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Drainage of Compressed Polymer Layers: Dynamics of a “Squeezed Sponge” Glenn H.Fredrickson* and P. Pincus Department of Chemical & Nuclear Engineering and Materials Department, University of California, Santa Barbara, California 93106 Received July 18, 1990. I n Final Form: October 12, 1990 We have performed a hydrodynamic lubrication analysis to describe the drainage of solvent from semidilute polymer layers compressed between two solid surfaces. For strongly overlapped layers, the dynamical response is expected to be similar for adsorbed or grafted polymers. We assume the solvent is good and employ the conventional sphere-plane substrate geometry. Under these conditions, the dissipative lubrication force that acts normal to the surfaces is very different from the classical Reynolds force for Newtonian liquids. In particular, we find that,the force between surfaces separated by h and approaching each other with a (constant) relative velocity h is proportional to qR21’3/2a5/2h-1/2h, where q is the solvent viscosity, r is the number of monomers per area in the grafted or adsorbed layers, R is the radius of curvature of the spherical substrate, and a is a monomer size. This unusual result arises from Brinkmanlike (plug) flow of the solvent through the polymer network affixed to the surfaces. Our analysis also provides expressions for the frequency-dependent elastic and dissipative components of the normal stress. For the case of impulsive squeezing of a polymer layer, the normal stress relaxation is found to have a power law decay over several decades, -t-l3/l*. These results should also have implications for the mechanical properties of swollen gels and for the rheology of concentrated colloidal suspensions. I. Introduction It is well-known that hydrodynamic interactions play an important role in establishing the rheological behavior of suspensions and polymer solutions.’ At low concentrations of particles or polymer in a viscous solvent, the long-ranged part of the interaction is most relevant, falling off with distance r as r-1. For suspensions a t higher concentration, however, the near-field hydrodynamic interactions dominate the rheological response.2 The latter interactions are commonly referred to as lubrication forces and serve to prevent solid body contacts between suspended particles. Lubrication forces, unlike conservative forces arising from surface charge or dispersion interactions, are dynamic forces in the sense that they vanish in the absence of relative motion between particles. A theoretical explanation of lubrication forces was first obtained by Reynolds.3 By considering a thin film of a Newtonian liquid confined between two weakly curved surfaces moving .toward each other with a (constant) relative velocity h , he was able to show that a repulsive hydrodynamic force acts to separate the surfaces
F, = - 6 i ~ R ~ ~ h (1.1) h where h is the instantaneous separation of the surfaces and q is the fluid shear viscosity. The geometrical factor 6xR2 depends on the shapes of the surfaces; here we have assumed one surface is a sphere of radius R >> h and the second surface is flat. ( h is measured as the distance of closest approach as shown in Figure la.) The lubrication force described by eq 1.1arises from viscous drag of the liquid on the surfaces as it is squeezed from the narrow gap. More precisely, the imposed motion of the surfaces sets up a transverse pressure gradient. It is this dynamic pressure gradient that drives the fluid out of the gap and provides the normal stress (lubrication force). The pressure gradient is opposed by the viscous drag of the ( 1 ) Doi, M.; Edwards, S.F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986. (2) Brady, J. F.; Bossis, G. J . Fluid Mech. 1985, 155, 105. (3) Reynolds, 0. Philos. Trans. R. SOC.London 1886, 177, 157.
