Langmuir 1992,8, 315Ck3154
3150
Solitary Waves in Asymmetric Soap Films W.Frey and E. Sackmann' Physics Department, Technische Universitiit Miinchen, Biophysik, 8046 Garching, Germany Received April 28,1992. I n Final Form: October 14,1992 We report hydrodynamic instabilitiesin thinlipid-water f i i on a hydrophilicsolidsubstrata (asymmetric soap f i b ) , stabilized vertically and in contact to a film balance in a surrounding of saturated humidity. A solitary wave is generated by pushing a lipid monolayer onto the water film. The experimental data fit to a long wave expansion theory by including the Marangoni effect. Introduction
Wetting of solid substrates plays an important role in technical processes such as lubrication, adhesion, covering, and cleaning of surfaces. Dynamics and equilibrium behavior of a thin film of a complete wetting fluid allows some insights into the interaction between the wetted solid surface and the wetting fluid.' If we define wetting as the replacement of one fluid by another, the spreading of surfactant, like a lipid, on an already existing film is a dynamic wetting process. In the equilibrium state this results in a surfactant-covered film, with the thickness determined by the DLVO forces? In the view of analogies to soap films we call this an asymmetric soap film.3 In addition, such films open a new possibility to study the dynamics of the formation of Langmuir-Blodgett films. They are also used as a model system to study the interaction of lipid layers with protein films adsorbed on a solid. This is related to fundamental problems of biocompatibility . In this paper only the dynamics of the spreading of a surfactant on an already existing film is reported. There are several reports on the spreading of liquid drops on a solid surface.*+ Marmur' suggested that gradients in surface tension are the driving force for the spreading of such drops. This process is the so-called Marangoni e f f e ~ t It . ~requires ~~ however a preexisting fluid f i i on the substrate.10 Safran was indeed able to describe the experimental behavior theoretically in terms of a Marangoni instability." The experimental and theoretical approacha in this paper are somewhat different. Starting with a thin liquid film, it is investigated under which conditions this film becomes unstable. Our description follows the long wave expansion for the stability analysis of thin liquid films flowing down a vertical wall.12 As already pointed out by Benney,13the instabilities may have solitary wave solutions under certain circumstances. But these solitary waves (1) de Gennes, P.G . Reu. Mod. Phys. 1985,57 (3-1), 827. (2) Israelachvili, J. N. Intermolecularand Surface Forces; Academic Press: New York, 1985. (3) Schneider, G.; Knoll, W.; Sackmann, E. Europhys. Lett. 1986,l (9), 449. (4) Lehlah, M. D.;Marmur, A. J. Colloid Interface Sci. 1981,82 (2), 518. (5) Neogi, P.J. Colloid Interface Sci. 1984, 105 (l),94. (6) Dodge, F. T. J. Colloid Interface Sci. 1988, 121 (l),154. (7) Marmur, A. Reu. Phys. Appl. 1988,23, 1039. (8) Scriven, L. E.; Stemling, C. V. Nature 1960, 187, 186. (9) Levich, V. G.Physico-ChemicalHydrodynamics; Prentice-Hak Englewood Cliffs, NJ, 1962. (IO) Troian, S.M.; Wu, X. L.; Safran, S. A. Phys. Rev. Lett. 1989,62 (13), 1496. (11) Troian, S. M.; Herbolzheimer, E.; Safran, S.A. Phys. Reo. Lett. 1990,65 (3), 333. (12) Kheshgi, H. S.;Scriven, L. E. Phys. Fluids 1987, 30 (4), 990. (13) Benney, D.J. J. Math. Phys. 1966, 45,150.
