sublayer were ignored.
All the sublayers used have thickness 1mm approximately.
First, experiments were performed in order to study the effects of elasticity on the spreading
behavior of the drop. This was done, by depositing the same drop of surfactant on a Newtonian
and a viscoelastic thin film. Both sublayers are aqueous solutions of Poly(ethylene oxide) of
different molecular masses.
The viscoelastic solution was created by mixing 1% in mass of
PEO of molecular mass 8 000 000 gr/mol with water. The chains of this polymer are very long
creating a viscoelastic network (see Fig. 2-32). Its physical and viscous properties are shown
in table 2.4. The Newtonian liquid is one of the liquids used in section 2.2.1.
Varying the
concentration or the molecular mass of the Poly(ethylene oxide) changes only the viscosity of
the solution, while its surface tension remains the same (see 2.4).
Different liquids were deposited on the viscoelastic and Newtonian sublayers. The physical
properties of all these liquids are presented in Table 2.6. Table 2.5, shows briefly the spreading
behavior of all the liquids on both sublayers. It is important to note that all the surfactants
are soluble into the viscoelastic and the Newtonian thin films.
Figure 2-33, shows a drop of food color spreading on a Newtonian and a viscoelastic thin
film. Small particles where uniformly deposited on the surface of the films, to visualize surface
55
100
10
C
I
0.1 +0.001
0.01
0.1
1
10
100
1000
shear rate [s-1]
Figure 2-32: Plot of Viscosity against shear rate for the visco-elastic aqueous solution of 1%
Poly(ethylene oxide) of molecular mass 8 000 000 g/mol
triton (x-100)
triton mixed with:
PEO viscoelastic
PEO Newtonian 7%
tiny fingers
spreads (slow diffusion)
fingers
creates a big hole
NO spread
fingers
water and blue methylene pure
SDS (chemical soap)
ethylene-glycol with
blue methylene
food die
see Fig. 2-33
fingers
thin-long fingers
lots of branching
clear thin fingers
fingers
behind a traveling thickened front lots of branching
I (see Fig. 2-34)
fingers
fingers
Table 2.5: Spreading behavior of different surfactants deposited on Newtonian and NonNewtonian sublayers. All the surfactants are soluble in both sublayers.
56
Figure 2-33: Drop of 100% pure surfactant spreading on a newtonian (a) and a viscoelastic (b)
thin film. Lucopotedes are deposited on the thin film's surface, in order to make its deformation
more clear.
flows. In the Newtonian case (Fig. 2-33a), the liquid is swept away from the vicinity of the drop
causing thinning of the film near the drop edge, while a thickened front travels away from the
drop, pushing the particles with it. Behind this front, fingers are developed. These observations
confirm previous experimental work (see e.g. [3], [13]). When the drop was deposited on the
viscoelastic liquid, it spread forming thin clear fingers (Fig. 2-33b and Fig. 2-34). However, it
appears that that the particles, and therefore the surface of the film are not displaced during
the spreading process. This is probably due to the high viscosity of the viscoelastic liquid.
Figure 2-34, shows a drop of food die spreading on a thin viscoelastic film. The fingers
produced are more pronounced and undergo a lot of branching, coalescence and tip-splitting
as they spread. Since the viscosity of the viscoelastic liquid is high, the fingers diffuse very
slowly and therefore can became much longer than the ones in the Newtonian liquid. Viscosity
difference between the drop and the thin film was found to be very important for the formation
of fingers. For example, triton has a very high viscosity (see Table 2.6).
When triton was
dropped on the Newtonian film, which has much smaller viscosity it sink without spreading at
all. When it was dropped on the non Newtonian liquid, which has a similar viscosity it formed
very small fingers. However, when triton was mixed with water, its viscosity decreased and full
fingers were formed.
More experiments were conducted with various fluids of different viscosities and surface
tensions. Focus was given to whether the drop is soluble in the thin film. The properties of
57
Figure 2-34: Drop of food die spreading on a thin viscoelastic sublayer.
the liquids are presented in Table 2.6. The outcomes of the spreading behavior of the drop are
shown in Table 2.7.
From these experiments, a series of interesting observations have been made. A sufficient
difference in surface tension must exist between the drop and the thin layer in order for the
instability to be initiated. If the surface tension difference is negligible then the drop spreads
circularly and slowly, without fingering (see ethylene-glycol spreading on polyacrylic acid and
castor oil spreading on polybutadiene). This observation is in accordance with previous studies,
in which it is proposed that the fingering instability is driven by the Marangoni effect (see
e.g. [20], [22], [27]). However, at very high surface tension differences the spreading observed
has a uniform circular edge.
