114
Langmuir 1990,6, 114-118
Elastic Percolation in Suspensions Flocculated by Polymer Bridging Yasufumi Otsubo Department of Image Science and Technology, Faculty of Engineering, Chiba University, 1-33 Yayoi-cho, Chiba-shi 260, Japan Received January 3, 1989. I n Final Form: June 1, 1989 The dynamic viscoelasticity was measured for silica suspensions flocculated by polymer bridging. When the particle and polymer concentrations exceed some critical values, the suspensions respond elastically to small deformations because a network structure of unbounded flocs is formed over the system. The dispersed particle is compared to the site and the polymer bridge to the bond in the percolation concept. Boundaries for elastic percolation were determined in terms of the site and bond occupancy, and the effect of particle size on the percolation boundary was examined. The theoretical percolation threshold depends on the coordination number of the lattice. A simple cubic lattice was adopted to analyze the percolation process. For suspensions containing sufficient adsorbing polymer, the critical behavior can be characterized by the site percolation. The critical site occupancy is about 0.35, and the elasticity exponent is about 4 irrespective of particle size. However, the bond percolation process cannot be explained by the theory on the same lattice. The critical bond occupancy experimentally determined for elastic percolation of flocculated suspensions is lower than the theoretical prediction and varies with the particle size. In bridging flocculation, a polymer chain adsorbs onto two or more particles to bind them together whereas each bond is regarded as independently distributed in the theoretical calculation. A series of several bridges formed by one polymer chain is not broken down to noninteracting bonds; this may be responsible for the low value of critical bond occupancy. Introduction Dilute suspensions of strongly flocculated particles consist of a collection of discrete flocs. The flocs formed by Brownian flocculation have been shown to be fractals.'" In more concentrated suspensions, the flocs cease to be discrete, and above some critical volume fraction a network structure is formed. When the network of unbounded flocs is built up over the system, the flocculated suspensions respond elastically to small deformations. The elasticity arises only from the attractive forces between particles. However, the three-dimensional network structure which is sufficiently strong to transmit the elastic forces through floc-floc bonds is essential for elastic responses of suspensions. The elasticity is attributed to a combination effect of attractive forces between particles and formation of network structure. Therefore, the appearance of elasticity is closely related to the structural changes from discrete flocs to unbounded flocs. Recently, percolation theory has been employed to understand the elastic properties of flocculated suspensions. Site percolation deals with the distribution of cluster sizes for particles distributed in an infinite lattice composed of sites linked together by bonds. When sites are occupied at random with probability ps and adjacent occupied sites are connected, there exists a critical probability p: above which unbounded clusters of connected sites occur. The main in the site percolation problem is to evaluate the mean cluster size as a function of the probability p s and to predict the transport properties of the percolating network above the critical probability p:. The elastic properties of percolating systems includ(1) Weitz, D. A.; Oliveria, M. Phys. Reo. Lett. 1984, 52, 1433. (2) Matsushita, M.; Sumida, K.; Sawada, Y. J . Phys. SOC.Jpn. 1985,54,2786. (3) Wiltzius, P. Phys. Reu. Lett. 1987, 58, 710. (4) Sonntag, R. C.; Russel, W. B. J . Colloid Interface Sci. 1986, 113, 399. (5) Stauffer, D. Introduction to Percolation Theory; Taylor & Francis: London, 1985. (6) de Gennes, P.-G. J . Phys. (Paris) Lett. 1976, 37, L-I.
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ing flocculated suspensions are extensively studied from both theoretica17-10and experimental"-'* points. Polymer molecules can be used to induce either stabilization or flocculation in colloidal suspensions. When a high molecular weight nonionic polymer causes flocculation, the process has been interpreted by a bridging mechanism in which the polymer chain adsorbs simultaneously onto two or more particles to bind them t ~ g e t h e r . ' It ~ is generally accepted that bridging flocculation occurs under conditions where the polymer chain is long enough and the surface cover e by adsorbed polymer is low. In previous papers:6' we have reported that silica particles with diameters of the order of 10 nm are easily flocculated by addition of polyacrylamide with molecular weights of about 5 X lo6 even over the complete coverage because the particle size is comparable to the loop length of adsorbed polymer. The flocculated suspensions show elastic responses to small strains.20In suspensions flocculated by polymer bridging, the particle-particle bond (bridge) is compared to the bond in the percolation concept. In a similar manner to site percolation, unbounded clusters are expected when the occu(7) Feng, S.; Sen, P. N. Phys. Reu. Lett. 1984,52,216. (8) Kantor, Y.; Webman, I. Phys. Reu. Lett. 1984,52, 1891. (9) Feng, S.; Sen, P. N.; Halperin, B. I.; Lobb, C . J. Phys. Reu. B 1984,30,5386. (IO) Mall, S.; Russel, W. B. J. Rheol. 1987, 31, 651. (11) Zosel, A. Rheol. Acta 1982,21, 72. (12) Buscall, R.; McGowan, I. J.; Mills, P. D. A.; Stewart, R. F.;
Sutton, D.; White, L. F.; Yates, G. E. J . Non-Newtonian Fluid Mech.