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solvent on the bounding surfaces; hence the force given in eq 1.1 is a dissipative one, proportional to the viscosity. Because the lubrication force diverges as h-’ as the surfaces are brought together, it is clear that solid body contacts in particle-fluid suspensions will be strongly supressed (provided there are no attractive static interactions among particles). Equation 1.1 has been subjected to numerous experimental tests over the past century, most recently by workers using the surface forces apparatus developed by I~raelachvili.~In particular, Chan and Horn5 and Israelachvili6 have presented an extensive experimental test of the Reynolds lubrication theory for thin films of nonpolar, Newtonian liquids confined between mica surfaces. They find that the theory and straightforward extensions thereof are capable of very accurately describing both steady and time-dependent squeezing experiments on such films. Until film thicknesses are reduced to 5-10 molecular diameters, the continuum hydrodynamic description is found to be quantitative. Thereafter, the frictional drag is enhanced up to the point where the film ceases to behave as a liquid. Recently, several research have repeated the above experiments, but with non-Newtonian (i.e., polymeric) liquids or with polymers adsorbed or grafted to the mica surfaces. The latter situation has particular relevance to the rheology of concentrated suspensions of sterically stabilized particles and is the case considered in the present paper. Intuitively, one expects that lubrication forces could be dramatically enhanced by the presence of polymer, since grafted or adsorbed chains serve as additional sources of hydrodynamic friction and thus can increase viscous dissipation. If the binding of these polymers to the surfaces is sufficiently strong, they should act as a fixed network (4) Israelachvili, J. N.; Adams, G. E. J . Chem. SOC.,Faraday Trans. 1978, 174, 975. (5) Chan, D. Y. C.; Horn, R. G. J . Chem. Phys. 1985,83, 5311. (6) Israelachvili, J. N. J . Colloid Interface Sei. 1986, 110, 263. (7) Israelachvili,J . N. ColloidPolym. Sei. 1986,264,1060. Israelachvili, J. N.; Kott, S. J. J . Polym. Sei., Part B: Polym. Phys. 1989, 27, 489. (8) Klein, J. J . Chem. Soc., Faraday Trans. 1 1983, 79, 99. (9) Montfort, J. P.; Hadziioannou, G.J . Chem. Phys. 1988, 88, 7187.
0 1991 American Chemical Society
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of hydrodynamic scatterers, much like a fixed particle bed or porous medium. Qualitatively, these expectations are manifested in the surface forces measurement~.~t~ At surface separations h that greatly exceed the equilibrium layer thickness L (Figure la), the lubrication forces are unchanged in form from eq 1.1,provided that h is replaced by h - 2LH in that expression, where LHis a “hydrodynamic layer thickness”. (Typically, L H L , where L is determined by static force measurements.) This experimental observation suggests that solvent flow is impeded within the layers (i.e. they are not freely draining) and the majority of the viscous dissipation occurs in the Newtonian liquid layer of thickness h - 2LH. The experiments show, however, a qualitative change in the force laws as h is reduced below 2LH 2L and the polymer layers interpenetrate and become compressed (Figure lb). The forces are much larger than would be predicted from the Reynolds formula with the solvent viscosity 17, yet it would seem to be inappropriate to simply replace 7 by a larger effective viscosity. It was also observed that the means of chain attachment, i.e. by grafting or adsorption, does not appear to influence the qualitative behavior in this regime. In spite of the scientific and technological significance of lubrication forces in compressed polymer layers, there has been quite limited theoretical analysis. Montfort and Hadziioannoug generalized the Reynolds formula to confined polymer melts by allowing for a phenomenological viscoelastic response. For solvent-filled layers, Kleins presented a simple analysis of lubrication forces under conditions of weak overlap and poor solvent. Here, we develop a systematic formalism to address this general class of problems and investigate in detail the behavior of layers containing good solvent and at greater levels of compression. In the present paper we study theoretically both the nonoverlapped ( h >> 2L) and the strongly overlapped ( h > 2L
-
-
(1.2)
where 5~ is the hydrodynamic screening length1pl0-l2for the network (that varies with separation h in the manner described below). In deriving this key result, we have assumed the following progression of length scales: EH h. For simplicity, both surfaces are assumed to have the same grafting density r (monomers per unit area) and the same type of polymer layer. A low molecular weight solvent fills the remaining space between the surfaces not occupied by polymer. The solvent is assumed to be a good solvent for the polymer, is taken to be incompressible, and has a shear viscosity that will be denoted by 7. We further assume that the polymer molecules attached to the two surfaces have been immobilized by grafting or strong adsorption. Except in the next section, where we consider nonoverlapping layers, it will not prove necessary to distinguish between these two means of attachment. A. Steady Squeezing of Nonoverlapping Layers. In the present section we treat the experimental situation depicted in Figure l a , namely R >> h >> 2L. We