Afsa-Sidduqui et al. found that there is a critical surfactant
concentration which exhibits stable spreading with the presence of a surfactant-coated "disk"
in the center. They suggest that this is due to the fact that micelles present in the bulk at
this high concentration act as monomer sources that continually replenish the interface thus
eliminating surface tension gradients and Marangoni stresses [25]. Troian et al. proposed that
transport of excess surfactant from the bulk to the surface may inhibit the formation of surface
tension gradients large enough to give rise to unstable flow [20].
Both soluble and insoluble surfactants were used. It was observed that when the surfactant
was insoluble, the drop spread as a uniform circular disk of liquid, as seen in an ordinary
58
viscosity
[Pa s]
Newtonian liquids
density
[gcm- 3 ]
polyacrylic acid
surface tension
at 200C
[mN/m]
54
comments
Molecular Weight=
450000 gr/mol
coating oil
1.5
1.226
30.85
red color
castor oil
decanol
1.02
0.96
35
28.5
transparent color
mixed with blue methylene
(good mixing)
mixed with blue methylene
(not good mixing)
mixed with blue methylene
(good mixing)
reacts chemically with some fluids
very viscous-4low Diffusion coefficient
decane
23.8
silicon oil
0.15
polybutadiene
triton (x-100)
triton mixed with:
water and
blue methylene pure
glycerol
1.075
20.6
0.89
1.07
a mixture made,
to decrease the viscosity of triton
(increase diffusion)
1.49
63.4
Table 2.6: Physical properties of liquids used in experiments
wetting (see silicon oil spreading on castor oil). However, for a soluble surfactant, the drop
instantaneously formed fingers which propagate from the nominal contact line (decane on castor
oil). Hence, diffusion of the surfactant drop in the thin film is an essential requirement for the
instability to occur during the spreading. Despite its importance, all the numerical studies that
attempt to model the fingering phenomenon until now, examine the stability of an insoluble
surfactant. Wagner et al. in their numerical study in 2004, accounted solubility effects for the
first time [10].
Afsar et al.
studied experimentally the effect of surfactant solubility and observed that
highly soluble surfactant forms fingers more pronounced and branched than sparingly soluble
surfactant [25, 26].
This has also been confirmed by Jensen et al. [14].
They suggest that
in the case of soluble surfactant, the surfactant will desorb from the surface to the bulk until
both the bulk and the surface concentrations are in local equilibrium. If the sorption is rapid
then an advancing pulse of fluid develops instead of the shocklike structure of the insoluble
case. If desorption is slow, initially the surfactant will spread as in the insoluble case. When
the desorption begins to occur, spreading rates will reduce and once the surface and bulk
59
S ublayer
ethyleneglycol
decanol
decane
silicon oil
polyacrylic acid
NO spread
miscible
castor oil
NO spread
immiscible
spreads with fingers
coating oil
NO spread
immiscible
spreads with fingers
miscible
fingers
miscible
spreads with fingers
miscible
spreads with NO fingers
miscible
spreads with NO fingers
immiscible
immiscible
castor oil
polybutadiene
spreads
NO fingers
miscible
NO spread
miscible
Table 2.7: Spreading Behavior
60
concentration are in equilibrium, the pulse of fluid develops. Therefore, surface deformations
in the soluble case are more severe than in the insoluble case [14]. However they claim that
solubility does not affect the spreading exponent and the t1 / 4 prediction made by Groterberg
et al. for the insoluble study remains valid [16].
61
Chapter 3
SIMULATIONS
3.1
Introduction
Several modeling studies have tried to explain the physical mechanism responsible for this
instability. Troian et al. first proposed a model that attempted to isolate the destabilizing
mechanism using a linear stability analysis, which exploited certain mathematical similarities
between the present instability and viscous fingering (Saffman and Taylor 1958). They modeled
an insoluble surfactant coated drop spreading on a clean thin liquid consisting of the same fluid
and assumed that the viscosity is uniform (due to low surfactant concentrations).
Using the
lubrication approximation, they developed a set of coupled fourth order equation to describe
the spreading behavior and showed that the fingering instability is driven by the Marangoni
effect [30]. A long-thin region is formed at the front of the drop (this agrees with the qualitative experimental observations), in which a surfactant-concentration gradient is established
that controls the spreading of the drop. Their stability, allowed variations in the surfactant
concentration but not in film thickness and predicted transient growth followed by decay.