can only propagate along the direction of gravity." Joosten16 reported that thermally exited solitons should be possible in soap films, but they have not been observed yet. The theory developed here includes the Marangoni effect to explain the formation and direction of motion of instabilities in asymmetric soap films. The theory assumes, that the DLVO disjoining pressure contains the main nonhydrodynamic forces. It explains the occurrence of solitary waves as a special case of these instabilities. The Korteweg de Vries (KdV) equation is achieved, when DLVO forces do not play the dominant role. The theory is compared with experiments and found to be in good agreement. Experimental Setup A Langmuir trough is combined with a polarizer compensator sample analyzer (PCSA)ellipsometer,'6which can be used in an integrating beam mode and in an imaging mode. In the F i t mode the intensity is measured by a photodiode and a lock-in amplifier which allow exact determination of all optical parameters of the sample before and after the experiment. The second mode allows the visual control of the dynamic behavior of the filmby a video camera and recording by a VCR for dynamic image processing." In this mode a halogen lamp is wed instead of a laser to avoid speckles. To keep the computation of the film thickness as simple as possible an s-polarization is used. The Filmbalance is connected by a small channel to a separated dipping basin, where the contact between the monolayer on the film balance and the solid substrate is made. Thisbasin is a part of a chamber to establish saturated humidity by a constant flow of saturated air. A cylinder of quartz glass allow optical measurements over a range of 6 mm in the vertical z-direction, centered at a positionof about 15mm abovethe meniscus. Figure 1showsthe experimentalsetup,which is described in more detail in a forthcoming paper. The solid substrate is a thermally oxidized silicon wafer with one-third of the surface made hydrophobic by deposition of octadecyltrichlorosilane(OTS)(Sigma). Dipping this substrate into the subphase of the film balance and pulling it out with a speed of about 5 mm/s creates a thin film of water with a sharp boundary at the transitionto the silane-coveredsurface (cf. Figure 2a). At the beginning no lipid covers the air-water interface of the film balance and the thin film,which is still in contact with the subphase. Spreading of dimyristoylphosphatidic acid (DMPA)solution (Sigma) at the +water interface of the film balance has no influence on the thin film,as long as the lateral pressure does not rise. Theory The theory of nonlinear stability analysis of a thin liquid film in the long wave limit was established by Benney.13 (14) Scott, A. C.;Chu, F. Y. F.; McLaughlin, D. W. R o c . IEEE 1973, 61, (lo), 1443. (15) Jmten, J. G. H. J. Chem. Phys. 1985,82 (5), 2427. (16) Azaam, R. M.A.; Bashara, N. M. Ellipsometry and Polarized Light; North Holland: Amsterdam, 1977. (17) Zilker, A.; Ziegler, M.; Sackmann, E. Phys. Reo. A, in presa.
0743-7463/92/2408-3150$03.00/0Q 1992 American Chemical Society
Langmuir, Vol. 8, No. 12,1992 3151
Solitary Waves in Asymmetric Soap Films
..... image mode -intensity mode Interference filter
Halogen lamp
Polarizer Compensator
Figure 2b shows a schematic view of the dimensions and flow profile of the liquid film in equilibrium (solid line), which shows a small disturbance (dashed line). The velocities in z and x directions are u and u, respectively, p is the pressure, p the density, Y the kinematic viscosity of the fluid, and g the acceleration of gravity. In the following the lower indices z, x , and t denote partial derivatives with respect to the coordinates and the time and overscore stands for equilibrium. In equilibrium the following equations of a continuous flow down a vertical Wall
a, + 0, = 0 = - - p , + g + Y ( i i , , + a,,) P D, + iiD, + DD, = - 1 p , + v(D,, + D,) P together with the boundary conditions
(2)
Po = P(h) + II(h); x = h have the solution
(5)
ii,
image lens (image mode) Analyzer Mirror (image mode)
Si diode
\\
\\
m E K l
+ iia, + DQ,
Lock-in amplifier
Figure 1. Optical setup of a PCSA ellipsometer with an integrated film balance. The ellipsometer can work in two modes: an intensity mode to measure the optical parameters of the substrate before and after the experiment and an image mode to observe the dynamics of the film and record the profile with a VCR for image processing. The film balance is divided into two parts, the main trough and the dipping baain, both interconnected by a small channel. Above the basin the humidity can be controlled in a small chamber. A quartz glass cylinder allows optical measurements. h0
8
1
ii = L 2v ( 2 h -~x 2 ) ;
This description can be extended to the nonequilibrium case in the presence of a surface tension gradient in the -2 direction. This results in a tangential stress pointing in the direction of the gradient. This effect is called the “Marangoni effect”. In order to account for our experimental situation, the possibility of a continuous flow, along the direction of gravitation, of liquid due to condensation of supersaturated air is taken into account. Following J ~ o s t e n , the ’ ~ interaction of the surface of the liquid with the solid substrate is accounted for by the disjoining pressure, rather than the complete Maxwell stress tensor.le (18)Felderhoff, B. U.J. Chem. Phye. 1968, 49 (1) 44.
