Langmuir 1993,9,3212-3218
3212
Hydrodynamic Modes of Viscoelastic Soap Films P. Sens,C. Marques, and J. F. Joanny* Institut Charles Sadron, 6, rue Boussingault, 67083 Strasbourg Cedex, France Received April 22,1993. I n Final Form: July 15,199P We discuss the dispersion relation of the undulation modes of a soap f i i containing a viscoelastic Maxwell fluid modeling a water soluble polymer. Two types of modes exist, bending modes and squeezing modes. In addition to the modes existing for a usual viscous soap f i i , we find Rayleigh waves driven by the bulk elasticity E of the liquid. The dispersion relation of the undulation modes strongly depends on the value of the surface elastic modulus of the surfactant layers B. We study in detail the limit where the surface elastic modulus vanishes and the limit where it ia of the same order of magnitude as the surface tension. In this last case the Rayleigh waves are observed if Eh >> ea where h is the thickness of the f i i . We also discuss briefly the existence of higher harmonica due to the coupling of the undulation modes to compressional elastic modes of the film.
I. Introduction
A large number of industrial processes require the simultaneous use of polymers and surfactants. The interplay between the properties of the polymer and that of the surfactant is in general very subtle and is difficult to describe theoretically. A typical example is the stabilization of foams or soapfilmsby addition of hydrosoluble polymers. In order to understand the role of a water-soluble polymer on the properties of a soap film, detailed experimental studies have been performed in particular by di Meglio and collaborators.'d They have shown that the drainage of the soap film is strongly affectedby the polymer and that the classical drainage law proposed by Frankel and Mysels4which describes well the drainage of simple soap films is qualitatively not followed. Tentative explanations based on the existenceof normal viscousstresses or on the slip of the polymer on the surfactant layer have been p r ~ p o s e d . ~ In order to test the hydrodynamic properties of soap films, one can also measure their undulation modes by quasielastic light scattering. A very thorough study of the undulations of soap films from both the experimental and the theoretical point of view has been performed by Joosten.- His hydrodynamic theory is in good agreement with the experimental results obtained by light scattering and shows the important role of both the surface elastic modulus and the direct interactions between the surfaces of the film characterized by the so-called disjoining pressure. In this work we want to discuss the effect of the polymer on the undulation modes of the soap film. As a first step, we consider the polymer solution confined between the surfaces of the soap film as a homogeneous viscoelastic medium. We thus ignore the effect of the polymersurfactant interaction which induces an adsorption or a ~
Abstract published in Advance ACS Abstracts, September 15, 1993. (1) Cohen-Addad, S.; di Meglio, J. M. Langmuir 1992,8, 324. (2) Cohen-Addad, S. These Universite Paris VI 1992 unpublished. (3) Cohen-Addad, S.; di Meglio, J. M.; Ober, R. C. R. Seances Acad. Sci., Ser. 2 1992, 325, 39. (4) Mysels, K.; Shinoda, K.; Frankel, S. Soap f i l m , studies of their thinning; Pergamon Press: London, 1959. (5) Bruinsma,R.; di Meglio,J. M.;Quere,D.;Cohen-Addad,S.Preprint. (6) Joosten, J. J. Chem. Phys. 1984,80, 2363. (7) Joosten, J. J. Chem. Phys. 1984,80, 2383. (8) Joosten, J. In Liquids at Interfaces;Les Houches summer school; Charvolin, J., Joanny, J. F., Zinn-Justin, J., Eds.; North Holland: Amsterdam, 1940.
depletion of the polymer close to the surfaces of the film (givingthus a contribution to the disjoining pressure) and we investigateonly the role of the elasticityof the polymeric liquid?JO We uae the simplest possible model of a viscoelastic fluid, the Maxwell model where the complex viscosity at a frequency w varies as = ET/(^+ iwT) (1) This model has a single relaxation time T and a plateau modulus E. The liquid behaves as a viscous medium with a viscosity vo = ETat long times and as an elastic medium at short times. The spectrum of the waves on the surface of a semiinfiiite viscoelastic Maxwell liquid has been studied in detail by Harden et al.11J2 who showed the existence of elastic Rayleigh waves in addition to the usual capillary waves. Our model can thus also be considered as a generalization of their results to liquid slabs of finite thicknesses. The paper is organized as follows. In the next section we derive the hydrodynamic equations for a viscoelastic soap film taking into account the molecular interactions between the surfaces (described here by the DLVO disjoining pressure) followingclosely the work of Jooetenfor viscous soap films. In section 111, we discuss the dispersion relation of the various undulation modes of the soap film both for bending modes and for squeezing modes. We present explicit calculations for the cases where the Gibbs surface elastic modulus vanishes and where it is large. The last section presents some concludingremarks and discusses some possible future directions. q(w)
11. Bending and Squeezing Modes The soap film is a thin liquid slab of thickness h stabilized by the adsorption of an ionic surfactant on its surfaces. In the following, the z axis is normal to the film and the origin is at the middle of the film. The equilibrium thickness of the film is due to the balance between the disjoining pressure characterizing the molecular interactions in the film and the hydrostatic pressure head Pl,= pgH that can be imposed between the film and a bulk reservoir (pis the liquid density,g the gravity acceleration, and H the height difference between the film and the (9) Bird, R.; Armstrong, R.; Hassager, 0. Dynamics of Polymeric Liquids; Wiley: New York, 1977. (10) Ferry, J. Viscoelastic properties of Polymers; Wiley: New York, 1980. (11) Harden, J.; Pleiner, H.; Pincus, P. Europhys. Lett. 1988, 7,383. (12) Harden, J.; Pleiner, H.; Pincus, P. J. Chem. Phys. 1991,94,5208.