1987, 24, 183. (13) Sonntag, R. C.; Russel, W. B. J. Colloid Interface Sci. 1987, 116, 485. (14) Heyes, D. M. J. Non-Newtonian Fluid Mech. 1988,27, 47. (15) Fleer, G. J.; Lyklema, J. J. Colloid Interface Sci. 1974, 46, 1. (16) Otsubo, Y.; Umeya, K. J. Colloid Interface Sci. 1983, 95, 279. (17) Otsubo, Y.; Umeya, K. J. Rheol. 1984,2& 95. (18) Otsubo, Y.; Watanabe, K. J . Non-Newtonian Fluid Mech. 1987, 24, 265. (19) Otsubo, Y.; Watanabe, K. J . Colloid Interface Sci. 1988, 122, 346. (20) Otsubo, Y.; Watanabe, K. J . Colloid Interface Sci. 1989, 127, 214.
0 1990 American Chemical Society
Langmuir, Vol. 6, No.I, 1990 115
Elastic Percolation in Flocculated Suspensions
pancy of bonds, pB,in a lattice whose sites are fully occupied exceeds a critical value pcB. In this study, the dynamic elasticity was measured for silica suspensions flocculated by bridging. Considering that the dispersed particles correspond to the sites and the polymer bridges to the bonds, the boundaries for elastic percolation are experimentally determined in the pspB diagram. The existing percolation theories completely ignore correlations between sites or bonds. However, in bridging flocculation, many bridges are formed by one polymer chain, and a series of bridges cannot be broken down to noninteracting bonds. The relation between percolation boundary and number of bridges formed by one polymer chain is examined by using suspensions of silica particles with different diameters.
Materials and Methods Materials. The suspensions were composed of silica, polyacrylamide (PAAm), glycerin, and water. The media were solutions of PAAm at concentrations of 0.05-0.5% by weight in a mixture of glycerin and water with a 50/50 mixing ratio. PAAm having a molecular weight of (5-6) X lo6 was obtained from Aldrich Chemical Co. and was used as received. The silica samples were Aerosil 130, 200, and 300 from Degussa Co., the particle diameters of which were 20, 15, and 10 nm, respectively. . suspenThe density of primary particles is 2.1 g - ~ m - ~Silica sions were prepared a t concentrations of 3-12% by weight. After the medium and the required amount of silica were mixed by hand, the suspensions were ultrasonicated for 10 min to eliminate entrapped air. The suspensions aged only for 1 day often showed an irreversible increase in viscosity with shearing time a t shear rates beyond a certain This behavior may be attributed to the flocculation by shear-induced bridging in which the floc-floc bond is formed by adsorption of a polymer chain extending from one particle to a particle in the other floc during collision. Aging has a significant effect on the viscosity behavior of suspensions when the period is less than 1 week. With an increase in the aging period, the initial viscosity increases and the critical shear rate decreases; this shows that the flocculation process is slow in a quiescent state. The suspensions aged for more than 10 days gave their final equilibrium responses. Although the settlement of flocs was not observed, the bridged particles seem to form large flocs. The rheological measurements were carried out for sufficiently aged suspensions. Methods. Dynamic viscoelastic properties were measured in the frequency range 1.5 X lo-* to 6.3 X lo1 s-’ by using a parallel disk geometry on a Rheometrics System IV rheometer. The diameter of the upper disk was 50.00 mm, and the gap distance between two plates (sample thickness) was 1.000 mm. In general, the storage modulus G‘ of flocculated suspensions is constant a t low strains and decreases with increasing strain. The nonlinearity reflects a breakdown of the network structure. Measurements in the linear ranges are required to discuss the elastic properties of suspensions in relation to percolation threshold. Adsorption of polymer on the silica surface was measured with suspensions dispersed in a 0.5 wt % solution of PAAm. For the determination of the concentration of nonadsorbed polymer, the particles or flocs were separated by centrifugation at lOOOg,and the viscosity of supernatant solution was measured. The amount of polymer adsorbed on the particles was calculated from the residual polymer concentration. In addition, the final sedimentation volumes enable us to calculate the bridging distance between particles in the flocs.