Over the years, this set of equations has been extended to include Marangoni stresses,
capillary effects arising form the surface deformation, bulk and surface diffusion of surfactant,
gravitational terms and disjoining forces that either promote or retard the film thinning [15,
16, 13, 18, 14]. These initial theoretical studies have shown that a thin liquid film is spontaneously pulled in the direction increasing surface tension, while at the surfactant leading edge
a thickened front is formed which advances rapidly over the uncontaminated liquid. The height
62
T
inI
in
rnI
P rop,
'Jb
4r
I k rli :r
Aqu)2 ;ph3
-i
l b
l
au
,Wr rwr
Figure 3-1: Schematic of the flow geometry, representing a drop laden with soluble surfactant
deposited on an uncontaminated thin liquid film, resting on a horizonal rigid support (taken
from [10]).
of this advancing rim is almost twice the undisturbed liquid thickness (see e.g. [15]). To accommodate this elevation, the liquid film thins near the point of deposition giving rise to the
liquid height profile shown in Figure 3-1 [16]. Flow variables (spreading velocity and shape of
interfacial profile) were found to depend on the thickness and viscosity of the sublayer and the
surface diffusion. Higher diffusion can smooth the shock at the leading edge and cause faster
spreading [15, 16]. Analysis of the surfactant front with negligible gravitational and diffusion
effects predicts that the spreading radius advances in time as t1 / 4 [16, 13].
In a series of numerical studies, Matar and Troian examined the stability of an insoluble
surfactant monolayer spreading on a dimensionless thickness equal to unity. They allowed disturbances in both film thickness and surfactant concentration. In the presence of Marangoni,
capillary and diffusion forces the stability analysis revealed that the flow was stable to disturbances [1]. Inclusion of weaker capillary and diffusion forces confirmed the same result [2].
A transient growth analysis was conducted which suggests that continued growth is only obtained under significant van der Waals forces; in the absence of which, large transient growth
is obtained followed by decay [1, 2, 3, 4].
Similarly, in their recent numerical work, Fisher and Troian [11], focus upon the monolayer
leading edge and show that growth followed by decay is obtained for constant thickness layer
despite an alternative method of perturbation growth. However, in the presence of van der
Waals forces, transverse perturbations of intermediate wave number grow exponentially in the
severely thinned region, leading to the formation of fingerlike patterns, consistent with the
63
experimental results [8].
In contrast to the previous models that implemented a finite surfactant source, Fisher and
Troian considered continuous supply of surfactant and showed that large transient growth and
asymptotic instability are possible for this case [12]. However, it is still unclear how this can
actually relate to the physical problem under consideration.
Whereas previous authors studied the spreading of a monolayer on a film thickness of order
one, Warner et al. examined the linear and nonlinear stability of a thick insoluble surfactant
deposition spreading over a film of much smaller thickness. They were the first to consider the
presence of adverse mobility gradients at the leading edge of the deposition. Full numerical
simulations of the nonlinear governing equations showed disturbance growth of the temporally
evolving base state, despite the absence of intermolecular forces. This growth is amplified by
increasing the initial thickness ratio of deposition to the thin film and decreasing the magnitude of capillarity and surface diffusion [9]. In their latest work, Warner et al. extended these
findings to account for solubility effects [10].
They assume rapid vertical diffusion and per-
formed cross-sectional averaging of the convective-diffusion equation governing the surfactant
bulk concentration. Linear stability analysis and direct numerical simulation showed that a
combination of surfactant solubility with high counteractions of surfactant further destabilizes
the flow. This agrees with previous experimental observations [25, 26].
3.2
Model Formulation
In this section, we present a brief formulation of the mathematical model that describes the
dynamics of the thin film and the drop. We consider the spreading of a drop of fluid on an
initially undisturbed, thin liquid layer of initial uniform thickness H. The drop, is bounded
from above by an inviscid gas (air) and the thin film rests on a horizontal flat solid substrate.
