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Faraday Discuss. Chem. Soc., The Concept of J. 1987, 83, 1-20 The Lennard-Jones Lecture Brownian Motion in Modern Statistical Mechanics M. Deutch and I. Oppenheim"? Chemistry Department, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, U.S.A. The statistical-mechanical bases for the fundamental equations describing Brownian motion are presented. These equations are the Langevin, Fokker- Planck and Smoluchowski equations. The conditions under which they are valid for single and many Brownian particle systems are delineated. The imposition of boundary conditions on these equations is discussed. The application of these results to reaction kinetics, diff usion-limited aggregation and cluster-cluster aggregation is briefly described. It is a great honour to present this lecture in memory of Prof.Sir J. E. Lennard-Jones, one of the greatest theoretical chemists of his time. 1. Introduction The modern statistical mechanical treatment of Brownian motion is based on the brilliant phenomenological studies of Langevin, Lorentz, Fokker, Planck, Smoluchowski, Kramers and Chandrasekhar. The concept of Brownian motion and its generalizations are the basis for almost all statistical-mechanical theories of time-dependent phenomena in fluids. These concepts have also stimulated a large number of experimental observa- tions and a somewhat smaller number of computer simulations. The theoretical description of Brownian motion is based on Langevin equations for the dynamical variables of the Brownian particles and Fokker-Planck equations for the distribution functions of the Brownian particles.In section 2 we discuss the molecular derivations and ranges of validity of the Langevin and Fokker-Planck equations and comment on the care that must be taken in writing down non-linear Langevin equations. In section 3 we present a brief descrip- tion of the forms of the friction coefficients and diffusion constants in many Brownian particle systems. In section 4 we consider the imposition of boundary conditions on Langevin and Fokker- Plank equations. The remaining sections describe the applications of Brownian concepts to reaction kinetics (section 5), diff usion-limited aggregation (section 6) and cluster-cluster aggregation (section 7). We include some brief comments on new directions for future research in section 8.2. Langevin and Fokker-Planck Equations The derivation of Langevin and Fokker- Planck equations for Brownian particles depends essentially on the fact that the masses ( M ) of the Brownian particles are much larger than the masses ( m ) of the bath particles in which the Brownian particles are immersed. An essential parameter in these derivations is E, where t Professors Deutch and Oppenheim are joint Lennard-Jones Lecturers for 1987 12 The Lennard-Jones Lecture Another important parameter is the ratio of the density of a Brownian particle ( p s ) to the density of the bath (pb). We start by considering a classical system consisting of one spherical Brownian particle of mass M, position R, and momentum P and radius 3 in a bath of N particles of mass m, position rN, and momenta p N , in a volume V.The density of the Brownian particle is and the density of the bath particles is N pb= m-. V The Hamiltonian for the system is (2.3) P2 2M -- = +Ho where U ( r N ) is the potential of interaction of the bath particles and @, the potential of interaction between the Brownian particle and the bath particles, is given by (2.5) We shall defer to section 4 the discussion of the effect of the walls on the motion of the Brownian particle. We define the projection operator P by: PB = c0(X, R ) B dX (B)o (2.6) I where F0 is the equilibrium distribution function for the bath in the presence of a fixed Brownian particle at position, R, i.e. where X = ( r N , p " ) is the phase point of the bath. The exact equations of motion for the momentum and position of the Brownian particle are:' p ( t ) = K ( t ) + 1 exp [iL( t - T)][VJP - PP/ MI (FK( T ) ) ~ dT J o li( t ) = P( t ) / M .Here, the fluctuating force K ( t ) is given by: K ( t ) = exp [ (1 - P)iLt]( 1 - P)iLP = exp [ (1 - P)iLt]F (2.9) where F, the force on the Brownian particle, is F = --VR@ (2.10) and iL, the Liouville operator for the system isJ. M. Deutch and I. Oppenheim 3 where iLo is the Liouville operator for the bath in the presence of a fixed Brownian particle and iL, describes the motion of the Brownian particle. The kernel ( F K ( T ) ) ~ is in general a function of P, so that eqn (2.8) is a non-linear equation in P and is also non-local in time. While eqn (2.8) is a suggestive form for P ( t ) , we emphasize that it is an exact equation and its reduction to a Langevin equation is not trivial.Eqn (2.12) is linear in P ( t ) and has a time-independent coefficient 6. The fluctuating force E ( t ) is supposed to be a stochastic Gaussian variable of zero mean and with delta function correlations in time. The extremely simple form of eqn (2.12) was very quickly objected to by Lorentz, who used hydrodynamics to argue that, at the very least, eqn (2.12) must be rewritten as P ( t ) = E ( t ) - lo‘ t( t - T ) P ( T ) dT (2.13) where ,$ is still independent of P and the stochastic properties of E must be changed accordingly. The questions that must now be addressed are: (1) under what conditions does eqn (2.8) reduce to eqn (2.13); (2) are there conditions under which eqn (2.8) reduces to eqn (2.12); and finally, (3) what is the form of (? Even after these questions are addressed, we must find an appropriate form for the initial distribution function of the system in order to use this equation to describe the average behaviour of the Brownian particle.In the following discussion we shall always assume that eqn (2.1) is valid. In order to reduce eqn (2.8) to a linear equation in momentum, the velocity of the Brownian particle must be small compared to the velocity of a bath particle. On the other hand, if the momentum of the Brownian particle is comparable to the momentum of a bath particle, eqn (2.8) is not a suitable starting point. Careful analyses of eqn (2.8) have been carried out by a number of authors’’2 and the conditions under which it reduces to either eqn (2.12) or (2.13) have been delineated.The kernel in eqn (2.8) has all of the slow time scales associated with the motion of the Brownian particle removed by the projected time dependence of K ( t ) , eqn (2.9). The slow time scales associated with the hydrodynamic modes of the bath and their coupling to the force on the Brownian particle are still present, however. Eqn (2.8) becomes linear in P when E << 1 and pb/pB( 1. Under these conditions it reduces to the form of eqn (2.13), i.e. P ( t ) = E ( t ) - @ ( t ) (2.12) where Fo( 7) = exp (iL,T)F. (2.15) The time dependence in eqn (2.15) is for the situation in which the Brownian particle is held fixed. In this case (K(t)K(7))O2.(FO(t - +?O. (2.16) If pb/pB<< 1, eqn (2.14) reduces further to (2.17) which is of the form of eqn (2.12) with [=- ( F O F ~ ( T ) ) ~ ~ T .3M I,: (2.18)4 The Lennard-Jones Lecture In this regime the value of 5 depends on the parameter of the intermolecular force and 1 is the mean distance %a2/i3cc 1, 5 = & a %?a2/ 13, where between bath where the subscript CE stands for Chapman-Enskog. For %a2/13 >> 1, 5 = tSE = 4nq%/ M where the subscript SE stands for Stokes-Einstein. a is the range particles. For (2.19) (2.20) If Pb/PB = 1, the long-time tail associated with the hydrodynamic modes of the fluid must be taken into account. The Laplace transform of eqn (2.14) is 6(s)=i(s)---(F* P so(s))ofi(s) 3 M (2.21) where s is the Laplace transform variable and A denotes a Laplace-transformed quantity.The kernel has the form: (2.22) ( F &( S))o = 4TYR (3pb kT)[ 1 -k 3 (S/ V) ' j 2 ] for small s, where Y is the kinematic viscosity of the fluid, Y = q/pb. The force autocorrelation function has a negative t - 3 / 2 tail. Excellent experimental confirmation of this result has been ~ b t a i n e d . ~ We note that the slow hydrodynamic modes of the bath change the zero-frequency value of the kernel completely. The fact that (=% instead of 3* is an immediate indication of the importance of long-time tails! Finally, we consider the situation in which the bath is much denser than the Brownian particle, i.e. Pb/PB >> 1. In this case, there is an initial very slow decay of the momentum autocorrelation function proportional to which for longer times goes over to the t - 3 / 2 hydrodynamic decay.In the remainder of this section, we shall consider the regime in which 5 is given by eqn (2.20) and the classical Langevin equation, eqn (2.12), for p ( t ) is valid. The Fokker-Planck equation associated with eqn (2.12) can be easily derived either from that equation and the properties of E ( t ) or directly using the well known techniques of elimination of fast variables which has recently been formalized by van K a m ~ e n . ~ The advantages of the latter approach are twofold: ( a ) no assumption about the initial form of the system distribution function must be made; and ( b ) the effect of the Brownian particle motion on the bath distribution function can be explicitly obtained. We start with the Liouville equation for the system described by the Hamiltonian of eqn (2.4) and the Liouville operator of eqn (2.11), i.e.where p = p(X, R, P, t ) is Brownian particle system. where b0 is given by eqn W(R, P, t ) , is defined by: b( t ) = -iLp( t ) (2.23) the distribution function for the N bath particles plus one We introduce the projection operator, P', defined by P'B=bo B d X 5 (2.7). The distribution function for the Brownian particle, W( t ) = p( t ) dX. (2.25) I In equilibrium, exp ( - P W j exp (-PH)dX dR d P Po = (2.26)J. M. Deutch and I. Oppenheim 5 exp ( - P P ~ / ~ M ) exp ( -PP2/2M) d P dR W, = (2.27) (2.28) is the conditional equilibrium distribution function for the bath given that the Brownian particle has momentum P and position R. We define y ( t ) by y ( t ) = P+p(t)=p"oW(t) (2.29) z( t ) = (1 - P+)P( t ) = Q+p( t ) (2.30) where the projection operator Q' is defined by eqn (2.30).It is useful at this stage to introduce the quantity P* f EP (2.31) which is assumed to be of order E'. The fundamental equations of motion are and P*= EF (2.32) and R = sP*/m. The Liouville operator iL, of eqn (2.11) becomes (2.33) and is of order E. We can now write: j ( t ) = -P+iLy( t ) - P+iLz( t ) i( t ) = -Q+i LQ+z( t ) - Q+iLy( t ) . Since P+iLo = iLoP+ = o (2.35~) (2.35 6) (2.36) eqn (2.35) become on the time scale T = st y( T) = -P+iLTy( T ) - P+iLTz( T ) (2.37a) i ( ~ ) = -- Q+iLoQ+z(T) - Q+iLTQ+z(T) - Q'iLTy(7). (2.37b) We assume that z( T ) , the fast variable, can be expanded in powers of E, and that y and i can also be so expanded; i.e.1 E Z ( T) = Zo( T) + EZ(*)( 7) + - - * (2.38 j The operator Q'iLoQ'= G has an inverse since the zero eigenvalue of iLo is removed by the projection operator. Thus we find from eqn (2.376) (2.39) e tc.6 The Lennard-Jones Lecture Substitution of these results into eqn (2.370) yields to order c2: where (2.41) Eqn (2.40) is the Fokker-Planck equation for W(R, P*, t ) . Eqn (2.40) and (2.41) are equivalent to the Langevin equation, eqn (2.17), with the friction coefficient 5 = P / M T if the initial distribution function for the system is assumed to have the form P(X, R, P, 0) = boW(R, p, 0 ) . (2.42) From eqn (2.39), we find that the conditional distribution function for the bath is given asymptotically by Clearly 6 = bo only when W = W,.The difference between 6 and io is due to the perturbation of the fluid produced by the motion of the Brownian particle. Once the Fokker-Planck equation for W(R, P, t ) , eqn (2.40), has been obtained, the Smoluchowski equation for the distribution function f ( R , t ) , of the position of the Brownian particle is easily obtained. We introduce the projection operator (2.44) where exp ( - ~ ~ * ~ / 2 m ) (2.45) exp (-PP*2/2m) dP* no(p*) = I We operate on eqn (2.40) with P, and (1 - P,), using the fact that PI W R , p*, t ) = no(P*)f(R, t ) and carry through a procedure similar to that above and obtain so that the diffusion constant, 0, is given by kT D=-. M t (2.46) (2.47) (2.48) It is straightforward to extend the techniques used to derive the exact equation, eqn (2.8), to obtain an exact equation for the time derivative of an arbitrary function of the position and momentum of the Brownian particle, G(R, P).' We define KG( t ) = exp [( 1 - P)iLt]( 1 - P)iLG = exp [( 1 - P)iLt]F- VPG(R, P ) (2.49) and find ( F exp [ (1 - P)~LT]F), VpG + KG( t ) .(2.50)J. M. Deutch and I. Oppenheim 7 Under the conditions that eqn (2.8) reduces to eqn (2.17), we find Eqn (2.51) is a non-linear Langevin equation and is a perfectly proper stochastic differential equation. Of course, the fluctuating force & ( t ) is not additive as in eqn (2.17), but is multiplicative in the sense that it depends on R and P for arbitrary G. Its stochastic properties are different from those of the fluctuating force in eqn (2.17). For example, but (2.52a) (2.52b) ( 2 .5 2 ~ ) The correlation function of the fluctuating forces is not time translationally invariant as it is for the fluctuating force in eqn (2.14), and the fluctuation dissipation theorem has a complicated form. The extensions of the Langevin equation and the Fokker-Planck equation to systems containing several Brownian particles is also straightforward.' We consider a system containing n identical Brownian particles with positions R", momenta P" and mass M, and N identical bath particles with positions r", momenta p N and mass rn. The development for Brownian particles of different masses is essentially identical. The Hamiltonian for the system is P" P" N N + v(R")+-+ U ( r N ) + Q ( r N , R " ) = - + V(R") + H,. P" P" H=- 2 M 2m 2 M (2.53) Here V is the interaction potential among the Brownian particles and CP is the interaction between the Brownian particles and the bath particles.The potential U is a function of the scalar distances among the bath particles, and the function Q, is a function of the scalar distances between the Brownian particles and the bath particles. The Liouville operator for this system is i L = i L, + i Lo (2.54a) where (2.54b) and We choose the same projection operator as in eqn ( 2 . 6 ) , but must take into account the fact that the denominator in eqn (2.7) is now a function of R". The exact equation for the time derivative of P,( t ) is:8 where The Lennard-Jones Lecture and F, = -VR,@. In the Langevin limit, eqn (2.55) becomes (2.56) (2.57) (2.58) The kernel (Sn exp (iLoT)fil)o is a function of the relative distances between the Brownian particles and the notation ( ) o ( t ) implies that the relative distances are those at time t.We define the friction tensor gjl(t) by and rewrite eqn (2.59) as (2.60) (2.61) Here the motion of particle 1 is coupled to the motions of the other particles, the friction tensor is an explicit function of time since it depends on R n ( t ) and the process is not temporally homogeneous. Thus while (2.62 ) Again, the fluctuation dissipation theorem becomes extremely complicated. The explicit forms of the various components of the friction tensor will be discussed in the next section. The Fokker-Planck equation for this system is 1 where erj = 4&(R"). Finally, the reduction of eqn (2.63) to a diffusion equation for f ( R " , t ) proceeds in the same fashion as for the one Brownian particle system [see eqn (2.44)-(2.47)].The result is the Smoluckowski equation f ( R n , t ) = V ~ ' l * D ( R n ) ' [ ~ R " + ~ ( V R " V - ( F " ) ~ ) ] ~ (2.64)J, M. Deutch and I. Oppenheim 9 where the spatially dependent diffusion tensor has the form kT D( R") = - [tj( R") 3-' M (2.65) and the friction tensor has 3 n x 3n components of the form where j and k denote individual Brownian particles and a and y are the x, y or z spatial components. In this derivation, it is assumed that VRV and (F")o are sufficiently small that the Brownian momentum distribution becomes of the equilibrium form on a faster timescale than its coordinate distribution. These terms in the Langevin, Fokker-Planck and Smoluchowski equations go to zero when the Brownian particles are further apart than ca.2% and are frequently neglected. The extension of these results to rigid non-spherical particles is straightforward and involves rotational as well as translational motion. If the Brownian particles have internal vibrational or rotational degrees of freedom, great care must be taken in obtaining the appropriate dynamical equations [see e.g. ref. (7) 1. 3. Brownian Motion-Many Particles The classical theory of Brownian motion for a single particle leads to the Einstein expression for the diffusion coefficient D = kT/ M.$ (3.1) where M is the mass of the Brownian particle and 6 is the friction coefficient. Within the framework of macroscopic hydrodynamics one can relate the friction coefficient to the viscous force acting on the particle.In three dimensions, the result for an impen- etrable sphere of radius 3 is 6 = CTvo3 ( 3 . 2 ) where qo is the solvent viscosity and C equals 6(4) for stick (slip) boundary conditions at the surface of the sphere. This result [eqn ( 3 . 2 ) ] when combined with eqn ( 3 . 1 ) leads to the famous Stokes-Einstein relation D = kT/ CMrv,%. ( 3 . 3 ) The hydrodynamic calculation leading to eqn ( 3 . 2 ) is valid for a single particle in an infinite incompressible fluid. Should any other fixed object such as a wall or another Brownian particle be present the result, eqn ( 3 . 2 ) , is modified in a major way. The reason that the modification is qualitatively significant, even for dilute systems, is that the velocity field surrounding a point particle (and hence any differential element of a more extended object) is of long range.Consider the perturbation to the velocity field Sv introduced by a point force Fo placed on the fluid, say at the origin. The linearized steady-state Navier-Stokes equation for this situation is 0 = q V'SV - vp + FoS ( r ) ( 3 . 4 ) with the incompressibility condition V * Su = 0. In three dimensions the solution to this equation is 6 ~ ( r ) = T( r ) Fo (3.5)10 The Lennard-Jones Lecture where T( r ) is the Oseen hydrodynamic interaction tensor [ I + F i ] 1 T ( r ) =- 8 v o r (3.6) which exhibits long-range r-l behaviour. Thus, if a pair of Brownian particles is present in solution, one fixed at the origin and the other fixed at r, each will experience a modified frictional force ,$[ uo + So] and the pair will exhibit a separation-dependent friction coefficient g ( r ) .If the particles are of finite size R there will be corrections to eqn (3.5) of order (R/4).' Evidently this circumstance necessitates major modification to the classical one- particle theory of Brownian motion. Oppenheim and Deutch6 have examined the modifications required to the Langevin equation to describe the case of several Brownian particles. One must expect the appearance of many-particle friction coefficients gij( R") and diffusion coefficients D,(R") which depend on the positions of the n Brownian particles. The structure of the resulting Langevin equation [see eqn (2.61)] is The relationship between the friction coefficient and the diffusion coefficients is best found by an argument due to Zwanzig.' For a fixed configuration the force on the ith particle is given by Fi = -([UP+ S V , ] = -(q (3.8) and which leads to where (3.9) (3.10) (3.11) While the expression for the friction coefficient gij will be complicated, the expression for the diffusion coefficient is not.The diffusion equation (or generalized Smoluchowski equation) is found from the continuity equation [see eqn (2.64)] (3.12) where f(R", t ) is the probability of finding the n Brownian particles at position R" at time t a n d j i is the flux j , = VYJ (3.13) In thermal equilibrium the force arises from thermal fluctuations so approximately one has FJ = --kTVjln$ The resulting diffusion equation is df=zcv;* Dil*[oj+p(v,v-(F,)03f (3.14) d t i j where (3.15)J. M.Deutch and I. Oppenheim 11 These results are the starting point for a number of important applications in the theory of Brownian motion. First, a number of researchers, particularly Felderhof," have included the influence of hydrodynamic interactions beyond the leading Oseen term. Secondly, several groups, beginning with Fixman,' have investigated the concentration dependence which occurs in suspensions for the diffusion coefficient and other transport coefficients. Not surpris- ingly because of the long-range nature of the hydrodynamic interaction one encounters non-analytic concentration dependence. Thirdly, while the foregoing discussion addresses the influence of hydrodynamic interaction on the translational friction and diffusion coefficient, it is evident that similar effects will be observed for rotational friction and the rotational diffusion coefficient in many-particle systems.This problem has been addressed by Wolynes and Deutch.12 It is also important to appreciate that the many-particle Brownian motion picture is the starting point for most descriptions of the dynamics of both flexible and rigid polymer systems. In these theories the individual segments are considered to be point centres of frictional force [although the criterion of small (size/separation) ratio is clearly not valid] in order to describe the perturbed hydrodynamic field which surrounds and penetrates the macromolecule. For example in the Kirkwood theory, the centre of friction diffusion coefficient is (3.16) In the free-draining limit, one neglects the hydrodynamic interaction and finds D = kT/ NtM.(3.17) In the non-free-draining limit, the hydrodynamic interaction dominates and one finds D = kT/6.nqoRH (3.18) where (3.19) Evidently, the result corresponds to the Stokes-Einstein result for a single sphere of radius RH ; the hydrodynamic interaction effectively excludes the solvent from the interior of the porous sphere. It is interesting to note that the hydrodynamic force on a typical polymer segment in a polymer can be estimated by FH(0), where FH( r ) T( r - r ' ) p ( r ' ) d r ' 00 (3.20) where p ( r ) is the distribution of polymer segments p( r ) == ( exp ( - d r 2 / 2 ( R 2 ) ) (3.21) and d is the number of dimensions.One finds FH(0) N l P ( - d - 2 ) v ( d ) where v ( d ) is the exponent defined by (R') == N'". For a random chain Y = 1/2, which is also the value for Y for a linear chain with excluded volume in d = 4. One concludes that there is an abrupt transition at d = 4, where the frictional force will dominate the effect of hydrodynamic interactions so that the diffusion coefficient for the polymer is given by eqn (3.17). Above four dimensions, the polymer structure is sufficiently open that the effect of hydrodynamic interaction is irrelevant.12 The Lennard-Jones Lecture 4. Boundary Conditions The imposition of boundary conditions on stochastic differential equations is an extraor- dinarily difficult task. In almost all cases the stochastic properties of the fluctuating force change in a dramatic fashion and it is not readily apparent either from intuition or direct calculation how to take this into account analytically." On the other hand, it is usually straightforward to impose boundary conditions on the Fokker-Planck equation and then to use those results to obtain the properties of the corresponding Langevin equations.We shall illustrate these considerations by treating an extremely simple system of one Brownian particle in a bath in a one-dimensional box with reflecting walls at X = 0 and a.I4 If the box were infinite, the Langevin equation would be: P( t ) = E ( t ) - t P ( t ) (4.1) where E ( t ) is a Gaussian random variable of zero mean and delta function time correlations. The corresponding Fokker- Planck equation is: P aw0 (4.2) Here, the superscript " implies that the system is infinite.The presence of the walls can be taken into account by introducing an additional term containing the wall potential into eqn (4.2). It is much simpler, however, to relate the distribution function for the finite system with reflecting walls, W(X, P, t ) , to the infinite system distribution functions by: -a W ( X , P , t ) = C [Wo(X+2na,P,t)+ W"(2na-X,-P,t)]. (4.3) n = - - a The notation W(X, P, t ) is shorthand for the conditional probability that the Brownian particle is at position X, with momentum P, at time t given that it was at position Xo, with momentum Po, at time t = 0. The normalization condition for W is: and it obeys the Fokker-Planck equation, eqn (4.2), for 0 d X d a and for all momentum.The average of any arbitrary function of X and P, G ( X , P ) , is obtained by multiplying by W(X, P, t ) and integrating over X from 0 to a and P from -a to +a. Thus G ( t ) = loa dX [-:dPG(X, P ) W ( X , P, t ) . (4.5) Explicit results for G ( t ) can be obtained, since W" is known, by substituting eqn (4.3) into eqn (4.5). It is easy to see from eqn (4.3), that the following boundary conditions obtain: dPW(X,P, t)P"+'=O at A-=O,a, 2=0,1,2 , . . . (4.6a) W ( X , P, t)P" d P = 0 at X = 0, a. (4.6 b ) A special case of eqn (4.6b) for 1 = 0 is a ax W ( X , P, t ) d P =--f(X, t ) = 0 at X = 0, a. (4.7)J. M. Deutch and I. Oppenheim 13 It also follows'that: P(t)= 1" d P 1" d X W ( X , P, t)P''=PZ'O(t) J--00 JO and P2'+'( t ) = P2'+l"( t ) + A21+1( t ) .(4.8) (4.9) Thus even powers of P have the same average as in the infinite system, whereas odd powers of P have different averages from those in an infinite system. The quantities A z r + , ( t ) follow explicitly from eqn (4.9) and (4.3). The form of Wo(X, P, t ) is: 1 27rM( FG - H2)"* GR2 - 2HRS + FS2 ( 2(FG-H2) - WO(X, P, t ) = where Po M R = x -xO-[-' -[I -exp ( - c t ) ] P - Po exp ( - [ t ) M S = kT F = - [25t - 3 + 4 exp ( - [ t ) - exp ( - 2 5 t ) ] G = - M [ 1 - exp ( - 2 t t ) l M t 2 kT (4.1 1 ) Thus, e.g. + MH[exp (-RznP1/2F) -exp ( - R : , / 2 F ) ] (4.12) where (4.13) PO R2,, = 2 n a - ~ ~ - t - ' - [ 1 - e x p ( - ~ t ) ] . M We write the stochastic differential equation for the finite system in the form: P( t ) = A( t ) - [P( t ) (4.14) where A ( t ) is a stochastic variable whose properties are characterized by its averages and correlation functions. The formal solution of eqn (4.14) is: P ( t ) - exp (--&)Po = exp [ -[( t - T ) ] A ( T) dT (4.15) The stochastic properties of B ( t ) are identical to the stochastic properties of P - exp (-&)Po = MS.Thus B'( t ) = [ P - exp (-tt)pOl'(t) = 1; d X j-:dP [P-exp (-[t)PO]'W(X, P, t ) . lot = B ( t ) . - (4.16)14 The Lennard-Jones Lecture In particular B( t ) = A,( t ) (4.17) which is given explicitly in eqn (4.12). Thus not only does the stochastic force not have zero mean, but it also has long time behaviour and it is non-Gaussian. It also depends on X , and Po, as we might expect. It follows from the forms of the As, that if X , is far from 0 and a, the walls can be neglected for all times of interest, i.e.( t == 1. However, if X , is in the neighbourhood of the wall, A , ( t ) decays as t-”*. The properties of Brownian motion near a wall have been experimentally determined using photon-correlation spectroscopy from an evanescent wave. l 5 The results confirm the considerations given here. We emphasize the point that boundary effects are much more easily obtained from Fokker-Planck or Smoluchowski equations than from Langevin equations. 5. Reaction Kinetics Another major manifestation of Brownian motion ideas is in the area of chemical kinetics.I6 The introduction of the ideas of Brownian motion to the subject are due originally to Smoluchowski and notably Kramers. The elaboration of the basic idea and analysis has found widespread application in modern theories of chemical reactions in solution.The prototype chemical reaction occurs by the following mechanism k , k - 1 A+B S AB k AB -& product. (5.2) In this mechanism the rate-limiting step for the overall reaction is the mass-transport step ( 1 ) of the reactants finding each other in solution under the condition when the chemical step (2) is fast, k, >> k-, . The basic contribution of Smoluchowski is the calculation of k , under steady-state conditions when the reactant particles encounter each other by diffusion. The result is k~ = 47T( DA + DB) ( RA + RB) (5.3) for d = 3. The result was generalized by Kramers and DebyeI7 to include the influence of effective intermolecular forces U ( r ) between A and B and by others” to include the hydrodynamic interaction which may be present as the reactants approach each other in an incompressible solvent Here D( r ) is the position-dependent relative diffusion coefficient.Since the effective rate equation is of the form d[A]/dt = -k,[AB] (5.5) the predicted time dependence is [A](t) = t-’ for all dimensions since k , is time independent. The effects of diffusion on chemical reaction rates has bien investigated recently by a number of authors.” As a simple example we consider a system consisting of a non-reactive solvent and reactive species A and B which undergo the reaction: k A+B -+ C. (5.6)J. M. Deutch and I. Oppenheim 15 Initially, A and B are present in equal numbers and are distributed uniformly from the macroscopic point of view.Elementary reaction rate theory yields the equation: where CA denotes the concentration of A per unit volume and k is the reaction rate coefficient. The solution to eqn (5.7) is: L O C,kt + 1 C,( t ) = ~ where C, is the initial concentration of A. For large t, eqn (5.8) predicts that CA( ?) ‘1 ( k ? ) (5.9) and the concentration of A decreases as ? - I , independent of the number of dimensions. This result is correct in the limit of large diffusion coefficients or if the reaction mixture is stirred continuously. It neglects the effects of local fluctuations in the densities of A and B. Because of these fluctuations, there will be regions in which there are more A(B) than B(A) molecules. The As and Bs which are close together will react quickly and we will then be left with regions containing only A or B molecules.Before the reaction can proceed, there must be a diffusive process bringing the isolated As and Bs in contact with each other. Thus we would expect a dimensional dependence for CA(t) and a slower rate of decay at least in some dimensions. One way of treating this problem is to consider the system broken up into cells each of which is large microscopically (i.e. each cell contains many A and B molecules) but small macroscopically. The probability distribution for the numbers of As and Bs in each cell will be initially Poissonian. We can then define a probability distribution for the number of As and Bs in each cell, P({A}, {B}, t ) , and assume that P obeys a master equation. The number of A(B) molecules in each cell changes as a result of the chemical reaction in each cell and as a result of diffusion from and to neighbouring cells.We denote the average number of A molecules in cell j by (A,}(t). Then ( A , ) ( t > = -K(A,B,)(t) = ( @ ( t ) . (5.10) The rate coefficient K is simply related to the rate coefficient k by K = M k / V (5.11) where M is the number of cells in the system. There is no diffusion term in eqn (5.10) because the system is macroscopically uniform. It is easy to see that (A,) is zero if either A, or I?, is zero. Eqn (5.10) can be rewritten as (A,)W = -K(rA,41+(A,)2) (5.12) where [A$?,] is the-factorial cumulant. It agrees with the phenomenological equation, eqn ( 5 . 5 ) , only when [AJB,] is zero.For a Poisson distribution all factorial cumulants beyond the first are zero. Thus at t = 0, [A,B,] = 0. As a result of the chemical reaction [A,BJ] becomes negative; the diffusion process tries to restore the Poisson distribution and make [A,B,] zero again. Thus, while the equation for (AJ) contains no diffusional term, the equation for [A,B,] does. This equation can also be obtained from the master equation for P. In each cell, the reaction stops when (A,)* = -[AJ41. ( 5 . 1 3 ) The quantity [A,B,], which has become negative, now tends to zero as the result of diffusion and increases to zero as -tCdi2, where d is the number of dimensions. Thus,16 The Lennard-Jones Lecture ( A j ) ( t ) = t-d/4 as long as this decay is slower than t-'.(Aj)(t) decays as t-'/4 in one dimension, t-'l2 in two dimensions, t-3/4 in three dimensions and t-' in four and higher dimensions. The coefficient of t - d / 4 is easily obtained from the set of equations for the factorial cumulants. These results have been confirmed in one and two dimensions by computer simu- 1ati0n.l'~ This striking new result illustrates the continued vitality of the ideas of Brownian motion. The results, of course, require experimental confirmation and careful iden- tification of those instances where the effect can be qualitatively important. We note that this new prediction does not include the effect of reverse reaction or of hydrodynamic interaction created by the velocity field of the reactions, either of which could lead to qualitative modification of this exciting result.6. Diff usion-limited Aggregation The past five years have seen an explosive growth in the study of the phenomena of diff usion-limited aggregation (DLA) and coagulation. Since several of the leading figures in this area will be presenting papers on various aspects of this subject, only brief comments will be offered here. The central result of DLA originally revealed through computer simulation studies of Whiten and Sander2' and Meakin21 concern the geometric shape of the aggregate formed when particles diffuse and stick to a cluster which grows from an initial seed particle at the origin of a d-dimensional space. The classical expectation for this physical situation is that the aggregate would grow as a compact object with a radius of gyration R related to the number of particle N according to In fact the object is found to have a For these structure one finds R = N ~ .(6.1) fractal structure22 with a highly ramified shape. (6.2) N R D ( d ' where D ( d ) is the fractal dimension. The DLA model is found to correspond to a number of physical situations including electrodeposition, dendritic growth, dielectric breakdown and to the important subject of fluid displacement processes. A variety of theoretical approaches have been proposed to explain the values of D ( d ) [D(2) =r 5/3, D(3) = 5/21 which have been found in the computer simulations. Prominent among these early efforts include the contributions of Gould et u Z . ~ ~ and M u t h ~ k u m a r , ~ ~ whc advocates the interesting formula d 2 + 1 d + l D ( d ) = - (6.3) and of Tokuyama and Kawasaki25 and Hentsche1.26 More recently Turkevich and SherL7 have argued that D is non-universal and depends upon the symmetry of the underlying lattice on which the aggregation occurs.They propose the expression (6.4) D = [ ( 3 ~ - 0 ) / ( 2 ~ - 0)] for d = 2, leading to the values D = 5/3 and D = 7/4 for square and hexagonal lattices, respectively. The current situation with respect to the non-universality has been reviewed by Meakin;28 suffice it to say that a complete theoretical explanation is not yet in hand. It is not surprising that this initial discovery has led to an enormous number of studies exploring various aspects of this growth process, including surface structuresJ. M.Deutch and I. Oppenheim 17 and various generalizations, e.g. the influence of non-Brownian trajectories, anisotropic diffusion, rotation etc. Evidently, there are consequences for the kinetics of the growh as well as the structure of DLA aggregates. Meakin and Deutch2’ have proposed R( t ) oc t [ 1 i ( 2 + D - d ) 1 , ( d - D ) < 2 (6.5) which yields the classical result for a compact object R’CC t, but predicts R cc t2’3 for d = 3. This scaling has been confirmed in computer simulation3’ for the early stages of aggregate growth. The topic of DLA illustrates that the fundamental ideas of Brownian motion and diffusion still have relevance to a variety of practical problems and basic theoretical questions that are as yet not fully answered. 7. Cluster-Cluster Aggregation The fractal objects described in the last section result from a precise mechanism of growth which is called diff usion-limited aggregation.Other mechanisms of irreversible growth also lead to fractal structures, although of a different character or universality class. An especially important mechanism is that of cluster-cluster aggregation (CCA). In this model each particle can act as a nucleating centre, and when two clusters of size i a n d j touch during their random diffusive motion, a larger cluster is formed of size ( i + j ) . The CCA model has recently received considerable attention because it is properly considered to be a more realistic description of the process of coagulation of many physical systems. The CCA model can be studied either by theoretical analysis or by computer simulation.A summary of recent work may be found in the book by Family and Landau.31 One theoretical approach is to study the Smoluchowski equation for coagulation which has the form -- dCk(t) - C 1 K,Ci( t)Cj( t ) - C,( t ) C KjkCj( t ) . d t i ; i = l (7.1) An exact solution to eqn (7.1) can be found only for special cases of the reaction kernal K , . For example, for a constant kernal K , = K subject to an initial monodisperse concentration Ck(0) = aIk, the solution is C,( t ) = ( K t / 2 ) , - ’ / [ 1 + ( K t / 2 ) l k ” which satisfies the conservation of mass condition 1 = CT=l kCk( t ) , although the infinite upper limit conceals a tricky mathematical point.32 The asymptotic time dependence for solutions of the kinetic equation, eqn (7.1), have been studied by a number of groups including those of Ziff and E r n ~ t .~ ~ The Smoluchowski equation, eqn (7.1), clearly has some limitations as a kinetic description of the reaction-diffusion system which is envisioned in the CCA model. It is a topological equation which does not take into account the dimension or geometry of the reacting system. With respect to the former, the dimensionality of the space only enters indirectly through the form adopted for the reaction kernal K,. If these rates are assumed to be diffusion-limited K , should have the form K , , = ( D , + D , ) ( R , + R , ) “ - ‘ . (7.3) Further, if the clusters and compact and not permeable to solvent flow, one should expect D, cc R f A 2 and R , = il’d. However, it is possible, and computer simulations confirm, that the clusters have an open fractal shape R, = nl’D with D < d.This not only complicates the form of K,, but also draws into question the use of the Stokes- Einstein expression for D,.18 The Lennard-Jones Lecture The Smoluchowski equation exhibits gel formation in the following sense. When the rate of change of the total mass defined by'* (7.4) does not vanish in the limit N+w, there is 'a loss of mass to infinity'. The conditions on the reaction kernal K , required for this to occur have been widely studied as has the resulting kinetic description for the gelation process. However, the Smoluchowski equation, eqn (7.1), has serious limitations for describing the gelation process at finite density. As clusters grow with their open fractal shapes there will be a point where in a fixed volume at finite density, the space will be spanned by a single connected cluster.The reaction-diffusion problem which is presented by such a system where large clusters of various sizes are moving about and irreversibly joining is highly complex. The final stages certainly involve concentration (and dimensionality) effects on both diffusion and reactions which are simply not included in eqn (7.1). In addition, there is the possibility of reverse reactions when the clusters fragment'4 which can have a pronounced effect on the kinetics. In full generality the appropriate reaction-diffusion equations has the form dC -= V - D - V C + K: CC - F C a t (7.5) where the vectors are of the form C = ( C , , Cz, .. . , C N ) . Not surprisingly several groups have turned to computer simulation of the CCA model.3s The simulation results for the CCA model have been analysed by a scaling form for C,( r)36 in two dimensions C,( t ) = t-"nPTf( n / t') (7.6) which involves the scaling function f ( x ) [ f ( x ) = 1 for x << 1; f ( x ) << 1 for x >> 11 and two dynamical exponents w and z. The constant mass condition requires that w = (2 - T ) Z . The weight-average cluster size S ( t ) behaves like S ( t ) = - n2Cn(r)dtoctZ W t ) while the number-average cluster size, A( t ) , behaves like 'T<1 A ( t ) = M ( r ) / [ c,( t ) dn { '' t"' T > 1' (7.7) (7.8) These scaling expressions have been found to be in good agreement with the simulation results.The aggregates formed by the CCA mechanism have a fractal structure. The distribu- tion of mass has the form p ( r ) K F ( ~ - ~ ) . (7.9) At small length scales, D has the value of the DLA mechanism since most of the irreversible growth occurs by the addition of solute particles. At larger length scales the CAA clusters develop a fractal dimension D of their own. Some efforts have been made to produce a theory for D(CCA)" and to compare the prediction to the simulation results of M e a k i ~ ~ ~ ~ for 2 < d < 6. An interesting feature of Meakin's simulation results is that the fractal dimension of CCA clusters is much less sensitive than that of DLA clusters to the dimensionality of the trajectory (diff usive or ballistic) which characterizes the movement of the species.Schaefer and coworkers39 have studied the structure of colloidal silica in water by light scattering and small-angle X-ray scattering and compare their observations with the CCA simulation results.J. M. Deutch and I. Oppenheim 19 8. Conclusions Significant progress has been made on the theoretical, experimental and computer- simulation studies of Brownian motion. Still, much work remains to be done on the effects of walls and on almost all aspects of many Brownian particle systems either with or without aggregation and chemical reaction. In particular, in most studies the solvent is assumed to play a passive role, but it has been shown that the dynamic properties of the solvent can be important, particularly for chemical reactions.The experimental tools now available, especially dynamic light scattering and photon correlation spectroscopy from evanescent waves, are extremely powerful and have already and will continue to yield interesting and informative results. Computer simulations have been extremely successful in the past and in some aspects have led the way in the description of long-time tail phenomena, aggregation and chemical reactions. They have an important role in future studies as well. It is clear from this meeting that the study of Brownian motion is a vibrant field and much of interest and significance remains to be done. A portion of this work was supported by the National Science Foundation under Grant CH E84- 10682. References 1 P. Mazur and I. Oppenheim, Physica, 1970, 50, 241.2 T. Keyes and I . Oppenheim, Physica, 1973,70, 100; 1975,81A, 241; I . A. Michaels and 1. Oppenheim, Physica, 1975, 81A, 221; M. Tokuyama and I. Oppenheim, Physica, 1978,94A, 501. 3 G. L. Paul and P. H. Pusey, J. Phys. A, 1981, 14, 3301; K. Ohbayashi, T. Kohno and H. Utiyama, Phys. Rev. A , 1983, 27, 2632. 4 N. G. van Kampen, Phys. Rep., 1985, 124, 69; N. G. van Kampen and I. Oppenheim, Physica, 1986, 138A, 231, and references therein. 5 J. Albers, J. M. Deutch and I. Oppenheim, J. Chem. Phys., 1971, 54, 3541. 6 J. M. Deutch and I. Oppenheim, J. Chem. Phys., 1971, 54, 3547; J. L. Aguirre and T. J. Murphy, J. 7 I. Oppenheim, U. Mohanty and K. E. Shuler, Physica, 1982, l15A, 1 . 8 J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics ( Prentice-Hall, Englewood Cliffs, 9 R.Zwanzig, Adv. Chem. Phys., 1969, 15, 325. Chem. Phys., 1973, 59, 1833. 1965). 10 See, e.g. B. U. Felderhof, P. Reuland and R. B. Jones, Physica, 1978, 93A, 465. 1 1 M. Fixman and C. W. Pyun, J. Chem. Phys., 1964, 41, 937. 12 P. G. Wolynes and J. M. Deutch, J. Chem. Phys., 1977, 67, 733. 13 See, e.g. P. Harnggi, K. E. Shuler and I. Oppenheim, Physica, 1981. 107A, 143. 14 I. Oppenheim and P. Mazur, Physica, 1964, 30, 1833. 15 K. H. Lau, N. Ostrowsky and D. Sornette, Phys. Rev. Lett., 1986, 57, 17. 16 See, e.g. D. F. Calef and J. M. Deutch, Annu. Rev. Phys. Chem., 1983, 34, 493, and references therein. 17 H. A. Kramers, Physica, 1940, 7, 284; P. Debye, Trans. Electrochem. Soc., 1942, 82. 265. 18 P. G. Wolynes and J. M. Deutch, J. Chem. Phys., 1976, 65, 450. 19 D. Toussaint and F. Wilcek, J. Chem. Phys., 1983, 78, 2642; K. Kang and S. Redner, Phys. Rev. A , 1985, 32, 435; P. Kraemer and I. Oppenheim, Physica A, to be published. 20 T. A. Whiten and L. M. Sander, Phys. Rev. Lett., 1981, 47, 1400. 21 Paul Meakin, Phys. Rev. A, 1983, 27, 604; 1983, 27, 1495. 22 B. €3. Mandelbrot, 7he Fracral Geometry of Nature (Freeman, San Francisco, 1982). 23 H. Gould, F. Family and H. E. Stanley, Phys. Rev. Lett., 1983, 50, 686. 24 M. Muthukumar, Phys. Rev. Lett., 1983, 50, 839. 25 M. Tokuyama and K. Kawasaki, Phys. Lett., 1984, IOQA, 337. 26 H. E. G. Hentschel, Phys. Rev. Lett., 1984, 52, 212. 27 L. Turkevich and H. Sher, Phys. Rev. Let& 1985, 55, 1026. 28 P. Meakin, Fractals in Physics, ed. L. Pietronero and E. Tosatti (Elsevier Science Publishers, 1986). 29 P. Meakin and J. M. Deutch, J. Chem. Phys., 1983, 78, 2093. 30 P. Meakin and J. M. Deutch, J. Chem. Phys., 1984, 80, 2115. 31 F. Family and D. P. Landau, Kinetics ofAggregation and Gelation (North-Holland, Amsterdam, 1984). 32 R. M. Ziff, in Kinetics of Aggregation and Gelation, F. Family and D. P. Landau (Nsrth-Holland, Amsterdam, 1984).20 The Lennard-Jones Lecture 33 R. M. Ziff, J. Stat. Phys., 1980, 23, 241; R. M. Ziff and G. Stell, J. Chem. Phys., 1980, 73, 3492; R. M. Ziff, M. H. Ernst and E. M . Hendriks, J. Phys. A, 1983, 16, 2293; E. M. Hendriks, M. H. Ernst and R. M. Ziff, J. Stat. Phys., 1983, 31, 519; R. M. Ziff, E. M. Hendriks and M. H. Ernst, Phys. Rev. Lett., 1982, 49, 593; M. H. Ernst, E. M. Hendriks and R. M. Ziff, Phys. Lett., 1982, 92A, 267. 34 F. Family, P. Meakin and J. M. Deutch, Phys. Rev. Lett., 1986, 57, 727; M. Kolb, J. Phys. A, 1986, 19, 263. 35 P. Meakin, Phys. Rev. Lett., 1983, 51, 1189; M. Kolb, R. Botet and R. Jullien, Phys. Rev. Lett., 1983, 51, 1123; R. Botet and R. Jullien, Phys. Rev. Left., 1985, 55, 1943; R. Jullien, M. Kolb and R. Botch, in Kinetics of Aggregation and Gelation (North-Holland, Amsterdam, 1984), p. 101. 36 T. Vicsek and F. Family, Phys. Rev. Lett., 1984, 52, 1669. 37 H. G. E. Hentschel and J. M. Deutch, Phys. Rev. A, 1983, 29, 1609. 38 P. Meakin, Phys. Rev. A, 1984, 29, 997. 39 K. D. Keefar and D. W. Schaefer, Phys. Rev. Lett., 1986, 56, 2376, and references therein. Received 13th April, 1987