imagine that the lower surface is held fixed, while the upper surface is pushed toward it at a constant velocity, V , = h < 0. If the compression is stopped,the force between the surfaces must vanish (provided there are no long-ranged interactions present) since the layers do not overlap. Hence, there is no static force in the present regime. For nonvanishing V,, however, there is a dynamic lubrication force that we now proceed to calculate. In the present paper we adopt a phenomenological, continuum description of the viscoelastic polymer layers. Such a description is suitable for describing forces and fields that vary over length scales large compared with a characteristic “mesh” size of the layers. We assume that the chains that constitute these layers are strongly overlapped but insist that the volume fraction of polymer within a layer be small. For unattached chains in solution, this is of course the classical semidilute regimelJO and the characteristic mesh size is the equilibrium concentration correlation length, f . This length scales with monomer concentration c aslo (good solvents) [ c-3/4a-5/4,where a is a characteristic monomer size. As a first approximation for grafted brushes, we can use the “step function” concentration profile of Alexander13 and take f evaluated at this average concentration to be the coarse-graining scale. A refinement of this approximation would be to use the parabolic brush concentration profile of Milner, Witten, and Cates14 and coarse-grain to the scale of f ( c ( z = 0)). The case of an adsorbed layer is more problematic, with a self-similar mesh size that varies 1inearlylOwith the displacement z from the surface, f t. Here, we cannot coarse-grain beyond the monomer scale. Regardless of our choice of mesh size, the resulting continuum description can be applied only to quantities that vary over distances exceeding that length. In the present section this condition leads to the restriction that L L Hshould be some large multiple of the coarse-graining scale, which is always met in adsorbed layers. Since the layer thickness of brushes increases rapidly with polymer molecular weight at fixed surface coverage, this restriction is also of no significance for grafted layers. Our phenomenological description of the solvent velocity field v(t,r,O)for slow steady squeezing of the polymer/ solvent layers shown in Figure l a is based on Brinkman’s
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-
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(13) Alexander, S.J . Phys. (Paris) 1977, 38, 983. (14) Milner, S.T.; Witten, T. A.; Cates, M. E. Macromolecules 1988, 21, 2610.
equation,15 commonly employed in the treatment of creeping flow through porous media. For the present situation the equation can be written
qv*v - ?&(C)V
-VP = 0
(2.2) which is to be augmented by the incompressibility condition
v-v = 0 (2.3) (Note that because of our diluteness assumption, the polymer momentum density does not enter eq 2.3.) In eq 2.2, P i s the pressure and f~ is a hydrodynamic screening length that is taken to depend on the appropriately coarsegrained local monomer concentration, c(r,z). A variety of analytical methods1J1J2J6J7have been employed to demonstrate that the hydrodynamic screening length of semidilute polymer solutions is of the same magnitude and has the same concentration scaling as the equilibrium correlation length, f . We assume this relationship to hold within grafted or adsorbed layers as well and employ the concentration dependences described above for f . Equations 2.2 and 2.3 can now be solved for the geometry of Figure l a , imposing no-slip boundary conditions at z = 0 and at z = h ( r , t ) , where the boundary is moving with velocity V , = h. (Note that V , < 0 under compression.) For simplicity, we treat the case of a step function brush, defined by the monomer concentration profile (for the bottom layer) co, z IL
(2.4) c(r,t) = 0, z > L As shown in Appendix A, the resulting hydrodynamic equations can be solved independently in the two polymer layers and in the Newtonian solvent region, with normal and tangential velocities and stresses matched18 at the boundaries. By integration of the normal stress (dynamic pressure) over the bottom surface, the hydrodynamic (lubrication) force can be easily calculated. As shown in Appendix A, this force is given for 2 L < h > [H, the term involving (hl - 2L)3/24 dominates the integrand and eq 2.5 reduces to the expected result
which is simply the Reynolds formula with h replaced by h - 2LH = h - 2L. Thus, for the step function brush we can identify the hydrodynamic layer thickness as LH = L. For 0 < h - 2L 5 [H, the integral in eq 2.5 leads to a force expression quite different than eq 2.6, corresponding to a situation where a nonnegligible fraction of the viscous dissipation occurs within the polymer layers. The analysis of Appendix A that led to the above results can be repeated for parabolic grafted brushesI4 or for adsorbed polymer layers.18 Because [H c ( z ) - ~is/ ~a function of the axial coordinate z in these cases, the solutions of the homogeneous part of eq A.3 are not simple exponentials,
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(15) Brinkman, H. C. Appl. Sci. Res. 1947, A I , 27. (16) Edwards, S. F.; Freed, K. F. J. Chem. Phys. 1974,61,1189. Edwards, S.F.; Muthukumar, M. Macromolecules 1984, 17, 586. (17) Shiwa, Y.; Oono, Y.;Baldwin, P. R. Macromolecules 1988,21,208. (18) Yang, S.-M.; Leal, L. G. PhysicoChem. Hydrodyn. 1989,11,543.