The geometry with relevant coordinates is illustrated in figure 3-2. The drop is uniformly coated
with soluble surfactant of initial concentration IF. The initial viscosity, and density of the thin
film (base) and the surfactant drop are p and o with subscripts b and s respectively. The thin
film is assumed to be incompressible and Newtonian. In our experiments it was observed that
the surfactant must exhibit some solubility in the bulk liquid in order for the instability to
64
zh(x t)
x,
'0 dx=
yx
x+dx/7
x
/
Figure 3-2: Schematic representation of the surfactant drop spreading on the thin liquid film.
occur. Therefore solubility effects will be taken into account. The concentration of absorbed
surfactant is unknown and is part of the solution of the problem. Since, equilibrium is not
established immediately the kinetics of the absorption must be taken into account.
3.2.1
Governing equations and scaling
A finite-area soluble surfactant is placed on a thin liquid film resting on a horizontal substrate.
The spreading of the drop on the thin liquid layer is caused by the initial difference between the
surface tension of the surfactant and the higher surface tension of the clean liquid layer. The
parameter H = y - ym = Ay, denotes the spreading coefficient, where 7Y is the surface tension
of the gas-liquid interface with no surfactant (clean liquid) and 7m the surface tension of the
gas-liquid interface with an absorbed monolayer of surfactant (coated liquid). The mismatch in
surfactant concentration at the intersection between the clean and the impure surface, develops
a large shear stress, which drives a Marangoni flow that spreads the surfactant towards regions
of higher surface tension. The surface tension, -y, is a linear function of concentration, F [38]:
= 7 - ar
70
where:
a
- const.
-[vi
65
(3.1)
We will assume that a/y 0 << 1.
In this study the spreading is considered to be dominated by Marangoni stresses in the
presence of other weaker diffusion and capillarity forces. Gravity, diffusion from the bulk to
the surface and disjoining pressure forces are neglected. A linear geometry is defined such that
x denotes the horizontal direction, z the vertical direction and y the transverse direction, as
shown in Figure 3-2.
Since the gradients along the y direction are much smaller than those
along x and z, a quasi-two-dimensional geometry will be considered, with very small variations
in the y direction. The starting points, assuming incompressibility of the fluids, are the Navier
Strokes and continuity equations in 2-dimensions:
au
-
1
+ (v - V)v =
_Vp + -V2v + g
p
%4t
p
inertial forces
V -V = 0
where V' = (u, w) is the velocity field with u and w components in the x- and z-direction
respectively. These equations need to be coupled with some appropriate initial and boundary
conditions. Some approximations can be made to simplify the system:
1. Since the depth of the layer (H) is observed experimentally to be small compared to
the horizontal extent of the surfactant distribution (L), and the slope of the free surface
is small, a lubrication model can be developed.
e is defined as follows and is
< 1 in
accordance with the lubrication approximation:
6=
H
L
< 1.
2. The Reynolds number measures the relative importance of the inertial forces and the
viscous forces and is defined by:
Re = ULp
From our experiments, Re << 1, hence the inertial forces can be ignored and an incompressible Stokes flow can be considered.
66
3. The films are very thin hence gravity forces can be neglected.
These approximations, can reduce the equations to a much simple form (see [36] and [37]):
v
Vp =
V-v=O
For the boundary conditions we impose no-slip and Dirichlet condition (i.e. no mass flux across
the boundary) at the wall (z = 0):
U= W =
0
-0
and the boundary conditions at the free surface (z = h(x, t)) are given by normal and tangential
stress balances and the Dirichlet condition :
n - or--
= p = -7k
du
dz
.I z=h
where h is the unit outward vector normal to the interface,
t is
the unit vector tan-
gential to the interface, k = how is the mean curvature of free-surface fluid and
tensor with y-derivatives neglected, is given by
-p +2p
p(2 + !u)
67
-p + 2pa
-, the stress
Mass and surfactant concentration conservation laws must also be satisfied. Mass conservation in the x-direction in a small control volume of differential length Ax (see Figure 3-2)
gives:
0 (hAxp)
jh(x) U
at
o
Oh
Ot
jh(x)
-
(x, t)dz
X~~td
0
Ox
(h(x)
x+xAx -
u(xt)dz) = 0
0
(3.2)
where, the horizontal velocity u has been averaged over the vertical z-direction. The surfactant
conservation equation is [10]:
OF(x,zt
0
z, t)
± z(, ,t]133
+ Oxa (F(x, z, t) -u(x, z, t)) = P [rX (x , Z, t) + rzz(X, Z, t)
at
(3.3)
where
UL
Pe = U
D
(3.4)
D is the diffusion coefficient. Note that since there is diffusion in the z-direction, the surfactant
concentration is a function of x, z and t.