ISSN:0301-7249    

DOI:10.1039/DC9878300001

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1957, 83, 21-31 The Generalized Langevin Equation as a Contraction of the Description An Approach to Tracer Diffusion Magdaleno Medina-Noyola Departamento de Fisica, Centro de Investigacion y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 Mkxico, D. F., Mexico An effective Langevin equation for a tracer Brownian particle immersed in a macrofluid of other diffusing particles is derived as a contraction of the description involving the stochastic equations for the local concentration and the local current of the macrofluid particles. The resulting Langevin equation contains the effects of the interactions with the other diffusing particles in a temporally non-local friction term plus a fluctuating force representing the random, diffusion-driven departures from spherical sym- metry of the distribution of macrofluid particles around the tracer.This fluctuating force satisfies a fluctuation-dissipation relation with the effective time-dependent friction. This program is fully developed here only in the absence of hydrodynamic interactions, although the formal aspects of its extension are also suggested. The results derived here, however, are found to provide a unifying framework to describe, for example, self-friction and electrolyte friction in suspensions of charged colloidal particles within the same theoretical scheme. 1. Introduction The theory of Brownian motion consists, to a large extent, of the many efforts to understand, derive, generalize and apply the well known Langevin equation: d v ( t ) dt M - - ( v ( t ) +f( t ) .This equation constitutes the simplest phenomenological model of the random motion of an individual Brownian particle, whose average behaviour is summarized by quantities such as the velocity autocorrelation function and the mean-squared displacement. In the classical theory’ f( t ) is postulated to be a purely random Gaussian stochastic vector. The stationarity of the equilibrium state leads then to the fluctuation-dissipation theorem (FDT), (J;( t ) J ( O ) ) = kT52S( t ) S , J , and to the conclusion that u( t ) is a stationary, Gaussian, Markov process. [ T above is the temperature, k , Boltzmann’s constant, M and 5 the mass and friction coefficient of the particle. S ( t ) is Dirac’s delta function and a,, is Yuronccker’s delta.] As a phenomenological model of equilibrium fluctuations, this theory has been extended by Onsager and Machlup.“‘ It has been e~tablished,”~ for example, that the Landau- Lifshitz stochastic version of the hydrodynamic equations,s and the linearized Boltzmann equation with added fluctuation^,^-^ are two examples of fluctuation theories that fit in such general scheme, i.e.they can be cast as multivariate, stationary, Gaussian, Markov processes. These two equations also constitute a good example of two phenomenological descriptions of the same fluctuating phenomenon, differing only in 2122 Generalized Langevin Equation and Tracer Difusion the degree of detail. Thus one would expect that projecting the latter into the subspace spanned by the hydrodynamic variables one should be able to derive the former.Such derivation has been carried and constitutes an illustrative example of the general idea of contracting the de~cription,~ in which the aim is to establish a connection between two phenomenological levels of description of the same fluctuation phenomenon. Another important realization of the same idea was suggested and developed by Fox and Uhlenbeck,’ who derived eqn (1,l) from the stochastic hydrodynamic equations plus the equation of motion of a suspended particle: This equation couples the motion of the particle, through the pressure tensor P integrated on its surface, with the stochastic hydrodynamic fields. In general, the contraction procedure introduces a memory effect in the contracted description.Latter of the program of Fox and Uhlenbeck have led to the derivation of a Langevin equation with memory, along with a corresponding FDT, and to the explicit determination of the memory function, from a purely hydrodynamic level of description. This program has been extended by Mazurgc to many Brownian particles in an incompressible fluid. A microscopic derivation exists, however, of the N-particle Langevin equation,” which reads In this equation, M,, v,( t ) and rr( t ) are the mass, velocity and position, respectively, of the ith Brownian particle, which is subjected to a random forcef;. GI, is the hydrody- namic friction tensor, and 4,J( I r, - 5 I) the pair potential, between particles i and j . Eqn (1.3), or its equivalent many-body Fokker-Planck equation,12 may be regarded as the fundamental ‘microscopic’ description of the dynamic properties of the macrofluid constituted by the N Brownian particles.Starting from this level of description, a formal projection to a hydrodynamic level has been carried out by Hess and Klein.13 The variables describing the state of the system at this level are the local concentration n ( r, t ) and the local current j ( r, t ) of Brownian particles. An essential feature of the macrofluid analogue of the Navier-Stokes equation is its temporal and spatial non-locality in the typical time- and length-scales probed in dynamic light-scattering experiments. l 4 Unfortunately no simple phenomenological laws, analogous to Newton’s law in the Navier-Stokes equations, are known for the macrofluid’s hydrodynamic description, except in certain limiting condition^.'^ Assuming, however, that such law could be derived, or approximated, one could in principle set up an analogous program, at the macrofluid level, of the program suggested by Fox and Uhlenbeck3 at the simple fluid level, i.e.one could derive an effective Langevin equation for a tracer Brownain particle immersed in a macrofluid of other diffusing particles as the contraction of a description based on the stochastic version of the hydrodynamic equations of the macrofluid. It is the purpose of the present work to discuss some of the formal aspects involved in such a program, and to present the results of its explicit application in the simple case in which hydrodynamic interactions are neglected.Thus, in sections 2 and 3 we discuss the general aspects of the program proposed. We then carry out this program explicitly in section 4, in the absence of hydrodynamic interactions. The result of its application to specific systems is then reviewed in section 5. 2. Contractions of Non-Markov Processes One of the main concerns in the derivation of the Langevin equation from fluctuating hydrodynamic^'^^-'^ has been the demonstration of the FDT for the former, given thatM. Medina-Noyola 23 a FDT holds at the hydrodynamic level. As Berman” has pointed out, this particular task may be carried out in a more formal and general fashion, by verifying that certain conditions, the FDT included, hold at the non-contracted description. In our derivation of the generalized Langevin equation of a tracer Brownain particle from the stochastic hydrodynamic equations of the surrounding macrofluid, we adopt a similar attitude.Thus in this section we develop the formal elements which will allow us to carry out such derivation in a fairly straightforward manner. Since the stochastic hydrodynamic equations of the macrofluid are known to exhibit memory, we must first establish the conditions under which a fluctuation-dissipation theorem holds for projections of non-Markov processes. These conditions are stated in the following two theorems, whose demonstration will be provided elsewhere.’’ Theorem A (the ‘stationarity’ theorem). Consider a general N-dimensional stochastic vector a ( t ) defined as the solution of the generalized Langevin equation: *= -lor G( t - t’)a( t’) dt’+f( t ) dt with random initial condition a ( 0 ) of zero mean, and driven by the stochastic vector f( t ) of vanishing average and statistically independent of a(O), i.e. such that ( a(0)fT( t ) ) = 0.Then the following three statements are equivalent. (The superindex T indicates transpose.) (i) The correlation function c( t, t’) = ( a ( t)aT( t’)) is time-translational invariant, i.e. c( t + T, t‘+ T) = c( t, t’) for all t, t’ and 7. [For simplicity, we shall generally refer to this property of a ( t ) as its stationarity condition.] (ii) The correlation function off( t ) is related with the memory function matrix G( t ) by the following fluctuation-dissipation theorem (f( t)f’( t‘)) = 6 ( t - t‘)G( t - t ’ ) ~ + 6 ( t’ - t)uGT( t’- t ) , ( t # t‘) (2.2) where 6 ( t ) is Heaviside’s step function and u = (a(O)a’(O)) is the covariance matrix of (iii) f( t ) is stationary [in the sense defined in (i) for the process a( t)], and G( t ) is a(0).such that the generalized Langevin equation, eqn (2.1), has the following structure: -- dt‘ L( t - t‘)u-’a( t’) +f( t ) d a ( t ) dt general (2.3) where o is an antisymmetric, time-independent matrix and L( t ) = (f( t)f’r(0)). Theorem B (the ‘contraction’ theorem). Let a( t ) be the N-dimensional stochastic vector defined by the conditions of theorem A. Let us assume, in addition, that a ( t ) is indeed stationary. Consider now a projection of a ( t ) defined by the n-dimensional stochastic vector a( t ) with components ai( t ) = ai( t ) , i = 1,2, .. , n < N. Assume, in addition, that (ai(0)aj(O)) = 0 for 1 s i d n < j s N. Then, the vector a( t ) also satisfies all the conditions of theorem A, and is also stationary. Furthermore, the generalized Langevin equation for a(t) may be written as dcllo= -lor r(t- t’)a(t’) dt’+cp(t) d t where q ( t ) is the new vector of fluctuating ‘forces’, and r(t) is given in terms of the memory matrix G ( t ) of a ( t ) by f ( z ) = e, * ( 2 ) - 6 I*( z)[ 2122 + e*Z( z)] - l & ( z ) . (2.5)24 Generalized Langevin Equation and Tracer Diflusion In this equation, I,, is the ( N - n ) x ( N - n ) identity matrix and the circumflex indicates a Laplace transform. G , t ) , GIz( t ) , . . . , are the submatrices of dimension n x n, n x ( N - n ) , .. . , of G ( t ) , defined by the following partition: The demonstration of theorem A follows essentially Kubo’s derivation16 of the (‘second’) Auctuation-dissipation theorem. The demonstration of theorem B is based on straightforward projection operator algebra to identify r(t) as given in the theorem, and on the explicit demonstration that the resulting vector cp( t ) satisfies the fluctuation- dissipation relation in statement (ii) of theorem A. We should mention that these theorems also hold for non-Gaussian processes, for variables a , ( t ) of arbitrary sym- metries (time-reversal etc.) and for stationary states other than the thermodynamic equilibrium state. Let us notice, however, that if a, ( t ) -+ y,a, ( t ) under time-reversal, with y, = f 1, then w = - I ‘ d , and LT( t ) = rL( t)r, with rj, = S,,y,.The latter equation is a generalization of Onsager’s reciprocity relations, which result if r = * I and L( t ) = 2 S ( t ) 9 . The previous theorems constitute the formal scheme in which we shall carry out the contraction procedure involved in the program we proposed before. 3. Fluctuating Hydrodynamic Equations for a Macrofluid As a simple illustration of the use of the theorems above, we formally derive the fluctuating diffusion equation for the local macroparticle concentration N n(r, t ) - C S [ r - r i ( t ) ] , = I (3.1) of a macrofluid in the presence of an external static field $(I-), from the macrofluid analogue of the stochastic Navier-Stokes equation. This we do as an illustration of the use of the theorems above, and because in the absence of hydrodynamic interactions, the motion of a tracer only couples directly with n(r, t ) , as discussed in the following section, and this will be the only case which we shall consider explicitly.The macrofluid analogue of the fluctuating Navier-Stokes equation describes the departures S j ( r, t ) of the local current: N j ( r , t ) = 1 u i ( t ) S [ r - r , ( t ) ] 1 1 1 (3.2) from its vanishing equilibrium average. The variable an( r, t ) = n( r, t ) - neq( r ) satisfies the exact continuity equation, which reads: d S n ( r, t ) d t = -V Sj(r, t ) . (3 3) The analogue of the Navier-Stokes equation can be constr~cted’~ from the require- ment that the vector a( t ) , with components a,( r, t ) = an( r, t ) and a, ( r , t ) = @,( r, t ) ( i = 1,2,3) may be cast as a stationary non-Markov process, satisfying the conditions and consequences of theorem A above.As a result, Sj(r, t ) is found to satisfy the following stochastic equation -0 d’r’a,-,’(r, r’)Sn(r‘, t ) J d S j ( r, t ) kBTneq( r ) a t M - - (3.4) dt’ \ d3r’6( r, r’; t - t’) .Sj( r’, t’) +f( r, t )M. Medina-Noyola where the stochastic vector f( r, t ) satisfies the fluctuation-dissipation theorem (J;(r, t)$(r’, O > > = [ G ( r , r’; t)li,jk,Tn”‘(r’)lM 25 and the inverse function of uoo( r, r’) = ( S n ( r, 0) Sn ( r ‘ , 0)) is indicated by 0;: ( r, r’). The tensor G(r, r’; t ) embodies all the essential dynamic information of the macrofluid, and its exact calculation is only possible in certain limiting cases.” Hence, approximations or assumptions will eventually have to be provided to define this missing piece of information. Since owing to time-reversal symmetry the static correlation (an( r, t)&( r’, 0)) vanishes, then the additional condition of theorem B is also satisfied for the vector a( t ) with components a,( r, t ) = Sn( r, t ) . Thus, according to theorem B, a n ( r, t ) is also a stationary non-Markov process, which obeys the following generalized diffusion equation: t)=-[o‘dt’[ d3r’G(r, r’; t - t ’ ) S n ( r ’ , t ’ ) + f ( r , t ) d t where G ( r , r‘; t ) = - V ; d’r’’H(r, r”; t)neq(r”) * V r ~ o & , l ( d ’ , r’). kBT M I (3.5) ( 3 . 6 ) The matrix Hij(r, r’; t ) , syqbolically denoted by [ H ( t ) ] , ( r , r ’ ) , is given by the inverse Laplace transform of [ z 1 + GZ2( z ) ] - ’ , where [ 13 j j ( r, r’) = 6,s ( r - r’) and [ G22( t ) ] j j ( r, r’) = [G(r, r’; t > l i j - In the next section we shall refer to the ordinary Fick’s diffusion law: I aSn(r, t ) at -- - -V, - Doneq(r)V, d’r’a,-d( r, #)an( r’, t ) +f( r, t ) (3.7) in which Do= k , T / i o is the free diffusion coefficient of a macrofluid particle.This equation results from the Markov limit of eqn (3.5), with the approximation Gij(r, r’; t ) = ( i o / / M ) 6 ( r - r ’ ) S j j 2 S ( t ) , which is the strict hydrodynamic limit of Gij(r, r’; t ) in the absence of hydrodynamic interaction^.'^ 4. Effective Langevin Equation Going back to our original project, we now need an equation of motion for a Brownian tracer, analogous to eqn (1.2), which expresses the force on that tracer in terms of the hydrodynamic variables.Such an equation may be suggested by the many-particle Langevin equation, eqn (1.3), as applied to the tracer. Let us assume that under certain approximations, the tensors cij may be substituted by a pairwise effective friction tensor cif(ri - q), depending only on the relative positions of the particles i and j . Under this condition, eqn (1.3) may be written, for the tracer, as +I [ v r + ( l r - r - l - ( t ) ~ ) ~ n ( r , t> d ’ r + f ~ t ) (4.1) with n(r, t ) and j ( r , t ) being defined by eqn (3.1) and ( 3 . 2 ) , <TD being the effective friction tensor, and + ( I - ) the potential of the interaction between the tracer and any of the other diffusing particles. cT is the solvent friction on the tracer.This equation, along with the stochastic hydrodynamic equations discussed in section 3, might provide the26 Generalized Langevin Equation and Tracer Difusion non-contracted description leading to the effective Langevin equation for the tracer after the elimination of the hydrodynamic variables. In this paper we only carry out a simplified version of this program, in which hydrodynamic interactions are neglected, i.e. GTD = 0. Thus there is no need to consider explicitly the variable j ( r , t ) , and we may base our hydrodynamic description on the generalized diffusion equation of the previous section. The equation of motion for the tracer may then be written as where we have elimiated rT( t ) from the integral by shifting the origin of the integration variables r to the centre of the tracer.Thus, the new local concentration n ’ ( p , t ) is defined as n’(p, t ) = n[ rT( t ) + p, t ] . Its equilibrium average, neq((p), is the local concentra- tion of diffusing particles in the static field # ( p ) . Because of its radial symmetry, n e q ( p ) does not contribute to the integral in eqn (4.2). Hence n’(p, t ) must be substituted in that equation by its fluctuation around neq(p), Sn’(p, t ) = n’(p, t ) - neq((p). The equili- brium equal-time correlation of an’( p, t ) must also be time-independent, and is essentially the two-body correlation function of the macrofluid in the static field + ( p ) . Thus definining croo(p, p’) = (8n‘(p, O)Sn’(p’, 0)), we must have that” V,n‘“p> = -P I %(P, P’)V,,+(P’) d3P’ (4.3) where P = l/(kBT).equation: The fluctuations 6n (p, t ) are now expected to satisfy a stochastic generalized diffusion = [V,,neq(p)] - u( t ) - [‘ dt’ d’p’ G’(p, p’; t - t ’ ) S n ’ ( p ’ , t’) + f ’ ( p , t ) . (4.4) a w p , t ) a t - 0 The linearized streaming term originates from the change of coordinates to the reference frame of the tracer. G’(p, p’; t ) determines the diffusive relaxation of the concentration fluctuations as described from such a reference frame, which is in fact diffusing. For the moment, however, we do not specify G’(p, 9’; t ) . Eqn (4.2) and (4.4) constitute our non-contracted description, to which we now apply the protocol developed above. Thus, we cast these equations as a generalized Langevin equation for the vector a( t ) , defined as aT( t ) = [a,( t ) , a2( t ) ] = [ u( t ) , Sn’(p, t ) ] .The corresponding static correlation matrix u is given by and the matrix G ( t ) , with analogous notation, by Using eqn (4.3), (4.5) and (4.6), we may verify that Thus, according to theorem A of section 2, a ( t ) will be stationary, provided that the following ff uctuation-dissipation relations hold: and 0 1 2 = G 1 2 u 2 2 = - [ G ~ I U ~ ~ ] ~ = - m21. ( f T ( t )~T(o)) = k,T!%2~(t)l (4.7)M. Medina- Noyola 27 These FD relations are, of course, assumed. Hence, we may now apply the contraction theorem, which implies, given that uI2 = uTl = 0, that the subvector a,( t ) = u( t ) is also a stationary process, satisfying the generalized Langevin equation M , d o o = -&( t) + f T ( t ) - Gint( t - t ’ ) u( t ’ ) dt’+fin‘( t ) .d t I,‘ The additional fluctuating force, f i n ‘ ( t ) , satisfies the FDT ( f i n ‘ ( t)fint(0)) = MTkB TG’”‘( t ) with Gin‘(t) given by Gint( t ) = - d’p d’p’[V+(p)]x’(p, p’; t)[V’neq(p’)] I I where ~ ’ ( p , p’; t ) is the solution of (4.9) (4.10) (4.1 1) with initial condition ~ ’ ( p , p’; t = 0 ) = S(p - p‘). Eqn (4.9) is the generalized Langevin equation for the tracer that we set out to derive. From symmetry considerations, one expects that Gin‘(?) = I A l ( t ) . Using eqn (4.3) and (4.10) we may show that there are two additional, equivalent expressions for Gin‘( t ) which, in terms of A&( t ) , read (4.12) and where C’( p, p’; t ) = (Sn’( p, t)Sn’( p‘, 0)) is essentially the van Hove function of the macro- fluid in the reference frame of the tracer, i.e.it is the solution of eqn (4.11) with initial value aoo(p, p’). The function [ ‘ ~ ; d ~ ’ ( t ) ] ( p , p’) in eqn (4.13) is the convolution of u,-d(r, r’) with the propagator ~ ’ ( p , p ’ ; I ) . More explicit results require the specification of G’(p, p’; t ) in eqn (4.4). Let us notice, however, that a general simplification results if we assume that G’(p, p’; t ) and o,,(p, p’) depend on p and p’ only through I p - p’l. This amounts to ignoring the inhomogeneity produced by the field + ( p ) of the tracer. In this case eqn (4.13) may also be written as (4.14) where hTD( k ) is the Fourier transform of [ neq(p)/ n - 11, and 1 -I- n h D D ( k) = goo( k)/ n is the static structure factor of the homogeneous macrofluid.Similarly, x’(k, t ) is the Fourier transform of the propagator x’( I p - p’ I; t ) of the local concentration fluctuations as described from the reference frame of the diffusing tracer. The results above still require the determination of the function G’(p, p’; t ) , or equivalently, of C’ or x’. The exact relationship between these quantities and the corresponding objects, G( r, r’; t ) , C and x, which describe the concentration fluctuations of the macrofluid from a reference frame at rest, is unknown. With the help of reasonable assumptions, however, such a relationship could be determined in an approximate fashion; doing that would complete the program we meant to carry out, namely, writing the effective generalized Langevin equation for the tracer, eqn (4.9), in terms of the28 Generalized Langevin Equation and Tracer Diflusion quantities which describe the dynamics of the macrofluid at a hydrodynamic level.As an example, let us adopt the approximation in which C’(p, p’; t ) = C’( 1 p - p’l, t ) . Its Fourier transform C’( k, t ) is then given by N C ’ ( k , t ) = ( i , j = l C exp{ik.[p,(t)--pj(O)]})-n2(2li):’d‘(k) =({exp [-ik*ArT(t)]} i , j = l exp{ik* [ r j ( t ) - r j ( 0 ) ] } ) ) - n 2 ( 2 r ) : ’ 6 ( k ) (4.15) where ri( t ) = rT( t ) +pi( t ) is the position of particle i in a fixed reference frame and ArT( t ) = rT( t ) - rT(0) is the displacement of the tracer at time t. If we assume that the average of the product in this equation may be approximated by the product of the averages, then we have that C’(k, t ) = x T ( k , t ) C ( k , t ) - n 2 ( 2 n ) ’ 8 ( k ) where X T ( k t , (exp rik ArT( 11) is the tracer-diffusion propagator.The only change produced in eqn (4.14) when this relationship is used is the substitution of the collective propagator ~ ’ ( k , t ) referred to the tracer, by the product of the normal (i.e. referred to a fixed reference frame) collective propagator x ( k , t ) times the. tracer-diffusion propagator XT(k, t ) . We shall call the resulting expression for Al( t) the mode-mode coupling approximation, in reference to the fact that such an expression was first derived by Hess and JSlein13 via mode-mode coupling arguments, for the case in which the tracer is one of the macrofluid particles (self-diffusion).Writing A[( t ) in terms of xr( k, t ) reflects the intrinsic non-linearity of the phenomenon of tracer diffusion. In our derivations, we have disguised such non- linearity by the change of the integration variable in eqn (4.2). One hopes, however, that an additional relation between A l ( t ) and xT( k, t ) could be provided by additional, independent approximations. For example, we might suggest the Gaussian approxima- tion14 for xT( k, t ) . Alternatively, Hess and Klein have suggested another closure relation, derived from their systematic application of the mode-mode coupling method.” On the other hand, although we aimed at establishing a connection betweeen tracer diffusion properties and the hydrodynamic description of the macrofluid, the end result, namely, writing A l ( t ) in terms of G ( r , r’; t ) or x ( r , r’; t ) , does not constitute a self- contained theory of tracer diffusion.The determination of these quantities requires additional approximations. The hope, again, is that simple approximations for x ( r, r’; t ) will lead to useful results for A[( t ) . In the following section we shall consider the results of the simplest version of the mode-mode coupling approximation,” suggested by Hess and Klein and referred to as the mean-field approximation. This is defined by approximating and xT(k, t ) = exp (- k 2 ~ ; t ) x ( k, t ) = [ -k2D0t/( 1 + nh,,( k ) ] where D; = kBT/ [; and Do = k , T / 5’. The mode-mode coupling relationship between x’ and x is, however, not the only possible relationship between these quantities.Other approximations could also be suggested based on similar considerations, or on other phenomenological considerations. In recent work we have considered the approximation that results from a modification of Fick’s diffusion law in eqn (3.7) in which we approximate G’(p, p‘; t ) by G’(p, p‘; t ) = V , * D*neq(p)V,a,-,’(p, p’)2S( t ) (4.16)M. Medina-Noyola 29 with D" = D;+ Do. The rationale for this approximation is that Do in eqn (3.7) describes the free diffusion of a macrofluid particle. When this phenomenon is observed from the tracer, whose free diffusion is described by D:, the apparent free diffusion coefficient of the macrofluid particles is described by D:+ Do. In the following section we shall present specific results derived from this approximation.As a final note, let us mention that all the results in this section can be extended to the case of a polydisperse macrofluid in a rather straightforward manner, by including an additional index to denote macroparticle species. 5. Applications The results of the previous section have been applied to specific systems. Here we mention some of these applications. 5.1. Self-friction The simplest application corresponds to the case in which the tracer is identical to the other diffusing particles. Let us illustrate this situation by considering the system whose direct interactions are defined by P + ( r ) = K exp[-z(x-l)]/x, 1<x= r / o = 00, 1 > x. This model has been used 1371472*721 to represent the direct interactions between charged colloidal particles in suspension.The correlation function of this model is calculated using the mean spherical appro~imation'~ (MSA), with the understanding that the rescaling prescription of Hansen and Hayter2' is employed when it is necessary. The mode-mode coupling approximation, within the mean-field closure for xT( k, t ) , has been applied before to this model by Hess and Klein.13 Recently, an extensive comparison22 of such results with the Brownian dynamics simulations of Gaylor et aL2l indicates that the mode-mode coupling approximation provides an accurate description of the self-diffusion properties at short times even at values of the coupling constant K , and of the volume fraction 7, near the freezing transition. For long times, a similar agreement is observed only at states not so close to the freezing transition. Similar results may also be obtained by using the simple Fick's diffusion law in eqn (4.16) with neq( r ) = n, although the quantitative accuracy of the mode-mode coupling approxima- tion is in general superior.It is also interesting to consider the results of the two approximations for a system with attractive, rather than repulsive Yukawa interactions. This corresponds to a solution of non-ionic micelles, with the Yukawa tail representing, in a crude approximation, the van der Waals direct interactions, and in the idealized conditions of no hydrodynamic interactions. The results of the two approximations discussed above are now illustrated with the long-time limit of ((ArT( t)')/6?, i.e. the self-diffusion coefficient D,, which in fig.1 is plotted as a function of the volume fraction for a system with z = 5. The two isotherms considered correspond to K = -0.5 and K = -3.0. This system shows a gas-liquid phase transition, whose critical isotherm corresponds to K , = -3.32, as calculated from the MSA using the compressibility equation. The results in the figure illustrate the fact that for systems with attractive Yukawa interactions, the two approxima- tions yield quite similar results for D,, although as the critical point is approached still closer, important differences appear). Finally, let us mention that the two approximations considered above coincide in the low concentration limit. For hard spheres, their result is given by D,/ Do = 1 -$q, rather than by the exact value 1 -27.The origin of this discrepancy is the homogeneous30 - I I I 1 Generalized Langevin Equation and Tracer Difusion Fig. 1. Long-time self-diffusion coefficient D,, normalized with Do, as a function of volume fraction q for the hard-sphere plus Yukawa model with screening constant z = 5, calculated from the mode-mode coupling approximation (-) and from the modification of Fick’s diffusion equation (- - -). The upper curves correspond to the isotherm K = -0.5 and the lower curves to K = -3.0. macrofluid approximation introduced for simplicity. One can show” that if the inhomogeneous Fick’s diffusion approximation in eqn (4.16) is employed, the correct result should follow. 5.2. Electrolyte Friction As mentioned before, the results of the previous section may be extended to the case in which the macrofluid contains various species of diffusing particles.Imagine that such diffusing particles are in fact the ions of an electrolyte solution in which a charged tracer diffuses. Then, A c ( t ) describes the effects of the interactions of the tracer with the electrolyte (in the absence, however, of hydrodynamic interactions). Our first a p p l i ~ a t i o n ~ ~ of the method discussed in this paper was precisely to this problem. Such an application was based on the multi-species version of eqn (4.16). It was that when the interionic correlation functions ajj( r, r’) were approximated by their Debye- Hiickel limit, the result derived before by S ~ h u r r ~ ~ for the static electrolyte friction, i.e.the time integral of A l ( t ) , is obtained. In further we have also calculated A c ( t ) in the Debye-Huckel limit, and have also extended Schurr’s results to include the effects of a finite size of the small ions using the primitive model and the mean spherical approximation.28 So far, the theories of electrolyte friction and of self-friction have been developed without a clear connection with each other. We should mention that the results of the previous section provide a simple approach that unifies the treatment of these two effects, which are simultaneously present in the phenomenon of tracer diffusion of charged colloidal particles. 29*30 6. Conclusions In this paper we have derived a generalized Langevin equation [eqn (4.9)] for a tracer Brownian particle immersed in a macrofluid of other diffusing particles.This derivation (section 4) proceeded as a contraction of the description involving the stochastic hydrodynamic equations of the macrofluid, and it ignored hydrodynamic interactions. The brief summary of applications in section 5 illustrates how the results thus derivedM. Medina-Noyola 31 may be explicitly applied, within reasonable approximations, to the description of tracer-diff usion in important specific systems. Our derivation in section 4 made use of other formal results developed before in sections 2 and 3. The theorems in section 2 provided the formal basis for the process of contraction. In section 3 it was indicated that the stochastic hydrodynamic equations may be cast as a stationary non-Markov process satisfying a generalized Langevin equation.These results, together with the equation of motion for the tracer in eqn (4.1), already suggest the main ingredients for an extension of the results presented here to include hydrodynamic interactions. It is our intention to pursue this work in such direction in the near future. The collaboration of the author with A. Vizcarra-Rendon and J. L. Del Rio-Correa was essential to the early development of this work, and is gratefully acknowledged. It is also a pleasure to thank Prof. R. Klein and the members of his group for many fruitful discussions on the dynamics of Brownian systems, and for their kind hospitality at Universitat Konstanz, where this paper was written. The author is greatly indebted with the Alexander von-Humbolt foundation for its support through a research fellowship, and to COSNET-SEP (Mexico), CONACyT (Mexico) and Bundesministerium fur Forschung und Technologie (Federal Republic of Germany) for additional support.References 1 S. Chandrasekhar, Rev. Mod. Phys., 1943, 15, 1. 2 L. Onsager and S. Machlup, Phys. Rev., 1953, 91, 1505. 3 R. F. Fox and G. E. Uhlenbeck, Phys. Fluids, 1970, 13, 1893. 4 R. F. Fox and G. E. Uhlenbeck, Phys. Fluids, 1970, 13, 2881. 5 L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1959). 6 M. Bixon and R. Zwanzig, Phys. Rev., 1969, 187, 267. 7 ( a ) E. H. Hauge and A. Martin-Lof, J. Stat. Phys., 1973, 7, 259; ( 6 ) M. G. Velarde and E. H. Hauge, 8 T. S. Chow and J. J. Hermans, Physics, 1973, 65, 156. 9 ( a ) D. Bedeaux and P. Maxur, Physica, 1974, 76, 247; ( h ) D. Bedeaux, A. M. Albano and P. Mazur, J. Stat. Phyr., 1974, 10, 103. Physica, 1977, 88A, 574; ( c ) P. Mazur, Physica A , 1982, 110, 128. 10 L). H. Berman, J. Stat. Phys., 1979, 20, 57. 11 J. M. Deutch and 1. Oppenheim, J. Chem. Phys., 1971, 54, 3547. 12 T. J. Murphy and J. L. Aguirre, J. Chem. Phys., 1972, 57, 2098. 13 W. Hess and R. Klein, Adu. Phys., 1983, 32, 173. 14 P. N. Pusey and R. J. A. Tough, in D-vnamic Light Scattering and Velocimetry: Applications of Photon 15 J. L. Del Rio-Correa and M. Medina-Noyola, to be published. 16 R. Kubo, Rep. Prog. Phys., 1966, 29, 255. 17 M. Medina-Noyola, unpublished results. 18 R. Evans, Adu. Phys., 1979, 28, 143. 19 D. A. McQuarrie, Statistical Mechanic5 (Harper and Row, New York, 1975). 20 J. P. Hansen and J. B. Hayter, Mol. Phrs., 1982, 46, 651. 21 K. J. Gaylor, I. K. Snook, W. J. van Megen, and R. 0. Watts, J. Chem. Soc., Faraday Trans. 2, 1980, 76, 1067. 22 G. Nagele, J. L. Arauz-Lara, M. Medina-Noyola and R. Klein, to be published. 23 J. M. Mendez, personal communication. 24 M. Medina-Noyola and A. Vizcarra-Rendon, Phys. Reu. A , 1985, 32, 3696. 25 J. M. Schurr, Chem. Phys., 1980, 45, 119. 26 H. Ruiz-Estrada, A. Vizcarra-Rendon, M. Medina-Noyola and R. Klein, Phys. Rev. A , 1986, 34, 3446. 27 A. Vizcarra-Rendon, H. Ruiz-Estrada, M. Medina-Noyola and R. Klein, J. Chem. Phys., to be published. 28 E. Waisman and J. L. Lebowitz, J. Chem. Phyr., 1972, 56, 3086. 29 W. D. Dozier, H. M. Lindsay and P. M. Chaikin, J . Phys. (Paris) Colloq., 1985, 46, C3-165. 30 S. Gorti, L. Planck and B. R. Ware, J. Chem. Phys., 1984, 81, 909. Correlation Spectroscopy, ed. R. Pecora (Plenum Press, New York, 1985), chap. 4, pp. 85-179. Received 15th December, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300021

