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Langmuir 1989,5, 1436-1438
A Two-Fluid Model for Surface Modes on Concentrated Polymer Solutions and Gels J. L. Harden,* H. Pleiner,? and P. A. Pincus Materials Department, College of Engineering, University of California, Santa Barbara, California 93106 Received May 9, 1989. I n Final Form: August 2, 1989 We present a coupled two-fluid model for surface modes on concentrated polymer solutions and swollen polymer gels. We use this model to derive a surface mode dispersion relation in the limit of perfect coupling, and we present a perturbation calculation of the form of these modes when the surface tension is a weak effect. Finally, we discuss the effects of chain diffusion on the form of the surface modes. In previous theoretical treatments of surface modes on complex liquids, bulk liquid properties have been modeled by single-component viscoelastic fluids.lP2 However, macromolecular systems such as polymer solutions and swollen polymer gels are multicomponent liquids. Any realistic treatment of surface modes on such materials should take explicit account of their two-component nature. Coupled two-component fluid models have been used to investigate shear modes in semidilute polymer s01utions.~'~ In this letter, we use such a coupled twocomponent model to study the surface modes on entangled polymer solutions and polymer gels. We assume that the unperturbed material occupies the semi-infinite halfspace, z < 0, and that dilute vapor or vacuum fills the rest of space. We characterize the boundary between material and vapor by a step function density profile and a surface tension y. Thus, we assume that the olymer and the solvent have the same surface tension! Thermal fluctuations or externally applied perturbations are assumed to induce small amplitude surface modes. We treat the entangled polymer network as being the superposition of an elastic continuum and a Newtonian fluid. This approximation entails a coarse-grain averaging of the microscopic distribution of polymer and solvent; for length scales much larger than the characteristic size of the polymer entanglements, detailed knowledge of the microstructure is unnecessary, and a continuum approximation is valid. We treat the interaction between polymer and solvent, an inherently nonlocal one on microscopic length scales, by an effective local interaction proportional to the difference between polymer network and solvent velocities. The proportionality factor is related to the reduced monomer mobility in the solvent and contains the detailed features of the nonlocal hydrodynamic interactions between polymer subunits and solvent molecules. The form of this factor has been discussed extensively in the polymer dynamics l i t e r a t ~ r e . ~ It suffices for our discussion to note that it scales as v /
'Present address: FB Physik, Univ. Essen, D4300 Essen 1,F.R.G.
(1) Tejero, C. F.; Rodriguez, M. J.; Baus, M. Phys. Lett. 1983, 984(7), 371. Tejero, C. F.; Baus, M. Mol. Phys. 1985, 54(6), 1307. (2) Pleiner, H.; Harden, J. L.; Pincus, P. A. Europhys. Lett. 1988, 7(5), 383. (3) de Gennes, P.-G.; Pincus, P. A. J. Chemie Physique 1977, 74, No. 5, 616. (4) de Gennes, P.-G. Macromolecules 1976, 9, 594. (5) Actually, the analysis presented holds in the more general case of ypol > Y~~~~~~~In this case, the polymer is slightly depleted near the InGface. Such a depletion layer is quite thin and acts to renormalize the effective surface tension. As long as the surface mode wavelength is much larger than this layer thickness, the mode structure should be unaffected by such interfacial structure.