Results By dynamic viscoelastic measurements, the storage modulus Gf and loss modulus G” are determined at different frequencies. When the degree of flocculation is low (at low particle and polymer concentrations), both moduli have comparable values. In highly flocculated suspensions, the elastic responses are predominant, producing values of the storage modulus more than 10 times those
Y
(3
lo2 u)
3 3 0
g
10’
QI
0
2
0
loo
sr lo-‘ 1
Angular
frequency
W
(s-’)
Figure 1. Frequency dependence of storage modulus for suspensions of 15-nm silica in a 0.5 wt % PAAm solution.
of the loss modulus. Since the research interests center mainly on the elastic properties, the results for the storage modulus only will be presented. Figure 1 shows the frequency dependence of the storage modulus for suspensions of 15-nm silica in a 0.5 wt 5% solution of PAAm. At particle concentrations of 3 wt % and below, the storage modulus rapidly and linearly decreases with decreasing frequency on the log-log plot. The medium and suspensions show almost the same viscoelastic behavior over the entire range of frequencies. When the particle concentration is increased beyond 4 wt %, the frequency-dependent curve shows a plateau, the height of which increases with particle concentration. It is well-known21’22 that the viscoelastic function of suspensions dispersed in polymer solutions shows a plateau in the Newtonian range of the medium. The plateau has been considered to be a manifestation of network structure of particles. An important feature of the concentration dependence of the storage modulus is the existence of a critical value for appearance of plateau. The critical concentration seems to be 3 wt % for suspensions dispersed in the 0.5 wt 70 polymer solution. In this study, it is assumed that the three-dimensional network of flocs is achieved; i.e., elastic percolation occurs a t a particle concentration where the storage modulus reaches 5 Pa in the range 10-2-10-1 s-’. The suspensions a t 4 wt 5% and above are considered to be percolating systems. But it must be stressed that the critical particle concentration depends on the polymer concentration. Figure 2 shows the frequency dependence of the storage modulus for 5 wt % suspensions of 15-nm silica in solutions containing PAAm at different concentrations. At low polymer concentrations, the suspensions do not show the plateau region, but the storage modulus rapidly decreases with decreasing frequency. At polymer concentrations above 0.25 wt %, the plateau due to the network formation is observed. The polymer concentration also has a critical value for elastic percolation. Another important point is that the storage modulus first increases with polymer concentration and then reaches a constant (21) Onogi, S.; Matsumoto, T.; Warashina, Y. Trans. SOC.Rheol. 1973, 17,175. (22) Umeya, K.; Otsubo, Y. J. Rheol. 1980,24, 239.
Otsubo
116 Langmuir, Vol. 6, No. 1, 1990 I
Angular
frequency
I
I
I
w
(s-'1
Figure 2. Frequency dependence of storage modulus for 5 wt % suspensions of 15-nm silica in solutions containing PAAm
loo
at different concentrations.