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1987, 83, 33-46 Many-sphere Hydrodynamic Interactions Peter Mazur Instituut-Lorentz, Rijksuniversiteit Leiden, P. 0. Box 9506, 2300 RA Leiden, The Netherlands A scheme for the evaluation of many-sphere hydrodynamic interactions including interactions with container walls is presented. For a system consist- ing of a suspension of N spherical particles in an unbounded incompressible viscous fluid general formulae for the many sphere mobility tensors were obtained with explicit expressions for three- and four-body contributions up to order R-' with R a typical interparticle distance. Explicit expressions were also obtained to order R-"RO"I, n + m G 3, for the three-body problem consisting of two suspended spheres at arbitrary positions within a spherical container of radius R,,.The influence of hydrodynamic interactions on transport properties, in particular their essential non-additivity and the influence of wall effects, is then discussed for self-diffusion and sedimentation. In this paper we present an analysis of many-sphere hydrodynamic interactions which is based on a method of induced forces. These interactions which play an important role in the quantitative understanding of the properties of suspensions, were traditionally studied by the method of reflections, inaugurated by Smoluchowski' for those situations in which the fluid can adequately be described by the quasi-static Stokes equation for incompresssible steady flow. However, owing to the increasing complexity of the prob- lem, essentially only the two-sphere case was analysed by these methods.132 It must be mentioned at this point that Kynch' had already, within the framework of the method of reflections, derived expressions for three- and four-body contributions to the mobility tensors, but his work seems to have remained largely unnoticed.For the discussion of the properties of dilute suspensions only pair interactions need to be taken into a c c o ~ n t . ~ It was usually assumed and hoped that pairwise additivity of hydrodynamic interactions might hold in practice in concentrated suspensions as well [see e.g. ref. ( 5 ) ] . The long-range nature of these interactions, however, was not an argument in favour of such an assumption. With this in mind we developed a systematic scheme, which is reviewed here,6 to treat the full many-body problem (see in this connexion also work by Muthukumar' and Yoshizaki and Yamakawa').We shall first present this scheme in sections 2-6. We then briefly discuss self-diffusion and sedimentation as phenomena for which many-body interactions play an essential role. We consider these properties on the short timescale, i.e. for a time regime such that the relative configuration of suspended particles does not change appreciably. Two points will receive special attention: ( 1 ) the essential non-additivity of hydrodynamic interactions and (2) the influence of very long range hydrodynamic interactions and of wall effects on transport properties. To conclude we mention a number of related problems. 1. Equations of Motion; Formal Solution Consider N macroscopic spheres with radii a j ( i = 1,2.. . N ) , which move with velocities u, and angular velocities oi through an otherwise unbounded incompressible viscous 3334 Many-sphere Hydrodynamic Interactions fluid. The motion of the fluid obeys the quasi-static Stokes equation for all Ir - Ril > a,, i = 1 , 2 . . . N I V P(r) = O v u ( r ) = 0 with Here P is the pressure tensor, p the hydrostatic pressure, u the velocity field and 77 the viscosity of the fluid; Ri denotes the position of the centre of sphere i, while Greek indices label Cartesian components of tensorial quantities. The force Ki and torque T, exerted by the fluid on sphere i are given by where S, is the surface of sphere i (to be precise: the surface of a sphere centred at R, with radius a,+& in the limit c - 0 ) and i, is a unit vector normal to this surface pointing in the outward direction.In order to solve the set of equations (1.1) and (1.2) and subsequently determine the forces and torques from eqn (1.3) and (1.4), boundary conditions at the surfaces of the spheres must be specified. We assume stick conditions u ( r ) = u, + w , A ( r - R, ) for I r - R, I = a,. (1.5) The problem posed by eqn (1.1)-(1.5) may be reformulated by introducing a force density 4 ( r ) induced on the spheres and extending the fluid equations inside the spheres. The fluid equations are then written in the equivalent form for all r (1.6) N V * P = c F , ( r ) v u ( r ) = O j = 1 with F,(r)=O for Ir-R’l>a,. to Inside the spheres the fluid velocity field and pressure field are extended according u ( r ) = u , + w , A ( ~ - R , ) f o r ( r - R , ( s a , (1.7) p (r) = o for 1 r - R, I < a,.(1.8) E ( r ) = a;*.m)~(Ir-R,l- a,). (1.9) The induced force density is then of the form The factor aL2 has been introduced here for convenience. Making use of eqn (1.3), (1.4), (1.6) and (1.9) one can express the force K, and torque T, which the fluid exerts on sphere i in terms of the induced surface forces J;. With Gauss’s theorem one has indeed (1.10) T, = - dn^,n^, A J ; ( ~ ^ ~ ) . ( 1 . 1 1 ) JP. Mazur 35 To solve formally the equation of motion of the fluid we introduce Fourier transforms of e.g. the velocity field: u ( k ) = dr exp ( - i k * r ) u ( r ) . (1.12) I The Fourier-transformed induced force density F ( k ) is defined in a reference frame in which the centre of sphere i is at the origin F j ( k ) = drexp[-ik.( r - R , > J F , ( r ) . (1.13) With these definitions, the equations of motion (1.6) in wavevector representation I become: qk*v( k ) = By applying ;he operator I - LL, N -ikp( k ) + C exp (-ik R i ) 4 ( k ) . (1.14) (1.15) where k^= k / k is the unit vector in the direction of k j = 1 k v ( k ) = 0. and I the unit tensor, to both sides of eqn (1.14) one obtains with eqn (1.15) q k 2 u ( k ) = c exp (-ik R,)(I - ii) ~ ; ( k ) (1.17) which has, assuming that the fluid unperturbed by the motion of the spheres is at rest, the formal solution u ( k ) = c q - 1 ~ - 2 exp (-ik R,)(I - ti) . ~ , ( k ) . (1.18) This equation will serve as the starting point for the calculation of the forces and torques exerted by the fluid on the spheres, and thus of the hydrodynamic interactions which are set up between the spheres by their motion through the fluid.i j 2. Irreducible Tensors ; Induced Force Multipoles and Velocity Surface Moments For the purpose of evaluating hydrodynamic interactions it is convenient to introduce irreducible force multipoles, defined in terms of the surface forces A(&) according to n Here b' is an irreducible tensor of rank I, i.e. the tensor of rank I traceless and symmetric in any pair of its indices, constructed with the vector 6. For I = 1,2,3 one has, see e.g. ref. (9) or (lo), n n b = b, b,bp = b,bp -$6,pb2 - b,bpby = b,bpb, -+(6,pb, + 6,,bp + 6p,b,)b2. According to eqn (l.lO), (1.11) and (2.1) and36 Many-sphere Hydrodynamic Interactions In eqn (2.4) F!*=) is the antisymmetric part of F(2) and E the Levi-Civita tensor, for which one has the identity L : L = -21.The tensors 6f satisfy the orthogonality and completeness conditions6," In the above equations (21+ 1 ) ! 1 = 1 , 3 , 5 . . . (21-1)(2/+ 1). The dot 0 denotes a full 1-fsd contraction between the tensors 7 a n d z with the convention that the last index of 6; is contracted with the first index of etc. A('71) represents an isotropic tensor of rank 21 that projects out the irreducible part of a tensor of rank I : For I = 0, 1 , 2 one has (2.7) A('+') 0 b' = 6' 0 A",') =z With relations (2.5) and (2.6) one shows that the surface-induced force density f;( 6,) has the following expansion in terms of irreducible force multipoles: This expansion, which is written in a coordinate-free way, is equivalent to an expansion in spherical harmonics, to which it can be reduced if polar coordinates are introduced.For the Fourier-transformed induced force density Fi (k), the expansion (2.9) leads to [cf: eqn (1.9) and (1.13)] a? E ( k ) = 1 (21+ l)!!i31(kai)F0 FI'+') / = 0 (2.10) withj,(x) the spherical Bessel function of order I. In deriving the expansion (2.10) from eqn (2.9) use has been made of the identity" n - (--l)Fj/(k). a' sin k ak' k (2.1 1 ) Next to the irreducible induced force multipoles defined above, we also introduce irreducible surface moments of the fluid velocity field. The irreducible surface moment of order rn is defined as (2m + l)!! 4?ra2 I d r F u ( r ) 6( I r - R, I - a , ) n- (2m + l ) ! ! i ~ u ( i ) " = r z l dkj,(k)krnu(k) exp (ik- R,), rn 3 0 .(2.12) I (2rn + l ) ! ! ( 2 d 3 = i" The numerical factor (2m + l)!! is introduced for convenience. The velocity surface moments are the coefficients of an expansions of the fluid velocity field at the surfaces of the spheres in irreducible tensors F. Using the boundary condition (1.5) as well as the orthogonality condition (2.5) one has u( r ) sf = u,, 3iiU( r ) sf = a,L wi ( z m + l ) ! ! ~ u ( u ) "=o, f o r m 3 3 . (2.13)P. Mazur 37 In the next section we shall relate the induced force multipoles to the surface moments of the fluid velocity field through a hierarchy of equations. It is this hierarchy which will then enable us to obtain expressions for the mobility tensors which relate the forces and torques on the spheres to their velocities and angular velocities. 3.Determination of Induced Forces ; Properties of Connectors To determine the induced forces on the surfaces of the spheres we employ eqn (2.12) in the following way: we substitute the formal solution (1.18), together with the expansion (2.10) for F j ( k ) into the last member, and use the results (2.13) for the left-hand side. One then obtains the following set multipoles: where the coefficients, the so-called given by of coupled equations for the irreducible force N c o wjSn2) = C C At,"' Q Fj"' (3.1) j = l m = l connectors which are tensors of rank n+m, are 3(2n - 1)!!(2rn - l)!! y in-" 4T2 A!".") = A A A x dk exp (ik R,)$"-'\/ - kk)k"'lk-2jfl-,( kaj)jm-l( kaj).(3.2) Here R, = Rj - R j ; for i Zj, R , = lR,l> a j + aj. In principle one can determine from the set (3.1) all force multiples in terms of the velocities uj and angular velocities oj and in particular derive expressions for the mobility tensors which relate the forces and the torques exerted by the fluid on the spheres to these quantities. Before establishing these expressions we shall list a number of properties of the connectors A',"."'. ( 1 ) One verifies by inspection that the tensor quantities satisfy a symmetry relation upon simultaneous interchange of the particle indices i and j , the multiple indices n and rn and complete transposition of the tensor indices. (2) One also verifies that the matrix of connectors is positive definite in the following sense: where FI"' is the generalized transpose of the tensor of rank n, F'").One shows that this inequality ensures that the energy dissipation caused by the motion of N spheres is positive, as it should be. (3) Consider now in more detail the self-connectors A$'9m'. Using standard ortho- gonality properties of spherical Bessel functions it follows that A$3m' = -a~lB(n*n)gn,m (3.4) where the tensors B'"."' of rank 2n are independent of the index i. The tensors B(","' have been calculated12 explicitly in terms of tensors A('>'). The first two are where the tensor S of rank 4,38 Many-sp here Hydrodynamic Interact ions (4) Next we discuss the behaviour of the connectors A',"."', i # j , as a function of the interparticle distance R,.Straightforward evaluation of the remaining integral then leads to the result ( 3 . 8 ) A',", m ) ~ G 11'. m ) R ;( n + m - 1 ) + H ',", m R ; ( n + m + 1 ) where the tensors G',"."' and Hf,"'), which only depend on the unit vector fl, = R , / R , , and the radii a, and a,, are given by (3.9) The arrow on d/dR in eqn (3.9) indicates a differentiation to the left. ations, and becomes The expression for H'"."') can easily be further simplified by carrying out the differenti- For the tensor G("3'n' the differentiations can in principle be carried out in a similar formal way. We list here the explicit results for the first few of these tensors G r,' 9' ) = :(I + fg?,, ) (3.12) ( 3 . 1 3 ) (3.14) In eqn ( 3 . 1 3 ) , GY3*') denotes the part of G!,'.') that is (traceless) symmetric in its last two indices.A similar notation is adopted for GfS,*') in eqn (3.14); the tensor D is traceless and symmetric in its first and last two indices and defined by Dapys = 2rffrpryrs -5t rary& + r a r d , , + rpr8Sffy + r p ~ y s f f s ) . ( 3 . 1 5 ) Further explicit expressions for Gif""", with n,m =s 3 and n + m d 5 , may be found in ref. (6). 4. Mobility Tensors In the linear regime considered, the velocities and angular velocities of the spheres are related to the forces and torques exerted on them by the fluid in a way described by the following set of linear coupled equations: In the above equations, p 7 and pRR are translational and rotational mobility tensors, respectively. The tensors p;R and i F couple translational and rotational motion.The mobility tensors account for the hydrodynamic interactions between the spheres through their dependence on their relative positions. The analysis given in the previous sections enables us to express the mobilities in terms of connectors and thereby calculate these quantities as series in powers of inverse distances between the spheres.P. Mazur 39 We shall now, for simplicity's sake, restrict ourselves to the case of free rotation for which the torques T, vanish: T, = a l e : F ~ 2 u ) = 0 (4.3) and evaluate only the mobility tensors p y . By elimination from the set of equations (3.1) of all force multipoles Fin' 2 3, one obtains the following series for this quantity: py= (6~7)-'( aF'6, +AY,')(l + 6,) + C 0 B(2F,2r)-1 0 A t ' , ' ) k # k J Here B ( m , m ) - ' , rn # 2, is the generalized inverse of B(m,m' when actiqg on tensors of rank rn which are irreducible in their first rn - 1 indices.As for B(2r32s)- it follows from eqn (3.6) that this tensor of rank 4 is given by (4.5) B(2.\,2.T)-' - 10 ( 2 , 2 ) --FA . Each term in the series (4.4) has, as a function of a typical interparticle distance R, a given behaviour which is determined by the upper indices of the connectors and their number. Thus according to eqn (3.8) a term in eqn (4.4) with s connectors, s = 1,2,3 . . . , gives contributions proportional to with 1,3 f o r s = 1 P = ( 3s-2+2q f o r s z 2 , q=O,1,2 ,.... (4-6) This implies that py cannot contain terms proportional to R-2 and R-5. Note also that each term in the expression (4.4) containing a sequence of s connectors involves the hydrodynamic interaction between at most s + 1 spheres.Therefore the dominant n- sphere contributions, n 3 2, are of order R-3n+5, where eqn (4.6) has been applied with s = n - 1 and q=O. Explicit expressions for the various terms in the expansion (4.4) can in principle be found, using eqn (3.2) and (3.8)-(3.10) and forming the necessary tensor products. Thus the three-sphere contributions of order R-' to ~7 is given by the product looR-2R - 3 - 2 ( 1 , 2 \ ) . 81 I k k l R,, G , ~ . G ~ ~ , ~ s ) : G ~ ; Y Into this product one then has to insert expressions (3.13) and (3.14) for the correspond- ing G-tensors. In ref. (6) all contributions to the tensors p'f;T, p:R and p;" up to order RP7 are listed explicitly.As a final remark we mention that the series (4.4) can be written in closed form in a matrix notation in which tensors A:,'"' are the n, rn, i , j elements of a generalized matrix.6 5. Wall Effects; The Spherical Container In the preceding sections, we assumed that the suspended spherical particles were moving in an unbounded fluid, and calculated the corresponding mobility tensors which accounted for the hydrodynamic interactions between the spheres. Characteristic of these interactions is their very long range, which is apparent from the explicit expressions for the mobility tensors p:, i Z j . To lowest order in the expansion in connectors these are given by40 Many-sphere Hydrodynamic Interactions As a consequence of this long-range nature, the influence of boundary walls can be of importance even in cases where the vessel containing the suspension is very large.We shall, therefore, discuss in this section an extension of the scheme developed for the evaluation of mobility tensors that includes the effect of a spherical wall bounding the ~uspension.'~ The solution of the problem of N spheres moving in a viscous fluid inside a spherical container may be obtained from the solution to the problem of N + 1 spheres in an unbounded medium studied above, by observing that the analysis given remains valid if one of the spherical boundaries, the container specified by the index i = 0, encloses the other N spheres ( i = 1,2, . . . , N ) and the viscous fluid, provided the induced force Fo on the container is chosen in such a way that F,( r ) = 0 for I r - R,I < a, (5.2) where Ro is the centre of the container and a, its radius, and that the velocity field has, in addition to the extensions (1.7), the extension ~ ( r ) = o for I(r - R,)J z a,.(5.3) The analysis of section 3 then leads to the following set of equations: with connectors A!",") ( i , j = 0 , 1,2 . . . N ) defined again by the integrals (3.2) with the additional conditions R , > a i + a j f o r i , j = l , 2 . . . N, i # j ( 5 . 5 ) Roj<a,-aj f o r j = l , 2 . . . N. (5.6) The particle-particle connectors, therefore, remain unchanged; the particle-container connectors are of a different type but can also be evaluated13 using properties of integrals over Bessel functions. Since the velocity u, and angular velocity oo of the container vanish, one can reduce the set of equations (5.4) for the N + 1 spheres (particles and container) to a reduced set of the form (3.1) for the particles alone, but now in terms of new connectors AEZ!, which incorporate the hydrodynamic interaction with the container, and are given by m A!.",~) 1J;S.C.= + c A$P) . A@"), 1 , j = 1 , 2 . . . N. (5.7) p = l Note that the 'self'-connectors A!,?!:! are not diagonal in their upper indices: different multipoles in the same sphere couple via the container wall. It is in terms of the new connectors that the mobility tensors p,, which are again of the form of eqn (4.19)-(4.22), must now be evaluated. We give the expression for the translational mobility tensors pij (omitting from now on the indices TT) for the case that particle i is concentric with the container, in an expansion to third order in the parameters a / a o and a / R ( a and R are a typical particle radius and an interparticle distance, respectively):P.Mazur 41 ExpIicit expressions for more general cases can be found in ref. (13). 6. The Fluid Velocity Field As we saw (section 5), one needs, for a proper discussion of phenomena such as sedimentation, an expression for the velocity field of the fluid at a point r, caused by the motion of the spheres. Within the linear regime studied and for the case of free rotation, u( r ) may be expressed in terms of the forces exerted by the fluid on the spheres in the following way: N ~ ( r ) = - C Sj(r) K,. j = l The tensors S j ( r ) defined above can be derived from the general expressions for the translational mobilities of N + 1 spheres by putting RN+1= r and taking the limit aN+1+ 0: S j ( r ) ~ lim ) I N + ~ , ~ ( R ~ + , = r ) , j = 1 , 2 .. . N. a’N+,+O This equation expresses the fact that the velocity field can be probed with the aid of an infinitesimally small sphere located at r ; u( r) in eqn (6.1) is the velocity of this test sphere. To lowest order in the expansion of the mobilities in connectors one has for an unbounded suspension as follows from eqn (6.2) and (5.1) for i = N + 1 ( i e . for i denoting the test particle) and j = 1 , 2 . . . N. In eqn (6.3) 6 denotes the unit vector pointing from r to the centre of sphere j . Note that if one puts R, = r, i = 1,2,.. . , N, in eqn (5.1), one has for a given sphere j , to lowest order (and for lRj - rl> uj + ai) 7 af A A )I~( Ri = r ) - Sj( r ) = - ( rjrj - 4). 87r771Rj - rl The left-hand side represents, per unit of force exerted on sphere j , the velocity of a sphere i at position r with respect to the fluid velocity at that point in the absence of sphere i. Note that the R-’ contribution to this relative sphere velocity cancels, but that a long-range R-3 contribution remains. 7. Diffusion To apprehend the influence of hydrodynamic interactions on properties of suspensions, we shall discuss in this survey two transport phenomena. We consider again suspensions of hard spheres of common radius a and mass rn which, except for their short-range hard-sphere interaction and their hydrodynamic coupling, do not exert any direct long-range forces ( e .g . electromagnetic) on each other. The phenomena we shall study here are diffusion, in particular self-diffusion and sedimentation. Another transport phenomenon, viscosity, will not be dealt with, but has been studied using the same methods by Beenakke~,’~ to whose work we refer. As a starting point for our discussion we write down the expression for the short-time wavenumber-dependent diffusion coefficient D ( k ) (cJ: Pusey and Tough”):42 Many-sphere Hydrodynamic Interactions Here kB is Boltzmann’s constant and T the temperature of the system; G ( k ) , the Fourier-transformed density correlation function of the suspended spheres, is given by N G ( k ) = N - ’ C (exp ( i k .( R j - - R j ) ) . ( j = 1 (7.2) The coefficient (7.1) characterizes diffusion on a timescale t such that t,<< t i < t,, with t R = ( 6 7 ~ 7 p - l the characteristic time for the velocity correlation of a Brownian particle, and t, a structural relaxation time in which the configuration of the particles changes appreciably. For typical suspensions t , =: lo-’ s and t , s s. We restrict ourselves to this short time regime because it is both experimentally accessible and theoretically more managable than the long time regime. The (short-time) self-diffusion coefficient is obtained from eqn (7.1) in the limit as k + 00 and is given by D, = lim D( k ) = kTN-’ 1 k*(p ;;) i. kAm I (7.3) The physical meaning of short-time self-diffusion can easily be understood by observing that D, characterizes self-diff usion on a timescale such that the root-mean-square displacement of a particular particle remains much smaller than the average distance between particles.It is so to say the diffusion of a particle in a cage formed by the surrounding particles. The hydrodynamic interactions with other particles play a role, but not yet the direct hard-sphere interaction. We have mentioned that to lowest order in the expansion in connectors the (transla- tional) mobilities pel, for i # j , contain terms of order R-’ and K3 [cf: eqn (5.1)]. Such terms might, when evaluating transport properties, give rise to complications, more specifically to divergent integrals. For self-diffusion where only )rli needs to be considered, the above long-range terms do not contribute.We shall show here that these terms, as known, cause no difficulties in the evaluation of D( k ) , not only in the limit as k -+ 00, i.e. for D,, but even for arbitrary values of k. From eqn (4.4) and (7.2) if follows that to lowest order in the expansion in connectors one has with the Stokes-Einstein diffusion coefficient In eqn (2.1) g ( r ) is the pair correlation function, (7.6) no = N / V the average density of spheres, and A(’,’)( r ) the monopole-monopole con- nector field defined as A(l,I)(,.) (7.7) r / ( R , = 4. A(1.1) Note that since A!.” is only defined for R,, > 2a, the connector field is also only defined in the range r > 2a. However, as g ( r ) = 0 for r < 2a any choice for the continuation of A(’,’)(r) for overlapping spheres leaves the integral in eqn (7.4) unchanged.With this in mind we may define A(”I’(r) for all r as [cf eqn (3.2)] A(’.’)( r ) = ( 2 7 ~ - ~ dk exp (-ik r)A(‘,’)(k) (7.8)P. Mazur with A","(k) = 6 n ( l - &)k-2[jo(ka)]2. Eqn (7.4) may then be written in the form 43 (7.9) d r exp (ik r ) [ g ( r ) - l]A('71)( r ) k" G ( k ) D ( k ) Do + n , L . d r exp (ik. r ) ( 2 7 ~ ) - ~ dk' exp (-ik' r ) A ' ' , ' ) ( k ' ) - i. (7.10) I I Performing the integration over r in the second term yields + n,c 0 aAi'31)( k ) k" d r exp (ik r ) [ g ( r ) - l]A'',')( r) k^ (7.11) (7.12) in view of eqn (7.9). Thus to lowest order in the expansion in connectors only a convergent integral remains [ g ( r ) -+ 1 for r -+ CO, sufficiently fast], which vanishes as k -+ 00.All contributions from higher-order terms in the expansion in connectors, or essentially higher-order terms in an expansion in interparticle distances, yield convergent contributions. This completes the proof that the long-range 1/R and l / R 3 hydrodynamic interaction terms do not give rise to any divergences of D( k ) , in particular of D ( k = 0), the collective diffusion coefficient of the suspension. Let us now consider the self-diffusion coefficient in somewhat more detail. If one inserts into the expression (7.3) for the self-diffusing coefficient the series (4.4), together with the explicit form of the connectors (see section 3), one can, in principle, evaluate D, as a power series in no = N / V (a so-called virial expansion). This has been done up to and including terms of second-order in the density." Up to this order only two- and three-body hydrodynamic interactions need to be considered, since the probability that a given sphere has s neighbours in of order n i .Furthermore, to this order one needs only a knowledge of the hard-sphere pair distribution g ( r ) function to first-order in no and of the three-sphere distribution function g ( R 1 2 , R I 3 , RZ3) to lowest-order. Thus one must insert into the relevant integrals (0 for r < 2a + +[8- 12r/4a + 4 ( r / 4 ~ ) ~ ] for 2a II r ~ 4 a for r > 4a 0 R12<2a or RI3<2a or R2,<2a g ( R12 R13 R23) = I 1 elsewhere. In eqn (7.13) 4 is the volume fraction of suspended spheres (7.13) (7.14) 47T 4 =-a'n,. 3 (7.15)44 Many-sphere Hydrodynamic Interactions Using the above expressions for the distribution functions, it was found that D,/D,= 1 - 1.734 + 0 .8 8 4 ~ + e(4’). (7.16) Only two-body hydrodynamic interaction contribute to the well known term of order # and are, therefore, the only ones to contribute at sufficiently low densities. At higher densities, however, the many-sphere hydrodynamic interactions may not be neglected: two-sphere contributions alone would have led to a value of -0.93#2 for the term of order #*, instead of the value of +0.88#2 in eqn (3.4). This illustrates dramatically the non-additivity of hydrodynamic interactions. We should mention here that in evaluating the coefficients in eqn (7.16), we have in the expansion of the mobility in inverse powers of intersphere distances neglected, both for the two- and three-body case, terms of order R-* and higher.It can be shown however that the terms neglected contribute at most a few percent. It is quite clear from the above results that in a concentrated suspension one fully has to take into account the many-body hydrodynamic interactions between an arbitrary number of spheres. A virial expansion is not appropriate at high densities. However, it is possible to resum algebraically contributions due to hydrodynamic interactions between an arbitrary number of spheres.12 If one then subjects, in combination with this resummation, the expression for D, to a fluctuation expansion, i.e. if one writes D, as a sum of contributions from density correlation functions of higher and higher order, one obtains already from the first two orders numerical values which for volume fractions # < 0, 3 agree reasonably well with experimental results.8. Sedimentation As we have seen, the long-range hydrodynamic interaction terms of order R-’ and K 3 do not contribute to self-diffusion and give rise to convergent integrals in the evaluation of collective diffusion. If one calculates, on the other hand, the velocity of sedimentation in an unbounded medium, this quantity diverges, a fact which is sometimes referred to as the Smoluchowski paradox. Pyun and Fixman16 have shown the way to avoid the difficulty caused by the 1/R divergence by considering sedimentation with respect to the mean volume flow. In this way one indirectly takes into account the backflow caused by container walls.However, even then the R-3 term still gives rise to a conditionally convergent integral and poses, as noted by Burgers,” the problem of a possible depen- dence of the sedimentation velocity on the shape of the vessel containing the suspension. Batchelor18 was able to assign a definite value to the integral in question using an argument based on general considerations of a physical nature (valid for the unbounded system). Ultimately the difficulties mentioned should be resolved by a direct and explicit evaluation of the influence of container walls on the mobilities of sedimenting particles. Such a calculation has been carried out for two geometries, first for the case of a plane wall19 and then also for the case of a spherical container.The latter case has been reviewed in section 5 . For a discussion of sedimentation one also has to consickr the fluid flow caused by the motion of suspended spheres (section 1.6). One is then, using the results found as outlined in the two sections mehtioned, in a position to evaluate the mean particle velocity up and mean fluid velocity uf of a homogeneous distribution of identical spheres, ai = a, = a, sedimenting inside a spherical container.20 These two quantities, calculated at the centre of the container and in the limit that its radius a, tends to infinity, may be written as conditional averagesP. Mazur 45 of= lim C S , ( r = R , ) ( R , , > a forallj O F . (8.2) a0-w ( 1 ) Here (- - - I R,, = 0) denotes an average over those configurations for which RIO = 0, while ( - - - I R,, > a for all j ) denotes an average over configurations for which no suspended sphere overlaps the centre of the container; F is the gravitational force (corrected for buoyancy on each of the particles).To linear order in the volume fraction 4 of suspended spheres calculation of up and of on the basis of eqn (8.1) and ( 8 . 2 ) and eqn ( 5 . 8 ) and (6.2) yields up = [ 1 - 3 . 5 5 4 + e( 4 ' ) ] ( 6 ~ ~ p - ' F of = 1124 + e( +2j](6T77a j-% ( 8 . 3 ) (8.4) We can now also determine the average volume velocity given by o, = &+,+ (1 - 4)uf. From eqn ( 8 . 3 ) and (8.4) it follows that o, = [ 3 4 + e ( 4 ' ) ] ( 6 ~ ~ p - ' F . (8.5) Since, because of incompressibility, the volume flux through any closed surface must vanish, this result, namely that there is a non-vanishing volume velocity at the centre, implies the existence of a vortex of convective flow in the spherical container.Finally, one may evaluate the mean particle velocity with respect to the average volume velocity. The result is V , - V , = [ 1 - 6 . 5 5 4 + e( +')IF. (8.6) This is the result found for this quantity by Batchelor for an unbounded system. It is also what is found for sedimentation perpendicular to and towards a plane wall, in which case o, vanishes, in the limit of an infinitely distant We therefore come to the following conclusions: the average local velocity of a sedimenting particle in an homogeneous suspension depends on the shape of the container, however far the container walls. However, this shape-dependence disappears for the sedimentation velocity with respect to the average volume velocity.These results illustrate once more the essential role played by hydrodynamic many- body interactions. They show that for sedimentation the 'three-body' hydrodynamic interaction of two-particles and the container can in fact, for non-zero values of 4, never be omitted from consideration, even for sufficiently dilute suspensions. The discussion of wall effects and of the non-additivity of hydrodynamic couplings in the previous section thus underscores the relevance and usefulness of the scheme developed and summarized in sections 2-6 for the evaluation of many-sphere hydro- dynamic interactions. The same methods can also be used for the treatment of a number of related problems.Thus they can be applied to the case that the suspended particles are liquid drops instead of solid hard spheres. In this way one extends to many drops the classic Rybczynski- Hadamard theory for a single drop.21 In conclusion, we mention that next to its application to hydrodynamic problems, essentially the same scheme can be used for the analysis of many-body dielectric or diamagnetic interactions in dispersions of dielectric (metallic) or superconducting spheres, respectively. References 1 M. Smoluchowski, Bull. Int. Acad. Polon. Sci. Lett., 1911, l A , 28. 2 See e.g. H. Faxin, Arkiv. Mat. Asiron. Fys., 1925,19A, N 13; J. Happel and H. Brenner, in Low Reynolds Number Hydrodynamics (Noordhoff, Leiden, 1973); J. M. Burgers, Proc. Kon. Ned. Acad. Wet., 1940, 43, 425; 646; 1941, 44, 1045; 1177; B. U. Felderhof, Physica, 1977, 89A, 373.46 Many-sphere Hydrodynamic Interactions 3 G. J. Kynch, J. Fluid Mech., 1959, 5, 193. 4 G. K. Batchelor, J. Fluid Mech., 1976, 74, 1; B. U. Felderhof, J. Phys. A, 1978, 11, 929; R. B. Jones, 5 P. N. Pusey and R. J. A. Tough, in Dynamic Light-scattering, ed. R. Pecora (Plenum Press, New York, 6 P. Mazur and W. van Saarloos, Physica, 1982, 115A, 21; P. Mazur, Helv. Phys. Acta, 1986, 59, 263. 7 M. Muthukumar, J. Chem. Phys., 1982, 77, 959. 8 T. Yoshizaki and H. Yamakawa, J. Chem. Phys., 1980, 73, 578. 9 S. Hess and W. Kohler, in Formeln zur Tensor-Rechnung (Palm und Enke, Erlangen, 1980). Physl’ca, 1979, 97A, 113. 1985). 10 U. Geigenmuller and P. Mazur, Physica, 1986, 136A, 316. 11 P. Mazur and A. J. Weisenborn, Physica, 1984, 123A, 209. 12 C. W. J. Beenakker and P. Mazur, Physica, 1983, 1204 398. 13 C. W. J. Beenakker and P. Mazur, Physica, 1985, 131A, 311. 14 C. W. J. Beenakker, Physica, 1984, 128A, 349 15 P. N. Pusey and R. J. A. Tough, J. Phys. A , 1982, 15, 1291. 16 C. W. Pyun and M. Fixman, J. Chem. Phvs., 1964, 41, 973. 17 J. M. Burgers, Proc. Kon. Ned. Acad. Wet., 1941, 44, 1045; 1177; 1942, 45, 9; 126. 18 G. K. Batchelor, J. Fluid Mech., 1972, 52, 245. 19 C. W. J. Beenakker, W. van Saarloos and P. Mazur, Physica, 1984, 127A, 451. 20 C. W. J. Beenakker and P. Mazur, Phys. Fluids, 1985, 28, 3203; 1985, 28, 767. 21 U. Geigenmuller and P. Mazur, Physica, 1986, 136A, 269. 22 U. Geigenmuller, to be published. Received 8th December, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300033

