Faraday Discuss. Chem. Soc., 1983, 76, 137-150 Mass-diffusion and Self-diffusion Properties in Systems of Strongly Charged Spherical Particles BY RUDOLF KLEIN AND WALTER HESS Fakultat fur Physik, Universitat Konstanz, D-7750 Konstanz, West Germany Received 6th May, 1983 Using the Fokker-Planck equation as the basic description, various collective and single- particle properties of highly charged spherical macroparticles in solution are derived. The dynamic structure factor S(k,t) for finite k is found to be non-exponential in t due to the viscoelastic properties of the liquid of interacting macroions. By using a mode-coupling pro- cedure the dynamical properties can be completely reduced to the static structure factor S(k), which is calculated according to the method developed by Hansen and Hayter.At short times the system exhibits elastic behaviour and the corresponding high-frequency elastic moduli have been calculated. The bulk modulus is found to increase with c 2 , where c is the con- centration. With regard to the single-particle properties, the velocity autocorrelation function is calculated and from it are obtained the mean-square displacement and the self-diffusion coefficient D, as a function of concentration and of the coupling strength between macroions. 1. INTRODUCTION Colloidal suspensions can be considered as binary liquids consisting of two very different components. The solvent is a simple liquid and the macroparticles are much larger and heavier than the solvent molecules. In concentrated solutions each macroparticle experiences not only frequent collisions with solvent molecules but also the presence of other macroparticles through direct deterministic interactions. In this paper, highly charged spherical particles are considered, where the main interactions are the hard core (excluded volume) and the fairly long-ranged screened Coulomb interactions.Such systems are realized by polystyrene spheres in aqueous solutions. - 4 It is known experimentally that these systems exhibit highly correlated behaviour for volume fractions of lop4 and less. Under such circumstances the neglect of hydrodynamic interactions seems to be a good approximation. Since the solution consists of two very different components the dynamical behaviour of the total system will take place on two separated time scales.Most experiments such as light scattering look only at the subsystem of macroparticles; therefore, it is sufficient to develop a theoretical description for the dynamics which is applicable on a time scale on which the fast dynamics of the solvent molecules plays no role. Such a description is provided by the Fokker-Planck equation, which is the equation of motion for the distribution function f(r, t ) = f(pl, . . . ,pN,rl, . . . ,rN,t) of the momenta pi and coordinates r j of the centres of mass of the spherical macroparticles. On the basis of the Fokker-Planck equation the solvent is described as a continuous medium which produces friction and mediates a hydrodynamic interaction bet ween different macro particles. Static light scattering shows a liquid-like structure factor S(k) with a pronounced first peak.Since the scattering arises practically only from the macroparticles one can speak of the subsystem of macroparticles as a ‘liquid of interacting Brownian138 STRONGLY CHARGED PARTICLES particles’. The main difference from an ordinary so-called ‘simple’ liquid is that each constituent of the Brownian liquid experiences friction due to the presence of the solvent. In this paper the dynamical properties of the Brownian liquid are developed in a similar way to the generalized hydrodynamics of simple liquids. For the mass- or collective-diffusion properties the dynamic structure factor S(k,t) is calculated by a two-step memory function procedure, the first memory function being a generalized diffusion function and the second the longitudinal dynamic viscosity of the liquid of Brownian particles. For the case of single-particle properties such as the velocity autocorrelation function and the self-diffusion coefficient it is the self-dynamic structure factor G(k,t) which is treated similarly.To calculate the second memory function using this procedure a mode-coupling method is used which reduces the dynamics essentially to static properties. The static property of most interest is the static structure factor