value above 0.3 wt 70. In the range 0.3-0.5 wt 70,the storage modulus is independent of PAAm concentration. The dynamic viscoelastic measurements were repeated using silica particles with different diameters. The results for the percolation threshold are summarized as follows: (a) the critical particle concentration is lower for suspensions containing polymer a t higher concentrations and (b) two critical concentrations for particle and polymer decrease with decreasing particle diameter. For further analysis, the value of the storage modulus at 1.5 X lo-' s-l is adopted as a measure to characterize the percolating suspensions. Based on scaling analyses, the elastic modulus G can be expressed by the following equation near the critical concentration C,:
G = k(C - C,)" where C is the particle concentration, k is a constant, and n is the critical exponent. Many author^"-'^^^^ have reported that the critical exponent for flocculated suspensions varies in the range 2.0-5.3 in spite of quite different values of C,. However, difficulties in the analysis are the definition and determination of percolation threshold. Presumably, the limitations of the rheometer may give another factor in controlling the threshold. From Figure 1,the critical particle concentration has been estimated to be 3 wt % for suspensions in 0.5 wt % PAAm solution. This value is the same for suspensions in 0.3 and 0.4 wt % PAAm solutions, but it is not constant for others. The suspensions prepared with more dilute polymer solutions have higher critical concentrations. In the present study, as the first approach, the critical particle concentration is assumed to be 3 wt % for suspensions of 15-nm silica. The validity of this treatment will be given later. Figure 3 shows the storage modulus a t 1.5 X lo-' s-' plotted against the particle concentration above C, for suspensions of 15-nm silica. A t 0.5 wt % PAAm, the plots are closely related by a straight line with a slope of 4.0. The critical exponent is within the range previously reported. As the PAAm concentration is decreased, the plots begin to depart from the line, especially a t higher particle concentrations. At a constant particle concentration, the storage modulus increases with PAAm con(23) Russel, W. B. Powder Technol. 1987,51, 15.
IO2
10'
c - cc
(wt%)
Figure 3. Storage modulus at 1.5 X IO-* s-l plotted against particle concentration above the percolation threshold for suspensions of 15-nm silica.
I Q
O.I
x\
t
x \ 2 x-x-x-*-
0
5 Si I i ca
c o n c e n t ra t i on
=i 10 ( w t %)
Figure 4. Diagram showing the regions of three different flocculated states for suspensions of 15-nm silica: percolation bound-
ary (-); saturation coverage boundary (- - -); nonpercolating system (x); percolating and completely bridging system ( 0 ) ;percolating and partially bridging system (0). centration and becomes constant above some critical value. In suspensions which show the power law dependence of G' on C - C, with an exponent of 4.0 a t relatively high PAAm concentrations, the bridging flocculation by polymer adsorption would be fully developed. Since further addition of polymer cannot contribute to an increase in elasticity, the degree of bridging is expected to reach the saturation. Also, the suspensions of 10-nm silica and 20nm silica show the power law dependence with exponents of 4.0 and 3.7, respectively, when dispersed in a 0.5 wt 5% PAAm solution. From the rheological data, the flocculated suspensions can be classified into three groups: (a) a nonpercolating system in which no elastic responses are detectable, (b) a percolating system in which the polymer bridging is fully developed, giving the saturated value of storage modulus, and (c) another percolating system in which the degree of bridging still does not reach the saturation (hence this percolating system has a lower storage modulus t h a n t h e former percolating system). For suspensions of 15-nm silica, a diagram showing the regions of three different flocculated states is given in Figure 4. To understand the floc structures, the adsorption and sedimentation experiments were carried out. The results
Langmuir, Vol. 6, No. 1, 1990 117
Elastic Percolation i n Flocculated Suspensions Table I. Experimental Data for Floc Structure silica diameter, adsorbance, bridging concentration, nm mg/g of silica distance, nm wt% 20 15 10
47 55
30
20 17 12
21
13
V
11 10
m
for 3 wt % suspensions dispersed in 0.5 wt % PAAm solutions are summarized in Table I. The bridging distance, i.e., the mean distance between the surfaces of particles which may be arranged in random packing in the sediment, is comparable to the particle diameter. The adsorbance is constant irrespective of polymer and silica concentrations, so the surface coverage of silica by polymer adsorption can be predicted from the initial composition of suspension. The broken line in Figure 4 indicates the boundary of saturation coverage. In suspensions whose compositions are given by points below the boundary, the adsorption or bridging does not reach completion, whereas for suspensions above the boundary line, the complete coverage of particles takes place and the excess polymer remains in the solution phase. The adsorption behavior shows good agreement with the rheology. Therefore, it is concluded that the storage modulus reaches the saturated value when the complete bridging of dispersed particles occurs at a given particle concentration.