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1987, 83, 47-57 Particle Diffusion in Concentrated Dispersions William van Megen and Sylvia M. Underwood’F Department of Applied Physics, Royal Melbourne Institute of Technology, Melbourne, Victoria, Australia Ronald H. Ottewill* and Neal St. J. Williams School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1 TS Peter N. Pusey Royal Signals and Radar Establishment, Malvern, Worcestershire WR14 3PS Three dynamic light scattering experiments on concentrated non-aqueous dispersions of spherical particles are discussed. The first two consist of measurements of the diffusion of tracer particles in dift’erent systems. In each case, host dispersions were rendered transparent by adjusting the refractive index of the dispersion medium to be the same as that of the particles.Trace amounts of particles of different refractive index, but of similar size and sterically stabilized by the same polymeric layer as the host particles, were added to the host dispersion. Thus the tracer particies provided the dominant incoherent light scattering. The measured correlation functions were analysed to provide particle mean-square displacements from which short- and long-time self-diffusion coefficients were obtained. In the third experiment the coherent scattering from concentrated dispersions of a single particle species was studied up to very high concentrations. Clear evidence of the glass transition, recently predicted, was found. 1. Introduction In this paper we describe three separate dynamic light scattering (d.1.s.) experiments designed to elucidate the nature of diff usion processes occurring in concentrated colloidal dispersions of spherical particles (polymer colloids).In the first two experiments the self-motion of individual particles, at dispersion volume fractions up to ca. 0.50, was studied by tracer techniques (incoherent scattering). In the third experiment collective particle motions (coherent scattering) were studied at higher concentrations where a long-lived amorphous or glassy phase was observed. In a system of identical interacting particles d.1.s. measures the collective particle motions. However, a simple and important dynamic property of such a system is the average motion of a single particle. An obvious way to measure this self-motion by light scattering is to match the refractive index of the dispersion medium to that of the particles and to add to the resulting transparent (non-scattering) ‘host’ dispersion a small concentration of ‘tracer’ particles.The latter should have a different refractive index, but the same size and interaction properties as the particles of the host dispersion. This principle is exploited in the first two experiments where systems different in several respects were studied in Bristol and Melbourne. At Bristol the system consisted of poly(methylmethacry1ate) (PMMA) tracer particles in a host dispersion of poly( vinylace- tate) (PVA) particles index-matched in a mixture of cis- and trans-decalin; the average diameter of both types of particle was 168 nm. At Melbourne PMMA particles of i Permanent address: Dulux Australia Limited, Clayton, Victoria 3168, Australia. 4748 Particle Diflusion diameter 660 nm, index-matched in a mixture of decalin and carbon disulphide, were used as the host particles; the tracer particles were silica of a similar size.In all cases, however, the particles were stabilized by a comb polymer with essentially a PMMA backbone and 'teeth' of poly( 12-hydroxystearic acid) (PHS). Because of the large difference in particle size between the two experiments different regimes of particle motion are probed, i.e. motions on small and large spatial scales relative to the particle diameters. The third experiment (Malvern) used PMMA/PHS particles (average diameter 250 nm), dispersed in a mixture of decalin and carbon disulphide. The refractive index was chosen to be slightly different from the particles in order to provide reasonably strong coherent scattering.2. Theory Dynamic light scattering measures the normalized time correlation function of the intensity, I, of scattered light, where t is the correlation delay time, g")( t ) = ( I ( o ) I ( t ) > / ( l > z (1) and the angular brackets represent an average over starting times '0'. If the light has Gaussian statistics the measured normalized dynamic structure factor is given by where C is a constant which is determined by experimental factors such as the finite size of the detector.' The assumption of Gaussian statistics applies to all systems discussed in this paper except the very dense metastable suspensions described in section 4.2.For a suspension of spherical particles the measured dynamic structure factor is2 N FM(Q, '1 = C ('j(Q)b,(Q) exp {'Q ['J(') - "k(')I))* ( 3 ) j,k-1 Here bk( Q ) is the scattering amplitude of the kth particle and r k ( t ) its position at time t, N is the number of particles in the scattering volume and Q = 4nn sin ( 8 / 2 ) / h 0 is the scattering vector (where n is the refractive index of the dispersion, A. the wavelength of the light in vacuo and 8 the scattering angle). In the 'ideal' case, where the particles differ only in their scattering amplitudes, there is no correlation between bJ and rJ so that eqn ( 3 ) reduces to2 where F( Q, t ) describes the dynamics of the spatial Fourier component (of wavenumber Q ) of fluctuations in the number density of particles; S( Q ) = F( Q, 0) is the static structure factor.W.van Megen et al. 49 If we further assume that the spheres are optically homogeneous, with particle j having refractive index nj, then (in the Rayleigh-Gans-Debye approximation') bj(Q)= V(nj-n,)f(Q) (8) where V is the volume of the particle, no the refractive index of the dispersion medium and f( Q ) is the scattering amplitude normalized such that f ( 0 ) = 1 ; thus the form factor P ( Q ) = f 2 ( 0). Therefore, in order to obtain F, in a d.1.s. experiment a refractive index matching condition is required such that which implies N b 2 ( Q ) - b O 2 = V * P ( Q ) W C (nj2-n;). (10) j = l In the particular case of only two types of particle with refractive indices n, and n2 at relative number concentrations a and 1 - a, index-matching [eqn (9)] implies a n l + ( l -a)n*= n, (11) b2(Q)-b(0)*= V2P(Q)a(l -a)(n1-n2)*.(12) so that Thus the refractive index of the suspension medium and the composition of the sus- pension can be adjusted to provide strong side incoherent scattering. Note that in this ideal situation a does not need to be small. However, in practice there can be distinct advantages in taking cy << I, i.e. performing a genuine tracer experiment. In this case we take b2(Q) = 0 (i.e. n, = no) so that the summation in eqn (3) runs only over particles of type 1; if the latter are present in trace amounts ( Nl << N ) eqn (3) reduces to N, FM(Q, t ) = C bf(Q)(ex~ [iQ Ar,(t)l) J = 1 where F,,,(Q, t ) is the self-dynamic structure factor of the j t h tracer particle and its displacement Ahv,( t ) = rj( t ) - rj(0).If the tracer particles are monodisperse the normal- ized (measured) dynamic structure factor reduces to F,(Q, t ) . The advantage of this approach is that F, is less sensitive to artefacts introduced by optical inhomogeneities and polydispersity of the particles. In particular, in this situation polydispersity of the host particles does not affect the light scattering directly and only enters through its effect on the average motion of the tracer particles. For a monodisperse system the self-dynamic structure can be written4 F,( Q, t ) = exp [ - Q2(Ar*( t ) ) / 6 ] x { 1 + a,( t ) [ Q2(Ar2( t ) ) / 6 I 2 + - - .} (14) where a2 etc. are functions which describe the non-Gaussian statistical properties of the displacement hr( t ) .From the mean-square displacements, (Ar2( t ) ) , one can obtain self-diff usion coefficients in the short- and long-time limits: 2 s Di = lim (Ar*( t ) j / 6 t (15) t+O and DL = lim (Ar2( t ) ) / 6 t . (16) t-0050 Particle Diflusion Table 1. Details of latices diameter/ nm system e.m. d.1.s. polydispersity designation PVA 128 166*4 2 5 ''0 SPVA/35 PMMA/2 - 660 * 10 4 O/* SMU/ 1 PMMA/3 - 250 f 10 1 5 */o SMU/6 silica 630* 10 6 */o SILA PMMA/l 141 170 f 4 1 5 */* SPSO/22 Here DH is the coefficient associated with (local) particle motion over distances much smaller than the particle radius, whereas D; describes motion over macroscopic dis- tances. D i is obtained from the initial decay of F,(Q, t ) at all scattering vectors whilst DL is obtained from the long-time behaviour of F, only'in the limit of small scattering vectors where non-Gaussian terms are unimportant [eqn ( 14)].2T5 The work described in section 4.2 is concerned with a suspension showing significant size polydispersity but little optical polydispersity.It has been argued6 that in this case FM(Q, t ) [eqn ( 3 ) ] can again be written as the sum of the averaged coherent and incoherent terms whose amplitudes are complicated functions of both the particle size distribution and concentration. However, for measurements near the main peak in the (static) structure factor, the case considered here, the interpretation of the data will assume that the strong coherent scattering dominates any incoherent scattering.Thus FM( 0, t ) will be assumed to provide a reasonable estimate of the full dynamic structure factor given in eqn (6). 3. Experimental Materials The PMMA latices were prepared by dispersion polymerization using the method previously described7,' and the PVA latices were prepared by a similar process.9 The particles were stabilized by a comb stabilizer which consisted of PHS 'teeth' grafted to a poly(glycidy1 methacrylate)-PMMA ba'ckbone.' Silica particles were prepared from tetraethylorthosilicate in an ethanol-water medium using the method of Stober et aL;" the particles were subsequently stabilized by physical adsorption of the same comb polymer as that used for the PMMA and PVA particles. The particle diameters deter- mined by d.1.s.in dilute suspension and, in some cases, electron microscopy (carbon replicas), along with their polydispersities (standard deviation of the size distribution divided by the mean), are listed in table 1. After preparation the particle dispersions were cleaned by centrifugation, removal of the supernatant and redispersing in the required filtered solvents; this process was repeated several times. In addition, latices of the smallest particles (i.e. PVA and PMMA/ 1) were filtered through Nucleopore membranes. Preparation of Optically Matched Dispersions The PVA particles, with an average refractive index of 1.471, could be precisely optically matched (rendered transparent) by a mixture of cis-decalin ( n = 1.481) and trans-decalin ( n = 1.469). Optimum match conditions were achieved by adjusting the liquid composi- tion until the intensity of the (laser) light scattered by the dispersion was a minimum.A similar procedure was followed for PMMA/2 and PMMA/3 using a mixture of decalinW. van Megen et al. 51 (mixture of cis and trans, n =: 1.48) and carbon disulphide ( n = 1.63) for an index- matching liquid. Owing to the significant (different) optical dispersions for PMMA and the decalin-CS, mixture, the matchpoint in these dispersions displays a greater wavelength dependence than for the PVA dispersions. Both PMMA/2 and PMMA/3 were optimally matched for the blue ( A =488 nm) laser line, but they still provided strong scattering at larger wavelengths (e.g. A = 647 nm). The (index-matched) dispersions were brought to the required particle concentration by centrifugation, removal of a weighed amount of clear supernatant and redispersal of the particles by a combination of rapid agitation and slow tumbling.The dynamic light scattering experiments were performed using standard photon correlation equipment. 4. Results and Discussion 4.1 Tracer Experiments The first of two tracer systems (Tl) was prepared by adding to the transparent PVA host dispersion, PMMA/l, particles at relative PMMA: PVA concentration 1 :43. The second tracer system (T2) was composed of ca. one silica particle for every 500 PMMA/2 host particles. In both cases the tracer particles were dispersed in the same liquid mixture as the host particles prior to their addition to the host dispersion. Since the host and tracer particles have about the same diameter (see table 1 ) and carry the same stabilizing polymer coating, it may be assumed that the (repulsive) interactions between the host particles and the host and tracer particles are essentially identical.In order to facilitate presentation of the results below, two further details need to be discussed. First, the PMMA/1 and PVA volume fractions were determined from weight-fraction analysis of the dispersions; the conversion from weight fraction to effective interaction volume fraction was accomplished taking the densities of the components and assuming a value of 9 nm for the stabilizer layer thickness. In the case of PMMA/2 this was accomplished by equating the core volume fraction (4J at which crystallization commenced with the known freezing volume fraction (& = 0.494) of hard spheres.This procedure, which is fully discussed in ref. (5) and (1 l ) , gives the effective volume fraction This gives an effective hard-sphere radius ca. 13 nm larger than the PMMA/2 core radius. The difference between this value (13 nm) and the expected average extended chain length of the adsorbed polymer (9 nm) may be due to some swelling of the'particle cores as a result of CS, absorption in the CS,-decalin mixtures. A similar procedure for this conversion to 4E is precluded for PVA owing to its small particle diameter and significant polydispersity. The second point concerns the non-Gaussian terms in eqn (14), i.e. the interpretation of F,. The wavenumbers at which system T1 (PMMA/l in PVA) was studied were below the location of the first maximum,12 Om, in the static structure factor and too small for the non-Gaussian terms to be appreciable.Therefore, in this case the Gaussian approximation to F,, namely, should be reasonable. By virtue of the larger particle diameters, system T2 (silica in PMMA/2) was studied at wavenumbers well beyond Om, thereby ensuring that the Gaussian part of F, has decayed to negligible values before appreciable growth of the non-Gaussian terms. Thus, eqn (18) has also been used in interpreting the data for system T2. Recent computations of F, by Brownian dynamics on concentrated disper- sions suggest that non-Gaussian effects are most noticeable at Q = Q,, . 1 3 Further support52 Particle Diflusion 0 / / / / / / I 20 40 60 delay time/ms Fig.1. Mean-square displacements in terms of the particle diameter 2R, for tracer system T1 (PMMA/l in PVA) showing free diffusion (-) and results for (effective) volume fractions & = 0.04 (- - -), & = 0.11 (. . . . .), & = 0.20 (- - -), C$E = 0.41 (- - -), & = 0.50 (- . -). The inset shows the short-time behaviour at C$bE = 0.20 (- . . -) and (bE = 0.41 (- - -). for the above assumption is provided by the fact that any wavenumber dependence of the measured F, was within the random errors normally expected for d.1.s. measurements. Fig. 1 and 2 show the mean-square displacements, in units of the particle diameter, derived from eqn (18) as functions of time for the systems T1 and T2. The results obtained are curved and show increasing deviation from free diffusion as #E is increased.The inset in fig. 1 shows the behaviour at shorter times for the system T1 at & = 0.20 and 0.41. As expected, in view of the different particle sizes in the two systems, and evident from the ordinate scales of fig. 1 and 2, quite different relative spatial scales are probed in the two cases. In fig. 2 the curvature of the plots is associated with the transition from short-time (local) motion to long-time (large-distance) diffusion. However, the asymptotic long-time behaviour has not been fully reached before F, has decayed into the noise. On the other hand, in the main part of fig. 1 the long-time regime should have been reached and linear dependence of the mean-square displace- ment found. The residual curvature evident in fig.1 is probably caused by the significant polydispersity of the tracer particles in system T1 (see table 1 ) . In order to obtain the short-time diffusion coefficients, Dk, defined in eqn (15) and plotted in fig. 3 , self-dynamic structure factors, measured at short times, were analysed by the method of c ~ m u l a n t s . ' ~ The resulting values of Dk for the two tracer systems are in good agreement; they also agree, within experimental error, with results (also plotted in fig. 3) obtained earlier from the large-wavenumber coherent scattering by dispersions of significantly larger PMMA particle^.'^ The theoretical prediction of Beenakker and Mazur,I6 based on a partial summation of the many-body hydrodynamic interactions, is also shown in fig.3 ; slight differences between experiment and theory are evident.W. van Megen et al. 53 0.12 0.10 0.0 6 0.04 0.0 2 20 40 60 80 100 delay time/ ms Fig. 2. Mean-square displacements for tracer system T2 (silica in PMMA/2) showing free diffusion (-) and results for (effective) volume fractions +E = 0.38 (- - . -), & = 0.41 (- - -), & = 0.44 (- . -), & = 0.49 (. . . .) and 4E = 0.54 (- - -). Note that the last two curves, for 4E = 0.49 and 0.54, represent results for coexisting disordered and crystalline phases. The long-time tracer diffusion coefficients DL, defined in eqn (16) and plotted in fig. 4, were obtained by fitting a single exponential to the long-time behaviour of F,. In the analysis of the data of system T1 this involved omitting the first quarter of the data, whereas for T2, roughly the first half was omitted.However, as mentioned above, the macroscopic diffusive limit is not fully reached for system T2, so that the values of DL obtained from these data are likely to be overestimated; such a trend is evident in fig. 4. The present results are in good agreement with the earlier data of Kops-Werkhoven and Fijnaut,17 also plotted in fig. 4. In addition, fig. 4 shows a trace summarising the short-time results D i of fig. 3. Clearly, local diffusion of a particle within its current nearest-neighbour cage (as represented by D i ) is, at high concentrations, many times faster than its motion (represented by D;) over distances comparable with or larger than its diameter. The approach of DL to zero at &, =: 0.55 is consistent with the onset of crystallization." 4.2.The Glass Transition In the previous section we discussed single-particle diffusion in both the fluid and crystalline phases of suspensions. Whilst at volume fractions beyond the melting con- centration (& = 0.545) the thermodynamically stable phase is crystalline, recent work' ' has shown that it is possible to prepare suspensions which remain in an amorphous (or glassy) state for very long times. Such behaviour has also recently been found in computer simulation 18-20 and theoretical models21'22 of simple atomic fluids. Since it is54 Particle Difusion I I I I I 4eff 0.2 0.4 0.6 Fig. 3. Short-time self-diffusion coefficients Dk (expressed in units of the free particle diffusion constant Do) defined by eqn (15), us.effective volume fraction, as obtained from tracer systems T1 (A) and T2 (0). Shown also are the data from ref. (15) (0) and the theoretical result of ref. (16) (- - -). 1.0 0.8 4" 0.6 Q --. 7A-l 0.4 0.2 0.2 0.4 0.6 Aff Fig. 4. Long-time diffusion coefficients, DL, defined by eqn (16), us. effective volume fraction, as obtained from tracer systems T1 (A) and T2 (0). Shown also are the data from ref. (17) (0). The dashed curve represents a trace through the results for D i in fig. 3.W. van Megen et al. 5 5 t \ ***. I . I 1 I I i I I 0.04 0.08 0.12 delay time, t l s Fig. 5. Logarithm of the intensity autocorrelation function us. time for dispersions of PMMA/3. Sample 1, q5,=0.196 (-- -); sample 2, q5,=0.262 (- - - -); sample 3, 4,=0.308 (, .. . .); sample 4, 4, = 0.331 (- - -); the 'coexisting liquid' of sample 5 , (- -), studied after phase separation had occurred (see text); sample 5, c$c = 0.349 (-). impossible to compress (or quench) atomic fluids sufficiently rapidly to bypass crystalliz- ation, concentrated suspensions of particles, in which the diffusive motions are slower by a factor of ca. lo', constitute possibly the only real systtms composed of spherical units in which the nature of the glass transition can be studied. Here we report measurements of the dynamic light scattering near the main peak in the structure factor ( i e . near Qm) by particle suspensions at concentrations up to & =: 0.60. The reason for choosing this particular wavevector is to allow us to compare results with recent molecular dynamics studies by Ullo and Yip2' on a fluid of slightly soft atoms.In order to measure the very slow motion occurring at high particle concentrations in a reasonable length of time we used PMMA particles of roughly half the size (PMMA/3, see table 1) of those used in earlier work." Unfortunately, to date it has not proved possible to prepare such small PMMA particles with narrow size distributions, and for the particles used here the polydispersity was ca. 15%. In fig. 5 and 6, we show plots of In [g'2'( t ) - 11 us. t at Q = 2.61 x 10' cm-' for a range of core volume fractions 4c, determined in the same manner as for PMMA/2 (section 4.1). In general, the correlation functions show a relatively rapid initial decay followed by a slower decay at long times, the latter becoming essentially zero at the highest volume fractions studied.This vanishing of the long-time decay rate is precisely the behaviour found in the theoretical treatments of Leutheusser*' and Bengtzelius et aL2* and in the computer simulations of Ullo and Yip." The interpretation given to this observation is that at the glass transition the particles become trapped by their local environments. While the particles still retain some freedom for local motions within their nearest-neighbour cages, as indicated by the rapid initial decay of the correlation functions, that part of the density fluctuation corresponding to longer-ranged particle translation has effectively been frozen in.56 Particle Diflusion \ \ \ \ \ 0.2 0.4 0.6 0.8- lD delay time, t / s Fig.6. Logarithm of the intensity autocorrelation fraction us. time for dispersions of PMMA/3. Sample 4, 4c = 0.331 (- - -); sample 5, 4c= 0.349 (-); sample 6, 4c = 0.378 (- . -); sample 7, 4c = 0.408 ( - . .). In the case where the density fluctuations are partly frozen in, the light scattered into a fixed detector no longer has Gaussian statistics. Thus, the scattered intensity consists of a fluctuating component associated with the local motions and an essentially constant component associated with the frozen density fluctuations. This leads to a reduction in the magnitude of the intensity fluctuations, ((12)/(1)*) - 1, measured in an experiment of finite duration. The zero-time value of g(2)(t)-l is a measure of ((12)/(1)2)- 1 [see eqn (l)].In fig. 6 we note that, whereas sample 5 shows the full modulation of the intensity, a marked reduction of this fluctuation magnitude is observed for samples 6 and 7. To be precise, the light scattered by sample 5 undergoes many (Gaussian) fluctuations in the duration of a measurement ( lo3 s). However, the reduced magnitude of fluctuations measured in the intensity scattered by samples 6 and 7 implies unambiguously, if not the existence of permanently frozen-in density fluctuations, at least the presence of fluctuation times much longer than lo3 s. For various reasons, for example the significant polydispersity of the samples, we will not, in this preliminary report, attempt a quantitative comparison with the molecular dynamics results of Ullo and Yip." However, our findings are in good qualitative agreement with their computer simulation data [compare fig. 5 with fig.3 of ref. (20)]. It remains to determine the effective volume fraction at which the glass transition occurs. In previous work on samples which showed a relatively rapid and reproducible crystallization" we determined freezing and melting concentrations and, by reference to a hard-sphere model, were able to obtain effective hard-sphere volume fractions &W. van Megen et al. 57 (see section 4.1). Probably owing to their large p~lydispersity,”,~~ the present samples did not exhibit the rapid crystallization characteristic of (almost) monodisperse systems. However, after several days, sample 5 had separated into a polycrystalline phase, occupying ca.80% of the total volume, and a fluid-like phase. Thus, taking freezing and melting volume fractions of hard spheres to be 0.494 and 0.545, re~pectively,’~ we calculate the effective hard-sphere volume fraction of sample 5 to be &=0.535. Then sample 6, which is the first to show relaxation times > lo3 s, has an effective volume fraction 4,=0.575. In view of the significant uncertainties in this determination of 4, for a polydisperse system, this result for the onset of the glass transition is in reasonable agreement with values found in previous experimental work“ and computer ~imulations,’~”~ which lie in the range 0.56 & 0.60. In conclusion the above observations clearly indicate the formation of a long-lived dense metastable amorphous state in particle suspensions (and, by implication, in a simple liquid).We are currently investigating this behaviour in greater detail with less polydisperse samples. References 1 E. Jakeman, in Photon Correlation and Light Beating Spectroscopy, ed. H. Z. Cummins and E. R. Pike (Plenum Press, New York, 1974). 2 P. N. Pusey and R. J. A. Tough, in Dynamic Light Scattering, ed. R. Pecora (Plenum Press, New York, 1985). 3 M. Kerker, The Scattering of Light (Academic Press, New York, 1969). 4 B. R. A. Nijboer and A. Rahman, Physica, 1966, 32, 415. 5 W. van Megen, S. M. Underwood and I. Snook, J. Chem. Phys., 1986,85, 4065. 6 P. N. Pusey, H. M. Fijnaut and A. Vrij, J. Chem. Phys., 1982, 77, 4270. 7 Dispersion Polymerization in Organic Media, ed. K. E. J. Barrett (Wiley, London, 1975). 8 L. Anti, J. W. Goodwin, R. D. Hill, R. H. Ottewill, S. M. Owens, S. Papworth and J. A. Waters, Colloid 9 S . Papworth and R. €3. Ottewill, to be published. Surf.‘, 1986, 17, 67. 10 W. Stober, A. Fink and E. Bohn, J. Colloid Interface Sci., 1968, 26, 62. 11 P. N. Pusey and W. van Megen, Nature (London), 1986, 320, 340. 12 R. H. Ottewill and N. St. J. Williams, Nature (London), 1987, 325, 232. 13 W. van Megen and 1. Snook, to be published. 14 D. E. Koppel, J. Chem. Phys., 1972, 57, 4814. 15 P. N. Pusey and W. van Megen, J. Phys. (Paris), 1983, 44, 258. 16 C. W. J. Beenakker and P. Mazur, Physica, 1984, 126A, 349. 17 M. M. Kops-Werkhoven and H. M. Fijnaut, J. Chem. Phys., 1981, 74, 1618. 18 L. V. Woodcock, Ann. N.Y. Acad. Sci., 1981, 37, 274. 19 C. A. Angell, J. H. R. Clarke and L. V. Woodcock, Adv. Chem. Phys.: 1981, 48, 397. 20 J. J. Ullo and S. Yip, Phys. Rev. Lett., 1985, 54, 1509. 21 E. Leutheusser, Phys. Rev. A, 1984, 29, 2765. 22 U. Bengtzelius, W. Gotze and A. Sjolander, J. Phys. C, 1984, 17, 5915. 23 E. Dickinson and R. Parker, J. Phys. Lett., 1985, 46, C229. 24 J. L. Barrat and J. P. Hansen, J. Phys. (Paris), 1986, 47, 1547. 25 W. G. Hoover and F. H. Ree, J. Chem. Phys., 1968,49, 3609. Received 5 th January, 1987

ISSN:0301-7249    

DOI:10.1039/DC9878300047

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1987, 83, 59-67 Brownian Diff usivities of Interacting Colloidal Particles Measured by Dynamic Light Scattering Aernout van Veluwen, Hendrik N. W. Lekkerkerker,* Cornelus G. de Kruif and Agienus Vrij Van ’t H o f Laboratory, University of Utrecht, Padualaan 8, 3584 CH Utrecht, The Netherlands In a system of interacting colloidal particles one can distinguish collective diffusion, short-time and long-time self-diff usion and exchange diffusion. The meanings of and connections between these diff usivities are established. It is shown how these various Brownian diffusivities may be obtained from dynamic light scattering measurements on monodisperse and polydisperse systems. Experimental results for both dilute and concentrated sterically stabilized silica dispersions are presented.The experimental concentra- tion dependence of the various diff usivities is compared with theoretical predictions. 1. Introduction With the large increase of activity in the study of concentrated colloidal dispersions in recent years there has arisen a considerable interest in the Brownian motion of interacting colloidal particles. 1,2 These motions are influenced by direct interactions and by hydrody- namic interactions, both of which depend on the spatial configuration of the particles. This makes the calculation of the Brownian diff usivity of interacting colloidal particles a challenging many-body problem in statistical mechanics and hydrodynamic^.^ For independent spherical colloidal particles the Brownian motion can be character- ized by the single-particle diffusion coefficient, which is given by the well known Stokes- Einstein expression kT 6 r q a Do=- where k is Boltzmann’s constant, T is the absolute temperature, q is the viscosity of the Newtonian solvent and a is the radius of the p a r t i ~ l e .~ In the limit of infinite dilution the quantity Do determines both collective diffusion as well as self-diff usion. Collective diffusion refers to the process by which concentration gradients gradually disappear owing to the Brownian movement of the particles. This process can be described by the diffusion equation dp, D,V2p dt where p is the number density of the colloidal particles and D, is the collective diffusion coefficient. Self-diffusion refers to the Brownian motion of a particle in a suspension of uniform concentration.A quantitative measure of self-diff usion is the mean-square displacement (hr2( t ) ) of this particle as a function of time. For non-interacting colloidal particles the mean-square displacement is linear in time for times larger than the relaxation time TB of the Brownian fluctuations in their velocities. For spherical particles this time T~ is 5960 of the order of Brownian Difusivities where m is the mass of the colloidal particle. For times larger than Tg the mean-square displacement is given by (At-’( t ) ) = 6Qt; t >> Tg (4) where D, is the self-diffusion coefficient. In the case of non-interacting particles both the collective diffusion coefficient D, and the self-diffusion coefficient D, are equal to Do.For interacting particles this is no longer the case, however, and one has to distinguish between collective diffusion and self-diffusion. In addition, for self-diffusion one has to take into account that for times larger than T~ = a2/ D the particle configuration can no longer be considered as effectively constant. As first noted by P ~ s e y , ~ this change in the particle configuration influences the mean-square displacement of the particles in such a way that it is now only linear for ‘short’ times T ~ < < t << T~ and ‘long’ times t >> T ~ , however with different values of the proportionality constant, i.e. different diffusion coefficients. Denoting the short- and long-time self-diffusion coefficients by DZhort and DPng we may write (Ar2(t))=6D:hor’t; T ~ < < t<< T~ ( 5 ) (Ar2( t ) ) = 6Dpngt; t >> T ~ .(6) In general Ofng< Dzhort, indicating that the changes in the particle configuration on average give rise to a hindering of the motion of a diffusing particle. In the case of polydisperse systems of colloidal pariicies one can distinguish between collective diffusion and exchange diff usion.6 Here exchange diffusion refers to the process where species of various kinds are exchanged at constant osmotic pressure. In case the particles in the polydisperse system differ only in marking then the exchange diffusion coefficient is precisely equal to DPng. Here we report dynamic light scattering (DLS) measurements of the short-time self-diffusion in a monodisperse system of large particles (radius a = 400 nm) and the collective and exchange diffusion in a polydisperse system of small particles (average radius d = 10 nm).Particularly the exchange diffusion appears to be a very sensitive function of concentration; for high concentrations (volume fraction 4 = 0.45) the exchange diffusion process turns out to be of the order of lo3 times slower than collective diffusion. This paper is organized as follows. In the next section we review briefly the connection between DLS and diffusion. In section 3 we present experimental DLS results on dispersions of large particles from which we obtain the short-time self-diff usion, and in section 4 DLS measurements on systems of small particles are analysed in terms of collective and exchange diffusion.The conclusions that can be drawn from this work are collected in section 5 . 2. Dynamic Light Scattering and Diffusion172 2.1. Short-time Self-diffusion By DLS on a monodisperse suspension of colloidal particles one can determine the dynamic structure factor F ( q , t ) of the colloidal particlesA. van Veluwen et al. 61 where rj(t) is the position of the j t h particle at time t and q is the scattering vector. From the time decay of this function information about the Brownian diffusion of the colloidal particles can be obtained. Specifically, one can define an effective diffusion coefficient as follows: where S ( q ) = F ( q, 0) is the static structure factor. particles can be determined by the N-particle Smoluchowski equation: On the time-scale we are considering here, i.e.t >> rB, the dynamics of the colloidal Here P ( r N , t ) is the probability that the N particles in the suspensions adopt the configuration rN ({rl, r 2 , . . . , r N } being their Cartesian coordinates) at time t, U ( r N ) is the interparticle potential and D, ( r ) is the generalized diffusion tensor. The generalized Einstein result D.. = kTb.. (10) relates D, to the mobility matrix b,, which specifies a given particle's (coarse-grained) drift velocity ui in response to forces 4 acting on the particles in the suspension N Using the N-particle Smoluchowski equation the time derivative appearing in eqn (8) can be calculated and one obtains 1 N In the limit where qa is large, i.e. well beyond the oscillations in the structure factor, eqn (12) simplifies considerably because lim S ( q ) = 1 qa+m and further the terms j # I in the summation on the right-hand side of eqn (12) go to zero, leaving only the N terms for which j = 1.Thus we obtain This is precisely the short-time self-diff usion coefficient. Thus the short-time self- diffusion coefficient can be measured by dynamic light scattering by using large particles ( a 2 0.4 pm) such that one can study the temporal behaviour of F ( q, t ) for values of q beyond the oscillations in the structure factor, i.e. qa b 12. In order to calculate the short-time self-diffusion coefficient from eqn (15) one has to know the configuration-dependent mobility function b l l ( r"). This quantity can be considered as a sum of one, two, three etc.particle contributions N N bll(rN)=b\:)+ b\i)(rL, r j ) + b\:)(rl, q , r k ) + ' * - . (16) j = 2 j , k = 2 j # k62 Brownian Diflusivities Averaging the relevant two-particle mobility functions over the two-particle distribution function to (lowest) order in the density Batchelor* obtained the first-order correction in 4 to DEhort: Dzho* = Do( 1 - 1.834). The contribution in +* to DZhort contains two distinct contributions: (i) the two-body hydrodynamic mobility function averaged over the second-order density contribution of the two-particle distribution and (ii) the three-body hydrodynamic mobility function averaged over the lowest-order density contribution of the three-particle distribution function. Using hydrodynamic mobility functions to order ( a / r)’ in the interparticle distance r, Beenakker and Mazur’ obtained (18) Dshort - - Do(1-1.73~-0.9342+1.80~2)+.* * where the first of the two terms of order $* is due to two-body hydrodynamic interactions, while the second results from three-body contributions. The difference in the contribution to order 6 in eqn (17) and (18) is due to the fact that in obtaining eqn (17) accurate numerical values for the two-sphere hydrodynamic mobility functions were used, whereas eqn (18) was obtained using analytical results to order ( a / r ) 7 . The higher terms in the density expansion are very difficult to calculate. Using a special summation technique Beenakker and Mazur ‘O,” succeeded in taking into account the many-body hydrody- namic interactions between an arbitrary number of spheres and obtained numerical values for the self-diffusion coefficient up to volume fractions of 0.45.2.2. Exchange Diffusion In dynamic light scattering from a polydisperse colloidal system one can distinguish two kinds of fluctuations: (i) overall density fluctuations of the colloidal species and (ii) concentration fluctuations at constant overall density. Pusey et aL6 have argued that these fluctuations modes can be related to the fluctuating variables that appear in the work of Kirkwood and Goldberg.12 Adapted to the situation of a colloidal dispersion considered to be in dialytic equilibrium with the solvent (component 0) which is kept at constant chemical potential po these variables can be written as r Here $+(””) Nk TJl, N, I, where V is the volume of the dispersion, II is the osmotic pressure, Nk is the number of particles of the kth species and P k is the number density of the kth species.It is assumed that the system consists of r colloidal species differing in size that can be considered as r components. It can be shown that fluctuations in the osmotic pressure are only caused by the variable A The fluctuation modes h 2 , A3,. . . , A, only give rise to fluctuations in the differences in the chemical potentials of the various species at constant osmotic pressure. Osmotic pressure gradients drive collective diffusion, whereas gradients in chemical potential differences cause exchange diffusion. The process of exchange diffusion, which involves the Brownian displacement of particles over distances larger than their own diameter, is closely related to long-time self-diff usion.A. van Veluwen et al.63 The general polydisperse case is too complex for a detailed calculation of the amplitudes and decay times of the various modes; however, particularly at high con- centrations, it may be argued that owing to the different relaxation rates there is little coupling between the collective diffusion mode and the exchange modes. Assuming this to be the case, the amplitude related to collective diffusion is proportional to (A:). Further, assuming that we are dealing with small hard spheres such that the scattering amplitude of the kth species is proportional to a; and the thermodynamic quantities can be described by the Percus-Yevick theory, one obtains for the total scattering intensityh713 (in the limits qa --* 0) Here L is a proportionality constant that does not concern us here and Under the conditions for which the total intensity is given by eqn (22) one obtains for the light scattering intensity associated with the fluctuations in the collective mode A l the following The remaining light scattering contribution, A- = (A,,, - A+), will relax according to the process of exchange diffusion.The relative weights of A+ and A- are mainly determined by the standard deviation of the size distribution: and are relatively insensitive to the type of distribution. For narrow distributions one obtains6 ( 2 5 ) A- 1 1 - I--= A+ Ato, Ato, [ 1+3g24(4-4)][ 1+3g2('742+84+3) -- (1 - @I2 (1 +w2 where 4 = l3 is the total volume fraction.3. Dynamic Light Scattering in Dispersions of Large Particles Short-time self-diff usion coefficients were measured in two dispersions, both containing silica particles of 440 nm radius. The silica particles in the first system were coated with y-methacryloxypropyltrimethoxysilane (TPM), whereas the particles in the second sys- tem were first coated with TPM, followed by a second coating reaction with trimethylhy- droxysilane (TMHS). Both types of particles are sterically stabilized, but, possibly because not all silanol groups have reacted with TPM, the first type of particle still carries a negative charge, as indicated by the fact that it shows electrophoretic mobility. The remaining silanol groups can be blocked by a second coating reaction with a small molecule (in this case TMHS). The particles of the second system turned out to be uncharged in tetrahydrofurfuryl alcohol (THFA), the solvent used in the experiments (see below).To avoid multiple scattering, which would complicate the results of DLS measure- ments, one has to select a solvent with a refractive index close to that of the particles. Furthermore, the particles have to be stable in this solvent. It appears that THFA satisfies both requirements. The optical matching point was situated between 30 and 35 "C.64 Brownian Difusiuit ies 1.0 0.8 6 0-6 s 5 * --. Q 0.4 0.2 0 I I I I I I I I I 0.1 0.2 0.3 0.4 I 4 5 Fig. 1. Normalised short-time self-diffusion coefficients for double-coated silica particles at 30 "C ( x ) and 35 "C (0). (. - .) is the O(4) result [eqn (17)], (- - .-) is the O(#*) result [eqn (18)) and (-) represents the many-body result of Beenakker and Mazur [ref. (10) and ( l l ) ] . 1.0 x X I I I I I I I I 1 I 0 0 .l 0.2 0.3 0.4, c 4 5 Fig. 2. Normalised short-time self-diff usion coefficients for single- (0) and double-coated ( X ) silica particles at 30 "C. The value of Do was obtained both by extrapolation of data taken at low concentra- tions in THFA as well as from measurements on dilute systems to which a trace amount of toluene had been added to enhance the particle scattering and thereby the signal-to- noise ratio of the measured autocorrelation function. In fig. 1 we compare the experimental data from the second system (double-coated silica particles) with the various theoretical results described in section 2.1.Volume fractions were calculated from weight fractions using the densities of the silica particlesA. van Veiuwen et al. 65 and solvent, 1.58 and 1.04 g ~ m - ~ , respectively. We note that the first-order correction in 4 describes the data surprisingly well up to 4 == 0.20. For the high concentrations the calculations of Beenakker and Mazur that take into account many-body hydrody- namics come close to the experimental points. Similar conclusions have already been reached by Pusey and van MegenI4 for the short-time self-diffusion of sterically stabilized poly( methyl methacrylate) (Pbf MA) particles. In fig. 2 we compare data obtained from the charged (single-coated) system with data obtained from the uncharged (double-coated) system.There appears to be remark- ably little difference between them. This indicates that the short-time self-diffusion is mainly determined by hydrodynamic interactions and is apparently affected little by direct particle interactions. 4. Dynamic Light Scattering in Polydisperse Systems of Small Particles In order to measure the exchange diffusion and mode amplitudes as discussed in section 2.2. We performed DLS measurements on polydisperse systems of small silica particles, sterically stabilized by a coating of stearyl chains, dispersed in cyclohexane. These particles are known to behave like hard sphere^.'^ We compare the data obtained from two dispersions differing only in size and degree of polydispersity of the particles. These systems will hereafter be referred to as systems I and 11.The two systems were characterized by DLS measurements on dilute systems. From a second-order cumulant expansion an apparent diffusion coefficient (or apparent radius) and polydispersity index were obtained. From these the number-averaged mean radius ii and standard deviation (T were calculated. In this way we found for system I the values 6 = 10 nm and (T = 0.3 and for system I1 ii -21 n p and u = 0.2. These values are consistent with the results obtained from the apparent radii of gyration as measured by static light scattering. As far as the number-averaged mean radius is concerned the above values are also consistent with the values obtained by electron microscopy (6 = 9 f 1 nm for system I and ii = 23 f 3 nm for system 11).On the other hand, we found that the standard deviations as obtained from electron microscopy and small-angle neutron scattering are significantly lower ((T = 0.10-0.15) than the values obtained from DLS. These differences may possibly be attributed to the presence of clusters of particles in the dispersion. This phenomenon would have a pronounced effect on scattering at low q (light scattering), but rather little influence on scattering at high q (small-angle neutron scattering) and electron microscopy. DLS measurements were made up to volume fractions of 0.55. The volume fractions were calculated from weight fractions using the densities of the silica particles and the solvent, 1.58 and 0.774 g cmP3, respectively. It turns out that with increasing concentra- tion the measured electric field autocorrelation function definitely consists of a fast- and a slow-decaying part.The amplitudes associated with these parts can be determined reasonably well from these measurements and are given in fig. 3. The decay times associated with the slow and the fast parts differ considerably (see table 1). Because the slow decay was definitely not single-exponential, we give in table 1 a range of decay times, expressed as diffusion coefficients, instead of one single exchange diffusion coefficient. The lower values represent the overall decay of the slow part of the field autocorrelation function, whereas the upper values were determined from the initial part of the slow decay, where in any case the fast mode has effectively fully decayed. The amplitudes displayed in fig.3 correspond to the latter fitting procedure. The decay time of the slow part increases rapidly with increasing concentration, whereas the fast mode shows little concentration dependence. However, there are some differences. First, we have observed that not only is the slowly relaxing part multi-exponential (as was expected from theory) but it appears that the fast decay These results are in qualitative agreement with the available66 Brownian Difusivities 1.0 0.8 0.6 0.1 o*2 t 0.1 0.2 0.3 0. L 0.5 ( 4 6 Fig. 3. Relative amplitudes of the slow mode as a function of the total volume fraction 4 for system I (0) and for system I1 (A). The theoretical curves (--..-) were calculated using eqn (25): ( a ) u = 2 , (b) u=3, (c) u=4.Table 1. Values of the decay constant of the fast mode (D,) and the range of decay constants of the slow modes (D-) for system I - D-/ lo-'' m2 s-' 4 D+/ lo-'* m2 s-' lower value upper value 0.15 21.3 0.20 22.3 0.24 23.0 0.34 27.7 0.40 26.5 0.46 31 1 3 0.45 0.9 0.3 0.6 0.045 0.12 0.016 0.08 0.001 (< 0.08) part is also not single-exponential. Furthermore, the dependence on overall concentra- tion of the relative amplitudes is more pronounced than predicted by theory for both systems studied. A possible explanation for this may be that the scattering power polydispersity is not only caused by size differences but also by differences in the refractive index of the particles. 5. Conclusions With DLS measurements various Brownian diffusion characteristics of interacting col- loidal particles can be investigated.Monodisperse dispersions of large particles allow us to probe the short-time self-diffusion. It appears that the short-time self-diffusion is not very sensitive to direct particle interactions. Polydisperse dispersions of small particles allow us to probe long-time diffusive behaviour. For concentrated dispersions of small particles, polydisperse in size, the measured electric field autocorrelation function is composed of two groups of modes with well separated decay times. The fast mode may be associated with collective diffusion and the slow modes with exchangeA. van Veluwen et al. 67 diffusion. Theoretically, this mode-decoupling is only expected for relatively high volume fractions. In view of this, and because of the uncertainty in the degree of the polydispersity, it is difficult to say whether the deviations of the measured mode- amplitudes from the theoretical predictions are of great significance.The collective diffusion shows little concentration dependence, whereas both short- time self-diff usion and exchange diffusion become slower with increasing concentration. However, whereas for volume fractions of 0.45 short-time self-diffusion becomes roughly a factor of 5 slower compared to the limit of infinite dilution, exchange diffusion becomes up to 1000 times slower, indicating that diffusion over distances larger than the diameter of the particles becomes extremely slow in concentrated dispersions. This may explain why dispersions of hard spheres at high concentrations so readily form a The silica spheres coated with TPM and TMHS were synthesized by Mr A.P. Philipse and Mr V. de Leeuw, and the silica spheres coated with stearyl chains (system 11) were synthesized by Mr G. Geels. We are grateful to them for providing these systems. Mr S. Coenen is thanked for assistance with the DLS measurements. We thank Mrs M. Uit de Bulten for typing the manuscript. This work is part of the research program of the Foundation for Fundamental Research of Matter (FOM) with financial support from the Netherlands Organisation for Pure Research (ZWO). References 1 P. N . Pusey and R. J. A. Tough, Dynamic Light Scattering Applications of Photon Correlation Spectroscopy, ed. R. Pecora (Plenum Press, New York, 1985), pp. 85-179. 2 J. M. Rallison and P. J. Hinch, J. Fluid Mech., 1986, 167, 131. 3 P. Mazur, Can. J. Phys., 1985, 63, 24. 4 A. Einstein, Investigations on the Theory ofBrownian Movement (Dover Publications, New York, 1956). 5 P. N. Pusey, J. Phys. A, 1975, 8, 1433. 6 P. N. Pusey, H. M. Fijnaut and A. Vrij, J. Chem. Phys., 1982,77, 4270. 7 R. J. A. Tough, P. N. Pusey, H. N. W. Lekkerkerker and C. Van den Broeck, Mol. Phys., 1986,59, 595. 8 G. K. Batchelor, J. Fluid Mech., 1976, 74, 1 . 9 C. W. J. Reenakker and P. Mazur, Phys. Lett. A , 1982, 91, 290. 10 C. W. J. Beenakker and P. Mazur, Phys. Lett. A , 1983, 98, 22. 1 1 C. W. J. Beenakker and P. Mazur, Physica, 1984, 126A, 349. 12 J. G. Kirkwood and R. J. Goldberg, J. Chem. Phys., 1950, 18, 54. 13 A. Vrij, J. Colloid Interface Sci., 1982, 90, 110. 14 P. N. Pusey and W. Van Megen, J. Phys. (Paris), 1983, 44, 258. 15 A. Vrij, J. W. Jansen, J. K. G. Dhont, C. Pathmamanoharan, M. M. Kops-Werkhoven and H. M. 16 C. G. de Kruif, P. W. ROUW, J. W. Jansen and A. Vrij, J. Phys. (Paris), 1985, 46, C3-295. 17 P. N. Pusey and W. Van Megen, Nature (London), 1986, 320, 340. Fijnaut, Faraday Discuss. Chem. SOC., 1983, 76, 19. Received 17rh December, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300059