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as in the step function brush. More extensive solvent flow occurs within the polymer layers in such cases. Thus, for layers with smoother (and more realistic) concentration profiles, we anticipate that LH < L, but still expect lubrication forces that follow eq 2.6 for h >> 2L. (It should be noted that Milnerlg has recently considered the case of solvent flow past a parabolic brush.) Rather than pursue these issues further, we now turn to consider the more interesting regime of compressed layers. B. Steady Squeezing of Compressed Layers. In the previous section we have seen that for polymer layers separated by distances large compared with the layer thickness, but small compared with the upper surface radius of curvature, the lubrication force is obtained by a simple modification of Reynolds classic formula. As the separation of the layers is reduced to be on the order of the characteristic mesh size [H, however, this expression breaks down. A t this point, and as the layers are further brought together to first touch and then weakly interpenetrate, the lubrication forces become much more complicated. In such instances the forces are not “universal” in the sense that they depend crucially on the means of layer attachment and on the details of the layer concentration profiles. Additionally, a static elastic force arises at contact, the form of which is not universal among different types of layers. If, however, we continue to compress the brush to the point where the layers become strongly interpenetrated, but the volume fraction of polymer in the gap is still small (see Figure Ib), then the situation simplifies again. In such instances the monomer concentration is approximately uniform in the z direction. However, c varies slowly in the radial direction due to the fact that the layer is less compressed for larger than at the point of closest approach of the two surfaces, r = 0. To describe this variation of c with r , we make use of our assumption that the amount of polymer in the gap between surfaces is conserved; i.e. only solvent is squeezed out. If we define a surface excess, r, as the number of monomers per unit area associated with the layers before their assemblage and compression, then the local monomer concentration in the compressed layer is simply c(r) = I’/h(r) (2.7) It follows that for the purpose of describing solvent flow in the layer, the hydrodynamic screening length that enters Brinkman’s eq 2.2 varies with r as
As mentioned above, a static (elastic) force also arises when the layers begin to interpenetrate. The origin of this elastic force is that the solvent quality is good and, therefore, monomer-solvent contacts are preferred over monomer-monomer contacts. In the moderately compressed regime just discussed, where h 0 by E
GH(t) = h o L , dh (h - l ~ , ) E ( h ) h -e~x p [ - t / ~ ( h ) ]
(3.8)
G’H(w)
+ iG”H(w)
(3.14)
These components are given by the expressions (ho~1,it~ gives rise to a slow, power law decay GH(t) ~4I ’ ( 1 3 / 1 1 ) E o ( ~13/11 )
The storage modulus G’H(Q) describes the elastic response of the polymer layer arising from radial displacement modes of the network a t frequency 0. In the low frequency limit, Q 0, G’,(Q) has the unusual scaling behavior -+
(3.13)
This peculiar behavior could in principle be probed in a step strain experiment.’pZ3 In such an experiment, one subjects an equilibrated layer, compressed to an initial separation of ho, to an additional (instantaneous and infinitesimal) compressional strain. Equation 3.10 is a prediction for the detailed time decay of the resulting stress (excluding the static component),normalized by the strain amplitude. From eq 3.13 we observe that such relaxation will have a “long-time tail” of the form t-13/11. I t is important to note that the above results have been obtained under the assumption that ho