(3)
o =0
(6)
P=Po-n
(7)
Here II is the disjoining pressure and PO the atmosphere pressure. We now consider fluctuations of this stationary state, indicated by a dashed line in Figure 2b. We may assume the velocities as irrotational and therefore express them by a stream function \k, with u = \k, and u = It is desirable to introduce dimensionless variables. Denoting these by primes we get: z = 1 0 ~ ’ ;x = hdc’; h = h&; a = Uoii‘; u = UOU’; u = Uo(ho/lo)v’;P = p g h p ; p = pghop‘; a, = l/l&l; a, = l/h&; at = Uo/l&. Together with the velocity at the surface of the liquid Uo = gho2/2v, we introduce the Reynolds number R = U&o/v, the Weber number W = a/pgho2and a shallow water parameter F = ho/lo. The fluctuations may then be described by the differential equation
*,.
*,,
= Rr[*,t
+ (a + *,I*,,
- (0, + *,J*J-
+ Co + O h 3 )
Figure 2. (a) Schematic view of the lipid-water film wetting a
solidsubstrate up to a height were silane defies a sharp boundary. The oxidized silicon wafter remains in contact to a film balance, so that the lateral pressure can be varied continuously. (b) Schematic view of the part of the thin film of thickness ho together with the flow profile that arises, if supersaturated air will condense onto the film. Small fluctuations can deflect the surface aa indicated by a dashed line. The x-direction is normal to the substrate and thez-direction pointsalong the direction of gravity.
(1)
(8)
where Co is an integration constant. The boundary conditions (bc) are as follows: x = 0 (solid surface)
*, = \k, = 0
3t
(9)
= h(z,t) (air-water interface)
+ (ii + \k,)h, + *, = 0 (a, = * , , ) ( l - p2hZ2) - p 2 ( 1 + 4h,)*,. = - 2 FW, h,
(10) (11)
wr3)(12) Equation 10 is the kinematic bc; eq 11 is the bc for the tangential stress and contains the Marangoni term on the right side. The gradient in surface tension is taken to point upward, i.e. in the negative z-direction. Equation 12 is the bc for the normal stress containing the Laplace pressure as the first term. The disjoining pressure was
Frey and Sackmann
3152 Langmuir, Vol. 8, No. 12,1992
expanded to the second order at ita equilibrium value. With the aid of the Navier-Stokes relationship, eq 12 can be written in the more useful form
......................
4500 J 2 30 4000
-
- 25 -
+ (a + qx)qxz - (si, + *,,)*,I - 2pII'h, pII"(h - l)h, - 3p2qXz2 - 2p2h,,(Ci, + q X x-)2p2h2'PZX2 +
q,,, = R p
E*,
20
3 a
l5
O(r3)(12a)
g
The Laplace pressure disappears from eq 12a and is of 0(~3).
Following Benney,13 the basic idea of the long wave expansion is to solve eq 8 for small p by the expansion with respect to eqs 9-12a
5
0
*(le,z,t)
= 3'0'(x,z,t)
+ p*'l'(z,z,t) + p2?t'2'(le,z,t) + 0(2)(13)
The kinematic bc (10) allows one to rewrite the boundary value problem, eq 8 into a initial value problem. To the order of O ( p 2 ) ,this finally results in
h, = - 2h2h, + p 2 W,hh, -
I
(ih3h,, + 1-
3000
2000
1000
lime
4000
lsec]
Figure 3. Development of the thickness of the thin f i (dots)-measured in the middle of the window of observation-as a function of time and lateral pressure (line)during compression. When the lateral pressure starts to rise, the thickness increases drastically. After reaching a maximum, the thickness relaxes to stable state, which is larger than the initial state.