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Faraday Discuss. Chem. SOC., 1987, 83, 69-73 Brownian Motion of Colloidal Crystals B. Ubbo Felderhof Institut f u r Theoretische Physik A, R. W. T. H. Aachen, Templergraben 55, 5100 Aachen, West Germany Robert B. Jones* Department of Physics, Queen Mary College, Mile End Road, London El 4NS We study the dynamics of colloidal crystals taking account of the retardation of the interactions due to the finite diffusivity of the counterions. It is shown that the retardation gives rise to electric dipole interactions proportional to the velocities of the macroions. As a consequence the damping rate of the overdamped transverse phonons tends to zero at small wavenumber. This agrees with observation in dynamic light scattering. 1. Introduction Charged spheres immersed in electrolyte solution can form colloidal crystals with a lattice distance much larger than the diameter of the spheres if the salt content of the solution is sufficiently low.' The dynamics of colloidal crystals differs from that of solid-state crystals due to the presence of the solvent fluid.The fluid causes friction which strongly dampens the motion. The long-range Coulomb forces between equally charged spheres are shielded by small counterions which diffuse fast in the ambient fluid. In this respect there is some similarity with the situation in metals where the conduction electrons move much faster than the ions, and the interactions between the ions may be treated in adiabatic approximation. We shall show that in colloidal crystals the finite diffusivity of the counterions gives rise to important effects.The dynamics of colloidal crystals has been studied experimentally in dynamic light-scattering experiments.* One observes dispersion curves of the damping rates of the overdamped longitudinal and transverse phonons very similar to the phonon disper- sion curves of solid-state crystals. Theoretically the lattice dynamics has been treated by analogy with solid-state theory, the only modification being the flow of the ambient fluid and the associated friction on the spheres. This theory was first developed by Hurd et al.,293 who assumed central-nearest-neighbour and next-nearest-neighbour interactions. In this theory the damping rate of the transverse modes TT(q) does not tend to zero as the wavenumber q tends to zero, in disagreement with the experimental data.In an earlier article4 we extended the theory by allowing more general potential interactions and showed that this approach cannot produce a damping rate T r ( q ) which tends to zero at small q. We suggested earlier4 that a qualitative feature is missing from the theory, namely the retardation of the interactions due to the finite diffusivity of the counterions. The idea was that in colloidal crystals the adiabatic approximation breaks down and that the instantaneous potential interactions must be replaced by retarded interactions. Here we show in a simplified model that the retardation has drastic effects and gives rise to long-range dipolar forces proportional to the velocities of the spheres. This makes a qualitative difference in the dispersion equation and leads to damping rates r,( q ) which tend to zero for small q, in agreement with experiment.6970 Brownian Motion of Colloidal Crystals 2. Retarded Interactions In order to derive the interactions we must consider the distortion of the Debye clouds about moving charged spheres. We begin by considering a static configuration of N spheres of radius a, with centres at (R, , . . . , R N ) and with charges { Z , e , . . . , Z , e } . Later we shall assume the positions {Rj} to be the sites of a regular lattice and the charges ( 2 , e ) to be identical, but at present we allow greater generality. We assume that on a macroscopic scale the centres {R,> are distributed uniformly in the container. Further we assume identical counterions of charge ze and diffusion coefficient D.In our model we replace the spatially varying density of the total Debye cloud in the static configuration by a constant no and describe the static potential by the linear Poisson- Boltzmann expression where K ? is the ionic strength K* = ( 4 m , z 2 e 2 / s k T ) and E is the dielectric constant of the solvent. Next we consider small oscillations of the spheres so that their centres have positions r,(t)=R,+s,,exp(-iot) j = 1 , ..., N. (2.2) Correspondingly, the perturbed counterion density varies harmonically with time. It satisfies the continuity equation -ion, + V * j , = 0 (2.3) where the current density is proportional to the gradient of the electrochemical potential n, pcL, = kT-+ ze+,. no This leads to the generalized diffusion equation n ze kT -ion, = DV'n, fL DO2+,.The electrostatic potential 4, satisfies the Poisson equation GV*C#I, = - 4 ~ z e n , + 417 1 pjw V 6 ( r - Rj) j (2.4) where p,, = Zjesj, is the dipole moment due to the displacement of the j t h sphere. The solution of the coupled equations (2.5) and (2.6) is given by with A2=K2-io/D. (2.8) The last term in eqn (2.7) shows that a moving sphere gives rise to a long-range electrostatic potential proportional to its velocity. The force of the ith sphere is given byB. U. Felderhof and R. B. Jones 71 where Eiu is the self-field acting on sphere i due to its own distorted Debye cloud and Ed, involves the positions of the other spheres. The self-field is given by K 2 I - 3& Es =-( K - >Pi- (2.10) and gives rise to ionic friction. The total friction coefficient of the sphere becomes K~ K - A 3~ iw &(w)=6mp0+- ----Zfe2= Jh+Ye,;(w) (2.11) where 7 is the viscosity of the solvent.The remaining electric field may be written where Uj., = -iosj, is the velocity of the j t h sphere. The tensor G, is found from the shielded Coulomb potential exp ( - A r ) r G,(A) = G(R; - R,, A ) , G ( r , A ) = VV (2.13) and F, is the dipole tensor -1 + 3 i i F , = F(Ri - R j ) F(r) =- (2.14) r3 * Owing to the relaxation effect the Debye cloud about a sphere lags behind when the sphere is moving, and this creates an electric dip01e.~ The effect is reduced by the electrophoretic effect, which has been left out of the above calculation and should be included as a convective flow term in eqn (2.5).However, despite the electrophoretic effect, long-range electric dipole interactions remain present, and these drastically affect the dynamics of the system. In the following we study the dynamics on the basis of the above equations. 3. Colloidal Crystal Dynamics We specialize to the case where the charges { z j e } are all equal and the positions {Rj} are located at the sites of a regular lattice. We have shown previously4 that the characteristic damping rates follow from the dispersion equation liwl - M(q, w ) H ( q , w ) l = 0. Miq, 0) = Mob, o)+Mc(q, 0) (3.1) (3.2) The 3 x 3 mobility matrix M( q, w ) is given by where M,(q, w ) is independent of the lattice structure and M,(q, w ) involves a lattice sum of hydrodynamic interactions.The first matrix Mo( q, o) incorporates a mean-field treatment of the fluid flow. It is given by where uc is the volume of a unit cell and a' = -iup/V, with p the mass density of the solvent. The second matrix in eqn (3.2) is well approximated by M c ( q , 0) = - 5 h ' V 3 K ( 4 ) (3.4)72 Brownian Motion of Colloidal Crystals where 4 is the volume fraction occupied by spheres and K ( q ) is a dimensionless lattice sum evaluated by Hurd et al.3 The matrix H(q, w > in eqn (3.1) involves the Coulomb interactions between spheres, as affected by the counterions. The matrix is given by H(q, w ) = -w2ml + D(q, o) (3.5) where m is the mass of a sphere and the dynamical matrix D(q, 0) may be written D(q, o) =E(q, w)+iwF(q, o ) - i ~ ~ ~ ( o ) l (3.6) where E(q, o) refers to the first term in eqn (2.12) and F(q, w ) to the second term.The matrix E(q, o) may be expressed as EV, (3.7) where S(q, A ) is given by the lattice sum where the prime indicates that the origin is excluded. The matrix F(q, w ) may be expressed as The dispersion equation (3.1) may now be cast in the form IIo’+iA(q, w ) w ‘ - B ( q , w)w-iC(q,w)(=O where the matrices A, B and C are given by with mf = pv, the mass of a unit cell of fluid and with R ( 4 ) = [I - 4’”K(q)]-’. (3.9) (3.10) (3.1 1) (3.12) For the b.c.c. lattice and special directions in q-space the dispersion equation decomposes into separate equations for purely longitudinal and purely transverse modes. The damping rates of the slowest modes are well approximated by (3.13) A striking difference with the corresponding result in the theory with instantaneous potential interactions is that BT( q, 0 ) tends to a non-vanishing constant at small y.This may be seen from the behaviour of the lattice sum (3.9) at small q, which is given by‘ (3.14)B. U. Felderhof and R. B. Jones 73 Q 3 Fig. 1. Plot of the damping rates rL(4) and IYT(4) of overdamped longitudinal and transverse phonons in a b.c.c. colloidal crystal for wavevectors q in the [ l , O,O] direction. The damping rates are given in Hz and the dimensionless wavenumber Q is defined by Q = q a / h , where a is the lattice distance. The matrix E ( q , 0) vanishes as q’ at small q. As a consequence the transverse damping rate rT( q ) vanishes at small q. This theoretical result agrees with the experimental data.’ In fig.1 we plot the dispersion curves for T , - ( q ) and Tr(q) as calculated in the approximation (3.13) for wavevectors q in the [ 1, 0,0] direction of a b.c.c. lattice. We have used the parameters 2 = 1350, sphere radius a, = 0.1 17 pm, diffusion coefficient D = 2.2 x lo-‘ cm’ s-’, lattice distance a = 1.45 pm, Debye parameter K a = 4.04 and fluid viscosity q = 0.01 poise. Where possible we have chosen the parameters in accordance with the experiment of Hurd et aL2 The behaviour of the damping rates rL,T(q) near the boundary of the Brillouin zone may be changed slightly in a more accurate determina- tion of the roots of eqn (3.10). 4. Discussion We have shown above that the dynamics of colloidal crystals is drastically affected by electric dipole interactions caused by the lag of the Debye cloud about the moving macroions. The damping rate of transverse phonons is predicted to tend to zero at long wavelengths. This is in agreement with light scattering experiments in which the Brownian motion of coiloidal crystals is observed.‘ Our theory has been simplified through our use of the linear Poisson-Boltzmann equation and the omission of the electrophoretic effect. We expect that an improved theory would lead to qualitatively the same results. References 1 P. Pieranski, Contemp. Phyy., 1983, 24, 25. 2 A. J. Hurd, N. A. Clark, R. C. Mockler and W. J . O’Sullivan, Phys. Rev. A, 1982, 26, 2869. 3 A. J. Hurd, N. A. Clark, R. C. Mockler and W. J. O’Sullivan, J. Fluid Mech., 1985, 153, 401 4 B. U. Felderhof and R. B. Jones, 2. Phys., Teil B, 1986, 64, 393. 5 F. Booth, J. Chem. Php., 1954, 22, 1956. 6 B. R. A. Nijboer and F. W. de Wette, Physica, 1958, 24, 422. Received 9th December, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300069

出版商:RSC    

年代:1987

数据来源: RSC

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Furuday Discuss. Chem. SOC., 1987, 83, 75-85 Brownian Motion of Charged Colloidal Particles surrounded by Electric Double Layers Gerhard A. Schumacher and Theodorus G. M. van de Ven" Pulp and Paper Research Institute of Canada and Department of Chemistry, McGill University, Montreal, Quebec, Canada H3A 2A7 Using photon correlation spectroscopy, diffusion coefficients of charged colloidal particles surrounded by electrical double layers have been deter- mined. The diffusion constant equals the value of a neutral sphere at high and low electrolyte concentrations, but is reduced by several per cent when the electrical double layer is comparable to the radius of the particle. The reduction in diffusion constant depends on the zeta potential of the particle and the sizes of the ions in the double layer.The diffusion of charged particles can be explained by the theory of Ohshima ef al. (J. Chem. Soc., Furaday Trans. 2, 1984, 80, 1299) for the friction coefficient of charged spheres, assuming that the friction coefficient of a charged sphere in Brownian motion equals the equilibrium friction coefficient of a sedimenting sphere. Many experiments have been described in the l i t e r a t ~ r e l - ~ in which the diffusion coefficients of small particles, such as enzymes, polyelectrolytes or colloidal particles with adsorbed polymer, have been measured as a function of pH or salt concentration. Variations in the effective particle radius obtained from diffusion constants were ascribed to changes in particle dimensions (swelling, shrinking) or to changes in configurations of the adsorbed polymers.In those studies what has been overlooked is that the very presence of a diffuse ionic double layer affects the diffusion of small particles. Although not at all obvious, there are reasons to believe that the Stokes-Einstein relation continues to be applicable to charged particles, and hence the diffusion coefficient or equivalent sphere radius can be related to the friction coefficient of a charged sphere, expressions for which exist in the l i t e r a t ~ r e . ~ To test these theories we performed photon correlation spectroscopy (P.c.s.) measurements on classical gold sol particles of ca. 20nm and polystyrene latex spheres of ca. 40 nm, as a function of salt concentrations for various salts. Salt concentrations were varied in the range 0.1 d K a d 3 for the gold sol and 0.3 s K a s 7 for the latex, a being the sphere radius and K-' the Debye length or double-layer 'thickness'. We used three different salts: sodium chloride, sodium benzoate and tetraethylammonium chloride.These salts were chosen because of the widely different sizes of their ions and because they demonstrate little or no surface activity, thus avoiding specific adsorption. As predicted by theory, the diffusion was found to be maximum at low and high Ka values and minimum around Ka = 1. The depth of the minimum depended mainly on the positive ion size. At the minimum, the diffusion constants were 3 4 % lower than the values at high or low electrolyte concentration, in which case the diffusion constant equals that for a neutral sphere.According to the theory, in the limit of low 6 potentials (eJ/kT<< 1, e being the unit electric charge and kT the thermal energy), the depth of the minimum at K a = 1 should depend equally on the size of the positive and negative ions, for a given zeta potential. However, as the potential increases, the depth of the minimum becomes primarily dependent on the counterion (the ion opposite in charge to the colloidal particles). Since both the gold and the polystyrene are negatively charged, one would expect the minimum depth to 7576 Brownian Motion of Charged Colloidal Particles be deeper when tetraethylammonium rather than sodium is the cation. Sodium benzoate would be expected to have a slightly deeper minimum than sodium chloride, since there is still a small contribution from the coion.Experimental observations were in excellent agreement with these predictions, proving conclusively that the decrease in diffusion constants is due to the presence of electrical double layers around the particles. Theory One can consider Brownian motion to be a random walk in which the particle executes jumps of a characteristic length. Charged colloidal particles will execute this random walk, however, in the presence of an ionic double layer. Since the particle is moving, the ionic double layer will be distorted from its spherical symmetry and this will give rise to a microscopic electric dipole moment, which will increase the drag on the particle and thus reduce its velocity. Estimating the time of a Brownian jump of a colloidal particle of radius a, mass m, friction coefficient f and diffusion constant D as r,=rn/f and the relaxation time of an ion (of diffusion constant Di) in the double layer after a jump of length 1 by rr = 12/ Di it follows that the double layer can be considered to keep its equilbrium shape during a jump when r,<< rj.( 3 ) Estimating 1 from 1 == U ~ T ~ and the particle jump velocity uo from mu: = 3kT, using the Stokes-Einstein relation Df = kT and Stokes law (for a spherical particle) f = 6 ~ 7 a (4) which, with some modification, also holds for ions (7 being the viscosity of the medium), condition (3) can be restated as For most Brownian particles this condition is satisfied. The effect of ionic double layers on the sedimentation velocity of charged spherical colloidal particles in a dilute suspension has been determined theoretically by Ohshima et aL4 The ionic double layer loses its spherical symmetry because of the particle motion (sedimentation), setting up a microscopic electric dipole, which decreases the sedimenta- tion velocity.Brownian motion of charged colloidal particles can be thought of as being analogous to the effect of ionic double layers on sedimentation, if the equilibrium distorted double layer can follow the Brownian particle throughout its random walk. We can then replace the relative sedimentation velocity with the relative diffusion constant, to obtain theoreti- cal expressions on which to base our experiments. The diffusion constant is predicted to be maximum at low and high K a values and minimum at KU =: 1 (see fig.1). Besides general numerical calculations, Ohshima et uL4 also derived an analytical expression for the relative sedimentation velocity u / u, ( oo being the Stokes sedimentation velocity) as a function of KU and zeta potential, valid at low zeta potentials ( e f / k T < < 1).G. A. Schumucher and T. G. M. van de Ven 77 1.00 0.98 6 0.96 1 0.94 0.92 0.90 I I 1 I I J - 2 -1 0 1 2 1% (.a) Fig. 1. Relative diffusion constant as a function of ~a for three potentials: e l / k T = ( a ) 3, ( b ) 5 and ( c ) 8. The points are numerical results taken from Ohshima et ai.4 The solid lines are smooth fits. The dashed lines indicate the prediction of eqn (10) valid for low potentials.Calculations are for KCI. Replacing v / v o with D/Do, where Do is the Stokes diffusion constant, we obtain N I = 1 where with and where is the scaled drag coefficient of the ith ion; J; is the drag coefficient of the ith ion; zi is the valency of ion i and n: the bulk concentration; E , is the dielectric constant of the medium and E~ the permittivity of free space. For 1 : 1 electrolytes eqn (6) can be written as D/ Do = 1 - f i a* f ( KU ) (10) where78 - 90 - 80 > -70 E \ h - 60 - 50 -40 Brownian Motion of Charged Colloidal Particles I I I log (NaCl/mol dm-3) Fig. 2. Zeta potential us. sodium chloride concentration for colloidal gold particles. The inset shows a typical doppler spectrum from laser doppler electrophoretic experiments (relative intensity, I, us.frequency shift, Av; E = 509 V m-', [NaCl] == lop4 moI dmP3). The indicated spread is estimated from the width of the laser doppler spectrum. is the average scaled ion drag coefficient, or [ c j eqn (4) and (9)] the average scaled ion size; <D is the reduced y potential Predictions of eqn (10) are included in fig. 1 (dashed lines) for an aqueous KC1 solution. Experimental Gold sols were prepared according to the procedure used by Enustun and Turkevich.' The extraneous ions were removed by allowing the sol to flow through ion-exchange resin (Amberlite MB-1) until the conductivity of the sol dropped to 8 x lo-' K1 m-', which corresponds to a salt concentration of ca. 6 x lop6 mol dmP3 1 : 1 electrolyte. Electrophoretic mobilities u were determined as a function of NaCl concentration (see fig.2) using laser doppler electrophoresis from the relation6 A Av 2nE sin (8/2) U = where A is the wavelength of light, hv the frequency shift, n the refractive index of the medium, E the electric field strength and 0 the scattering angle. The zeta potentials, 5, were calculated from u, from the relation u ( & Ka) given by Oshshima et al.7 As can be seen from fig. 2, the measured [ potentials are in the range - 40 to - 90 mV, corresponding to @ in the range - 1.5 to -3.5. Electron micrographs of the gold sol (plate 1) were taken using a Philips EM400T transmission electron microscope. The micrographs were analysed to obtain the particle size distribution (see fig. 3). The histogram was obtained from measurement of 198 particles. From this distribution the mean diameter of the gold particles was found toFaraday I>iscuss.Chem. Soc., 1987, Vol. 83 Plate 1. Electron micrograph of gold particles. G. A. Schumacher and T. G . M. van de Ven Plate 1 (Facing p . 78)G. A. Schumacher and T G. M. van de Ven a/nm IS 20 25 I I I 1 30 20 10 0 - ._ 5 a cr 0 30 20 10 2 .o 2.5 D/ lo-" m2 s-l 1 3.0 79 Fig. 3. ( a ) Particle size distribution of gold sol and ( b ) particle diffusion constant distribution where the diffusion constant has been calculated from the size by means of the Stokes-Einstein relationship. The dashed curve is a Gaussian fit. be 20.2 nm (the standard deviation was 10%). The distribution is, however, skewed and therefore it is not completely justified to approximate the distribution as Gaussian.The distribution of diffusion constants, which were calculated from the size distribution using the Stokes-Einstein relationship, is quite nicely approximated by a Gaussian fit [see fig. 3 ( 6 ) ] . The standard deviation of this distribution was found to be 9%. Polystyrene latex was provided to us by Prof. R. H. Ottewill at the University of Bristol. The latex was prepared using the procedure of Ottewill and Richardson' and was subsequently extensively dialysed against distilled water. It was then stored over mixed-bed ion-exchange resin. The surface charge density u was measured conduc- tometrically and found to be -45.3 mC m-'. Using the Gouy-Chapman theory for flat plates, this surface charge density corresponds to a surface potential, $o, of - 166 mV at K = 5 x lo7 m-' and T = 298 K.Measured zeta potentials have been shown' to be 2 to 3 times smaller than the surface potential calculated from Couy-Chapman theory at salt concentrations of mol dm-3, the difference decreasing as the salt concentration increases. This suggests a zeta potential for the latex of ca. -60 to -80 mV. The reason for the difference between measured [-potentials and potentials calculated from the Gouy-Chapman theory is the following. For low potentials, the Debye-Huckel theory predicts that for flat plates,80 Brownian Motion of Charged Colloidal Particles Here E = E , E ~ . For spherical particles the equivalent equation is 1+Ka a (T=E:- *O. At K a = 1, for a given surface charge density, t,hO from eqn (15) will be half the value for t,ho calculated from eqn (14).This indicates that the difference between the calculated surface potentials and the measured zeta potentials is largely due to the use of the flat-plate approximation, which is not valid for spherical particles: Electron microscopy data for the latex particles gave an average diameter of 32 nm with a standard deviation of 18%. Diffusion constants of both the gold sol and the latex particles were measured by dynamic light scattering l o using a Brookhaven Instruments Corporation photon correla- tion spectrometer with a BI-2030 digital correlator and a Spectra Physics 120 helium-neon laser. All measurements were carried out at 25 f O . l "C and with a scattering angle, 8, ranging from 45 to 150". The gold sol was not diluted for the light scattering measure- ments, whereas it was necessary to dilute the latex. The volume fractions for both the gold sol and the latex were estimated as 2 x Samples were filtered through a 0.2 p m millipore filter before measurement of the diffusion constant.The normalized time correlation function of the electric field of scattered light, g(q, T), was assumed to be in the form of a cumulant expansion where only the first (r) and the second (p2) moments were calculated. The diffusion constant is related to r through the following expression where q is the absolute value of the scattering vector q : 4nn h q = - sin (:) . Results and Discussion Photon Correlation Spectroscopy (P.C.S.) Measurements The ratio, Q, of the second moment, p2, to the square of r was found to be < 0.1 in all cases for the polystyrene latex and between 0.1 and 0.2 for the gold sol.The second moment can be related to the standard deviation of the particle size distribution through the following: p2= ( r - T ) 2 G ( r ) d r (19) I where G(T) is the distribution of r values, which we have assumed to be Gaussian [cJ: fig. 3 ( b ) ] . Using the average sizes of the latex and the gold sol which were obtained from electron microscopy (see Experimental section), we found Q=0.03 for the latex and Q = 0.01 for the gold sol, using eqn (19). These values are low enough'"' to analyse g(q, 7) as a single exponential: d q , 7) = exp (- r4 (20) which assumes that the scattering of light is due to a monodisperse solution of scatterers.The calculated value of Q for the latex corresponds to the experimentally observed0.: 0.: 3 m v 0. - O.! -1 .I h t- ef; W -1.' E: - -2. G. A. Schumacher and T. G. M. van de Ven 81 1 1 I I 1 . 25 50 75 100 125 TIPS Fig. 4. ( a ) Experimental time correlation function for gold sol (not normalized). The inset is an expansion of the data at low sampling time. ( b ) Plot of the natural logarithm of the time correlation function against T. values, whereas for the gold sol the observed Q value is higher than the calculated one. This can be attributed to the presence of 1-4% doublets5 found in the gold sol, which will broaden the size distribution and hence increase Q. Although in the presence of doublets the gold sol diffusion constant distribution [fig. 3( b ) ] will no longer be Gaussian, an estimate of the broadening of the distribution can be made by calculating the standard deviation of this new distribution (with doublets), which was found to be 20%. This result is in agreement with the P.C.S.data. A typical experimentally observed time correlation function for the gold sol is shown in fig. 4(a). If these data are to be analysed according to eqn (20), then a plot of In [g(q, T)] us. T should yield a straight line. A typical example is shown in fig. 4(b). As Q increases, eqn (20) is no longer valid and must be replaced by eqn (16). However, from fig. 4(6) we can see that the P.C.S. data from the gold sol are easily approximated by a single exponential, which means that despite having 1-4% doublets in the gold sol, effects of polydispersity can be ignored.The first 2 ps of fig. 4 ( a ) are expanded (inset) to show that there is a decay of much smaller order than that for translational Brownian diffusion of the gold sol. The timescale of this decay leads us to believe that this is due to rotational Brownian diffusion, which for particles of this size should be of the order of a microsecond. As can be seen from plate 1 the gold particles are not perfectly spherical. The small percentage of doublets82 Brownian Motion of Charged Colloidal Particles 60 90 120 scattering angle/" 150 Fig. 5. Angular dependence of the measured diffusion constant for gold sol (a) and for polystyrene (0) latex. The error bars for the gold sol data indicate reproducibility, whereas for the latex they refer to the error in calculating z'.(Data for distilled water as medium.) could also be responsible for this effect. Including the datum point at 2 ps in the analysis did not significantly alter the results and therefore the complete time correlation function was used when calculating I'. For a dilute monodisperse suspension of scatterers, the diffusion constant measured should be independent of the scattering angle. Realistically, however, there is always some degree of polydispersity and therefore one expects some curvature in a plot of. diffusion constant us. scattering angle. Measured diffusion constants as a function of scattering angle are shown for both the latex and the gold sol in fig. 5 . The diffusion constant for the latex remains roughly constant with angle, while for the gold sol it begins to deviate at 90".The measured diffusion constant of the gold sol is constant over a substantial range, bringing us again to the conclusion that effects of polydispersity can be ignored. This reasoning applies even more to the latex. For the analysis of the variation of diffusion constant with salt concentration, a scattering angle of 135" was chosen. All values of K a are determined using the radius calculated from P.C.S. measurements. In the case of the gold sol the average diameter obtained from P.C.S. measurements, 22 nm, is slightly higher than that obtained from electron microscopy, 20 nm, and in the case of latex, P.C.S. measurements give a diameter of 40nm, whereas electron microscopy give a diameter of 32 nm.The difference in diameters found from P.C.S. and electron microscopy for the gold sol is small. The difference in diameters for the latex is larger, but not uncommon,'"l and can be attributed mainly to the difference in size when the latex is solvated (P.c.s.) or unsolvated (electron microscopy), and also due to the inherent differences in the types of measurement being made. The Effect of Ionic Double Layers on Brownian Diffusion Fig. 6 and 7 show the change in diffusion constant as a function of Ka for the gold sol and the latex, respectively. In both cases the depth of the minimum is smallest for sodium chloride, largest for tetraethylammonium chloride and somewhere between these two for sodium benzoate. Eqn (10) predicts that the depth of the minimum should beG.A. Schumacher and T. G. M. van de Ven 83 concentration/ mol dm-3 2.3 d IV1 2 . 2 "E + .- I \ 4: 9 2 . 1 0 0.5 1 .o 1.5 Ka Fig. 6. Plot of the experimentally determined diffusion constants us. KU for gold sol for three different salts: NaCl (e), NaC,H5C02 (0) and (CH3CH2)4NC1 (0). For the last two salts no data were collected for KU b 1.5 because of the occurrence of coagulation. The triangle corresponds to the measurements in distilled water (ca. lo-' mol dm-3). A typical error bar is shown to indicate the reproducibility of the results. proportional to the average of the scaled drag coefficients, m,, of the constituent ions for 1 : 1 strong electrolytes. Table 1 gives the scaled drag coefficients, m,, for various ions, from their limiting molar conductances. I 2 * l 3 Also included are the effective radii of the ions, from which it can be seen that condition (5) holds even for the largest ion and smallest particle.From the data in table 1, the reduction in diffusion constant should be in the ratio of 0.213 : 0.287 : 0.328 for NaCl : (CH3CH2)4NCl : NaC6H5C02. This is, however, not what is observed. Eqn (lo), however, is only applicable in the limit of low potentials (a<< 1). Fig. 1 is calculated using KCl as an electrolyte, in which case the scaled drag coefficients of K+ and C1- are essentially equal. Eqn (10) provides a good fit for e l / kT = 3 with KCl as electrolyte. However, for an electrolyte with ions of unequal drag coefficients, eqn (10) may no longer provide a good fit. This hypothesis is based on another approximate analytic expression from Ohshima et aL,4 in the limit of large Ka, where the ions can have different mobilities and all values of y are applicable.In this case it is no longer the average of the m, for the constituent ions which is important, but predominantly rn, of the counterion, typically of the order of 6 : 1 for the potentials of the gold sol and the latex. If we then use this result to interpret fig. 6 and 7, we can see that the minimum depth is correctly predicted within experimental error. The gold sol and the latex are negatively charged and therefore tetraethylammonium chloride should have the largest effect, due to its large positive ion. Sodium benzoate will have a larger effect than sodium chloride since there is still a small contribution from the anion.We can then re-express eqn ( 1 1 ) for when the condition e l / kT << 1 is not met, as rn=WTZ,+(l-a)m-. (21) From our data we estimate that for our experimental conditions LY eO.80.84 Brownian Motion of Charged Colloidal Particles 1.25 - 'm E : 9.20 0, 4 N l --. 1.15 concentration/mol dm-3 5. 1 I I I I I I I I 1 2 3 4 5 6 K a Fig. 7. Plot of the in distilled water ( (W, NaC,H5C02 diffusion constant us. KU for polystyrene latex for three different salts: NaCl (a), and (CH3CH2)4NCI (0). The triangle corresponds to the measurements ca. lop5 mol dm-3). A typical error bar is shown to indicate reproducibility of the results. Table 1. Limiting molar conductances, scaled drag coefficients and sizes for selected ions at 25 "C limiting molar scaled drag effective conductance coefficient, ion radiusa ion / lo2 K' m-' equiv-' m, /nm 50.11b 73.52b 32.3' 32.3' 73.34h 0.257 0.28 0.175 0.19 0.399 0.43 -0.169 0.18 0.399 0.43 a Calculated from eqn (9), assuming the friction coefficient of an ion is given by f; = 47rr)uetf (corresponding to the slip boundary condition). Ref.(12). Ref. (13). Using eqn (10) with this modified fi, we can solve for the reduced zeta potentials of the gold sol and the latex. The reduced zeta potentials, @, (zeta potentials, 5 ) for the gold sol are -3.2*0.1 (- 82*3 mV) at lop4 mol dmP3 salt concentration and -3.1 f 0.1 (- 80* 3 mV) at mol dm-', where the potentials given were averaged from the potentials calculated for each individual salt. The error given is the standard deviation.Similarly, the reduced zeta potential, a, (zeta-potential, 5) for the latex is -2.7f0.3 (- 69 k 8 mV) at Ka = 1.0. These results are in good agreement with the zeta potentials measured for the gold sol and with the zeta potential for the latex, estimated above. From the diffusion experiments one can even deduce the sign of from the dominant contribution of the counterion to fi.G. A. Schumacher and T. G. M. van de Ven 85 Conclusions We have shown conclusively that the double layer around a colloidal particle contributes to the diffusion constant of the particle, especially when the double layer thickness is comparable to the radius of the particle. When K a = 1, the double layer can reduce the diffusion coefficient of a colloidal particle by several per cent, depending on the zeta potential and the reduced friction coefficients of the ions in solution. The experimental results are in general agreement with the Stokes-Einstein relation Of = kT, with f given by the equilibrium friction coefficient of a sedimenting charged sphere, expressions and numerical results for which are given by Ohshima et al." For gold sol and latex particles of 20 and 40 nm, respectively, of reduced potentials Q> of ca.3, it was found that the data can be explained by the asymptotic theory valid for @<< 1, provided the mean drag coefficient given by eqn ( 1 1 ) is replaced by the modified eqn (21), i.e. the contribution of the counterions is much larger than the contribution of the coions. The agreement between the zeta potentials of the gold sol and latex particles with the values obtained from the diffusion coefficients [using eqn (lo)] indicates that zeta potentials of small particles can be obtained from measurements of diffusion constants (without the need for an external electric field).Alternatively, the method can be used to measure the friction coefficients (and hence the sizes) of ions in solution. In this respect it is interesting to observe that diffusion constant measurements have gone full circle. Early measurements by P e r ~ i n ' ~ were intended to measure the Boltzmann constant k or, equivalently, Avogadro's number, from which the sizes of atoms can be calculated (thus proving their existence). With the present method atomic dimensions (of ions) can be measured directly. Special thanks are in order to Dr K. Takamura for measuring zeta potentials of the gold sol, to Prof. R. H. Ottewill and associates for giving us the polystyrene latex, to Dr R. M. Fitch for providing an initial gold sol and to J-F. Revol and Dr P. Pradkre for taking electron micrographs. References 1 S . Sasaki, Colloid Polym. Sci., 1984, 262, 406. 2 T. Raj and W. H. Flygare, Biochemistry, 1974, 13, 3336. 3 J. W. S. Goossens and A. Zembrod, Colloid Polym. Sci., 1979, 257, 437. 4 H. Ohshima, T. W. Healy, L. R. White and R. W. O'Brien, J. Chem. Soc., Faraday Trans. 2, 1984, 80, 5 B. V. Enustun and J. Turkevich, J. Am. Chem. SOC., 1963, 85, 3317. 6 K. Takamura, R. S. Chow and D. L. Tse, in Proc. Inf. S.ymp. on Flocculation in Biotechnology and 7 H. Ohshima, T. W. Healy and L. R. White, J. Chem. Soc., Faraday Trans. 2, 1983, 79, 1613. 8 R. H. Ottewill and R. A. Richardson, Colloid Pol-ym. Sci., 1983, 260, 708. 9 M. E. Labib, Ph.D. Thesis (McGill University, Montreal, 1979), p. 175. 1299. Separation Systems (to be published). 10 B. J. Berne and R. Pecora, in Dynamic Light Scattering (Wiley-Interscience, New York, 1976), chap. 1-5. 11 M. M. Kops-Werkhoven and H. M. Fijnaut, J. Chem. Phys., 1981, 74, 1618. 12 D. A. MacInnes, in The Principles of Electrochemistry (Dover, New York, 1939, 1961), chap. 18, p. 342. 13 M. Tissier and C. Douhkret, Solution Chem., 1978, 7, 87. 14 J. Perrin, Les Atomes (Fklix Alcan, Paris, 1913), chap. 4. Received 12rh December, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300075