S(k). Following the work of Hayter and Penfold and of Hansen and Hayter we have calculated S(k) for the polystyrene solutions investigated by Gruner and Lehmann.3 The solution is described as a one-component plasma of macroions of finite size. The counterions are treated as a uniform neutralizing background, which determines the screening of the ion-ion interactions, taken to be a repulsive screened Coulomb interaction.The attractive van der Waals interactions are assumed to be negligible since the Coulomb interactions are in most cases larger than kBT. The input for the rescaled mean-spherical approximation (MSA) method developed by Hansen and Hayter is the Verwey-Overbeek two-particle potential where E is the dielectric constant of the solvent, a is the radius of the macroions, $o is the surface potential and K is the Debye-Huckel screening parameter. The value of t,bo is determined by fitting the height of the main peak of S(k) as obtained from the MSA calculation to the experimental result at one concentration.Keeping this value fixed for all other samples which were investigated by Gruner and Lehmann, good agreement was obtained with the experimentally determined S(k), with the exception of small scattering angles. The theoretical results are always smaller for k -+ 0 than the extrapolations of the experimental curves to k -, 0. Since later results are quite sensitive to the values of S(k) at long wavelengths, we have corrected7 the long- wavelength part of S(k) by a phenomenological m e t h ~ d . ~ In this way static structure factors are available, if the system parameters in eqn (1.1) and the volume fraction q~ are specified. In the next section the theory for the collective properties is outlined. It is shown that S(k,t) deviates from a simple exponential function outside the hydrodynamic regime because of the frequency and wavevector dependence of the dynamic longi- tudinal viscosity of the liquid of interacting macroions.Satisfactory agreement with experimental results from dynamic light scattering is obtained. The viscoelastic behaviour of the system is further illustrated by calculating the short-time or high- frequency elastic moduli. In the third section, one-particle properties such as the velocity autocorrelation function of a tagged particle and the self-diffusion coeffi- cient as a function of concentration and of the interaction strength are obtained.R. KLEIN AND W. HESS 139 2. MASS- OR COLLECTIVE-DIFFUSION PROPERTIES As mentioned in the introduction, the transport equation of a Brownian liquid is the Fokker-Planck equation a -fTr,t) = fif(r,t) (2.1 ) at where O(r) is the Fokker-Planck operator where pi denotes the forces which all other macroparticles exert on macroparticle i through direct interactions and lo is the friction coefficient at infinite dilution.The first part of eqn (2.2) is identical to the Liouville operator of a simple liquid and the second part originates from integrating out the fast solvent variables in the Liouville equation. O , The basic phase-space variables are the concentration fluctuations N ~ ( k ) = 1 exp( - ik ri) - (1 exp( - ik ri))o (2.3) i = 1 1 where the subscript zero indicates an equilibrium average, and the current fluctuations Because of the non-hermitean character of 6 the time derivatives of phase-space functions are given by operating with the hermitean adjoint operator Q+.One obtains a+P(j) = -ik - j ( k ) (2.5) In the last line the pressure fluctuations kB T 1 S(k) N $(k) = - P(k) ; S(k) = - (P(k) i'( -k))* and the fluctuating force density f ( k ) = ik a(k) - coj"(k) have been introduced, where a@) is the viscous stress tensor of the 'liquid of inter- acting Brownian particles'. The latter is formally identical to the expression obtained in the theory of simple liquids.12 The main difference between our Brownian liquid, in which every particle experiences friction by the solvent, and a simple liquid is the presence of the second tezm in eqn (2.8). It originates from the second part (the 'Fokker-Planck part') of SZ in eqn (2.2). For a study of collective properties of a system a quantity of central importance is the dynamic structure factor S(k,t), which can experimentally be obtained from scattering experiments. It is defined by140 STRONGLY CHARGED PARTICLES I S(k,t) = T(P(k) exp(fit)t( -k))oO(t).