0743-7463/93/2409-3212$04.00/0 (0 1993 American Chemical Society
Langmuir, Vol. 9, No. 11, 1993 3213
Hydrodynamic Modes of Viscoelastic Soap Films reservoir). If we ignore the steric forcesdue to the polymer, the molecular interactions in the film are well accounted for by the classical DLVO theory: the disjoining pressure II = IIvdw + IIDLis the sum of a van der Waals and an electrostatic ~0ntribution.l~ In the absence of retardation effects, the van der Waals disjoining pressure is attractive and decays with the thickness of the film as nvdw = -A/6rh3 (2) where the Hamaker constant A is positive and of the order of the thermal excitation kT. The electrostatic contribution to the disjoining pressure is repulsive and due to the interaction between the electrical doublelayers on the surfaces of the film. In the limit where the two surfaces overlap only weakly, it can be written as
,I = 64kTna2exp(-rh) (3) Here, n is the concentration of monovalent salt in the f i , K - the ~ associated Debye screening length (r2= 2e2n/ ckT, where e is the elementary charge and c the dielectric constant of water), and u is a dimensionless number characterizing the electrostatic potential of the surfaces $0, u = th(e$0/4kT). The equilibrium thickness of the film is in general in the range 5-90 nm. The film is stable only if the disjoining pressure is a decreasing function of thickness, i.e. if aII/ah < 0. 1. Hydrodynamic Equations. We now discuss the dynamics of small undulations of the surfaces of the film around the equilibrium configuration of a flat film of thickness h. Each point on the two surfaces has a displacement ,$ and we shall only consider the hydrodynamics of the film at linear order in 5. The velocity field v(r,t) inside the f i is given by the momentum conservation equation where we neglect the convective terms and the gravity effects t3v - V(u + T) - pV W
pat-
(4)
where p is the mass density of the liquid. The stress tensor u is the sum of the pressure and of the viscoelastic stress. The viscoelastic stress is more easily expressed in terms of the Fourier components in time
u(r,w) = -PI + q(w)[Vv(r,o)+ T ~ v ( r , w ) ~ (5) where 1 is the complex viscosity given by eq 1. The electrostatic interactions are described via the electrostatic Maxwell tensor T and W is the van der Waals potential, i.e. the difference in van der Waals energy between a molecule at position r in the film and a molecule in an infinite film. For an undulation of pulsation o, the momentum equation can be recast as piwv = 7(o)v2v- VP (6) where P is an effective pressure calculated by Joosten6 which contains the effect of both the van der Waals and the electrostatic interactions. The momentumequation must be supplementedby the incompressibility condition div v = 0 (7) On the surface of the film the molecules follow the velocity and the surface displacement is such that v =
a,yat.
~~~~
~
~
(13) Israelachvili, J. Intermolecular and surface forces; Academic Press: London, 1985.
The exact boundary conditions on the stress are rather complicated and can be found in ref 6, we use here approximate boundary conditions replacing the effective pressure P in the film by pII where II is the disjoining pressure calculated for the local thickness of the film and p the actual pressure. The spirit of this approximation is the sameas the Derjaguin approximationused in colloid science, it assumes that the film is locally flat and that the interactions are local in the directions parallel to the film. Our boundary conditions are thus valid in the limit where the wave vector of the undulation q is small enough that qh 1, the two surfaces of the film are not coupled and the undulation modes are the simple capillary modes of each surface studied in refs 11 and 12. In doing so, we also neglect the contribution of the van der Waals14and electrostatic interacti~nsl~ to such quantities as the surface tension or the surface elastic modulus of the film. In the following these quantities must thus be understood as the total surface tension or the total surface elastic modulus including the van der Waals and electrostatic contributions. On the upper surface of the f i i , the generalized pressure can then be written as
P = p o + TC - n + a,,a~,/az
(8)
wherepo is the external pressure. The second term on the right-hand side is the Laplace pressure, y is the surface tension, and C the local curvature. The tangential stress at the surface of the film is due to the Marangoni effect and the corresponding boundary condition can be written as16J7 ,,(aU,/az
+ au,/ax) = esa2t,/aX2
(9)
The transverseundulation waves that we want to discuss have two types of eigenmodes, squeezing modes in which the two interfaces move antiparallel with respect to the z axis and bending modes in which the two interfaces move parallel with respect to the z axis (see Figure 1); we now discuss separately the velocity field for the two types of modes in the long wavelength limit qh