出版商:RSC    

年代:1987

数据来源: RSC

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GENERAL DISCUSSION Prof. B. J. Ackerson (R.S. R.E., Malvern) addressed Dr Medina-Noyola: You have presented a theory for the diffusion of particles which treats one particle explicitly and the other particles as an effective fluid. In this theory a memory kerneI G ( p , p ’ ) appears which is treated in an approximate way using microscopic expressions, mode coupling theory etc. Is it possible to use fluctuation dissipation arguments to find or approximate G(p, p’)? Can one assume that the effective fluid will behave like a normal hydrodynamic fluid and produce a ‘Stokes drag’ on the tagged particle interacting with the fluid with a potential cp in the steady state? Presumably this drag would be in terms of an effective viscosity which would have to be eliminated self-consistently. Dr M.Medina-Noyola (Cinvestav, Mexico City) replied: Let me refer to the memory kernel G(r, r’; t ) in eqn (3.5) of my paper. This function contains essentially the same information as the dynamic correlation function (8n(r, t ) s n ( r ’ , 0)). I do not know of any fluctuation-dissipation arguments which, by themselves, lead to the determination of either of these quantities. However, as I have shown in my paper (section 3), this type of argument can be employed to define the general structure of G(r, r’; t ) , namely that in eqn (3.6). Although this only seems to substitute our ignorance concerning G( r, r ’ ; t ) by our lack of knowledge of the generalized friction tensor, G( r, r’; t ) , which describes the dissipative relaxation of the local macrofluid current, the fluctuation- dissipation theorem assures us, however, that this quantity may be expressed as the correlation function of a random force [see the equation following eqn (3.4)].This fact can be used in the construction of approximate expressions for G(r, r’; t ) . Very much in the spirit of your suggestion, Hess and Klein’ proceeded along these lines in the implementation of the mode-mode coupling approximation for the collective properties of a colloidal suspension. Such a method allows the self-consistent determination of G(r, r’; t ) , which is indeed related to an effective viscosity of the macrofluid. I would say, however, that in the time- and length-scales which we are interested in, this macro- (or effective) fluid cannot be assumed to behave as a normal hydrodynamic fluid.We may, on the other hand, calculate the analogue of the Stokes drag on a tagged particle with the stationary-state method that you describe. I will refer to it as the relaxation-effect (R.E.) method, as opposed to the fluctuation-theory (F.T.) approach that I employed in my paper. The equivalence between both methods is expected, at least in principle, on the basis of the (‘first’) fluctuation-dissipation theorem. Let me briefly illustrate this in the following way. Consider eqn (4.2) when the tracer is subjected to a constant force Fext: MT du(t)/dt=-{$u(t)+ [V+(p)]n’(p, t ) d3p+fT(t)+Fext. (A. 1 ) I I An average of this equation over all possible realizations of f T ( t ) leads to MT d(u(f))/dt = -S”ru(t))+ “v+(P)l(n‘(P, t ) ) d3P+Fext (A.2) which, under stationary-state conditions, reads F e x t = ~ $ ( ~ ) s h - [V+(p”{n‘(p))’’d’p (A.3) i.e.the external force needed to pull the particle at a constant velocity (u)” must balance 8788 General Discussion the friction produced by the solvent and the friction originating from the collisions of the tracer with the other diffusing particles. ( n ’ ( r ) ) s c contributes to the integral in eqn (A.3) only through its departure, An(p), from its radial equilibrium value, n e q ( p ) = ng(p). An (p) vanishes with Fxt, and is the solution of a (linearized) diffusion equation, which we shall write as t , = [Vneq(p)] - ( u ( t ) > + 1 d’p’D*V neq(p)Va&p, p’) An(p’, t ) . (A.4) a t This is essentially the average of eqn (4.4) over all the realizations of the random diffusive fluxes f’(p, t ) , and with G’(p, p’; t ) approximated by its modified Fick’s diffusion expression, eqn (4.16).The stationary solution reads Anss(p) = - %(p, p’)[V’neq(p’)] * (u)’” d ” p ’ (A.5) I where 3 is the Green’s function of eqn (A.4) under stationary conditions, i.e. it is the solution of d3p’ D*V n e q ( p ) V a i d ( ( p , p’)%(p’, p”) =S(p-p”). (A.6) I Substituting eqn (AS) in eqn (A.3), we get Fex‘= {LOTI+ d3p d”p’[V+(p)]%(p, p’)[V’neq(p’)]} - (u)”” (A.7) I I A . E { 607.1 - Glnt( z = 0)) - (2-))”. From this equation it is clear that the total friction coefficient on the tracer is the sum of the solvent friction, I[;, plus the contribution due to the direct interactions, which we denoted asAGin‘(z = 0).The reason for this notation is clear if we compare the expression for W”(z = 0) in eqn (A.7), which is the result of the R.E. method, with the corresponding result of the F.T. approach, i.e. the z = 0 limit of the Laplace transform of Gint ( t ) defined in eqn (4.10) and (4.1 l ) , together with eqn (4.16). From this comparison it is clear that both approaches should yield in principle the same results. Let me mention that if the external force F‘“‘ in eqn (A.l) is considered to be time-dependent, the R.E. method may still be applied along the lines just indicated, leading to the determination of a time-dependent friction function. Such a quantity is also given by eqn (4.10), (4.11) and (4.16). Furthermore, the use of Fick’s diffusion law, eqn (4.16), was meant to illustrate the relationship between the two methods.Other approximations for this diffusion equation, which might include memory effects, could also be considered. To the extent that the two approaches to calculate the friction coefficient are based on the same diffusion equation, identical results are to be expected. Let me finally remark that in establishing the equivalence between a result of the theory of spontaneous fluctuations, and a result derived with a linear response approach, the proper external field in the latter must be identified. In our example, since we are describing tracer diffusion, such an external field corresponds to a force acting only on the tracer, and not on the other diffusing particles. A more general discussion of the equivalence between the two theoretical approaches mentioned above has been made by Hess and Klein’ in the context of the properties of colloidal systems.1 W. Hess and R. Klein, Ado. Phys., 1983, 32, 173. Prof. I. Qppenheim ( M U , Cambridge, MA) then asked: Does it make a difference whether the tagged particle is heavy or light as compared to the other particles? Where does it enter that the other particles are big or small?General Discussion 89 Dr M. Medina-Noyola (Cinuestau, Mexico City) replied: The results for the friction function in eqn (4.9)-(4.13) are based, essentially, on eqn (4.2)-(4.4). The latter equations only define the structure of the coupled equations of motion of the tracer and of the local concentration of the surrounding particles, and are hence rather general.If the tagged particle is not much heavier than the solvent particles, one might have to replace the friction term -l:u(t), in eqn (4.2) by a temporally non-local friction. I would expect, however, that the Markovian limit in that equation should continue to be a very reasonable approximation for times comparable with the relaxation time of the local structure of the other particles around the tracer. On the other hand, considerations concerning the size of the other particles should enter when approximating the propagator ~ ’ ( p , p’; t ) or the relaxation kernel G’(p, p’; t ) appearing in the equations just referred. The applications mentioned in my paper, however, are based for simplicity on a Fick’s diffusion level of description for the dynamics of such particles, both, when they are as big (self-diffusion), or much smaller (electrolyte friction), than the tracer.Prof. H. N. W. Lekkerkerker (Utrecht, The Netherlands) said: I would like to raise the following point with Dr Medina-Noyola. In your paper you mention that the mode-mode coupling approximation and the modified Fick diffusion law [ eqn (4.16)] coincide in the low concentration limit. For hard spheres their result is given by D , = Do( 1 -$q) rather than the exact value D, = Do( 1 - 277). You say that you can show that if the inhomogeneous Fick’s diffusion approximation in eqn (4.16) is employed the correct result should follow. In view of the fact that the two approximations coincide in the low concentration limit, does this mean that you can also obtain the exact result from the mode-mode coupling approximation? If not what is wrong with the mode-mode coupling approxima- tion? Indeed can one trust the mode-mode coupling approximation at all if even for the simplest case it gives wrong results? Dr M.Medina-Noyola (Cinuestau, Mexico City) replied: Let me stress, first of all, that the most general results in my paper are eqn (4.9)-(4.13). These results still require a specific definition of G’(p, p’; t ) , i.e. of the diffusion equation in the reference frame of the tracer, and it is here that additional approximations must enter. We must say, however, that in the limit of infinite dilution of diffusing particles, and if the tracer is one such particle, the exact expression for G’(p, p’; t ) may be written, based on the two-particle Smoluchowski equation, as eqn (4.16) with neq(r) = n exp [-gb(r)/kT] o-iJ(r, r‘) = S( r - r ’ ) / neq( r ) .(L.1) (L.2) and Under these conditions, the properties of the hard-sphere (H.S.) system are known exactly after the work of Hanna et al.’ and of Ackerson and Fleishman.* In particular, D,/ Do = 1 - 277. As explained in my answer to Prof. Ackerson, the use of the general results in eqn (4.9)-(4.13) yields equivalent results, regarding the calculation of A l ( t ) , to the use of the R.E. method. In fact, under stationary conditions, eqn (A.4) (of my response to Prof. Ackerson), with eqn (L.l) and (L.2), is identical to the equation that had to be solved in the derivation of the exact result for D , at infinite dilution via the R.E.method by yourself and Dhont3 (for hard spheres) and by Van den Broeck4 (for H.S. plus square-well potential). This explains my statement that the use of the inhomogeneous Fick’s diffusion approximation, i.e. eqn (4.16), with eqn (L.1) and (L.2), should yield the exact infinite-dilution limit. Unfortunately, except in this limit, even the writing of G’(p, p’; t ) must involve approximations. Eqn (4.16) still constitutes an interesting approximation, but eqn (L. 1)90 Genera 1 Discussion and (L.2) must be replaced by respectively, where g( r ) is the radial distribution function of the diffusing particles around the tracer, and c(l)(r, r‘) is the direct correlation function of such particles in the field of the tracer, i.e. effectively a three-body correlation function.Thus, even within Fick’s diffusion approximation, some simplifications would be desirable. The ‘homogeneous fluid approximation’ is one such simplification, which replaces eqn (L.3) and (L.4) by neq( r ) = n ( Lh5) and (L.6) - 1 oo0 = S ( r - r ’ ) / n - c ( l r - r’l) where c ( r ) is the bulk direct correlation function of the macrofluid. This leads,’ for example, to the following expression for D,: - 1 d3r[g( r ) - 1j2] . (L.7) It is not a simple matter to estimate a priori, and in a general fashion, the degree of accuracy of this type of approximation. As stated above, the result in eqn (L.7) originates from our inability to calculate exactly the properties of systems away from the infinite- dilution limit. Hence, applying it to infinitely dilute systems has only the purpose of establishing a comparison with exact results, and in this sense we have pointed out that eqn (L.7) yields DJD” = 1 - (4/3)77 for the hard-sphere potential to linear order in volume fraction.On the other hand, with situations explicitly different from the infinite dilution limit in mind, one could think of introducing the homogeneous fluid approximation at the outset, without specifically defining G’(p, 9’; t ) otherwise. This led to eqn (4.14) of my paper. Fick’s diffusion law provides one possible closure for this equation. The mode- mode coupling approximation (MMCA) provides still another. The version of the MMCA considered in my paper has already built-in the homogeneous-fluid approxima- tion, and hence, similar comments as above should be made concerning its infinite- dilution limit.If, however, one is particularly interested in incorporating the exact low-concentration results, it is then clear that the homogeneous-fluid approximation has to be relaxed, in the MMC or in other approaches. We should mention that a related problem concerning the applicability of the MMCA at low densities has been faced in the field of simple liquids, where approaches have been developed‘ that essentially interpolate between the exact low-density results and MMC-type results at high densities. Whereas it is surely advisable to implement analogous developments for colloidal systems, it is also interesting to have an idea of the accuracy of the current MMC theory when applied to those conditions for which it was suggested in the first place, i.e.for systems in which the interaction of the tracer with the macrofluid’s collective fluctuations, rather than with individual particles during binary collisions, is the dominant relaxation mechanism. Here another interesting limit should also be considered, which is rigorously satisfied by the mode-mode coupling approximation, namely, the weak coupling limit.’ One could mention, for example, that the results of the mode-mode coupling theory, as applied to coulombic systems,’ are consistent in the Debye-Huckel limit with Onsager’s limiting law. On the other hand, away from the weak coupling limit, i.e. for highly coupled colloidal systems, we do not have simple exact results toGeneral Discussion 91 1 0.8 0.6 13 \ h * v 0 .L 0.2 0 1 1 1 2 3 .4 5 tlms Fig. 1. Time-dependent self-diffusion coefficient, D( t ) = (ArT( t))2)/(6t), in units of the free- diffusion coefficient, Do = 9.5 x lo-” m2 sC1 as a function of time, for a suspension of particles interacting through a hard-sphere plus a highly repulsive Yukawa tail. The dots are the computer- simulation data of ref. (8) and the solid line indicates the MMCA results. guide us, other than computer simulations. Hence, in fig. 1 I present a comparison of a MMC-mean-field approximation, defined by’ eqn (4.14), with x ’ ( k t ) = x A k t ) X ( k , t ) = exp ( - k 2 D D , t ) exp [ - k ’ D ” t / S ( k ) ] (L.8) as applied to the hard-sphere plus Yukawa system defined in section 5, with Brownian- dynamics simulation results’ for a screening parameter z = 0.149, volume fraction 7 = 4.4 x and coupling constant K = 801.That this is indeed a highly coupled system is indicated by the fact that for the same values of z and 7, it freezes at K =924. It seems to me that this comparison indicates that the MMCA is a quantitatively useful and reliable approximation, at least under those conditions for which it was conceived. 1 S . Hanna, W. Hess and R. Klein, Physica A, 1982, 111, 181. 2 B. J. Ackerson and L. Fleishman, J. Chem. Phys., 1982, 76, 2675. 3 H. N. W. Lekkerkerker and J. K. G. Dhont, J. Chem. Phys., 1984, 80, 5790. 4 C. Van den Broeck, J. Chem. Phys., 1985, 82, 4248. 5 A. Vizcarra-Rendon, H. Ruiz-Estrada, M. Medina-Noyola and R. Klein, J. Chem. Phys., 1987, 86, 2976. 6 G. F. Mazenko and S .Yip, in Statistical Mechanics. Part B : Time-dependent Processes, ed. B . Berne 7 G. Nagele, M. Medina-Noyola, R. Klein and J. L. Arauz-Lara, to be published. 8 K. J. Gaylor, I. K. Snook, W. J. van Megen and R. 0. Watts, J. Chem. SOC., Faraday Trans. 2, 1980,76, 9 W. Hess and R. Klein, Adv. Phys., 1983, 32, 173. (Plenum Press, New York, 1977)- chap. 4, pp. 181-231. 1067. Dr B. Cichocki ( R . W. T. H. Aachen, Federal Republic of Germany; Warsaw University, Poland) said: I have a comment related to Dr Medina-Noyola’s paper and Prof. Lekkerkerker’s remark. First I would like to stress that one should be very careful when applying kinetic theory methods originally derived for fluids to suspensions. An example is the Mori-Zwanzig projection operator technique.Insufficiently careful application of the technique to the generalized Smoluchowski equation has led to the well known controversy between the configuration space description of suspensions and the descrip- tion on the Fokker-Planck level.’ This point has recently been clarified.292 General Discussion Fig. 2. Dynamical events of two hard spheres: ( a ) in gas, ( b ) in suspension. Another example is the mode-mode coupling theory. To explain the discrepancy mentioned by Prof. Lekkerkerker let us consider two hard spheres in a dilute gas and in a dilute suspension. In the gas, when we neglect the presence of other particles, two hard spheres can collide only once [fig. 2 ( a ) ] . In a suspension they can collide many, even infinitely many, times [fig.2( b ) ] . This fact indicates clearly the difference between both systems. In the mode-mode coupling theory of suspensions, as presented by Hess and Klein' and also by Dr Medina-Noyola, the final expressions have the structure corresponding to the so-called one-ring events, i. e. with two collision operators separated by two one-particle propagators. This means that in the low-density regime the two- particle dynamics is approximated by that part in which two particles can collide only twice. This approximation is the cause of the discrepancy between the exact result for the self-diff usion coefficient for dilute hard-sphere suspensions and the result derived within this theory. One can hardly imagine that this theory leads to better results for higher densities. A proper way to construct an approximation of the mode-mode coupling type is first to perform the so-called binary collision expansion3 to obtain an adequate description of two-particle dynamics and next to apply the mode-mode coupling ~ c h e m e .~ Then in the low-density regime one reproduces the exact results for the self-diffusion coefficient of hard spheres. 1 W. Hess and R. Klein, Adu. Phys., 1983 32, 173. 2 B. Cichocki and W. Hess, J. Chem. Phys., 1986, 85, 1705; Physica A, 1987, 141, 475. 3 R. Zwanzig, Phys. Rev., 1963, 129, 486. 4 B. Cichocki, to be published. Dr M. Medina-Noyola commented in conclusion: I wish to point out that in deriving the main results of my paper I made no use of methods pertaining specifically to kinetic theory. I think, however, that a description of the dynamics of colloidal systems within the framework of kinetic theory will be most useful.Prof. B. U. Felderhof ( R W H , Aachen, Federal Republic of Germany) turned to Prof. Mazur: In your treatment of hydrodynamic interactions you advocate the use of Cartesian tensors. I would like to point out that this may be advantageous to low order in the method of reflections, but that in higher orders the use of spherical tensors may be preferable. In fact this approach has led to the higher-order terms in the work of Schmitz and myself on hydrodynamic interactions between two spherical particles with general scattering properties.' For the case of hard spheres with stick boundary conditions Jeffrey and Onishi* have derived hundreds of terms in the two-sphere mobility series in this way.The point is that in this manner one employs the results on spherical harmonics derived in quantum mechanics which make maximal use of the group theoretical symmetry properties.General Discussion 93 1 R. Schmitz and B. U. Felderhof, Physicu, 1982, 116A, 163. 2 D. J. Jeffrey and Y. Onishi, J. Fluid Mech., 1984, 139, 261. Prof. P. Mazur (Leiden, The Netherlands) replied: I am not in the least advocating the use of ‘Cartesian tensors’, but I myself use irreducible tensors because I find them convenient. Somebody else may of course prefer to use spherical harmonics. The difference between both formulations may be compared to the difference between abstract vector notation and a notation using components relative to a given basis: it is largely a matter of convenience, and sometimes also of taste, which formulation one prefers.The formulation we chose helped us to calculate explicitly three-, four- etc. sphere hydrodynamic interactions which as we have shown, and as was confirmed experi- mentally, play a dominant role in more concentrated suspensions, owing to the essential non-additivity of these interactions. In that light it is of secondary importance whether, and in which way, one calculates another hundred terms for the two-sphere problem. Now it is certainly so, in principle, that one might also calculate explicitly many-body hydrodynamic interactions with the help of spherical harmonics, or spherical tensors, but it had not been done, and, as far as I am aware, still has not been done. Of course we should not forget that Kynch some thirty years ago had exhibited expressions for three- and four-body interaction terms, which remained unused because they were not well understood.He did not use spherical harmonics: but that does not mean anything. Prof. Felderhof then continued: In your formulation of the sedimentation problem you employ a large spherical container. One is interested here in a transport property, namely the friction coefficient giving the difference of the local average particle and average fluid velocity in terms of the applied gravitational force. Since the transport property is a local one and is independent of the shape of the sample I would prefer a formulation for general sample shape. Prof. Mazur replied: I beg to differ concerning what I was interested in when I studied sedimentation in a spherical container. Had T been interested solely in the value of the mean mobility, or friction coefficient, as a local property, and up to linear order in the volume fraction, I would have been perfectly happy with, in particular, Batchelor’s theory for this quantity.However, I was also interested in a possible dependence of the sedimentation velocity in a homogeneous suspension on container shape. Burgers noted such a possibility but states that it ‘does not appear to be readily acceptable’.’ For inhomogeneous suspensions it is well known that huge buoyancy-driven convection can occur in sedimentation, in particular near inclined boundary walls. To give an answer, however, to what I would like to call for brevity’s sake, Burger’s problem, it seemed necessary to us within our ‘microscopic’ formulation of the problem, to take into account (long-range) hydrodynamic interactions with container walls.We were able to perform this calculation for a spherical container, as well as for sedimentation towards a plane wall. For the spherical container we then found in homogeneous suspensions what we call essential convection under the influence of gravity. It is true that this essential convection is small compared to, and rapidly masked in a spherical container by, buoyancy-driven convective flows; and that the sedimentation velocity with respect to the non-vanishing volume velocity has Batchelor’s value, as one would hope. Again, however, as far as I am aware, even though one has sometimes wondered about essential convection, nobody has as a result of theoretical analysis explicitly mentioned its existence, or calculated its magnitude for a particular shape, whatever the formulation of the problem.1 J. M. Burgers, Proc. Kon. Ned. Acad. Wet., 1941, 44, 1177.94 General Discussion Prof. Felderhof commented: The situation is analogous to the theory of the dielectric constant in electromagnetism. Some years ago I formulated the sedimentation problem by analogy with electromagnetism, emphasizing the role played by the local field.' 1 B. U. Felderhof, Physica A, 1976, 82, 596, 611. Prof. Mazur replied: An analogy with the theory of the dielectric constant certainly exists for properties of suspensions, in particular for the effective viscosity. I know that you have very skillfully exploited this analogy when considering, for instance, collective diffusion.However, I have the impression that, owing to hydrodynamic interactions with container walls (and stick boundary conditions on such walls) one has in the case of sedimentation, a different situation from the one encountered in the conventional problem of shape dependence, or shape independence, of quantities in dielectrics. Dr M. La1 (Unilever Research, Port Sunlight Laboratory) asked Prof. Mazur: Can the method of induced forces discussed in the paper be extended to the evaluation of the hydrodynamic interaction between non-spherical particles as a function of interpar- ticle separation and orientation? Prof.Mazur replied: Formally the method can be applied also to ellipsoidal particles. I have not succeeded, however, in evaluating the resulting integrals for more than one particle. Prof. M. Fixman (Colorado State University) turned to Prof. Mazur: I have a comment and a question. The comment concerns the slow convergence of power-series expansions in the treatment of hydrodynamic interaction. The truncation of the expansion sometimes has the consequence, which is particularly unpleasant in Brownian simulations, that the friction matrix has negative eigenvalues for some part of the configurational space. This can be avoided, and an incidental improvement of convergence obtained, if the problem is given a variational formulation.' The question concerns the expansion of the diffusion constant in powers of the volume fraction. The convergence of the analogous dielectric expansion is improved if the Lorentz-Lorenz function rather than the dielectric constant itself is expanded, and this choice may be physically motivated.Is there a corresponding alternative for the diffusion constant? 1 See e.g. J. Rotne and S . Prager, J. Chem. Phys., 1969, 50, 4831 and M. Fixman, J. Chem. Phys., 1982, 76, 6124. Prof. Mazur replied: The analogue of the Lorentz-Lorenz (or Clausius-Mossotti) formula for the hydrodynamic case is Saito'e expression for the effective viscosity of a suspension, which is also a mean-field-type result. Thus a Saito function can be defined corresponding to the Clausius-Mossotti function. If one compares the virial expansions for the dielectric constant I of a dispersion of metallic spheres, and for the effective viscosity T of suspension of hard spheres, to the virial expansions of ECM, the Clausius-Mossotti value of E and of ~ s , the Saito value of 7, respectively, one finds to second order in the volume fraction 4 ~=10(1+3++4.51+~+* * - )General Discussion 95 In both cases the mean field results give the linear order correctly.In second order there are deviations arising from the fact that the mean-field theory does not handle adequately the two-sphere correlation problem. Now it is not clear to me at all what the 'mean-field' or Clausius-Mossotti-like formula for (self-)diffusion should be. One might be tempted to assume a Stokes- Einstein-like formula with a viscosity given by the Saito formula.One then has DMF= Do( 1 - 4)( 1 +$+), with Do the one-particle Stokes-Einstein diffusion coefficient. The virial-expansion of this quantity leads to DMF=Do(1-2.54+3.75$*+* * * ) while in reality one has (with a 5% error in the coefficients). D = Do( 1 - 1.73 4 + 0,8 8 4 * + - * . ) Thus already the linear term is ca. 50% out, and the second-order term is out by a factor of 4. However, perhaps, if one defines with DMF the function which corresponds to the Clausius- Mossotti function, this would nevertheless provide the alternative to which you are referring. Dr R. B. Jones (Queen Mary College, London) said: I have a comment followed by a question for Prof. Mazur and also for Prof. Bossis and Prof. Brady, who are present in the audience. In the successful application of Prof.Mazur's many-body hydrodynamic interactions to tracer diffusion in highly concentrated suspensions,' the lubrication theory results for almost touching spheres are not included because the mobility expansions in the quantities a / & , where a is a particle radius and R, is an interparticle distance, are cut off at low order. On the other hand, in a recent simulation study of the viscosity of a sheared concentrated suspension due to Prof. Bossis and Prof. Brady,' the lubrica- tion theory results for two-body hydrodynamic interactions at close separation were included through the use of accurate friction coefficients but the many-body hydrody- namic interactions were not included. As it seems desirable to treat all of the transport coefficients in the same framework I wish to ask Prof.Mazur as well as Prof. Bossis and Brady if one should include both the lubrication theory two-body results and the many-body interactions in order to get a consistent picture of the transport properties of a concentrated suspension? 1 C. W. J. Beenakker and P. Mazur, Physica A, 1983, 120, 388. 2 J. F. Brady and G. BOSSIS, J. Fluid Mech., 1985, 155, 105. Prof. Mazur replied to Dr Jones: I am not completely sure I understand your question in as far as it refers to the work of Beenakker and myself on self-diffusion. It is only in the calculation of the first few virial coefficients that the mobility expansions are cut-off at low order in R,' and in that case the prime objective was to establish the relative importance of three-body interactions.For more concentrated suspensions we completely resum, within the framework of a fluctuation expansion, many-body hydro- dynamic interactions. The lowest order in this fluctuation expansion thus contains the two-body problem to all orders in the inverse interparticle distance, but not only the two-body problem, also the three-, four- etc. sphere problem. Of course we pay a price for this: particle correlations are not correctly taken into account, and for that fact one corrects in the next order of the fluctuation expansion. In this correction one then limits oneself to contributions of low order in R,' of renormalized interaction terms. So I would say that we have precisely done to a degree (within the limitations encountered in such a problem) what Dr Jones suggests should be done.Perhaps this contributes to the fact that our treatment is relatively successful, as he is kind enough to note.96 General Discussion Prof. J. F. Brady (Caltech, Pasadena) added: Concerning the need to include both many-body interactions and lubrication forces in treating the hydrodynamic interactions in suspensions, the answer is that it depends on the quantity to be studied, i.e. some properties are more sensitive to the near-field and others to the far-field physics, and this sensitivity changes as the volume fraction of particles increases, but in general both must be included. As a simple example, consider a linear chain of closely spaced spheres. For a sufficiently large number of spheres the chain should behave as a slender body, for which the behaviour is well known.If the same force is applied to each particle in the chain, then the far-field many-body interactions will generally suffice to give the qualitatively correct instantaneous mobilities. This is because the solution for a translating slender body has, apart from the very ends, a constant force density along its length. If, however, the forces are not the same on all spheres, so that there would be a tendency for relative particle motion, then lubrication forces are essential. If you push only one sphere at the end of the chain, clearly all particles will move with the same translational velocity because of the lubrication forces and the connectivity of the chain; far-field interactions alone will not capture this effect.In order to obtain the proper behaviour, both many-body interactions and lubrication forces are needed. We have recently devised a computationally efficient scheme for calculating both far-field many-body interactions and near-field lubrication forces.' The results of this method for finite numbers of interacting particles compare very favourably (< 1 % error) with all available exact (usually numerical) calculations, and the method also reproduces slender-body theory. The procedure is readily incorporated into our Stokesian dynamics method for simulating infinite suspensions,' and a general development of Stokesian dynamics along with several applications will appear in the near f ~ t u r e . ~ 1 i. Diirlofsky, J.F. Brady and Or. Bossis, J. Fluid Mech., 1987, 180, 21 2 J. F. Brady and G. Bossis, J. Fluid Mech., 1985, 155, 105. 3 J. F. Brady and G. Bossis, Annu. Rev. Fluid Mech., 1988, 20, 111. Dr E. Dickinson ( University ofLeeds) turned to Dr van Megen: My question relates to the curves corresponding to effective volume fractions 0.44, 0.49 and 0.54 in fig. 2 of your paper. Increasing the volume fraction from 0.44 to 0.49 in the disordered phase leads to a substantial reduction in mean-square displacement over the whole range of quoted delay times. On the other hand, further increasing the volume fraction by the same amount, and at the same time changing the state of the system from liquid-like to crystalline, leads to very little change in the mean-square displacement over the same range of delay times.Since the curvature of the plots is attributed to the transition from short-time to long-time diffusive motion, I am a little surprised that curves for volume fractions 0.49 and 0.54 are so close, in view of the fact that the short-time diffusion coefficient is very dependent on volume fraction at these particle densities, and that the long-time diffusion coefficient should be effectively zero for a dense crystalline colloidal system. Does Dr van Megen have any comment on these observations? Dr W. van Megen (RMIT, Melbourne) replied: At a given volume fraction one would expect the local mobility, or diffusion coefficient D i , of a particle in the crystalline phase to be larger than that in the disordered phase. This feature, which is supported by calculations of D; for charge-stabilized dispersions [see fig.1 of ref. (l)], expiains, at least qualitatively, the similarity of the mean-squared displacements (m.s.d.) at short times in the coexisting disordered (& = 0.49) and crystalline (b, = 0.54) phases. A better indication of the difference between the m.s.d. in the coexisting disordered and crystalline phases at longer times may be seen in fig. 6 of ref. (2), where further details of this experiment are discussed. Whilst the true long-time limiting behaviour has not been reached, owing to the large wavevectors at which tracer system T2 wasGeneral Discussion 97 studied, the m.s.d. in the crystalline phase does appear to be approaching a constant value. I might add that this value is roughly consistent with the Lindemann melting criterion. 1 I.Snook and W. van Megen, J. Colloid Interface Scr., 1984, 100, 194. 2 W. van Megen, S. M. Underwood and 1. Snook, J. Chem. Phys., 1986, 85, 4065. Prof. A. Vrij (Utrecht, The Netherlands) then asked: The first five measured points of the short-time and long-time self-diffusion coefficients shown in the fig. 3 and 4, respectively, are rather different. Theory tells us, however, that the slope of Ds/Do against the volume fraction is nearly the same: -1.83 and -2.1 1 for short-and long-time self-diffusion, respectively. The points in fig. 3 follow the slope (-1.83) rather closely, but this can obviously not be said about the points in fig. 4. Have the authors an explanation for the different behaviour shown in fig.4? Prof. R. H. Ottewill ( University of Bristol) and Dr P. N. Pusey (R.S.R.E., Malvern) replied: We agree that, for monodisperse hard spheres, the predicted concentration dependences, to first order in 4, of DS and DL are very similar. That this is apparently not found to be the case for the data of fig. 4 is probably due to the significant polydispersity of the tracer particle PMMA/l (see table 1 of our paper). In particular, even in the limit 4+0, the short- and long-time slopes of In FM(Q, t ) [eqn (13)] will, because of polydispersity, be different. Thus in fig. 4 the data for DL/Do should not have been extrapolated to unity but to a lower value, implying thereby a weaker concentration dependence and a better agreement with theory. Dr A. K.Livesey ( D A M P , Cambridge) considered both the Bristol and the Utrecht papers: I note that you (both) report a certain degree of polydispersity in your samples and yet make no attempt to fit your data using an algorithm which is capable of fitting broad distributions or otherwise accounting for this polydispersity. Would you like to comment further on your fitting procedures? Dr A. van Veluwen (Utrecht, The Netherlands) replied: The basic-reason for not analysing our measured correlation functions as if they were the Laplace transform of a spectrum of decay rates, is that such a description may be inadequate. Whereas for a dilute polydisperse sample it can be shown that the electric field autocorrelation function is a multiexponential decaying function, such a demonstration has not been given, as far as we know, for a concentrated interacting polydisperse system.The general polydisperse case is too complex for a detailed calculation, and the theoretical argument that the electric-field autocorrelation function of a concentrated sample will be composed of two decoupled modes, associated with collective and exchange diffusion, still needed experimental verification. This is essentially what we have done, and one does not need a very sophisticated data-fitting algorithm to conclude that the correlation function consists of a fast and a slow decaying part with well separated decay times. However, the time dependence of these modes is not necessarily (multi)exponential, and therefore analysing the correlation function that way may be misleading.Therefore apart from all the difficulties associated with Laplace inversion, the so-found relaxation time distribution may lack physical significance. Prof. B. U. Felderhof ( R . W. 7'. H., Aachen, Federal Republic of Germany) said: in fig. 2 of your paper you plot the normalized short-time self-diffusion coefficients of charged and uncharged silica spheres as a function of volume fraction and find that the experi- mental curves coincide. I do not understand the remark at the end of paragraph 3 of your paper, where you write that this indicates that the short-time self-diffusion is determined mainly by hydrodynamic interactions and apparently is affected little by90 General Discussion direct particle interactions. Is it not true that in the case of charged spheres the Debye clouds keep the particles separated and that therefore to a good approximation the hydrodynamic interactions may be neglected? Dr W.van Megen (R.M.I.T., Melbourne, Australia) added: I am also surprised by the independence of DEho* on the presence of electrostatic interactions between the particles (fig. 2 of your paper). Intuitively one expects the additional electrostatic repulsion between the singly coated silica particles to increase the mean interparticle spacing, ie., the first peak in the radial distribution function moves to a larger spacing, resulting in an increased local mobility or Dghort. This picture is supported by calculations of Diho* for charge stabilized dispersions, based albeit on a empirical screened pair mobility tensor.' 1 I.Snook and W. van Megen, J Colloid Inlet-fuce Sci., 1984, 100, 194. Dr van Veluwen replied: At very low volume fractions this appears to be the case. However, at the high volume fractions we used, the mean interparticle separation is inevitably small and so hydrodynamic interactions are important. If direct particle interactions had a pronounced influence on the short-time self-diff usion coefficient, we would have found different values for this quantity for the two systems considered. This not being the case indicates that the short-time self-diffusion coefficient is not very sensitive to these direct particle interactions, even though such interactions give rise to different particle functions. We may conclude from this that the detailed form of the particle distribution functions has no significant effect on the short-time self-diff usion coefficient.This conclusion can also be drawn from the work of Beenakker and Mazur.' 