(2.9) Results for S(k,t) for a system of particles described by a Fokker-Planck equation are now derived using a Mori-Zwanzig projection operator formalism. Taking the time derivative of eqn (2.9) and using eqn (2.5) d i at N - S(k,t) = - k - ( j ( k ) exp@t)?( -k))oO(t) + S(k) &I). (2.10) Defining the Laplace transform by fa, 1 f(k,z) = dt exp( - z t ) S(k,t) = A (?(k) [z - a]- ?( - k))o J O N eqn (2.10) becomes 1 zS(k,z) = S(k) - k * - ( J k ) [Z - 61-l ?(--k))o. (2.12) N Since according to eqn (2.5) the time derivative of the concentration fluctuations vanishes in the long-wavelength limit (k -+ 0), which exp_resses the conservation of the number of Brownian particles, a projectiqn operator P, will be introduced which projects an arbitrary phase-space function A(k) on the concentration fluctuations, which are the slow variables of our system 1..1 P,A(k) = ~ ( 4 k ) ?( - W o (c(k)- (2.13) N S(k) Using in eqn (2.12) the operator identity [z - 61-1 = [z - sz Q,y (1 + SiP,[z - a]-1) where & = 1 - p,, leads to (2.14) i N zf(k,z) = S(k) - k - (i(k) [Z - Si Qc]- ?( -k))o 1 - k--(Ak)[z - SiQc]-lSiPc[~ - 61-l P(-k))o. (2.15) N The se_cond term vanishes since Qc projects on the subspace orthogonal to ?(k) and since j ( k ) is orthogonal to t(k). Inserting eqn (2.13) in eqn (2.15), the last term factorizes, one term being $(k,z) according to eqn (2.1 1).Therefore S(k) S(k,z) = z + &k,z) k2 (2.16) wherej,,(k) is the component ofj’(k) parallel to k. The hydrodynamic limit of D(k,z) defines the collective- or mass-diffusion coefficient D, = D(0,O). (2.18)R. KLEIN AND W. HESS 141 In this limit the dynamic structure factor S(k,t) becomes a simple exponential func- tion of the time S(k,t) + S(0) exp( - D,kzt). (2.19) The function D(k,z) for non-zero k and z generalizes the diffusion coefficient to a wavevector- and frequency-dependent diffusion function. Since the light-scattering experiments on polystyrene spheres also cover the non-hydrodynamic regime, this generalized diffusion function is important. It can be further analysed by applying the Mori-Zwanzig technique to the Forrelation function in eqn (2.17) with a projec- tion operator Pj which projects on j ( k ) Using the same procedure as for s(k,z), one obtains where Q -= 1 -A Pi.Since cu_rrent _and c_oncentration fluctuation are orthogo_nal, QC Pj = Pj and Qc Qj = 1 - P, - Pj = Q. Using the definition eqn (2.20) for Pj in the second term in eqn (2.21) and an operator identity similar to eqn (2.14) in the third term yields (2.22) where r , , ( k , z ) is a generalized friction function and eqn (2.22) can be considered as the generalization of the Stokes-Einstein relation to finite k and z. In the hydro- dynamic limit and for non-interaEting particles [S(O) = 13, eqn (2.22) becomes D, = Do = kBT/co, where io = c(0,O). For interacting systems the mass- diffusion coefficient becomes, from eqn (2.18) and (2.22), and the dynamic structure factor becomes, from eqn (2.20) and (2.22), (2.24) (2.25) where cT(k) = [kBT/rn S(k)]* is the isothermal velocity of sound.142 STRONGLY CHARGED PARTICLES The dynamical friction function [ , ( k , z ) can be further simplified. Using eqn (2.3)-(2.8) in eqn (2.23) (2.26) The total friction function consists of two contributions, the first term is the single- particle friction due to the presence of the solvent and the second term is just the contribution from the longitudinal viscosity of the 'liquid of Brownian particles' (2.27) where k has been chosen in the z direction.It follows from eqn (2.25) and (2.26) that a measurement of the dynamical structure factor gives information about the viscos- ity function of the liquid which consists of the interacting macroions.For poly- styrene spheres of diameters of the order of several hundreds of Angstroms the relaxation time zB = m/C0 is of the order of lop8 s, whereas the shortest correlation times of a dynamical light-scattering experiment are ca. 10- 5-10-6 s. Therefore, eqn (2.25) can well be approximated by Experimental results for S(k, t ) are often characterized by following short-time expansion (2.28) cumulants, which is the S(k,t) = S(k) exp n! (2.29) The first two cumulants of eqn (2.28) are pcL1(k) = D"(k) k 2 ; D"(k) = Do/S(k) (2.30~) P 2 ( 4 = - P 1 W k2 VlI(k,O)/(~ lo). (2.30b) Therefore, the initial slope of loglS(k,t)/S(k)l against t determines the static structure factor, which is a well known r e s ~ l t .' ~ Including hydrodynamic interactions, D"(k) is