Discussion In the site percolation process, two adjacent occupied sites are necessarily connected; that is, the bond occupancy ( p B ) is 1.0. Above the boundary line in Figure 4, complete coverage of particles takes place, and hence p B = 1.0. The elastic behavior of suspensions which contain sufficient adsorbing polymer (0 in Figure 4) would be explained by the site percolation theory. It can be seen that the critical particle concentration C, = 3 wt % used in the analysis for suspensions of 15-nm silica is the percolation threshold for the site problem. On the other hand, it is assumed in the bond percolation process that all sites are fully occupied in the lattice, i.e., p s = 1.0. If the particle concentration a t which the site occupancy reaches 1.0 is determined, the elastic behavior of such suspensions can be analyzed as a function of pB through the bond percolation theory. However, it is difficult to directly translate the particle concentration into the site occupancy only from the rheological data. From the sedimentation experiments, the maximum particle concentration in flocs is 11 wt % for suspensions of 15nm silica. Therefore, it is considered that an 11 wt % suspension with fully developed polymer brid ing has the limiting floc structure with p s = 1.0 and pi = 1.0. In practice, the particle concentration can be increased beyond this value in the absence of PAAm because different floc structures are possible. Let us now consider the percolation process for p s < 1.0 and pB < 1.0. Referring to Figure 4, the elastic percolation can take place when both p s and p B are increased to some levels. This process is referred to as site-bond percolation. Once the limiting floc structure is known, the particle concentration can be converted into the site occupancy p s and the polymer concentration into the bond occupancy p B . For instance, an 8 wt % suspension in a 0.2 w t % solution of PAAm corresponds to p s = 0.73 and pB= 0.42 (the concentration of polymer solution must be 0.48 wt % to achieve the complete coverage of particles). Figure 5 shows a boundary for elastic percolation of suspensions of 15-nm silica. All the plots above the saturation coverage line in Figure 4 lie on the abscissa
a J
0.5 -
0
U C
0
m
-
0 0
1.0
0.5 Site
occupancy
Figure 5. Percolation transition on the pS-pB diagram for suspensions of 15-nm silica: nonpercolating system (X); percolating system (0).
0' 0
"
'
'
'
'
'
0.5 Site
'
'
J
I .o
occupancy
Figure 6. Effect of particle diameter on the percolation boundary: 10 nm (- - -): 15 nm (- - -); 20 nm (-).
of p B = 1.0. The pS-pB diagram demonstrates that the critical probability for the site problem, p:, is about 0.35 and that for the bond problem, pcB,is about 0.1. Many authors have discussed the critical percolation probability for various lattices. For the site percolation of simple cubic lattice, the critical probability p: has been estimated to be 0.325 by Monte Carlo methods24 and 0.307 by series methods.25926Since the theoretical and experimental values for p: are in good agreement, the simple cubic lattice would be an appropriate model to interpret the site percolation process for sample suspensions. Computer simulations also give information on the critical behavior of the cluster size for the bond problem of simple cubic lattice. The critical probability pcB has been estimated to be 0.254 by Monte Carlo methods2' and 0.247 by series method^.^^'^' However, the experimental value is very low compared with theoretical predictions. For the suspensions studied, the elastic percolation occurs a t very low bond occupancy. If the particles build up a floc structure whose coordination number z is 6, the average number of bridges, when formed at random with probability, pcB = 0.1, is 0.6 for (24) Frish, H. L.; Sonnenblick, E.;Vyssotsky, V. A.; Hammersley, J. M.Phys. Reu. 1961,124,1021. (25) Sykes, M. F.; Essam,J. W. J . Math. Phys. 1964,5,1117. (26) Sykes, M.F.; Essam, J. W. Phys. Reu. 1964,133,A310. (27) Vyssotsky, V. A,; Gordon, S. B.; Frisch, H. L.; Hammersley,L. M.Phys. Rev. 1961,123,1566.