1 C. W. J. Beenakker and P. Mazur, Physica A, 1984, 126, 349. Dr W. van Megen ( R . M.I. T., Melbourne, Australia) said: In addition to your finding that the singly coated 440 nm radius silica spheres show an electrophoretic mobility, have you any other evidence for the presence of screened electrostatic forces between the particles? The crystallization transition, for instance, can be easily measured' for an optically matched dispersion of near-micrometre sized particles such as yours, provided that the polydispersity is not too large. Further, the location of this transition is very sensitive to the range of the repulsive force between the particles.Have you observed this transition in either of your 440 nm silica systems? 1 P. N. Pusey and W. van Megen, Nature (London), 1986, 320, 340. Dr van Veluwen replied: Another indication of the presence of long-range forces between the single-coated particles is the observation that the sediment of these particles is significantly less dense than the sediment of the double-coated particles. The formation of crystallites is not only influenced by the direct particle interactions and polydispersity, but also by the sedimentation velocity. Silica particles have a higher density than the latex particles you refer to. This implies that in the solvent we used the large silica particles will have settled under gravity before crystallization occurs.After some time (up to 1 year?) crystallization takes place in the sediment, both for the single- as well for the double-coated particles. Of course this influence of sedimentation on the formation of crystallites is less pronounced for smaller particles. Indeed single-coated silica particles of 80 nm radius already show a crystallization transition at volume fractions of 4 = 0.25. This, together with the concentration dependence of the static structure factor, is clear evidence for the existence of long-range repulsive forces.General Discussion 99 Dr J. G. Rarity (R.S.R.E., Malvern) remarked: The data shown in your fig. 3 show a behaviour similar to the pure optical polydispersity case. Is it possible that your data could be better explained by a model that takes both optical and size polydispersities into account? What effect does the compressibility of the stabilizing polymer coating have on your calculation of volume fraction? Dr van Veluwen agreed: It is indeed likely that our experimental data can be better explained by a model that takes both optical and size polydispersity into account.The compressibility of the stabilizing polymer coating is so small that it has no significant effect on the calculation of the volume fraction. Prof. P. Mazur (Leiden, The Netherlands) said: Some years ago one did not have sufficient low-density data at one’s disposal for the short-time self-diff usion constant in a hard-sphere suspension to determine from these data the coefficient of the quadratic term in an expansion of this quantity in powers of the volume fraction.Now the situation is different: there are many low-density points available down to volume fractions of ca. 0.025. My question is: has anyone tried to determine this coefficient, in addition to comparing theoretical curves to experimental ones? We found theoretically for this coefficient, taking into account three-body hydrodynamic interactions, the value 0.88, as compared to -0.93 calculated on the basis of two-body interactions alone. Dr van Veluwen replied: From our measurements it follows that the coefficient of the quadratic term in the expansion of the short-time self-diffusion coefficient in powers of the volume fraction must be positive. Furthermore, if we try to determine this coefficient using the expression ,ihon/ D~ = 1 - 1.83+ + K ~ $ ~ we find K2 = 0.7k0.3, in agreement with your calculations. Prof.M. Fixman (Colorado State University) asked: The initial response of a flexible surfactant is probably to provide a slip boundary condition. Is it feasible to go to very fast time resolution and see such an effect? Prof. R. H. Ottewill replied: This suggestion is an interesting one, and this may well be possible. It would, however, require rather different measurements with a very fast time resolution at low concentrations. It certainly would be interesting to try this. Dr van Veluwen added: One would expect the phenomenon you mention to show up at the timescale at which the velocity of a colloidal particle relaxes, i.e.m/6vqa- s. We do not think it is feasible to probe the motion at such short timescales with dynamic light scattering. Dr A. van Veluwen and Prof. H. N. W. Lekkerkerker (Leiden, The Netherlands), said: van Megen et al. have argued that the non-Gaussian terms in the self-dynamic structure factor are unimportant in the limit of small values of the scattering vector Q. We would like to present experimental evidence that for values of Q which are not very small, but are still below the location of the first maximum in the static structure factor, the non-Gaussian terms do become appreciable. The system studied consisted of an optically matched host dispersion of sterically stabilised silica particles in cyclo-octane, to which a small amount of sterically stabilized poly(methy1 methacrylate) (PMMA) tracer particles had been added. Both types of particles have a diameter of 160 nm.100 General Discussion J f l It2 1 t3 t4 1 t5 0 0.5 1 1.5 2 2.5 3 delay time, t/ms Fig.3. Self-dynamic structure factor F, plotted against delay time t for 4 = 0.145 at three different scattering angles: (-) 8 = 40", Q = 0.98 x lo-* nm-l, (- - -) 8 = SO", Q = 1.85 x lo-' nm-', (- - -) 8 = 150", Q = 2.78 x nm-'. The arrows on the abscissa indicate the times at which -In F,/(Qa)' has been plotted as a function of Q2 (fig. 4). 0.LO r I I I 1 I I I 01 2 4 6 Q2/ 1 0-4 nm-* Fig. 4. Plot of -In F,( Q, t ) / ( Q c z ) ~ as a function of the square of the scattering wavevector Q' for the delay times t , to t , as indicated in fig. 3.Straight lines are first-order cumulant fits, according to eqn ( 1 ) . ( a ) tl=0.5ms, ( b ) r,=l.Oms, (c) t3=1.5ms, ( a ) r4=2.0ms, ( e ) tS=2.5ms. In fig. 3 we present -In F,(Q, t)/QLa' as a function of delay time t for various scattering angles. Here a is the radius of the particle. The volume fraction of the sample was # = 0.145, and the relative concentration of tracer to host particles was 1 : 50. If the non-Gaussian terms are unimportant, all these functions should coincide. This being not the case indicates the presence of non-Gaussian terms. Indeed we can write downGeneral Discussion 51 4 3 a 2 2 1 0 delay time, t/ms Fig. 5. Non-Gaussian behaviour of F, expressed as a2 plotted against t. a cumulant expansion of the self-dynamic structure factor: 0' 3 60 - ln K(Q, t ) - (ArLt)') Q2( 3(Ar(t)4) - 5(Ar(t)2)2 101 Here Ar( t ) = r( t ) - r ( 0 ) is the displacement of an individual particle.In order to extract the first non-Gaussian term we plot -In F,( Q, t ) / Q2a2 as a function of Q2 for various values of t (fig. 4). From the intercept at Q=O we can extract the mean-square displacement (Ar( 1 ) ' ) and from the slope [3(Ar( t)") - 5(Ar( t)2)2]. Following Rahman' we express the departure of F,(Q, t ) from a Gaussian form in terms of the function ct2 defined as In fig. 5 we plot a2 as a function of t. Note that the values of a2 experimentally obtained here are much higher than those obtained by Rahman in a molecular-dynamics simulation for liquid argon. 1 A. Rahman, Phys. Rev. A, 1964, 136, 405 Dr W. van Megen (R.M.I.T., Melbourne, Australia) commented on Prof.Lek- kerkerker's data: The non-Gaussian terms in the self-dynamic structure factor (or the self-space-time correlation function) have been calculated by computer simulation for a variety of systems at densities just below crystallization (freezing). The work of Rahman,' on a liquid of atoms interacting with the short-ranged Lennard-Jones pair potential, contrasts with the study of Gaylor et a1.2 for a dilute dispersion of particles interacting with a long-ranged weakly screened Coulomb pair potential. However, both systems show qualitatively similar non-Gaussian effects: the first non-Gaussian term a2, [as defined in ref. ( l ) ] has a maximum of ca. 0.1. Similar observations have also been made on liquid argon by Skold et al.' More recent Brownian-dynamics simulations on102 General Discussion a system of particles with a pair potential of intermediate range at a volume fraction of 0.25 (for which freezing occurs at 0.26) yield qualitatively similar results for a? (see fig.6). The addition of tensorial hydrodynamic interactions, represented by an empirical pair mobility as discussed in ref. (4), has little effect on the non-Gaussian terms. The magnitude of your value of cr2 and its apparent reluctance to converge to zero at short times are in stark contrast to the results discussed above. I find this even more surprising in view of the volume fraction of your (near) hard-sphere dispersion of only 0.15, which is well below the hard-sphere crystallization transition.1 A. Rahman, Phys. Rev. A, 1964, 136, 405. 2 K. Gaylor, 1. Snook and W. van Megen. J. Chem. Phys., 1981, 5, 1682. 3 K. Skold, J. M. Rowe, G. Ostrowski and P. D. Randolph, Phys. Rev. A, 1972, 6, li07. 4 I. Snook, W. van Megen and R. J. A. Tough, J. Chem. Phys., 1983, 78, 5825. Dr R. B. Jones (Queen Mary College, London) also commented: I wish to add to Prof. Oppenheim’s informally made remarks about the theoretical expectation of long- time tails and hence non-Gaussian behaviour in the intermediate scattering function. These non-Gaussian effects are contained in principle in the memory function and hence can be obtained from a careful calculation of F(k, t ) at high concentrations. Prof. Felderhof and I presented details of such a calculation based on two-body dynamics (without hydrodynamic interactions) at the 1983 Faraday Discussion on Concentrated Suspensions.It should be possible to obtain analytic expressions for the order k4 terms from our formalism to see how they compare with your experimental data. For low- density hard-sphere suspensions the memory function has been studied in detail by Prof. Ackerson’ and m y ~ e l f , ~ . ~ and there are clear non-Gaussian effects seen at intermedi- ate times. 1 B. J. Ackerson and L. Fleishman, J. Chem. Phys., 1982, 76, 2675. 2 R. B. Jones and G. S. Burfield, Physica A, 1982, 111, 562. 3 R B Jones, J. Phys. A, 1984, 17, 2305. Dr P. N. Pusey (R.S.R.E., Malvern) then added: I have some comments on Prof. Lekkerkerker’s very interesting data showing large non-Gaussian effects in the displace- ment of tracer particles in concentrated suspension.(1) As you pointed out, computer simulations of simple liquids show much smaller non-Gaussian effects than you find in particle suspensions. An important difference between the dynamics of atoms in a liquid and particles in suspension is the presence of hydrodynamic interactions in the latter case. As you know, we have recently shown that hydrodynamic interactions can cause non-Gaussian contributions in suspensions which are quite different in nature from those caused by direct interactions.’ While these theoretical calculations apply only at short times it seems possible that hydrody- namic interactions, with their complicated many-body nature, could cause large effects at longer times. These would not necessarily show up in the computer simulations of van Megen and Snook, since these authors used an effective pair form for the hydrody- namic interaction tensor.(2) Presumably you would still expect the non-Gaussian contributions to become small at short times. However, your data do not show this trend. (3) In view of the delicate manipulations required to derive these effects from the data it is clearly important to discount possible experimental artefacts. I do not see how your results could be caused by polydispersity of the particles. However, I wonder if you have considered multiple scattering and/or possible ‘breakthrough’ of coherent scattering which could, I think, cause spurious effects in the direction you observe. 1 R. J. A. Tough, P. N.Pusey, H. N. W. Lekkerkerker and C. van den Broeck, Mol. Phys., 1986, 59, 595.General Discussiori 103 0.4 a 0.2 0 0 X X X X X X x . ' X X ' 2 4 t/ms f Fig. 6. First (lower points) and second (upper points) non-Gaussian terms of the self-dynamic structure factor for a system of particles at d, =0.25. The pair potential in units of kT is 100 exp [ -20( r - 1)]/ r, with r expressed in units of the particle diameter. Freezing for this system occurs at d, = 0.26. The crosses exclude and the dots include the effects of tensorial hydrodynamic interactions. volume fraction Fig. 7. Di/ Do plotted against volume fraction for short-time self-diffusion of poly(methy1 methacrylate) particles in dispersions of refractive-index-matched poly( vinyl acetate). @, Experi- mental results; theoretical calculations from Beenakker and Mazur: (.- - a ) three-body, (- - -) multi-body and (-) two-body. Prof. Lekkerkerker and Dr van Veluwen replied: ( 1 ) We fully agree with you on this point. (2) We simply do not reach such short times in our experiments. An interesting question is, on what timescale would one expect the maximum of the non-Gaussian effect to occur. (3) We paid much attention to these problems. The refractive index and tracer concentration were chosen such that both coherent scattering and multiple scattering could be completely neglected. Besides, it can be argued that a possible 'breakthrough' of coherent scattering at large k values and/or multiple104 General Discussion scattering would lead to effects opposite to those observed, that is to say a negative value of a2 instead of a positive one.Prof. R. H. Ottewill (University of Bristol) said: In the work of Neal Williams and myself' we have shown that poly( vinyl acetate) particles can be conveniently matched by refractive index by using a mixture of cis- and trans- decalin. This, as reported in the paper, has enabled self-diffusion measurements to be made over a wide range of volume fractions using poly( methyl methacrylate) particles at tracer concentrations. The results so obtained can be compared with calculations of the expected results from the theories of Beenakker and Mazur2 as shown in fig. 7. As can be seen, the best agreement between experiment and theory for the short-time self-diffusion is obtained using their multibody approach.1 R. H. Ottewill and N. St. J. Williams, Nature (London), 1987, 325, 232. 2 C. W. J. Beenakker and P. Mazur, Physica A, 1984, 126, 349. Dr I. Markovie and Prof. R. H. Ottewill (University ofBristoZ) said: We should like to point out that in addition to photon correlation spectroscopy it is possible to determine the short-time self-diffusion coefficient of small colloidal particles by the use of the neutron spin-echo technique.' In recent work, using calcium carbonate particles stabilised by an alkylaryl sulphonic acid2 with an average particle diameter of 10nm we have investigated the small angle neutron scattering over a wide range of volume fractions and scattering vectors ( Q), thus obtaining information about the structure factor, S( Q), as a function of Q.It was then possible to determine the diffusion coefficient of the particles at high Q values, where S( Q) + 1, by neutron spin-echo. The sample times using the latter technique are of the order of nanoseconds and hence the diffusion coefficient obtained was the short-time self-diff usion ~oefficient.~ The results are plotted in fig. 8. Also plotted in this figure are the results obtained from tracer measurements of poly( methyl methacrylate) particles in dispersions of poly( vinyl acetate) matched by refractive index4 and results obtained on poly(methy1 methacrylate) particles in a dispersion medium of dodecane and carbon disulphide near the refractive index match point.5 The results presented cover a range of particle diameters from 10 nm to 1 ,urn, two orders of magnitude, by two different techniques.The agreement between the various sets of experimental data is remarkably good. Moreover, the results are in quite reasonable agreement with the theory of Beenakker and Mazur6 for the dependence of the self-diff usion coefficient on volume fraction with allowance for many-body interac- tions. 1 J. B. Hayter, in Lecrure Nores in Physics, ed. F. Mezei (Springer Verlag, Berlin, 1980). 2 I. MarkoviC and R. H. Ottewill, Colloid Polym. Sci., 1986, 264, 65; 454. 3 I. MarkoviC and R. H. Ottewill, Colloids SurJ, 1987, 24, 69. 4 R. H. Ottewill and N. St. J. Williams, Nature (London), 1987, 325, 232. 5 P. N. Pusey and W. van Megen, J. Phys. (Paris), 1983, 44, 258. 6 C. W. J. Beenakker and P. Mazur, Physica A, 1984, 126, 349.Dr I. Markovic and Prof. R. H. Ottewill (University ofBristoZ) (communicated): In many of the papers presented at this meeting measurements of a physical quantity, e.g. the self-diff usion coefficient, are plotted against volume fraction. In these experiments the particles used are often, using a general term, sterically stabilised, i.e. the particles have a hard core surrounded by a shell of polymeric or surface-active material, and in order to obtain effective colloid stability this layer is solvated; this means that there is penetration of the solvent molecules between the chains of the stabilising moeities. In recent work we have examined the determination of volume fraction by various experi- mental techniques for this type of system and have been forced to conclude that theGeneral Discussion 105 Fig.166 0.1 0.2 0.3 0.4 0.5 volume fraction 8. Di/ Do against volume fraction for various particles. A, poly(methylmethacry1ate) diameter nm;4 0, poly(methy1methacrylate diameter 1.18 ~ m ; ~ 0, calcium carbonate diameter 10 nm;3 (-) calculated from the multi-body equation6 precise determination of this is very difficult, if not impossible experimentally, without making a number of assumptions. Some of the methods which can be used include the following: ( 1 ) The weight/weight or weight/volume fraction can for many cases be determined precisely. However, for conversion to volume fraction a density is needed. This can also be determined accurately, hut, sn iisirig ihis to obtain the voliiiiie fraction, the quantity obtained is the ‘dry’ volume fraction and not the solvated volume fraction occupied by the particles in the dispersion. If there is non-ideal mixing of dispersion medium and the stabilising chains it becomes even more complicated, (2) Experimental techniques such as photon correlation spectroscopy and viscosity on very dilute dispersions are often used to obtain a ‘hydrodynamic radius’.Unfortu- nately this is not necessarily the same as the radius of the particle required to determine the actual volume of the solvated particle, since solvent outside the stabilising chains may be included. (3) In a number of cases the chemical formula of the stabilising molecule is known and hence measurements on molecular models can give an estimate of the likely extension of the molecule in the solvated state.This could be reasonable if the chains are monodisperse or fully extended. (4) In the case of concentrated monodisperse systems the theoretical freezing volume fraction (0.494) has been used to estimate the actual volume fraction. This, however, depends upon how accurate the theory of hard spheres is, and even if this is accurate it only fixes the volume fraction at one concentration. Thus the thickness obtained at this high concentration may not be valid for dilute systems. In a recent examination of poly( methyl methacrylate) particles by small-angle neutron scattering we have obtained some evidence for the compression of the stabilising layer106 General Discussion of poly(l2-hydroxystearic acid) as the concentration was changed from low to high values, suggesting that the volume of each particle was dependent on volume fraction.' It should be stressed that for large particles, e.g.1 pm, with a stabilising layer of 10 nm, the experimental error will be small even if the core volume fraction is used but for small particles; e.g. with a radius of 10 nm and the same thickness of stabilising layer, the accurate determination of volume fraction is non-trivial. 1 I. MarkoviC, R. H. Ottewill, S. M. Underwood and Th. F. Tadros, Langmuir, 1986, 2, 325. Prof. B. J. Ackerson (R.S.R.E. Malvern) addressed Prof. Felderhof: At a previous Faraday Discussion on Concentrated Dispersions Hess and Klein argued that the Fokker-Planck equation was the proper starting point for the calculation of light scattering functions in concentrated colloidal suspensions.' They argued that the general- ized Smoluchowski equation may lead to erroneous results.However, Cichocki and Hess2 have reinterpreted the memory function obtained for the generalized Smoluchowski operator. As a result the Fokker-Planck and generalized Smoluchowski equation approaches are believed to be equivalent descriptions on timescales of interest to us. However, in your present work on colloidal crystals you find a large effect due to counterion motion. The counterion distortion depends on the colloidal particle velocity and presumably the correlated velocities between different colloidal particles. Does this dependence on particle velocities mean that we must necessarily consider a Fokker-Planck approach in analysing the particle correlation functions or is the general- ized Smoluchowski equation still valid in general? 1 R.Klein and W. Hess, Faraday Discuss. Chem. Soc., 1983, 76, 137. 2 B. Cichocki and W. Hess, J. Chem. Phys., 1986, 85, 1705. Prof. B. U. Felderhof ( R . W. T. H., Aachen, Federal Republic of Germany) replied: As you know I have always been an advocate of the use of the Smoluchowski equation. I am not yet sure what effect the dipolar forces will have in the liquid state. I expect that it will still be possible to use the Smoluchowski equation on the timescale we see in light-scattering experiments, say ca. lOP4s. This is much longer than the momentum relaxation time of ca. lo-* s. Hence it should be possible to work in configuration space and not be necessary to go back to the more complicated Fokker-Planck equation, which describes the time evolution in phase space.What effect, if any, the dipolar forces have on the form of the Smoluchowski equation we hope to find out in the near future. Prof. M. Fixman (Colorado State University) said: You suppose that in the absence of ion retardation effects the friction constant associated with the relaxation of a transverse mode vanishes at q = 0; this makes the relaxation rate finite because the force constant also vanishes at q = 0, and the relaxation rate is, in rough terms, the ratio of force constant to friction constant. However, Hurd et al. (your references) make the point that their experiments were done in thin samples, and that a boundary condition on fluid velocity then changes the friction constant to a non-vanishing value at q = 0.They rationalize the vanishing relaxation rate in this way rather than by retardation effects, and suggest that the friction constant would really vanish for a large system. Although they did not consider the retardation effects, and they are surely significant for some problems, there is a physical reason to doubt their ability to make the friction constant finite for transverse modes at q = 0. If the transverse mode really consists of a coherent motion of solvent and colloidal particles, which is the reason that the friction constant vanishes if retardation is neglected, then convective terms that you discard would cause the ion atmosphere to move in phase with the solvent and colloidal particles.In other words, the electrophoretic effect would reduce the retardation force to zero.General Discussion 107 Prof. Felderhof responded: We do not think that the samples used by Hurd et af. were sufficiently thin for boundary conditions on the fluid velocity to have an appreciable effect. We believe that they have seen an effect at small wavenumbers characteristic of bulk samples. We claim that in bulk samples retardation affects the dispersion curves in the way we have discussed. Even for long wavelengths the actual flow pattern in a unit cell is much more complicated than you suggest. See for example Hasimoto's calculation' for viscous flow past a cubic array of spheres. The situation is analogous to that in a lattice of electric dipoles, the local crystalline field being quite different from the average Maxwell field.In our problem a complete calculation taking account of the lattice structure would have to include the effect of convection on the small ions, as well as the electrical force density acting on the fluid. The equations to be used are known [see our ref. ( 5 ) ] but the explicit calculation will be difficult. As we mention in the paper, we expect that the electrophoretic effects reduce the retardation force, but do not make a qualitative difference. Although the electric dipole moments may be small, their interactions are of long range and seem to have a drastic effect on the crystal dynamics. 1 H. Hasimoto, J. Nuid Mech., 1959, 5 , 317. Dr M. Medina-Noyola (Cinuestav, Mexico City) then addressed Dr van de Ven: In your experiments you measure the contribution AS"' to the friction on a single polyion originating from the spontaneous fluctuations of its electrical double layer around spherical symmetry.The theory of Ohshima et af.' that you employ in your analysis calculates the friction of an isolated polyion as measured in a sedimentation experiment. 1 hat in principle the results of such a theory are indeed applicable to your light-scattering measurements is expected from fluctuation-dissipation arguments such as those in my response to Prof. Ackerson's question on my paper. Nevertheless, a theory describing the electrolyte friction effects as the consequence of spontaneous fluctuations may also be elaborated as a particular application of the general approach presented in my paper.In a simple ~ e r s i o n ~ - ~ which neglects the coupling of the diffusive fluxes of the electrolyte ions with the hydrodynamic flow of the solvent, such a theory contains Schurr's results' for ALe' in the Debye-Huckel limit. Owing to the neglect of hydrodynamic interactions, our results are expected to be quantitatively less accurate than those of Ohshima et af., but the main qualitative features that you observe in your experiments seem already to be contained in such a simpler theory. Our methods, however, allow us3 to calculate A C ( t ) , and not only its time integral A[e'. In the Debye-Huckel approximation, the long-time asymptotic expression for A l ( t ) reads AL( t ) = ( Q * / / & F ) exp ( - ~ * D ' t ) l ( D0t)3'2 where Q is the charge of the polyion, F the dielectric constant of the solvent, Do the free-diffusion coefficient of the electrolyte ions and K is the inverse Debye screening length.If we take K - ~ = 20 nm and Do = 1.5 x m2 s-.', which correspond to conditions considered in your experiments, one estimates a relaxation time T~~ 2 0.27 ps. As seen in fig. 4 of your paper, in your experiments you seem to reach the microsecond regime. I wonder, then, to what extent the time dependence of A C ( t ) could be observed in measurements of the type that you have performed. I H. Ohshima, T. W. Healy and L. R. White, J. Chem. Soc., Faraday Trans. 2, 1983, 79, 1613. 2 M. Medina-Noyola and A. Vizcarra-Rendon, Phys. Rev. A, 1985, 32, 3596.3 H. Ruiz-Estrada, A. Vizcarra-Rendon, M. Medina-Noyola and R. Klein, Phys. Rev. A, 1986, 34, 3446. 4 A. Vizcarra-Rendon, H. Ruiz-Estrada, M. Medina-Noyola and R. Klein, J. Chem. Phys., 1987, 86, 2976. 5 J. M. Schurr, Chem. Phys., 1980,45, 119.108 General Discussion Dr T. G. M. van de Ven (McGill University, Montreal, Canada) replied: It is as yet not clear why we observe a short relaxation time of ca. 1 ps. It could be for the reason you mentioned. Another likely explanation is that it is due to rotary Brownian motion of a small fraction of doublets which are known to be present. Yet another possibility is that the effect is an experimental artifact. Dr P. N. Pusey (R.S.R.E., Maluern) said: Is it possible that the initial rapid decay of the correlation function [inset of your fig.4(a)J is caused by ‘after-pulsing’ in the photomultiplier tube? This is a well known artifact in PCS’ which arises from the ionization of a residual gas atom by a pulse of electrons, the drift of this positive ion back to the cathode resulting in a second, spurious pulse of electrons being detected a short time (ca. 1 ps) after a real pulse. Problems with after-pulsing can be largely circumvented by cross-correlating the outputs of two photomultipliers, both of which observe the same scattered light field. This is a procedure worth implementing if correlation delay times s 1 ps are important. I do not think it is necessary to invoke solvation of the latex to explain the difference in particle radii obtained by PCS and by electron microscopy.It seems to be caused largely by polydispersity. Consider the case of small particles where the intensity scattered by a particle goes as the sixth power of its radius a. Then it is easily shown that apcs, the radius derived by using in the Stokes-Einstein expression the average diffusion coefficient obtained from the first cumulant, is given by * -- aPCS = a ‘ / / ’ . ( 1 ) Similarly the second cumulant is A useful, not widely appreciated, property of a narrow relatively symmetrical distribution P ( a ) is that its moments are given by a universal function of its standard deviation ( T : ~ *2+. . . (3) regardless of the detailed form of the distribution; here - a 2 a a2 = -2- 1. Substitution of eqn (3) into eqn ( 1 ) and (2) gives aPCS =r a( 1 + 5a2) (4) and Q = (T2.( 5 ) For the polystyrene latex electron microscopy gave a = 32 nm and (T = 0.18. Thus eqn (4) predicts aPCS = 37 nm in reasonable agreement, when all uncertainties are accounted for, with the measured value of 40 nm. Eqn ( 5 ) predicts Q = 0.03, in excellent agreement with the value calculated by the more laborious procedure of assuming a Gaussian distribution of particle size. 1 C. J. Oliver, in Photon Correlation and Light Beating Spectroscopy, ed. H. Z. Cummins and E. R. Pike 2 J. C. Brown, P. N. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A, 1975, 8, 664. 3 P. N. Pusey, H. M. Fijnaut and A. Vrij, J. Chem. Phys., 1982, 77, 4270. (Plenum Press, New York, 1974).General Discussion 109 Dr T. G. M. van de Ven (McGill University, Montreal, Canada) replied: After-pulsing is one of the possible explanations for the rapid initial decay in the autocorrelation function.As discussed before, other explanations are possible. I agree with your comments on particle size. In the case of gold sols, the distribution in particle size is narrow, but is complicated by the presence of doublets. Here the difference between particle size from electron microscopy and PCS is small, but the second cumulant is large. In the case of latex particles it is of interest to remark that the radius from electron microscopy provides a better fit between our experiments and recent numerical calculations, kindly made for us by Dr R. W. O'Brien. Assuming latex particles of radius 40nm, the theory predicts no minimum in the diffusion coefficient in the electrolyte concentration range 10-4-10-3 mol dm-3 (for @ = -3), while for a radius of 32 nm a minimum is predicted for K a == 0.5.Furthermore, the numeri-cal results predict the observed ion size effects: for the three salts used the friction coefficient is predicted to be maximum for (CH3CH2)4NCl, followed by NaC6H5C02H and NaCl. The main difference between the theory and observation is, besides the minimum being at K a = 0.5 rather than K a = 1, that the observed effects are larger than the predictions. This implies that zeta potentials determined from diffusion coefficients are larger than- potentials determined from electrophoresis. Dr D. Weitz (Exxon, Annandale, NJ) said: As suggested by Dr Pusey, the effects of phototube after-pulsing may lead to the short time decay observed in these experi- ments (fig.4), and this should certainly be checked. However, I would like to suggest that rotational diffusion may also be quite important. This is due to the particular optical properties of gold colloids. Since gold is a nearly-free-electron metal, it possesses a pronounced electronic plasma resonance in the optical frequencies. This results in the characteristic wine-red colour of the unaggregated colloid. However, upon aggrega- tion, the resonances of the particles are shifted, which is reflected by the change in colour of the aggregates. In particular, a dimer will possess two distinct resonances: one when the incident electric field is aligned normal to the axis joining the two spheres, and a second when the field is aligned parallel to this axis.The former occurs at nearly the same frequency as that of an isolated sphere, while the latter occurs at a lower frequency. In fact, for gold, this lower-frequency resonance is quite close to the wavelength of the HeNe laser used in these experiments. As a result, at this wavelength the scattering from a dimer will be considerably more intense than from a monomer. Furthermore, the polari~ibi!iiy of a dimer will be quite anisotropic. Indeed, one calculation of this effect' (performed for silver but also applicable to gold), suggests an anisotropy of as much as two orders of magnitude for excitation along the axis of the two spheres compared to normal to it, when the exciting frequency exactly corresponds to the lower frequency resonance. While one can not expect quite so large an anisotropy here, it should nevertheless be significant.The combination of the large anisotropy and increased magnitude of the polarizability would lead to a substantial contribution from rotational diffusion in the dynamic light scattering. However, I would expect this effect to be caused by the dimers in the solution, rather than by the slight asphericity of the monomers, as is suggested in your paper. Finally, I note that this effect is due to the particular optical properties of gold colloids, and one can not predict the asymetry using the calculations for the much larger fractal clusters presented in fig. 4 of our paper. 1 P. K. Avarind, H. Metiu and A. Nitzan, SurfSci., 1981, 110, 189. Dr D.S . Home (Hannah Research Institute, Ayr) said: You rightly point out that previous studies of pH or salt effects on the diffusion coefficients of colloidal species neglected frictional effects associated with the electrostatic double layer. However, I think it would be equally wrong to dismiss the possibility that such changes, particularly110 General Discussion with proteins, were not the results of conformational or configurational changes in these molecules without independent evidence of the absence of the latter effects. I say this as one who in the past has ascribed increases in diffusion coefficient to configurational collapse of protein molecules on the surface of casein micelles following the introduction of ethanol (Horne and Davidson, Colloid Polym.Sci., 1986, 264, 727). On the model you propose, introduction of alcohol would vary the medium dielectric constant and through this the Debye length of the double layer. As well as varying the ethanol level, the studies of Horne and Davidson also considered the effect of added salt. If the increases they observed in the diffusion coefficient were due only to electrostatic effects on the double layer then the results of those experiments should all fall on the same line when plotted against K . I cannot make this happen. The increases in D, however, are influenced by the ionic strength of the medium, and this I think is a manifestation of secondary electrical effects on the interactions between the surface molecules or hairs rather than the primary effect on the diffusional motion you are suggesting here. Dr T.G. M. van de Ven (McGill University, Montreal, Canada) replied: I fully agree that conformational or configurational changes in proteins or polyelectrolytes can cause changes in their effective diffusion coefficients. All we are saying is that the effects of the surrounding double layers must be taken into account as well. Prof. B. J. Ackerson (R.S.R.E., Maluern) also addressed Dr van de Ven: I am pleased to see that you are able to go to sufficiently low colloidal particle concentrations to see only self-diffusion effects and still have enough intensity to perform the experiment. This is evidenced by seeing the measured diffusion coefficient approaching the Stokes value at high and low salt concentrations.The largest self-diffusion reduction is observed at Ka = 1. Evidently, at high salt concentration the colloidal particle charge is screened and not many ions participate in the drag reduction. At low salt concentration there are not many ions to participate in the drag reduction. Thus the maximum effect occurs at some intermediate salt concentration. However, dynamic light scattering measures a mutual diffusion constant. The effect of the small ions on mutual diffusion must be considered, in general, to be sure one knows what is being measured in a given experiment. The initial motion of the colloidal particle produces an asymmetry in the ion cloud surrounding the particle. The net charge separation between the particle and cloud retards the particle motion and lowers its self-diffusion in general. The net charge separation also produces a dipole which couples to other colloidal particles and modifies their motion. Unlike the self-diffusion effect, this dipole coupling continues to increase as the salt concentration decreases. The mutual diffusion coefficient is decreased by this dipole coupling, while it is increased by direct interactions, as the ionic strength is reduced. At finite colloidal particle concentrations a measurement of the first cumulant of the time decay of the intensity correlation function will have contributions from both self- and dipole-coupling effects, even after multiplying by the static structure factor to eliminate the direct interaction effect. Presumably at low enough concentrations self-diffusion is measured as shown in your results. However, at higher concentration the effect of dipole coupling may be to give a much more pronounced reduction in the measured effective diffusion constant which does not ever increase with decreasing salt concentration. The effect of dipole coupling may explain the differences seen between your self-diffusion measurements and the ‘self-diffusion’ measurements of Gorti et al.’ 1 S. Gorti, L. Plank and B. R. Ware, J. Chem. Phys., 1984, 81, 909. Dr T. G. M. van de Ven (McGill University, Montreal, Canada) replied: You point to an interesting singular limit, namely Ka + 0. For a fixed concentration of particles,General Discussion 111 no matter how small, as KU -+ 0 the double-layer thickness approaches infinity and double layers will start to interact. Hence it is not surprising that the dipole coupling continues to increase as the salt concentration decreases. However, we do not think that in our experiments these effects were present. At the lowest salt concentration, KU = 0.3, the double layer is about three times the particle radius, while at volume fractions of ca. the average distance between the particles is ca. 300 particle radii. At higher (but still low) volume fractions, dipole coupling can become important. Prof. V. Degiorgio (Pauia, Italy) commented: Since your experiment is performed at a fixed particle concentration and it is known that the diffusion coefficient of solutions of charged particles depends very strongly on concentration at very low ionic strength, it is possible, in principle, that the measured D contains at low salt concentration some effect of interparticle interactions. It would be interesting to evaluate whether such an effect is significant in your case. Dr J. G. Rarity (R.S.R.E., Malvern) also remarked: You interpret the short time data shown in fig. 4 of your paper as a rotational signal from the anisotropic polarisability of a small number of doublets in your system. A theoretical treatment of this type of contribution to the correlation function is given in the book by Berne and Pecora.' Your data imply that depoiarised scattering has a greater magnitude than the polarised scattering. For this to be true, with only 4% doublets, implies a strong resonant enhancement of scattering along the doublet axis and large phase differences between resonant and non-resonant scattered components. A measurement of the depolarised scattering ( I V H ) would quickly confirm or disprove your theory. 1 B. J. Berne and R. Pecora, in Dynamic Lighr Scattering (Wiley-Interscience, New York, 1976), chap. 7.