changed by a k-dependent factor. If Do k2/pu,(k) agrees with the determination of S(k) from static light-scattering experiments, hydrodynamic interactions are unim- portant. This seems to be the case in the experiments. Since ,u2(k) and all higher cumulants are of higher order in k2 the dynamic structure factor S(k,t) for small scattering angles is a simple exponential of t. The initial value q,,(k,t = 0) of the longitudinal viscosity enters the second cumulant; it is essentially determined by the total interaction potential UN exp[ -ik(zi - zj)] )o - - kBT). (2.31) SW The short-time behaviour of correlation functions can also be characterized by a moment expansion or a continued fraction representation. This method can be used to show that the system of interacting macroions exhibits elastic behaviour at short times.Following Zwanzig and Mountain who treated the visco- elastic behaviour of simple liquids, we have introduced high-frequency elastic shear and Schofield,R. KLEIN AND W. HESS 143 and bulk moduli G"(k) and K"(k) for the Brownian fluid. They are given by a combination of the first three moments of the transverse and longitudinal current correlation functions. One obtains 1 - cos kz a2 U(r) k2 ___ ax2 d3r) (2.32) 4 1 - cos kz d2U - G"(k) + K"(k) = c 3kBT + c g(r) 3 ( 1 k2 dz2 - d3r). (2.33) The moduli are therefore given in terms of the radial distribution function g(r) and the two-particle potential U(r).Using eqn (1.1) and the results for the structure factor as discussed in the first section, one obtains from eqn (2.32) and (2.33) the curves displayed in fig. 1 and 2. The high-frequency moduli are qualitatively similar to those of simple liquids; l 7 they decrease with increasing wavevector and are in- creasing functions of concentration. kais Fig. 1. High-frequency elastic moduli as a function of ka,, = ak/q1'3for four volume con- centrations: q = 0.5, 1.5, 3.0 and 7.5 x Full lines: g"(k) = G"(k) 47ca3/(3kBT); dashed lines: e"(k) = E(k) 4na3/(3kBT). The longitudinal modulus E(k) = (4/3)G"(k) + K"(k) can be obtained from the second cumulant, since eqn (2.30b) can be rewritten as l 6 This procedure was used by Griiner and Lehmannls to determine E(k -+ 0) as a function of concentration.They obtained E(k -+ 0) - c2, which is in agreement with our result from eqn (2.33), shown in fig. 2. The shear modulus G"(k + 0) is found by us to be proportional to c4I3. Going back to the time dependence of S(k,t) it is convenient to define a mean relaxation time (2.34)144 STRONGLY CHARGED PARTICLES 101 1 OC n s 8u -0 5 lo-' h = a0 1 6 ' 1 o - ~ I 10-4 1 o - ~ (P Fig. 2. Long-wavelength limit of high-frequency elastic moduli em(0) and g"O(0) as a function of concentration. Note eoo(0) - (p2 and g"(0) - As a measure of the deviation of S(k,t) from a simple exponential function of t we introduce (2.35) where in the second line use was made of eqn (2.28). The experimental determination of A(k) therefore gives the time integral of the dynamic longitudinal viscosity func- tion.The latter has been calculated7# l 6 from eqn (2.27) by a mode-coupling ap- proximation. In the spirit of this approximation one replaces Q in eqn (2.27) by a projection operator which projects onto bilinear products of the slow variables, which are the concentration fluctuations. As a result, qll(k,t) can be completely expressed by the static structure factor and the unknown dynamic structure factor so that the mode coupling expression together with eqn (2.25) and (2.26) give a closed set of equations. From an approximate numerical solution of these equations, using the phenomenologically corrected static structure factor, A(k) has been calculated for four different concentrations. The results are shown in fig.3 together with theR. KLEIN AND W. HESS 145 'I 0 0.5 1 1.5 2 2.5 k / k m Fig. 3. A(k) plotted against k/k,, eqn (2.35). Curves are theoretical results for cp = 0.1, 1.5, 3.0 and 4.5 x Data points are the experimental results of Gruner and Lehmann3 for 40 between 1.5 x and 7.5 x experimental results of Gruner and Lehmann.3 Noting that no adjustable parameter enters except the surface charge, which was already fitted to the static structure factor at one concentration, the agreement is quite satisfactory. The experimental results do not show a definite concentration dependence of A(k) within the investigated range of volume fractions cp. We have calculated A(k) as a function of cp for those values of k for which A(k) reaches its maximum. This value 1 1 I 1 1 0 0.2 0.4 0.6 0.8 1 1029 Fig.4. A(k x 2kJ3) as a function of concentration.146 STRONGLY CHARGED PARTICLES is roughly at k = (2/3) k , = 5.5/(2ais), where k , denotes the position of the main maximum of S(k) and 4, = (3/4~c)'/~is the ionic sphere radius. The result in fig. 4 shows that A(5.5/2ais) increases from zero to a nearly constant value of ca. 0.7 in the concentration range which was investigated e~perimentally.