h2h,)(211' + II"(h -
4500
1)) p2(16h2hZ3 + 2h4h,,,J (14)
4000
"
1
"
~
'
'
"
'
~
According to our experimental situation, it was already considered in eq 14 that the Rsynolds number R for a water film of thickness 2 pm is of the order O ( p 2 ) ,so that cubic terms of order O(Rp) were omitted. Also it is assumed that the gradient in surface tension is nearly linear, so that W,, is of the order O ( p 2 ) . Assuming now small fluctuations h = 1 q and neglecting again terms of order O(e), eq 14 can be rewritten as
'
"
~
'
'
'
~~ "1 "~ '~ ~~ "~ ' ~ ~ ~ ~ ' " " ~ ' ' ' '
I
1 --
B
J
H
+
1000
2
tt = - 2(1- ~W,)tl,- 441 - ~W,)stl,- 2~ tl,, 2 p (jvrzII' + (€11; + ctltlZ2)(2II' + I') + O(w2) (15)
i
If the disjoining pressure varies only slightly with z, the second line in eq 15 can be dropped and the Korteweg de Vries (KdV) equation for a solitary wave obtained. This has the solution
One special feature of the KdVequation is that the solitary wave is unidirectional, for the radicand has to be positive. This means that UO,the velocity of the wave, has to be negative if W, = 0. This corresponds to a propagation downward, that is in the positive z-direction. The occurrence of such solutions has been established by other authors.13J4 It is possible to find a solitary wave running upward, if the gradient in surface tension, pointing in the negative z-direction, obeys the following condition p W , > 1 +UO 2
If this condition is fulfilled,the wave always is an elevation.
Experimental Results After the substrate is pulled out of the subphase of the film balance, so that the lower part is still penetrating the water phase, the thin film is given time to relax to a stationary state. This state is characterized by a thickness independent surface profile along the z-direction (cf. Figure
I
,.,,,,,.,I,,
0
2000
,,
/ - - J , , , -, __.___ , ,,,,,,,,I,, ,,, , ,,, 4000 hmt,
0000
8000
10000
IWI
Figure 4. Development of the thickness of the thin film (dots) as a function of time and lateral pressure (line) during compression. The compression was stopped several times and the film was given time to relax. Every step of compression can induce a hydrodynamic instability. The sharp rise and drop in thickness at about t = 5700 s belong to a solitary wave moving upward. 2). The thickness of the thin film is mainly determined by condensation of supersaturated water vapor due to a small temperature gradient. In the stationary state the condensing water has to flow down the wafer. Spreading of a DMPA solution does not effect the thin water film, as long as the lateral pressure is immeasurable. Compression of the lipid monolayer also does not change the thickness of the thin film until the lateral pressure starte to rise. Figure 3 shows an example of the change in thickness as a function of the continuously varied lateral pressure measured in the film balance. The increase in thickness at the end of the fluid-gas coexistence of the monolayer is clearly an instability, for the film relaxes after reaching a maximum thickness. This final value is greater than the initial value, indicating that the thin water film is now covered by a lipid monolayer. The flow profile has changed from a half parabolic of a free surface to a parabolic of a rigid surface. The f i i has to become thicker, if the flow is kept constant. To investigate the dynamic behavior of the asymmetric soap film in more detail, the barrier was stopped at several lateral pressures and the film was given time to relax. Figure 4 shows the corresponding behavior of the film. Already an increase of the pressure to about 0.1 mN/m leads to an increase in film thickness. Every subsequent
Langmuir, Vol. 8, No. 12,1992 3153
Solitary Waves in Asymmetric Soap Films
3200
3600
4000
3200 3600 fllmthickness [nm]
4000
Figure 6. Picture of an interference pattem of the thin film showing a solitary wave. On the right side the profile ie shown, which was re~0nst~ct8d by image proceseing. 1500 6
2000
2500
3000
3500
4000
4500
3000
3500
4000
4500
E
E
I
5
54
s
3500
-E
I
e
3000
5
5
-E
2500 6
4
c
2
d
d .c
:3
3.-3
P)
5
.E
0)
5
2
.E 2
EUJ
Ecn
r Q)
r Q)
.-
.1
1
0
0
1500
2000
2500
3000
3500
film thickness [nm]
2500
film thickness [nm]
Figure6. Developmentof the film profde at variousti”(inseconds). Duringthe eharp rbe~in Figure 4 the compreaeionwas stopped in a separateexperiment. On the left aide the compreaeionwas stopped at 0.7 mN/m and the film thick” increaseson a largevertical scale. On the right side the compresaion was continued up to 1.2 mN/m and a solitary wave is observed. The intermediate state is stable and the delay of time between t = 959 on the left side and t = 0 on the rightdde is 76 8. Two procesees can be distinguished (cf. Figure 6): a step of compression again leads to a quick increase and
a much slower relaxation. Only the sharp riseat a pressure of about 0.6 mN/m is different. Figure 5 shows an interference image of the film taken directly after the compression was stopped (time t * 5900 s in Figure 4). On the right side the profile has been reconstructed from the interference pattem. The peak in thickness in Figure 4 corresponds to a solitary wave and the decrease is not a relaxation but the end of thiswave. Its velocity, its height, and ita shape are constant over the window of observation. It is possible to stop the process in the middle of the sharp rise. As can be seen from Figure 6, this intermediate state is stable, as the thickness remains constant.