Scaling.
To simplify the system of equations and boundary conditions, the equations are rescaled and
a lubrication approximations is applied. We use tildes for the dimensionless variables.
The
following scaling is adopted:
H
L
-.
w=
h(x, t)
-
hxLt
(3.5)
w
The non-dimensional velocity of the drop of surfactant or Marangoni spreading
velocity is determined by a force balance to be:
-
U
u=U
U
=-
Ast
68
(3.6)
eA-Y
A,~
Balance in the z-direction between pressure and gravity gives the dimensionless spreading pressure:
=
H2
P = -P.
H
(3.7)
The resulting non-dimensional groups are the Capillary and Marangoni numbers. The capillary
number reflects the ratio between viscous forces and surface tension forces.
3.2.2
Dimensionless
Ca =2 7
(3.8)
Ma = .
II
(3.9)
momentum equations and boundary conditions:
Using the characteristic scales (3.5-3.7) we non-dimensionalize the Stokes equation and drop the
terms involving the thin flow parameter e. Thus, we obtain the simplified equations of motion
(see Appendix C):
_
da
-
_
di
,eff (X,Z,t) 82
AS
=5
(3.10)
(3
These equations, will be solved for pressure and velocity which will be averaged across the depth
of the film. Using the scaling from Equations (3.5-3.7) the normal stress balance (at i
=
I)
gives:
P= -Ca-hz
the tangential stress balance (at i
=
-=
(3.11)
h) is:
-MaVtr
0_3
A
pleff (X, h, t)
(3.12)
where Vtr is the projection of the surfactant concentration gradient on the surface (see Appendix D). The Dirichlet condition at i =
Iz,
t) and i = 0 becomes:
69
(3.13)
and the no-slip condition at i
=
0 becomes:
u = w~ = 0.
(3.14)
Calculation of effective viscosity.
Since the two fluids (the sublayer and the drop of surfactant) are miscible we consider them as
a single layer (mixture) whose surfactant concentration and viscosity
with time. Here
[pef
eyff (x, z, t), is changing
is the effective viscosity of the mixture. Initially, when the drop is first
deposited on the sublayer, the two liquids are completely unmixed and the conditions of the
mixture are as follows:
=
1
Aeff(Xh,0)
=
[,,
F(x,h,0)
O)
=
[,
]F(x,0,0) =0
Aeff(X,,
The effective viscosity of the mixture is a function of the viscosity of the surfactant
(p),
the viscosity of the base (pb) and the surfactant concentration of the mixture (F(x, h, t)). The
calculation for the effective viscosity is analogous to the calculation of the effective resistance
of two resistors connected in parallel:
1
R
1
_1
-ot
R1
R2
Similarly:
1
1 -r(x,z,t)
T(x,z,t)
Peff (X, z, t)
PA
AS
eff
(X 'Itz7'
=
[PsIb
(X, z, t)Ab + (1 - r(x, z, t))[t,
70
surfactant
Z
clean liquid
x
z=0, u=O
Figure 3-3: Approximation of the velocity profile.
Integral method.
Our objective is to satisfy the mass and surfactant conservation equations derived from averaging the velocity over the z-direction. To do this we make an ansatz for the velocity and the
concentration profile. The approach is based on integral methods [40] and follows the ideas
developed by von Karman and Polhausen for the usual boundary layer flow around a body
[36]. As seen in Figure 3-3, there are two liquids that flow with different linear velocities. We
approximate this velocity profile by a second order polynomial:
fi(x, t) = c(x, t) + a(x, t), + b(x, t)z2
(3.15)
where a, b and c are functions of x and t. For the concentration profile we must impose Dirichlet
boundary conditions at i = 0 and i = h, to ensure no flow across the boundaries. Since the
bottom liquid is surfactant free and the top liquid is a surfactant, the concentration should
initially vary from 0 (bottom liquid) to 1 (top liquid) (see Figure 3-4).
The two liquids are
miscible hence the gradient of the surfactant concentration across the liquids interface is continuous. Using this information, the simplest assumption for the concentration profile is the
cubic:
zt)
l'l2
= g +d,; + ej2 + kj3
where g, d, e and k are functions of x and t.
Applying the Dirichlet condition (Equations 3.13) gives the approximation of the concentration profile:
71
z=h
r=1
drFdz=0
surfactant
z
clean liquid
z=0
r=0
X
drfdz=0
Figure 3-4: Approximation for the concentration profile.