~ The weak increase with cp above cp = 1.5 x explains the observation of Griiner and Lehmann that A(k) seems to be independent of concentration. However, fig. 4 shows that this does not hold at concentrations lower than those used in the experiments. 3. SINGLE-PARTICLE PROPERTIES Recently various attempts have been made to measure single-particle properties in colloidal systems.Therefore we'present here some results of calculations of such properties, which were obtained on the same basis as the collective results. The dynamics of the tagged particle (index I ) is described by the correlation function (3.1) G(k,t) = ( 2 , ( k ) exp(fit)2,( -k))" O ( t ) where 2,(k) = exp( -ik r , ) . From it one can calculate the mean-square dis- placement and the velocity autocorrelation function 1 a 2 1 v(l) = - ( o l ( t ) u1(0)) = - lim - - G(k,t). 3 k - 0 at2 k2 The self-diffusion coefficient is given by D, = lim W(t)/t = 2-03 (3.3) (3.4) Using procedures similar to those which resulted in eqn (2.16) and (2.22), one ob- tains for the Laplace transform of eqn (3.1) (3.5) 1 z + D,(k,z) k2 G(k,z) = Therefore, the velocity autocorrelation function is simply V(t) = Ds(O,t) and the self- diffusion coefficient D, = B,(O,O) = kBT/~,,(O,O).The projection operator formalism gives the longitudinal dynamic self-friction function as L(k4 = 5" + A L , , ( W AL,,(k,z) = p(filz exp( -ik rl)[z - Flz exp(ik rl))O (3.7) where pl denotes the direct interaction forces acting on the t_agged particle and Q1 is the projector onto the subspace perpendicular to E,(k) and jl(k). Since we are againR. KLEIN AND W. HESS 147 restricting ourselves to times larger than the relaxation time zB ;5: simplifies to s, eqn (3.6) D,(k,z) = Do - AB,(k,z) Therefore, the velocity autocorrelation function becomes V(t) = 2 Do d(t) O(t) - AD,(O,t) and the mean-square displacement becomes dt’ ( t - t’) AD,(O,t’).Thus for small times for large times. W(t) = (3.9) (3.10) (3.11) Since AB,(O,O) 3 0, the self-diffusion coefficient D, is always smaller than the free- diffusion constant Do. Eqn (3.10) and (3.11) show how the motion of the tagged particle is slowed down by interactions with other particles. It remains to calculate ATs,,(kLz). This was again done l9 in a mode-coupling approximation by approximating Q by bilinear products of the one-particle fluctu- ation i , ( k ) and of the concentration fluctuations of the other particles. The result of this calculation gives AC,,(k,t) in terms of S(k), G(k,t) and S(k,t) (3.12) where c,(k) = [S(k) - l]/[c S(k)] is the direct correlation function. The integral has been evaluated with the mean-field correlation function GMF(k,t) = exp( - Dok2t) and SMF(k,t) = S(k) exp[ - Dok2t/S(k)].With this result various single-particle pro- perties have been calculated. From eqn (3.8) and (3.9) the dynamic part AD,(O,t) of the velocity autocorrelation function is obtained. The result is shown in fig. 5. With increasing concentration its initial value increases and its decay becomes more rapid. Moreover, the velocity autocorrelation function is not a simple exponential in time. The mean-square dis- placement follows from eqn (3.10). Its initial slope according to eqn (3.11) is Do. With increasing time the neighbouring particles are felt and W(t) grows more slowly than linear. This deviation from the initial behaviour starts earlier for the more concentrated system. The quantity of most practical interest is the self-diffusion coefficient D,, which is calculated from V(t) using eqn (3.4).In fig. 6 D, is shown as a function of con- centration for the type of polystyrene spheres investigated by Griiner and Lehmann.3 One finds a rather sharp decrease at very low concentrations, but for the interval of concentrations investigated by these authors our calculation predicts an only slightly decreasing value of D,/Do between 