large scale increase in thickness and a localized wave running upward. The first is the same for all step of compression, while the second can only be observed at the beginning of the coverage with the lipid. The single wave is also connected with a large scale increase of thickness. However, while the velocities of the long-scale processes depend on the velocity of compression, the speed of the solitary wave is constant. The velocity of this wave is about 36pm/s, while the large scale velocity is in the range of 30-45 pm/s. The height of the wave alone is about 700 nm and does not appreciably depend on velocity of compression. The half-width-full-maximum (hwfm) is
3154 Langmuir, Vol. 8, No. 12, 1992
P
I
Frey and Sacknuann
-
5600
6000
6400
time [sec]
6700
6800
6900
7000
7100
time [sec]
Figure 7. Movement of the wave front (line) of the two steps of compression shown in Figure 6,together with lateral pressure (dots). The velocities in the first step and at the beginning of the second step are about the same as the velocity of the solitary wave. It does not depend on the speed of compression. This is not the case in those compression steps where the film is already covered with a lipid monolayer. (Linear lines are fits to determine the velocity.)
about 2 mm and again does not depend on the velocity of compression. According to the theory described above, the speed, the hwfm, and the height allow the determination of the gradient of surface tension a, and the amplitude e of the disturbance. The gradient of surface tension at about 15 mm above the meniscus is 0.6 mN/m over these 15 mm, which is the threshold necessary for the formation of a solitary wave. This is exactly the pressure at which the wave is observed. The amplitude of the disturbance is about lo4, which is of the order of O(p2)and justifies the neglecting of terms of order O(e2). Conclusions Formation and direction of the solitary waves can be well explained in terms of hydrodynamic instabilities driven by the Marangoni effect, both qualitatively and quantitatively. There is another kind of hydrodynamic instability which is of larger scale. This seems to be of the order O ( p ) only. Therefore every gradient of surface tension causes an instability in this case. There are two possibilities to explain why a gradient in surface tension occurs when a lipid monolayer is compressed. Firstly, a change in flow profile is caused by replacing a free boundary by a rigid one, formed by the lipid monolayer. The flow profile changes from half parabolic to a parabolic and this leads to an increase in thickness, if the liquid flow is kept constant. Secondly,
a gradient in the lateral density of the lipid monolayer along the -z direction (cf. Figure 2) is caused by a shear field, which is associated with a gradient in the surface potential. It appears that indeed both effects can be observed. Clearly, the f i s t mentioned effect cannot explain the large scale instabilities observed after changes in lateral pressure of the lipid-covered film.On the other side this explanation is most likely the reason for the shorter scale solitary wave. One indication for this conclusion is that it is only observed at the beginning of a compression. The lateral electrostatic gradient is considered responsible for the response of the film thickness to any change in lateral pressure. Due to the viscoelasticity of the lipid monolayer, the gradient in packing density and thus the gradient in electrostatic potential are expected to depend on the speed of compression. One therefore expects that the velocity of the large scale instability in the -z direction depends on the speed of compression in agreement with the experiments. Acknowledgment. We wish to acknowledge the helpful discussions with H. E. Gaub, W. Knoll, and J. G. H. Joosten. We want to thank A. Zilker for his help with electronic and computational problems. We are grateful to the Wacker Chemie Compound (Burghausen) for providing oxidized silicon wafers as a gift. This work was supported by the Deutsche Forschungegemeinechaft (SFB 266 C1) and the Fond der chemischem Industrie.