'(T, i, t) = g - 3kh_2 + kU3.
2
(3.16)
The velocity profile (3.15) can be expressed as a function of g and k (the coefficients of the
concentration profile), by using the momentum equations (3.10) and boundary conditions (3.11),
(3.12), (3.14). Now we have three unknowns h(x, t), g(x, t) and p(x, t), hence three equations
are necessary. We use the average mass conservation equation (3.2) as the first equation. The
other two equations are given by satisfying the surfactant conservation equation (3.3) at two
points:
= 0 and
= h. This is the collocation method [39]. Substitution of the velocity and
concentration profile in the three equations gives:
[
at +x
2
Ma [ 2
) V+
+Ca
1 [d2 g
Pe [dX2
Ot
at
d)-
(gV +I - Vd) - V
Og
Og
~p
2 at
(2
3p
h2
2
(2V
g 1_gpV+ +
+
a
Mah
L
.g _. \+
]
2dd
hxXX
0
]gV
d 2g
d
Cah2g
2
2ih)
~7
where:
p
V
-
=
kh 3
(1 -)
=
pb
72
1
const.
Pe d
Icd 2p
3p]
2p
2
2
=
The concentration is then found by:
T,,t) = g--P
23
(j =h
2
+
+P
(_)3
h
These spatially one-dimensional fourth-order partial differential equations, will be solved numerically. Capillarity appears in its standard lubrication form. The Marangoni term represents
the destabilizing surface stress.
Physical significance of each term:
The Capillary number reflects the balance between the surface tension forces acting on the
interface and the pressure jump along the interface. Despite the fact that capillary forces are
of the order e 2 (see Equation 3.8) capillary contributions are not neglected because the freesurface curvature includes terms as small as of the order 62 . Moreover, capillary terms act as
smoothing parameters at the regions where shocks tend to appear and provide suitable physical
regularization.
The Peclet number represents the surfactant transport by Marangoni convection to that by
surface diffusion and is typically very large. Therefore, the Peclet number term has a very small
magnitude (since it includes 1/Pe). However, the Peclet number term can not be neglected since
it contributes in the accurate location of the surfactant front and improves the smoothness of
the numerical profiles.
3.3
Results and Discussion
Consider a 0.018 ml drop of pure ethylene glycol spreading on a aqueous solution of 8% PEO
of 2mm thickness. From the liquids' properties given in section 2.2.1, II = A-Y = 14 - 10-
3
N/M, y = 13 Pas, and D = 10-12 m 2 /t. The dimensionless numbers can be calculated using
equations (3.4), (3.8) and (3.9):
Ca = O(10-3)
Ma =
Pe =
0(1)
O(107).
73
A time sequence of results is presented in Figures 3-5 and 3-6. Due to difficulties with the
numerical implementation, the capillary number used is one order of magnitude lower than the
real one. Therefore, in these results the surfactant diffuses faster than in reality. However, the
error is small since we are interested in the relative difference between the Ca and Pe numbers,
which is huge. The results show that after 20 minutes the surfactant drop has almost completely
spread and diffused in the sublayer (see 3-6). This is consistent with our experiments. Figure
3-8, is captured 20 minutes after a drop of surfactant is deposited on a 2 mm film of 8% PEO.
Since the viscosity is big, diffusion is very slow and the drop needs two hours in order to diffuse
completely. The same procedure was repeated for a sublayer of IL =0.2 Pa s and t = 2mm.
The results are presented in Figure 3-7. This viscosity is much smaller and the surfactant drop
diffuses and spreads much faster.
Two interesting observations can be made. First, as soon as the drop is deposited, the
liquid film thins near the point of deposition and a thickened front is formed at the leading
edge which advances over the uncotaminated liquid, giving rise to the profile shown in Figure
3-9. This agrees with previous numerical and experimental studies ([15],
[16], [10]).
Second,
the surfactant tends to accumulate at the edge of the drop behind the thin region front (see 0.1
second at Fig. 3-5). This indicates that the fingers are formed in the thinned region, where the
surface tension difference is very big (see [20]).