0.7 and 0.6. Fig. 7 shows D, for a fixed concentration as a function of a dimensionless coupling parameter y7 which essentially measures the surface potential, y = (fl/2) E a $; exp(2alc). Numerical simulations of single-particle properties by Gaylor et aL20 show similar qualitative behaviour. A detailed comparison is, however, not possible, since148 STRONGLY CHARGED PARTICLES lo” 10‘ 6 6 d d T ’O 1 16’ 1 I 1 t Fig.5. Normalized dynamic part of the velocity autocorrelation function as a function of z = Dot/(2a,J2 for four volume concentrations: cp = 0.1, 0.5, 1.5 and 7.5 x 1 0 9 0.5 9 0 0.2 0.4 0.6 0.8 1 I O * q 10-3. Fig. 6. Normalized self-diffusion coefficient as a function of volume concentration for poly- styrene spheres of diameter 900 A, surface potential I,!I~ = 73 meV and screening length K - = 5000 A.R. KLEIN AND W. HESS 149 0 1000 2000 3000 Y Fig. 7. Normalized self-diffusion coefficient as a function of the dimensionless couplin para- meter y for polystyrene spheres of diameter 900 A, screening length c1 = 5000 1 at a volume concentration of cp = 5.6 x lop4. the simulations have been performed for different system parameters. Earlier theoretical work on self-diffusion properties of strongly correlated macroparticle systems is based on the Smoluchowski equation, which is the strong-friction limit of the Fokker-Planck equation.Using the Mori-Zwanzig technique and a mode-coup- ling approximation together with an experimentally measured structure factor, we have given results for W ( t ) and D,. For the particular systems whose S(k) was used, the value of D, turned out to be lower than the value obtained on the basis of the Fokker-Planck equation. 4. CONCLUSIONS The generalized hydrodynamic theory for a Brownian liquid, developed on the basis of the Fokker-Planck equation, leads to the following picture of the dynamics of systems of charged colloidal particles: the strongly correlated system has a short- range structure [reflected in S(k)], consisting of particle clusters of some lifetime Z.For times smaller than Z the system exhibits elastic behaviour, whereas for longer times it is liquid-like. This viscoelastic behaviour is reflected in the time dependence of the viscosity functions. By assuming that the interactions among the macroions are of the form of eqn (1. l), the collective diffusion properties such as S(k,t) and the high-frequency elastic moduli can be calculated and are found to agree with experi- mental results. This agreement suggests a similar treatment for the single-particle properties, the results of which are given in section 3. We thank Dr. J. Hayter for providing us with the computer program to calculate S(k)- J.C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J . Phys. A , 1975,8, 664. P. S. Dalberg, A. Bere, K. A. Strand and T. Sikkeland, .I. Chem. Phys., 1978, 69, 5473.150 STRONGLY CHARGED PARTICLES F. Griiner and W. Lehmann, J. Phys. A , 1979, 12, L-303. P. N. Pusey and R. J. A. Tough, in Dynamic Light Scattering and Velocimetry: Applications of Photon Correlation Spectroscopy, ed. R. Pecora (Plenum Press, New York, 1982). J. B. Hayter and J. Penfold, Mol. Phys., 1981, 42, 109. J. P. Hansen and J. B. Hayter, Mol. Phys., 1982, 46, 651. R. Klein and W. Hess, in Ionic Liquids, Molten Salts and Polyelectrolytes, ed. K . H. Bennemann, F. Brouers and D. Quitmann (Springer-Verlag, Berlin, 1982), pp. 199-21 1. E. J. W. Verwey and J. T. G. Overbeek, Theory of the Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948). H. Minoo, C. Deutsch and J. P. Hansen, J. Phys. (Paris) Lett., 1977, 38, L-191. l o J. M. Deutch and I. Oppenheim, J. Chem. Phys., 1971, 54, 3547. T. J. Murphy and J. L. Aguirre, J. Chem. Phys., 1972, 57, 2098. l 2 J. P. Boon and S. Yip, Molecular Hydrodynamics (McGraw-Hill, New York, 1980). l 3 B. J. Ackerson, J. Chem. Phys., 1976, 64, 242. l4 R. Zwanzig and R. D. Mountain, J. Chem. Phys., 1965, 43, 4464. l 6 W. Hess and R. Klein, Ado. Phys., to be published. l 7 A. Z. Akcasu and E. Daniels, Phys. Rev. A, 1970, 2, 962. F. Gruner and W. Lehmann, J. Phys. A , 1982, 15, 2847. l 9 W. Hess and R. Klein, J. Phys. A , 1982, 15, L-669. 2o K. J. Gaylor, I. K. Snook, W. J. van Megen and R. 0. Watts, J. Phys. A , 1980,13,2513; J . Chem. 21 W. Hess and R. Klein, Physica, 1981, 105A, 552. P. Schofield, Proc. Phys. SOC. London, 1966, 88, 149. SOC., Faraday Trans. 2, 1980, 76, 1067.