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Compression of Grafted Polyelectrolyte Layers Y. Rabin,t*jG. H. Fredrickson,*~~J and P. Pincus* Materials Department and Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, California 93106 Received December 3, 1990. In Final Form: April 5, 1991 We calculate the repulsive dynamic force between moving surfaces bearing grafted, overlapped, p$yelectrolyte layers in good solvent. This force arises due to the viscous drag excerted on the 'spongelike" brushes by the solvent which is being pushed out during the steady compression. The results are compared with the static osmotic force produced by the confined counterions. We find that while the static force decreases with the addition of salt, the dynamic force is nearly independent of salt concentration. The applicability of our results to adsorbed polyelectrolyte layers is discussed. (a) h >> 2L Following our recent work on the dynamics of grafted layers of neutral polymers in good solvents192and on the statics of polyelectrolyte brushes: we study the dynamics of compression of polyelectrolyte layers grafted to surfaces. For concreteness we adopt the approximate sphere-plate geometry of the surface force apparatus (Figure 1). In order to compare the hydrodynamic forces to their equilibrium counterparts, we consider first the static force (b) h cc 2L between polyelectrolyte brushes. Although this problem was previously considered by one of the present author^,^ we would like to repeat the main arguments in order to make the present contribution self-contained and to adapt the original calculation to the geometry considered in the present work. Unlike the neutral brush case, one might expect long-range electrostatic interactions between Figure 1. (a) Geometry of the surfaces and polymer layers nonoverlapping polyelectrolyte brushes in the absence of employed in the present paper. The lower surface is flat, while the upper is slightly curved, taken to be a sphere with a large added salt. However, in this case, the counterions are radiusof curvature,R. We employ a cylindricalcoordinatesystem strongly localized within the brush and, to a good with z measured normal to the bottom surface and r being a approximation, strong interactions between the charged radial coordinate in the plane of the flat surface. The position brushes appear only at overlap. The repulsive interaction of the upper surfaceis given in this coordinatesystem by z = h(r) can be viewed as the entropic cost of localizing the coun= h + +/2R, where we assume h / R > 2L,and rodlike polyelectrolyte segments as the charge concentration within the brushes increases with compre~sion.~ end-grafted polymers. (b) The case of moderately compreaaed layers, where the chains attached to different eurfacee strongly In the zero-added-salt case the interchain spacing is of overlap,but the volume fraction of polymer remains small (semithe same order of magnitude as the thickness of coundilute layer). terion "atmosphere" around the segments4 and the entropic cost of counterion localization can be simply the zero-salt result by the volume fraction of the above obtained from the local counterion osmotic pressure P(r), "atmospheres", ac(r)/$(r). This yields P(r) = Tuc2(r)/ that depends on the radial distance r from the point of K2(r) where K2(r) = a[c(r)+ 2c,] and where we took the closest approach between the brushes. We obtain P(r) = Bjerrum length to be the same as the monomer size u and Tc(r),where Tis the temperature and where c(r) = r / h ( r ) , suppressed numerical factors. The staticforce Fs between with I' the manomer surface density and h(r) the local the layers is obtained by integrating the local pressure separation between the surfaces, is the local monomer over the surface normal to the overlapped brushes, Le., number concentration which is equal to the local counover values of r in the range defined by h < h(r) < 2L, terion number concentration (we assume that all the where h is the minimal distance between the curved monomers on the polyelectrolyte chain are fully ionized). surfaces and L is the brush thickness. We obtain In the presence of salt, at a uniform concentration cg, the average distance 1 = [ac(r)]-ll2 (where a is a microscopic F, = TRI'log [ ( 2 L / h ) ( l +2c,h/I')/(l+ 4c&/I')] (1) length scale, of monomer dimensions) between the charged chains exceeds the Debye screening length K-'(r) that where R is the radius of curvature of the surface (we characterizes the thickness of the counterion "atmospheren. assumed R >> 2L > h). Notice that for large salt The osmotic repulsion can be estimated by multiplying concentrations, 4cb/I' >> 1, the static force behaves as Fs = TRI'2/2cSh. In the limit of zero added salt the above + Permanent address: Department of Physics, Bar-Ilan Univerexpression reduces to sity, Raman-Can59200, Israel. Materials Department. Fs TRI'lOg (2L/h) (2) 8 Department of Chemical and Nuclear Engineering. (1)Rabin, Y.; Alexander, S. Europhys. Lett. 1990, 13, 49. which is several orders of magnitude larger than the cor(2) Fredrickson, G. H.;Pincus, P. Langmuir 1991, 7, 786. responding estimates for neutral brushes,&'FsO = (3) Pincus, P. Macromolecules 1991,24,2912. (4) Witten, T. A.; Pincue, P. Europhys. Lett. 1987,3, 315. T R ~ ( ~ c z justifying ~ / ~ ) ~ the / ~neglect , of the latter for all