74
5
*
0.001s
OS
4
N
0.8
4
0.7
3
3
2
2
0.6
0.5
1
0.4
0.3
1
0.2
0
0.005
0.01
0.015
0
0.005
0.01
0.1
x [M]
x [im]
5
0.01 s
N
4
0.1 s
4
N
3
3
2
2
1
1
0
0.005
0.01
0.015
00
0.005
0.01
0.015
x [in]
5
5
1s
5s
N
3
3
1
1
0
0.005
0.01
x in]
0.015
00
0.005
0.01
x[i]
0.015
Figure 3-5: Time sequence of surfactant droplet spreading on a 2mm film. The film is an
aqueous solution of 8 percent PEO (in mass) with 1, = 13 Pas. The colorbar shows the surfactant
concentration.
75
5
0.9
5
50s
10s
N 4
0.
N 4
0.7
0.6
).4
).3
0.2
x.-[in]
U.UUO
UU
Ul-
x
in]
5
2 min
N
4
3
4
20 min
3
0
0.005
0.01
[m]
0.015
x [i]
0.01
0.0
5
2 hrs
N
24 hrs
3
0
0.005
0.01
x [i]
0.015
U
U.UUO
u.u,
X
[MJ
U.U-1
Figure 3-6: Time sequence of surfactant droplet spreading on a 2mm film. The film is an
aqueous solution of 8 percent PEO (in mass) with it = 13 Pas. The colorbar shows the surfactant
concentration.
76
Os
N
0.01 ms
*
0.9
0.8
N 4
0.7
3
0.8
0.5
2
1
0
01
x
x
[in]
[m]
-
5
0.1 MS
0.001
N
4
N
5
4
3
3
2
1
01
U-UUD
0
x
0.01
0.015
x [mm]
5
5
0.01 S
N
0.005
[m]
1s
N
4
4
3
0
0.005
0.01
x m]
0
0.015
0.005
0.01
[M]
0.015
Figure 3-7: Time sequence of surfactant droplet spreading on a 2mm film with viscosity /y = 0.2
Pas. The colorbar shows the surfactant concentration.
77
Figure 3-8: Surfactant drop spreading on thin film. The picture is captured 20 min after the
drop is deposited. The film is an aqueous solution of 8 percent PEO (in mass) with pi = 13 Pas.
so-
h [i]W
4.54.0-
25-
20-
I-
I
0
I
I
5
I
10
15
x,0
I
20
1
25
0
30
x [M]
Figure 3-9: Profile 0.01 second after the drop is deposited on a 2mm film of aqueous solution
with 8% PEO. The spreading is characterized by the propagation of a thickened front at the
leading edge together with an accompanying trailing thinned region.
78
Chapter 4
CONCLUSION AND FUTURE
WORK
The instability of the spreading of a surfactant deposition on the surface of a thin liquid layer
has been investigated both experimentaly and numerically. Experiments were conducted in
order to study the effects of varying the film thickness and viscosity on the characteristics of
the fingering pattern. It was found that the that the normalized wavelength, A/t, scales with
Marangoni number, Ma=Ayt/pD, to the -1 exponent for any Marangoni higher than 4.3- 10 7 .
For Marangoni smaller than 4.3. 10
7
the normalized wavelength scales with Ma to the -0.4 but
is also a function of the Prandtl number, Pr=v/D. Hence bulk diffusion plays a significant role
in the spreading behavior only for small Marangoni numbers.
Experiments were repeated with liquids of different physical properties in order to identify
the crucial parameters responsible for the formation of fingers. It was found that high surfactant
solubility and surface tension difference between the drop and the thin layer must exist in
order for the instability to be initiated. If the surface tension difference is negligible then the
drop spreads circularly and slowly, without fingering. However, at very high surface tension
differences the spreading observed has a uniform circular edge.
To further understand the mechanism that give rise to the fingering instability, we have
formulated a mathematical model using lubrication theory, which explains the behavior of the
film thickness and surfactant concentration. The results of this study agree with the trends
79
observed in the experiments.
Further research on the fingering instability, should focus on the validating the scaling
law describing the wavelength of the instability. More experiments should be conducted in
order to understand the behavior for a bigger range of Marangoni numbers. In order to reduce
experimental inaccuracies, the experimental setup should be improved, especially by controlling
the humidity of the environment. Moreover, using a high speed camera to study the onset of
the instability, can give interesting results. A further step would be to investigate the onset of
the instability under a microscope.
Regarding the numerical part of this study, future developments are expected to develop in
more details our mathematical model. Such an extension will enable one to compare directly
the numerical results with the experimental outcomes and prove or disprove its validity.