t ', where q is the solvent viscosity and 5 is the characteristic mesh size of the polymer network. To see this, consider a unit volume AV of polymer network. At a given instant, one may view a cross section of this volume as a cross section of l/S close-packed capillaries of diameter 5. In response to an applied pressure head, solvent flowing through an individual capillary will experience Poiseuille flow conditions and hence a resistance to flow that scales as q/C, '. The effective total resistance to flow of solvent through a unit area of the array is given by the parallel combination of the resistance of each capillary.6 Thus the total friction between a unit volume of polymer network and solvent should scale as [ (q/ F 4 ) - i ( 1 / ~2 ) 1 - ~ a / 2.~ Under these assumptions, the coupled equations of motion in the incompressible limit have the followingform:
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ai,
p , , t = -VP
+ qv20 + fl(i,,&);V.6 = 0
where P is the solvent hydrostatic pressure field, i, is the solvent velocity field, ii is the polymer network displacement field, 71 is the solvent viscosity, E is the shear modulus of the polymer network, ps and pp are the solvent and polymer densities in the medium, and f, is the local polym5r-solvent coupling term. As mentioned previously, f I has the form f, = x(&- a) with x q/[ '. To obtain the form of the surface modes, we must solve these coupled equations subject to boundary conditions at the material-vapor interfaces and the requirement that surface modes decay exponentially into the medium.' For a general two-component system, one must specify an interface for each component and satisfy appropriate boundary conditions at each interface. In the case of entangled polymer solutions and gels, there is usually a strong affinity between polymer and solvent molecules relative to that between polymer and vapor; vapor is almost always a very poor solvent for polymer. Hence, we expect the polymer-vapor interface to fluctuate in phase with the solvent-vapor interface; i.e., a single interface well characterizes the boundary between material and vapor in these systems. This assumption greatly simplifies our calculation, and the houndary conditions adopt the form
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(3) (6) An argument due to S. Alexander communicated to us by A. Hdoerin. Levich, V. Physiochemical Hydrodynamics; Prentice Ha.Englewood Cliffs, NJ, 1962; Chapter XI.
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0 1989 American Chemical Society
Langmuir, Vol. 5, No. 6, 1989 1437
Letters where u$) and u$) are the polymer and solvent contributions to the total stress tensor, y is the surface tension, is the z component of the local surface displacement from equilibrium, and :V is the transverse Laplacian operator. Equation 2 expresses the requirement that the normal stress at the boundary must balance the Laplace pressure of the perturbed interface, while eq 3 expresses the condition of vanishing shear stress at the interface. The form of the polymer and solvent stress tensors follow from analogy with those of linear elastic solids and viscous Newtonian fluids, r e s p e c t i ~ e l y ~ * ~
(4)
where E is the shear modulus of the polymer network, ui is the ith component of the polymer network displacement field, q is the solvent viscosity, and ui is the ith component of the solvent velocity field. In principle, one may obtain solutions to these equations of motion for arbitrary values of the coupling parameter x. However, for reasonable values of frequency w and wavelength A, the coupling term dominates over the elastic and viscous terms in the equations of motion. Consider, for exam le, the viscous energy dissipation term qV2ii. Since V u CIA2, and x ? I t 2 we , see that lqV2BJ> [. This is consistent with our hypothesis of the magnitude of A; the applicability of a continuum analysis requires us to restrict our attention to fluctuations of wavelength much greater than 5. However, this is not a strong restriction; for typical concentrated polymer solutions and gels, [ ranges from several hundred to several thousand angstroms, a length scale much smaller than typical liquid surface mode wavelengths. A similar analysis of the strength of the elastic term in the equations of motion shows IEV2iil lo-'), this regime is easily realized in practice, and our simple Voigt model analysis is adequate. The analysis that we have presented provides a theoretical framework for studying surface modes on entangled polymer solutions and swollen polymer gels. Well into the semidilute range of polymer concentrations, the surface tension is a weak perturbation and the surface modes are essentially surface elastic waves a t long wavelengths and overdamped liquid modes at shorter wavelengths. Future studies will (i) extend these ideas to less concentrated solutions where the perfect coupling approximation breaks down and capillary and chain relaxation effects become important and (ii) consider situations in which adsorption induces polymer concentration gradients near the surface.
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Acknowledgment. The work of J.L.H. and P.A.P. was supported in part by the Department of Energy under contract No. DE-FG03-87ER45288. The work of H.P. was supported by the Deutsche Forschungsgemeinschaft. In addition, we are grateful to one of the referees for insightful comments regarding the surface tension and interfacial structure of concentrated polymer solutions. (11) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca. NY, 1979: Chapter VIII.