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but the highest salt concentrations. Notice that the logarithmic dependence of the static force on the distance between the surfaces differs from the result derived in ref 3 where the limit relevant for colloidal stabilization, R < L, was considered. It is important to emphasize that up to this point we did not need to make any assumptions about the state of interpenetration between the brushes, a subject that is still open to debate. We now turn the dynamics of grafted polyelectrolyte layers and consider the following situation: The brushes are moving toward each other with a relative velocity V, (V, < 0 under compression) such that the instantaneous minimal separation between them is h. For nonoverlapping layers, h >> 2L, the arguments of ref 2 apply and one recovers the modified Reynolds expression for the hydrodynamic lubrication force, FH = -6rR2qVz/(h- 2L). When the brushes are subjected to steady compression under conditions corresponding to strong overlap, h > 1, we obtain le10.819
[H = d[log ( ~ c , / c ) / u c ] ” ~ (6) a result that differs only by a salt-concentration-dependent logarithmic correction from the zero-salt expression, eq 4. Notice that as salt concentration is increase further, eventually it reaches the point where the Debye screening length becomes comparable to monomer dimensions; at this point the nonelectrostatic contributions to the osmotic pressure become important and the results of ref 2 should apply. Substituting the above expression into eq 3 and neglecting the slow spatial variation of the logarithmic term, we obtain
FH = -rvVfi2aI’[log (2L/h) -l]/log ( ~ c , / ( c ) ) (7) where ( c ) is the average concentration in the compressed polyelectrolyte layer. We conclude that in the case of overlapped polyelectrolyte layers the hydrodynamic force between the moving surfaces is only weakly affected by the added salt which reduces the zero-salt result by a factor log ( 2 4 ( c ) ) , a result that is quite surprising in view of the strong effect of added salt on the double-layer forces between charged surfaces.13 The reason for the weak saltdependence of the hydrodynamic force can be traced back to the fact that hydrodynamic screening length [H is determined by the geometrical “mesh” size which does not depend on the effective “rod”length lei, as long as this length remains much larger than the average distance between the “rods”. Therefore, [H remains unaffected when le1 initially increases and then decreases upon addition of salt, until one reaches the high-salt regime in which the persistence length becomes smaller than the interchain spacing. Finally we would like to comment on the applicability of our results to other situations, e.g., to adsorbed polyelectrolyte layers. Since the static forces are dominated by the counterions, they depend only on the local polyelectrolyte concentration and not on factors such as the conformationof the polyelectrolyte chains or their degree of interpenetration. Therefore, one may expect that our static force results should be directly applicable to compression of adsorbed polyelectrolyte layers. Since the kinetics of entanglement may be quite different for grafted and adsorbed polymers, the application of our dynamic results to the adsorbed case requires more careful consideration. In both cases direct comparison with existing (12) Shaqfeh, E. S. G.; Fredrickson, G. H. Phys. Fluids A 1990,2,7. (13) Israelachvili, J. N. Intermolecular and Surface Forces;Academic
Press: San Diego, CA, 1989. (14) Luckham, P. F.;Klein, J. J.Chem. SOC.,Faraday Trans. 198480, 865.
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experiments on adsorbed polyelectrolytelayers is hindered by the fact that in all the studies we are aware of, compression of the layers appears to affect the strength of adsorption, presumably as a result of electrostatic interaction between the polyelectrol@s and the charged surfaces.14 In order to avoid this complication, experiments Will have to be done on surfaces that do not have strong electrostatic interactions with the grafted or the adsorbed polymer layers.
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Acknowledgment. We acknowledge important discussion with J. Israelachvili,Y. Kamiyama, J. Klein, and E.Shaqfeh. Y.R. thanks P. Pincus for hoepitality during his stay in uc santaBarbara. G.H.F. is grateful to the National ScienceFoundation for support under PYI Grant DMR-9057147. P.P. acknowledgesfinancialsupport from IBM SURS grant and from DOE under Grant DE-FO387ER45288.