80
Appendix A
Laplace Equation
The basic form of Laplace's equation is:
AP =
R1
(A.1)
+
R2
where: AP is the change in pressure across the interface (between the droplet and the surrounding medium), y is the surface tension, and R 1 and R 2 are the radii of curvature. If gravity is
the only outside force acting on a droplet, the pressure difference can be written as:
AP = AP, + Ap g z
where, AP is the pressure at a fixed reference point, Ap is the difference in density between
the droplet and the substance surrounding it, g is the local acceleration due to gravity, and z is
the vertical distance from the reference point. Since the drop considered is axisymmetric with
its apex at the origin of the coordinate system:
1
R1
-
b
R2
where b is also known as the curvature, hence from A.1 and A.2:
AP, = 2b-y
81
(A.2)
Thus, the shape of the drop is related to by, Ap, and g. Thus, we can work backwards from a
droplet's shape and the known force of gravity to its surface tension.
82
Appendix B
Wavelength versus thickness plots
The wavelength of the fingers in a given flower (spreading on a particular thickness and viscosity)
changes with the radius (see Fig2-15). However, it is important to find a single wavelength that
characterizes each flower. There are different ways one can define this wavelength: the average
of the wavelengths at all the radii, the wavelength at a radius
rminrmax
the wavelength at rmin
or the wavelength at rmax.
Plots B-i show plots of these wavelengths versus thickness for all the viscosities.
83
Wavelength at a radius at the middle of the fingers
0.0035
-- 5%
6%
0.003
x-8%
-9-/0
0.0025
C
0.002
0.0015
0.001
0
0.005
0.01
0.015
0.025
0.02
0.03
a.
thickness [m]
Wavelength at biggest radius
0.0045
0.004
A
a,
C
0.0035
0.003
4-- 9%
- -8%
6%
-x-5%- 7%
--
.3 0.0025
0
0
0.002
0.0015
0.001
0
0.005
0.01
0.015
0.02
0.025
0.03
b.
thickness [m]
Average wavelength
0.0035-
-*-
7
5%
-
6%
0.003
+8%
-
0.0025
---%
0.002
~0.0015
0.001
0
0.01
0.005
thickness [m]
0.015
0.02
C.
Figure B-1: Wavelength versus thickness plots. a: the wavelength is measurd at a radius
(rmax+rmin)/2, b. the wavelength is measured at rmax, c. this is an average wavelength of the
wavelengths measured at all the radii.
84
Appendix C
Dimensionless momentum equations.
Rescaling the fluid equations using (3.5), the reduced Stokes equation, can be obtained for all
the directions:
y-direction:
A two-dimension system is studied and it is assumed that there is no variation
in the y-direction.
z-direction:
N-S equation:
Vp = IefVV
Op
(2
=zbpeff
Kx
2
+
02w
W2
since no variation in y-direction.
Substitute the non-dimensional values:
e2 11 O2 i-'
HI-OP
F2
d,
=
dz
x-direction:
bsL2
Lef O,2
= p
2
+ ytH
z2)
peff (Q(e) + 0(e 2))
- 0
=>
p
=
constant along z.
Use Navier-Stokes, Eq. in the x-direction, where u is the velocity field in the
x-direction:
85
(0 2U
Op
Ox
H L5
=
d-
Ateff ( 5--
+
Aef f (L22
OP
-
O(E2) +
7jjjLef f
2U since no variation in y-direction.
+
At8 H 2 Oj2)
6
-~
At8O2)
S__ ef f (X, Z, t) O 2 U
86
__
*
Appendix D
Boundary condition at the interface
Using the scaling (3.5), (3.6) and (3.7) the vertical and horizontal dynamic boundary conditions
become:
normal stress balance:
=P = -7k
H
->.
H~
h=4,
L27h7)
-Qy
0
P2 0
-Ca (1
a r V 27
70 )
_
since ' < 1:
s0
=
-=h(xt)
-Ca - I22
tangential stress balance:
i - C- - il
He 1
[eff
=
H
V->
-avtr
L
6i2
7H
i
pef f=
=
-MaVtF
z=h(x,t)
87
P
pef f(x, h, t)
where Vtr is the projection of the surfactant concentration on the surface
Vtr(x, z, t) = i - VP(x, z, t)
1+e2
( )( )
dr(x,z,t) 1
a
L
dr(x,z,t) 1
1
1
d H
az L
()
di
neglecting the 0(e2) terms:
Vt(xz,t)
=
dFr, z, t) 1
d.
L
88
dh dr(x, z, t) 1
dz
d;
L
H
Appendix E
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