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Langmuir 1990, 6, 118-121
each particle. According to the empirical equation” for the bond problem, the percolation threshold is zpCB= 3 / 2 for various three-dimensional lattices. It is surprising that one bridge per particle is not necessary for elastic percolation in suspensions flocculated by polymer bridging. Boundaries for elastic percolation can be determined through the same procedure for suspensions of silica particles with different diameters. Figure 6 shows the effect of silica diameter on the percolation boundary. The curves for 15-nm and 20-nm silica particles are very similar. As mentioned above, the critical particle concentration varies with particle diameter. But the critical occupancy p: for the site percolation process is independent of diameter. On the other hand, the critical bond occupancy pcBfor 10-nm silica particles is higher than that for larger particles. This is also a significant difference between theory and the actual bridging process. In bridging flocculation, a polymer chain adsorbs onto two or more particles whereas each bond is regarded as distributed at random in theoretical calculation. The particle diameter causes changes in the bridging distance and the curvature of adsorbed segments and in turn the interac-
tions. Since one polymer chain can make many bridges in series, the bonds are strongly correlated due to longrange interactions. Provided that the silica particles are arranged in a simple cubic lattice in the sediment and all bonds are fully occupied, the average number of bridges which are formed by one polymer chain can be calculated from the adsorbance and molecular weight. The results are as follows: 1260 bridges for 10-nm, 135 bridges for 15-nm, and 63 bridges for 20-nm silica particles. In addition, the minimum floc sizes are estimated to be 420 particles for 10nm, 41 particles for 15-nm, and 21 particles for 20-nm silica. Although the assumption that the polymer chains are distributed a t random in the flocs will be reasonable, a series of bridges formed by one chain cannot be broken down to noninteracting bonds. This may be responsible for the low value of pcBand the dependence on the particle diameter. To understand the elastic percolation in the polymer bridging process, information on bond distribution in the flocs is important, because from a statistical point the density of bonds which belong to the same polymer chain may decrease with increasing radius. Registry No. SiO,, 7631-86-9; PAAm, 9003-05-8.
(28) Ziman, J. M.J . Phys. C 1968,I, 1532.
A Basic Approach to Surface Diffusion on Heterogeneous Substrates V. Pereyra and G. Zgrablich* Instituto de Investigaciones en Tecnologia Quimica, Universidad Nacional de Sun Luis-Conicet, Casilla de Correo 290, 5700 Sun Luis, Argentina Received January 19, 1989 We report on a general formulation of the problem of surface diffusion of interacting adsorbed particles on heterogeneous surfaces and the effect of heterogeneity degree on the diffusivity. The problem of surface diffusion of adsorbed atoms and molecules has received the attention of many researchers in the last 2 decades, and its importance is growing due to its implications in the material science area. The main effort has been dedicated to study diffusivities on perfectly homogeneous which are characterized by a well-defined single activation energy, E,, for the jumping process of noninteracting particles. Adsorbate-adsorbate interactions make this activation energy coverage dependent, and this is taken into account through appropriate approximations for the description of the adsorbed lattice gas. In spite of the importance of heterogeneous surfaces, given that they are closer represen* Author t o whom correspondence should be addressed. (1) (2) (3) (4) (5) (6) (7) (8)
Bowker, M.; King, D. A. Surf. Sci. 1978,71,583. Bowker, M.; King, D. A. Surf. Sci. 1978,72,208. Reed, D.A,; Ehrlich, G. Surf. Sci. 1981,102,588. Reed, D.A.; Ehrlich, G. Surf. Sci. 1981,105,603. Murch, G.E. Philos. Mag. 1981,43A,871. Sadiq, A.; Binder, K. Surf. Sci. 1983,130,348. Naumovets, A. G.; Vedula, Yu. S. Surf. Sci. Rep. 1985,4,365. Zhdanov, V.P. Surf. Sci. 1985,149,L13.
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tations of “real” surfaces, very little is known about how to explicitly incorporate the principal characteristics of heterogeneous substrates (i.e., adsorptive energy distribution and correlation function) into the diffusion coefficient. Several existing models for diffusivity on heterogeneous surfaces”13 are far from achieving this goal. Two formulations, however, may be somewhat closer. First, in ref 14 the continuous time random walk (CTRW) model of Scher and Montroll15 is used, and the diffusion coefficient is averaged with an activation energy distribution. No coverage dependence is given through this (9) Smith, R. K.; Metzner, A. B. J. Phys. Chem. 1964,68,2741. (10) Horiguchi, Y.; Hudgins, R. R.; Silveston, P. L. Can. J. Chem. Eng. 1971,49,76. (11) Sladeck, K. J.; Gilliland, E. R.; Baddour, R. F. Ind. Eng. Chem. Fundam. 1974,13,100. (12) Ponzi, M.; Papa, J.; Rivarola, J. B.; Zgrablich, G. AIChE J. 1977,23,347. (13) Horas, J. A.; Marchese, J.; Rivarola, J. B. J. Chem. Phys. 1980, 73,2977. (14) Klafter, J.; Silbey, R. Surf. Sci. 1980,92,393. Montroll, E. W. Phys. Reu. 1975,B12,2455. (15) Scher, H.;
G 1990 American Chemical Society