ISSN:0301-7249    

DOI:10.1039/DC9878300087

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Furuduy Discuss. Chem. SOC., 1987, 83, 113-124 Computer Simulations of Diff usion-limited Aggregation Processes Paul Meakin Central Research and Development Department, E. I. du Pont de Nemours and Company, Wilmington, Delaware 19898, U.S.A. The diff usion-limited aggregation (DLA) model of Witten and Sander has been used to model a wide variety of physical processes. Here the way in which our picture for the structures generated by DLA models in two dimensions has evolved during the past few years is described. Results from very large scale square-lattice simulations are presented and it is shown how simulations with noise reduction are helping us to understand the effects of anisotropy on the DLA process. It now appears that in the asymptotic (large-mass) limit clusters generated on regular two-dimensional lattices are self-similar fractals with a non-universal fractal dimensionality which is close to but not equal to 1.5. Results are also presented for DLA on two- and three-dimensional percolation clusters.A wide variety of non-equilibrium growth and aggregation phenomena can be described as stochastic processes in which the growth probabilities are determined by a scalar field ( 4 ) which obeys Laplace’s equation (V2@ = 0). Such processes can be simulated using random walks and can often be thought of in terms of particles undergoing Brownian motion. In many important physical realizations the Laplacean field has absorbing boundary conditions ( 4 = 0 ) on the growing structure and a fixed value ( 4 = 1, for example) on a distant surface which encloses the growing objects.Some physical systems which can be described in these terms are given in table 1. The diffusion-limited aggregation (DLA) model was developed by Witten and Sander’ to represent the growth of colloidal aggregates. In this model (illustrated in fig. 1) particles are added, one at a time, to a growing cluster or aggregate of particles via random-walk trajectories. The particles are imagined to be launched from infinity. Since they must cross a circle enclosing the cluster at a random position if growth is to occur, they are in practice launched from a randomly selected point on a circle whose radius is only slightly larger than that of the cluster (fig. 1). In the case of a lattice based model the particles are then transferred to the nearest lattice site and made to follow a random walk on the lattice.Two typical trajectories ( t , and t 2 ) are shown in fig. 1. Trajectory t , eventually moves the particle to an unoccupied surface site (an unoccupied site with one or more occupied nearest neighbours) and growth occurs. Trajectory t2 moves the particle a long distance from the cluster. This trajectory is terminated and a new trajectory is started at a random position on the launching circle to reduce computer time requirements. In fig. 1 the trajectory is terminated at a radius of 3Rm,, where R,,, is the maximum radius of the cluster. In more recent simulations using improved algorithrn~’~~ a termination radius of lOOR,,,,, would be more typical. The termination radius should be sufficiently large so that once a particle has reached the termination circle its position on return to the launching circle will be unbiased, In practice a radius of 3R,,, seems to be sufficient for two-dimensional clusters containing ca.lo4 particles or sites4 and a radius of 100Rm,, is more than sufficient for clusters containing 4 x lo6 sites5 A typical cluster generated using a two-dimensional off -lattice version of this 113114 Diflusion Jim it ed Aggregation Processes Table 1. Some examples of diff usion-limited aggregation process controlling field ref. dielectric breakdown electrodeposition fluid-fiuid displacement in Hele-Shaw cells or porous media materials random dendritic growth dissolution of porous solidification and electric potential or electric potential concentration pressure pressure temperature and/or concentration 29 39-41 19, 21, 33-35 36 37, 38 Fig.1. A simple square-lattice model for diffusion-limited aggregation. This figure shows an early stage in the simulation. At this stage15 (shaded) sites have been occupied including the original seed or growth site (black). Two typical trajectories starting at random positions on the launching circle are shown. Trajectory t , reaches an unoccupied surface site (growth site) and this site is occupied. Trajectory t2 reaches the termination circle which in this case has a radius of 3R,,, where R,,, is the maximum radius of the cluster. This trajectory will be terminated and a trajectory started at a random position on the launching circle. model is shown in fig.2. Considerable interest has developed in DLA and related non-equilibrium growth models, since they lead to the formation of clusters or aggregates having a fractal geometry.' This interest has been sustained by the relevance of the DLA model to a variety of important processes (table 1). The DLA model does not provide a realistic representation of colloidal aggregati~n.~'~ However, it does provide a means of simulating stochastic growth processes controlled by Laplacean (harmonic)P. Meakin 115 4 1000 DIAMETERS b Fig. 2. A 50 000 particle DLA cluster grown using a two-dimensional off-lattice version of the model illustrated in fig. 1. fields. For example, the model illustrated in fig. 1 simulates such a growth process with absorbing boundary conditions on the growing aggregate and a constant field on the termination circle.The growth probability is determined by the local value of the field in the growth sites (or by the field gradient at the surface). Despite the apparent simplicity of the diffusion-limited aggregation model, progress towards a satisfactory theoretical Understanding has been slow. In the absence of such an understanding the interpretation of simulation results is unreliable. However, the results of computer simulations have stimulated and motivated the development of new theoretical ideas9-'* which seem to be leading to a better theoretical understanding of DLA. Our picture for the structure of DLA clusters has undergone considerable evolution during the past few years. The purpose of this paper is to describe some of the recent .advances in our understanding of the structure of DLA aggregates and to indicate some new research directions.Results from Small-scale Simulations The first results from DLA simulations' were obtained using relatively crude algorithms which are quite well represented by fig. 1. Using only slightly improved methods4 it was possible to generate clusters containing up to ca. lo4 particles or occupied lattices in embedding spaces or lattices with (Euclidean) dimensionalities in the range 2-6. These were interpreted in terms of self-similar fractal structures with a charac- teristic fractal dimensionality which was dependent on the dimensionality of the embed- ding space but was insensitive to model details such as sticking probabilities or lattice116 Difusion-limited Aggregation Processes Table 2.Fractal dimensionalities ( 0 ) for a variety of diffusion-limited aggregation modelsa D model 5 0 '/o 75% 2 0 , S(NN) = 1.0 2 0 , S(NNN) = 1.0 2 0 , S(NL) = 1.0 2 0 , S( NN) = 0.25 2 0 , S(NNN) = 0.1 3 0 , S(NN) = 1.0 3 0 , S(NN) = 0.25 3 0 , S(NL) = 1.0 4 0 , S(NN) = 1.0 5 0 , S(NN) = 1.0 6 0 , S(NN) = 1.0 1.73 f 0.06 1.7 1 * 0.08 1.70 f 0.09 1.70 f 0.06 1.70 f 0.10 2.5 1 f 0.06 2.47 f 0.1 5 2.52 f 0.13 3.34f0.10 4.20 f 0.1 1 ca. 5.3 1.70 f 0.06 1.72 f 0.05 1.7 1 * 0.07 1.72 f 0.06 1.69 * 0.08 2.53 rfr 0.06 2.49f0.12 2.50 f 0.08 3.31 f 0.10 4.20 * 0.16 ca. 5.35 structures. At this stage it was generally believed that DLA clusters were homogeneous, self-similar fractals with a broad range of univer~ality.'~ Some of the results used to support this picture are shown in table 2.For two-dimensional clusters using a variety of DLA models (including both lattice and non-lattice models) the radius of gyration (R,) grows with increasing cluster mass ( M ) according to R,- M~ (1) for clusters of small sizes. The exponent appears to have a universal value of ca. 0.585 corresponding to a fractal dimensionality ( Dp = 1/p) of 1.70- 1.7 1 .14 The Effects of Anisotropy The effects of lattice anisotropy first became apparent for two-dimensional square-lattice DLA when improved algorithms273 were developed which allowed clusters containing ca. lo5 sites to be grown. At this size most square-lattice DLA clusters fit into an almost diamond-shaped envelope and it appeared that this might be the asymptotic shape for DLA.This idea motivated the development of a theory for DLA' based on the idea that the asymptotic shape of the cluster is related in a simple way to the structure of the lattice and that the maximum growth probability can be obtained by solving Laplace's equation with simple boundary conditions corresponding to this asymptotic shape. This theory predicted that the fractal dimensionality of DLA clusters should be sensitive to lattice details. While the theory of Turkevich and Scher' is no longer believed to be correct in all of its details, it has motivated much of the subsequent work on DLA, and the idea that the fractal dimensionality of DLA can be obtained from the strength of the leading singularities in the growth probability measure has survived.It still seems probable that the strength of these singularities is related to the shape of the clusters but this shape is not related to the lattice structure in an obvious way. At sizes of lo5 lattice sites or greater it becomes apparent that the overall shape of square-lattice DLA clusters evolves beyond a diamond shape towards a cross-like shape. Fig. 3 shows four maps of the regions in which growth has occurred during the addition of the last 200 000 sites to clusters containing 4x lo6 sites5 This provides a picture of the active zonesI5 (regions where growth is occurring) for the clusters, and clearly illustrates the formation of a cross-like shape. In any event clearer illustration of theP.Meakin 117 h. c -1 I6000 LATTICE UNITS I 6 0 0 0 LATTICE UNITS 4 I6000 LATTICE UNITS 4 > I 6 0 0 0 LATTICE UNITS Fig. 3. Four maps of the active zones for 4 x lo6 site square lattice DLA clusters. Each element in the map consists of a 50 x 50 block of lattice sites and an element is filled if growth has occurred in any of its 2500 lattice sites during the addition of the last 200000 sites to the clusters. effects of lattice anisotropy on square-lattice DLA is provided by the angular distribution of mass (fig. 4). Even for clusters containing lo4 sites the angular distribution is more concentrated along the lattice axes than would be expected for a uniform diamond shape. If clusters are grown on a square lattice with unaxial anisotropy in the sticking probabilities, the cluster has a needle-like shape.The length of the needle in the 'easy' growth directon is given by - M"II (2) and the width of the needle in the 'hard' growth direction (smaller sticking probability) is given by W - Mv-. (3) Ball et a1.l' found that zq = 2/3 and v,. = 1/3. This means that the axial ratio ( Z / w ) diverges as the cluster mass increases and the mean cluster density approaches a constant value (since v,+ vIl = 1). After these results were obtained, it was generally assumed118 Difusion-limited Aggregation Processes Fig. 4. The angular distribution of mass for two-dimensional square-lattice DLA clusters. (a) shows the angular distribution at 5% increments in the cluster mass for clusters containing up to 1.2 x 10“ sites and ( b ) shows similar results for clusters containing up to 4 x lo6 sites.that lattice models for DLA would behave in a similar fashion [i.e. the lengths and width of the cluster arms could be described by eqn (2) and (3)]. A variety of simulation results were found to be consistent with the idea that vIl = 2/3.’4.’6-’8 Results for Y, were much more ambiguous but indicated that 1/3 < v- s 2/3. A value for the exponent vl smaller than vll would indicate that DLA clusters were not self-similar fractals but must instead be described in terms of self-affine fractal geometry.6 The Effects of Noise Reduction A variety of simulation results and have shown that DLA is sensitive to anisotropy and that the shape of DLA clusters is controlled by a ‘competition’ between noise and anisotropy.In the high-noise/low-anisotropy limit the clusters have an irregular shape and the dependence of the radius of gyration on cluster mass is given by eqn (1) with the radius of gyration exponent ( p ) having a universal value of 0.585P. Meakin 119 I I400 LATTICE UNITS I500 LATTICE UNITS I300 LATTICE UNITS I ( d ) 4 I650 LATTICE UNITS Fig. 5. Square-lattice DLA clusters grown with noise reduction parameter (m) of 1 (ordinary DLA), ( b ) 3, (c) 10 and ( d ) 100. for two-dimensional systems. A number of methods have been developed for enhancing anisotropy16-18’22 or reducing noise2716723-25 in DLA simulations. In the most commonly used noise-reduction method the DLA simulations are carried out in the normal fashion except that growth does not occur until a surface site has been reached rn times by the random walkers.After a surface site has been contacted by a random walker the random walk is stopped and a new walker is started from the launching circle after the ‘score’ associated with the contracted site has been incremented by 1. After a surface site has been contacted rn times by random walkers it is filled and any new surface sites are identified and given a score of zero. All of the old surface sites retain their scores, which continue to accumulate as the simulation proceeds. Fig. 5 shows four 50 000 site square-lattice DLA clusters grown with noise reduction parameters (rn) of 1 (ordinary DLA), 3, 10 and 100. The cluster generated with rn = 3120 Diflusion-lim ited Aggregation Processes 750 DIAMETERS < 'CI 1000 DIAMETERS I000 DIAMETERS < + 750 DIAMETERS Fig.6. Results obtained from two-dimensional off-lattice models for diff usion-limited aggregation in which the particles are added in a small number N of directions which are fixed in space. In this figure results are shown for N = ( a ) 4, ( h ) 5, ( c ) 6 and ( d ) 7 with noise reduction parameters ( M ) of 100. shows the effects of lattice anisotropy at least as clearly as the 4~ lo6 site clusters grown with rn = 1. This suggests that the noise-reduction procedure allows us to approach the asymptotic ( M + W) limit for square-lattice DLA without generating enormously large clusters. However, this conjecture has not been firmly established. If this association between large rn and large M is valid then the noise-reduced DLA simulations lead to the following results for square-lattice DLA.( 1 ) In the limit M --+ 00 the clusters are statistically self-similar with a fractal dimensionality which is close to but not equal to 1.5. (2) The axial ratio between the length and the width of the cluster arms ( 1 / w ) approaches a constant limiting value as M + 00. The results have been obtained previously from smaller-scale simulations using the noise-reduced DLA andP. Meakin 121 from related models.16 (3) An angle of ca. 30" can be associated with the tips of the branches of the square-lattice DLA clusters. It appears that it is this angle which determines the maximum growth probability and the fractal dimen~ionality.~~~' An angle of greater than 0" (a finite axial ratio) is consistent with a fractal dimensionality ( D ) of more than 1.5.Similar results have been obtained for growth on a hexagonal lattice with six-fold symmetry and on a hexagonal lattice with growth in three of the six possible directions. In the latter case the results are consistent with vIl = v, (a finite axial ratio in the limit M -+ a) or with vII > vl and vIl - Y, d 0.15. Using lattice models the effects of n-fold anisotropy can be explored where N = 2, 3 , 4 and 6 . A modified off-lattice model (with noise reduction) has been used to explore N = 5 , 7 and 8 also. In this model the vector joining the centre of the contacted particle in the cluster to the centre of the newly arrived particle is-rotated into the closest of N equally separated directions, To implement noise reduction a record is kept of how many times an incoming particle has been rotated to each of the N directions for every particle in the cluster.Fig. 6 shows some of the results obtained for 50000 particle clusters with noise-reduction factors of 100 for the cases N=4, 5 , 6 and 7. For N = 3 , 4 or 5 the clusters always grow with N well defining arms. For N = 6 there are generally fewer than 6 arms using this model, but lattice model simulations with large noise- reduction parameters (m) usually give six well defined arms. It seems that N = 6 is 'marginal' in the sense that the results depend on model details. For N = 7 and 8 results are quite irregular and for N 3 8 they strongly resemble off-lattice DLA clusters. Related Models A variety of models more or less closely related to DLA have been explored during the past few years.This work has been motivated by a need to describe more accurately the events occurring in real processes and by a desire to obtain results which may be used to stimulate and evaluate theoretical work. Much of the present interest in DLA has been the result of its relevance to processes such as fluid-fluid displacement in porous media. Since porous media themselves may have a fractal structure over a significant range of length scales there is reason to investigate DLA on fractal substrates. M e a k i r ~ ~ ~ . ~ ' has simulated DLA on two- and three-dimensional percolation clusters and on a variety of two-dimensional fractals at a finite concentration of random walkers.For two-dimensional percolation clusters at the percolation threshold ( D = 1.89) a fractal dimensionality of 1.40 f 0.05 was obtained, and for three-dimensional percolation clusters ( D = 2.5) a fractal dimensionality of 1.75 f 0.10 was obtained. Murat and Aharony28 have explored the zero concentration limit of this process using both a dielectric breakdown29 version of DLA in which Laplace's equation is solved numerically in the percolation cluster and a model in which the Laplacean field is simulated using random walkers. They find that D = 1.30*0.05 for both two-dimensional models. In a related model3' the growth of the DLA cluster is restricted to a percolation cluster, but the random walkers are not restricted to sites contained in the percolation cluster (they are, however, not allowed to enter sites which are already occupied by the growing aggregate).This process has been simulated in two ways. In the first version of this model the simulation is carried out using a simple modification of the square-lattice DLA model (fig. 1) in which sticking is allowed only at vacant sites which are both nearest neighbours to occupied sites in the growing cluster and are part of an underlying percolation cluster. In the second version the DLA simulation is carried out in the normal fashion except that after a particle (site) has been added to the cluster a random number X uniformly distributed over the range 0 < X < 1 is generated. If X is smaller than a fixed probability, P, the newly added site is sticky and can add more sites.If X a P the newly added site is considered to be 'dead' or 'poisoned' and can add no122 Digusion-Zim i ted Aggregation Processes -500 LATTICE UNITS -& Fig. 7. A cluster of sites grown using the ‘poisoned DLA’ model in which the probability that a site would remain alive (sticky) after addition to the cluster was 0.59. This model is closely related to DLA on a square lattice with growth restricted to a percolation cluster (P, = 0.5927). more sites to its perimeter. For P < P, (the percolation threshold probability) only small clusters grow. As P - P, it becomes possible to grow quite large clusters (fig. 7) and at P = P, this model becomes equivalent to growth on a percolation cluster with unrestricted random walkers. At P = P, a percolation cluster substrate is ‘grown’ as the simulation proceeds by blocking sites with a probability of 1 - P,.If these blocked sites are not included in this cluster the two models become equivalent. This model leads to the formation of clusters which have fractal dimensionality of ca. 1.75. This is not much different than the value of ca. 1.71 found for small square-lattice DLA clusters. However, in other respects the two models are quite different. ( 1 ) Since the underlying percolation cluster is not anisotropic, the clusters are not sensitive to the anisotropy of the square lattice. (2) For square-lattice DLA the dimensionality Dmin, which indicates how the minimum path measured on the cluster between two points on the cluster scales with their separation r measured in the Euclidean embedding space, has a value of 1:31 (4) 1 - yDmln* For DLA with growth restricted to a percolation cluster Dmin must be at least as large as D m i n for a percolation cluster; a value which is believed to be greater than l.32 (3) In contrast to ordinary square-lattice DLA the seed or growth site is not necessarily near the centre of the cluster.If the random walkers are allowed to enter sites which are already occupied by the growing aggregate (from unoccupied sites which are not part of the percolation cluster) then the percolation cluster becomes uniformly filled ( D = 1.89). In this case the trapping of random walkers at the sites on the percolation clusters which are adjacent to filled sites on the aggregate is not sufficient to prevent random walkers from penetrating into the interior of the aggregate.Summary The evolution of square-lattice DLA clusters from a more or less irregular circular shape to a cross-like shape results in different effective exponents (v, and vII) which describeP. Meakin 123 the growth of the width and length of the cluster arms. However, simulations carried out using modified DLA algorithms in which the effects of noise are reduced indicate that the axial ratio (Z/w) reaches a constant value and consequently that there is no divergence of length scales (I/(( = vl in the M -B 00 limit). Consequently, it now seems that the asymptotic geometry for DLA is self similar. Since DLA is relevant to a wide variety of physical processes a large variety of models more or less closely related to DLA but designed to better represent real systems have been developed.Results for DLA on percolation clusters are presented. In this case the substrate has no anisotropy and the clusters grown on percolation clusters (with random walkers both restacked to the percolation cluster and unrestricted) give clusters which are different from ordinary DLA in several important respects. The work described here would not have been possible without contributions from many colleagues. I would particularly like to thank R. C. Ball for his collaboration in much of the work presented here. References 1 T. A. Witten and L. M. Sander, Phys. Res. Lett., 1981, 47, 1400. 2 R. C. Ball and R. M. Brady, J. Phys. A , 1985, 18, L809. 3 P. Meakin, J.Phys. A , 1985, 18, L661. 4 P. Meakin, J. Phys. A, 1983, 27, 604; 1495. 5 P. Meakin, R. C. Ball, P. Ramanlal and L. M. Sander, Phys. Rev. A, submitted for publication. 6 B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, San Francisco, 1982). 7 P. Meakin, Phys. Rev. Lett., 1983, 51, 1119. 8 M. Kolb, R. Botet and R. Jullien, Phys. Rev. Lett., 1983, 51, 1123. 9 L. Turkevich and H. Scher, Phys. Rev. Lett., 1985, 55, 1026. 10 R. C. Ball, R. M. Brady, G. Rossi and B. R. Thompson, Phys. Rev. Lett., 1985, 55, 1406. 11 T. C. Halsey, P. Meakin and I. Procaccia, Phys. Rev. Lett., 1986, 56, 854. 12 R. C Ball, in Statistical Physics, ed. H. E. Stanley (North Holland, Amsterdam, 1987). 13 T. A. Witten, in Workshop on Dynamics of Macromolecules, Institute for Theoretical Physics, University of California at Santa Barbara, J.Polym. Sci., Polym. Symp. 73, ed. S. F. Edwards and P. A. Pincus, p. 7. 14 P. Meakin Phys. Rev. A, 1986, 33, 3371. 15 M. Plischke and Z. Racz, Phys. Rev. Lett., 1984, 53, 415. 16 J. Nittmann and H. E. Stanley, Nature (London), 321, 663. 17 J. D. Chen and D. Wilkinson, Phys. Rev. Lett., 1986, 55, 1892. 18 P. Meakin, Z-Y. Chen and P. Evesque, unpublished work. 19 E. Ben-Jacob, R. Godbey, N. D. Goldenfeld, J. Koplik, H. Levine, T. Mueller and L. M. Sander, Phys. 20 M. Matsushita and H. Honda, J. Phys. SOC. Jpn, 1986, 55, 2483. 21 J. Feder, T. Jossang, K. Jorgen and U. Oxaal, in Fragmentation, Form and Flow in Fractural Media, 22 J-D. Chen and D. Wilkinson, Phys. Rev. Lett., 1985, 55, 1892. 23 C. Tang, Phys. Rev. A, 1985, 31, 1977. 24 J. Kertesz and T. Vicsek, .I. Phys. A, 1986, 19, L257. 25 B. R. Thompson, preprint. 26 P. Meakin, in Kinetics ofAggregation and Gelation, ed. F. Family and D. P. Landau (North Holland, 27 P. Meakin, Phys. Rev. B, 1984, 29, 4327. 28 M. Murat and A. Aharony, Phys. Rev. Lett., 1986, 57, 1875. 29 L. Niemeyer, L. Pietronero and H. J. Wiesmann, Phys. Rev. Lett., 1984, 52, 1033. 30 P. Meakin and Y. Termonia, unpublished. 31 P. Meakin, I. Majid, S. Havlin and H. E. Stanley, J. Phys. A , 1984, 17, L,975. 32 S. Havlin and R. Nossal, J. Phys. A , 1984, 17, L427. 33 L. Patterson, Phys. Rev. Lett., 1984, 52, 1621. 34 J. Nittmann, G. Daccord and €3. E. Stanley, Nature (London), 314, 141. 35 H. Van Damme, F. Obrecht, P. Levitz, L. Gatineau and C. Laroche, Nature (London), 1986,320, 731. 36 G. Daccord and R. Lenormand, Nature (London), in press. Rev. Lett., 1985, 55, 1315. Ann. Isr. Phys. SOC., 1986, 8, 531. Amsterdam, 1984), p. 91.124 Difusion -1im ited Aggregation Processes 37 W. T. Elam, S. A. Wolf, J. Sprague, D. U. Gubser, D. Van Vechten, G. L. Barz Jr and P. Meakin, 38 H. Honjo, S. Ohta and M. Matsushita, J. Phys. Soc. Jpn, 1986, 55, 2487. 39 M. Matsushita, M. Sano, Y . Hayakawa, H. Honjo and Y. Sawada, Phys. Rev. Lett., 1984, 53, 286. 40 M. Matsushita, Y . Hayakawa and Y. Sawada, Phys. Rev. A, 1985, 32, 3814. 41 R. M. Brady and R. C. Ball, Nature (London), 1984, 309, 225. Phys. Rev. Lett., 1985, 54, 701. Received 8th December, 1986

ISSN:0301-7249    

DOI:10.1039/DC9878300113

出版商:RSC    

年代:1987

数据来源: RSC

摘要:

Faruday Discuss. Chern. SOC., 1987,83, 125-137 Computer Simulations of Cluster-Cluster Aggregation R. Jullien,* R. Botet and P. M. Mors? Physique des Solides, Universite' Paris- Sud, Centre d 'Orsay, 91405 Orsay, France We briefly introduce a simple extension of the diff usion-limited aggregation model of Witten and Sander, the cluster-cluster aggregation model, in which all clusters diffuse and grow by sticking at their first contact. This model is particularly well adapted to describe aggregation experiments on colloids or aerosols. Three recent extensions of the model are emphasized which take into account possible restructuring, cluster polarizability or indirect aggregation by polymers. In these three cases the computer simulations are compared with experiment. Since the discovery by Forrest and Witten' of the fractal structure2 of smoke aggregates, several theoretical models3 have been built to simulate aggregation phenomena.The first was introduced by Witten and Sander.4 This model considers a particle-cluster mechanism in which individual diffusing particles stick one after another onto a single growing aggregate. This model has been recognized to be quite well appropriate to describe 'field-induced' aggregation experiments, such as filtration5 or electrodeposition,6 as well as many other experiments which are not truly aggregation, such as dielectric breakdown7 or viscous fingering.' It appears, however, that the Witten-Sander model leads to more compact structures than those observed in standard aggregation experiments such as the experiments realized with colloids or aerosols.To simulate these experiments, an alternative model, the cluster-cluster aggregation model, where the growing mechanism is governed by col- lisions between diffusing clusters, was introduced independently in the U.S.A.9 and in France." In this paper we briefly recall the definition cluster-cluster aggregation and review the different versions of the model. We then describe the intrinsic anisotropy properties of aggregates grown by such mechanism and describe more precisely recent extensions which have been motivated by experiments. These extensions consider possible readjust- ing effects, electrical polarizability effects and flocculation mechanism induced by the presence of polymers. The Cluster-Cluster Aggregation Model and Its Different Versions In its original v e r s i ~ n ~ , ' ~ the cluster-cluster model starts with a collection of equal-sized individual spherical particles randomly disposed in a box.Then these particles are allowed to undergo a diffusing motion simulated by a pure random walk (with consider- ing periodic boundary conditions at the edges of the box). When two particles come into contact, they irreversibly stick together to form a rigid dimer which is also able to diffuse in the box. This dimer can stick to other dimers or single particles etc. After each collision the two colliding clusters form a new, rigid, larger cluster. The mechanism can be pursued until a unique large aggregate remains alone in the box. t Permanent address: Instituto de Fisica, Universidade Federal do Rio Grande do Sul, 90049 Port0 Alegre, RS Brazil.125126 Computer Simulations of Cluster- Cluster Aggregation In these simulations a parameter a is naturally introduced, which characterizes how the velocity (or preferably the diffusivity) of a cluster varies as a function of its number of particles, i : vi - i". For realistic cases of sufficiently negative a values, i.e. when small clusters move faster than large clusters, and in the limiting case of very low initial particle concentration, it has been shown that the aggregates exhibit a fractal structure and their fractal dimension is independent on a in a large range of a values. This fractal dimension depends only on the dimension of space and, for pure brownian motion, it has been estimated to be D == 1.44 in d = 2 and D ==: 1.78 in d = 3 , in good agreement with the experimental values for colloids" and aerosols.' In contrast to the geometrical characteristics, all the kinetics strongly depend on a.The time dependence of the mean cluster size, as well as the entire shape of the cluster-size distribution,'* can be satisfactorily described by an old kinetic equation due to Smolu~howski,'~ at least for space dimensions larger than d = 2, where concentration fluctuations can be neg1e~ted.l~ An idealized version of the cluster-cluster model, the hierarchical model, has been i n t r ~ d u c e d ' ~ in which only clusters of rigorously the same number of particles are allowed to stick together. This version, which gives the same quantitative results but is less consuming of computer time, has permitted many systematic extensions.In par- ticular, the model has been extended to high space dimensions.16 Extensions to larger a values show that, beyond a critical value, the aggregation mechanism is dominated by particle-cluster collisions: one large cluster finally dominates by absorbing all other small This abrupt change from cluster-cluster to particle-cluster aggregation, when increasing a, can be related to the change of analytical properties in the solutions of the Smoluchowski equation." Extensions to larger con- centrations, which lead to more compact aggregates, have been introduced to simulate the sol-gel transition." Other extensions introduce a modification of the cluster trajectory.Instead of considering a brownian trajectory (the fractal dimension of which being d , = 2 ) , one can consider straight-line trajectories (d, = I).*' This ballistic model is a goad simulation of the molecular regime of aerosols. Such a modification only slightly increases the fractal dimension of the resulting aggregates. To take into account long-range attractive forces one can also consider straight-line trajectories without impact parameter: now the clusters move along a line passing through their centres of m a s 2 ' The fractal dimension is again slightly increased. Another extension of the cluster-cluster model completely eliminates the role of the trajectory: this is the chemically limited cluster-cluster aggregation This model can be built by introducing a sticking probability, and by letting this probability tending to zero.In this limit the clusters have all the time to investigate all the sticking possibilities and they finally choose one at random. This model can be considered as the cluster- cluster counterpart of the Eden particle-cluster growth model, in which particles are added at random on the surface of a growing cluster.23 The interest of such model is that it is experimentally realized in colloids when the electrostatic repulsion is not completely screened. The fractal dimension is then D==2 in d = 3 , slightly, but sig- nificantly, larger than the value D =r 1.78 obtained in the pure diffuse case (with a sticking probability equal to one). Such change of the fractal properties of aggregates has been effectively observed by Weitz et al." in their experiments on gold colloids.To summarize the influence of the nature of the trajectory on the resulting shape of the aggregates, we show in fig. 1 typical three-dimensional clusters grown using the hierarchical model. Cases ( a ) , ( c ) and ( d ) correspond to Brownian trajectories, linear trajectories with impact parameter and linear trajectories without impact parameter,R. Jullien, R. Botet and P. M. Mors 127 Fig. 1. Typical aggregates showing the influence of the nature of cluster trajectory on the shape of the resulting clusters obtained by cluster-cluster aggregation process in three dimensions: ( a ) Brownian, ( b ) chemical, ( c ) linear with impact parameter and ( d ) linear without impact parameter.To give an idea of their relative sizes, all the aggregates have the same number (4096) of equal-sized particles. Table 1. Fractal dimension of the cluster-cluster model as a function of the space dimension in the case of the Brownian, ballistic (random straight-line trajectories) and chemical versions of the model as well as in the case of linear trajectories without impact parameter" d 2 3 4 5 6 ( d , = 2) Brownian 1.44 1.78 2.05 2.27 2.6 ( d , = 1) ballistic 1.51 1.91 2.22 2.47 2.7 chemical 1.55 2.04 2.32 linear without impact parameter 1.56 2.06 2.53 2.97 3.46 a The absolute error is of the order 0.05. respectively. Case ( b ) , which corresponds to chemically limited cluster-cluster aggrega- tion, is shown for comparison. The corresponding fractal dimensions, which are numeri- cally obtained by averaging over many aggregates and by extrapolating to infinite sizes, are given in table 1.The Intrinsic Anisotropy of Cluster-Cluster Aggregates There is a great difference in shape between particle-cluster and cluster-cluster aggre- gates, and this difference is not only due to the difference in their fractal dimensions. This can be seen directly for the typical two-dimensional clusters shown in fig. 2. While particle-cluster aggregates exhibit a roughly spherical shape with a well defined centre around which the growth mechanism takes place, cluster-cluster aggregates exhibit instead an oblong, ellipsoidal, shape, in general. Since the conception of the cluster- cluster model, it has been recognized that clusters grown by this process are a n i s ~ t r o p i c .~ ~ However, it is only recently that a systematic quantitative study of these anisotropy properties has been made.25128 Computer Simulations of Cluster- Cluster Aggregation Fig. 2. A typical Witten-Sander aggregate (left) compared with a typical cluster-cluster aggregate (right) containing the same number of particles (4096), in two dimensions. Table2. Anisotropy ratio Al between the largest and the smallest eigenvalues of the radius of gyration tensor, in dimension 2, estimated from finite-size simulations in the case of three versions of the cluster-cluster model Brownian linear chemical A1 5.7 f 0.2 5.2 f 0.2 4.7 * 0.2 D 1.44 f 0.05 1.51 k0.05 1.55 f 0.05 The anisotropy can be quantitatively investigated by diagonalizing the tensor of the radii of gyration: ( ~ u b ) ’ = C ( r i u - rju>(rih - rjb)/(2N) i,j where a and b refer to the coordinates and i and j to the particles in the aggregate.The usual radius of gyration (squared) is simply the trace of the tensor: R’ = c ( R J 2 . U Using the hierarchical procedure in two dimensions, where 1000 independent clusters of 1024 particles have been built, the tensor has been diagonalized, and the largest and lowest eigenvalues and (RJ’ have been averaged over all the clusters of the same number of particles. It is found that the ratio varies very slowly with cluster size. The results for the extrapolation to infinite size are given in table 2 in the case of the Brownian, ballistic and chemical models. When comparing the three cases, one observes a systematic decrease in Al when increasing the fractal dimension.In all cases the anisotropy ratio is quite large. The same calculation performed on particle-cluster aggregates gives an extrapolated value A, =; 1. Thus particle-cluster aggregates can be considered as isotropic in the sense that the eigenvalues of the tensor are degenerate. This does not exclude other more subtle effects such as anisotropy induced by the underlying lattice26 or self-affinity proper tie^,^^ as recently found in the Witten-Sander model. The calculations have been extended to higher dimensions in the case of the cluster- cluster model with linear trajectories. The results for all the d - 1 anisotropy ratios:R. Jullien, R. Botet and P. M. Mors 129 Table 3.Anisotropy ratios Ai = (( R ( ) * ) / ( ( R,)') between the ith and the lowest eigenvalues of the radius of gyration tensor, estimated from finite size extrapolations in the case of the cluster-cluster aggregation model with linear trajectories, for space dimensions up to d = 4 . The anisotropy ratio A: for a two-dimensional projection of a three-dimensional aggregate is also given. d A3 2 5.2 + 0.02 3 10.0 f 0.3 2.5 f 0.3 4.5 f 0.3 4 14.0 f 0.5 4.2 f 0.4 2.0 f 0.3 (the eigenvalues being given in decreasing order) for space dimensions d = 2, 3 and 4, are reported in table 3. For a given d, all the eigenvalues are different. Moreover, the larger ratio A, grows rmghly linearly with d. This last result is supported by simple analytical considerations.2s For a comparison with experiment, and in particular for the analysis of two- dimensional micrographs, it is of interest to have some information on the anisotropy properties of the projection of three-dimensional clusters onto a plane.The ratio A\ for such a projection has been calculated and is also reported in table 3. A; is of the order 4.5, slightly smaller than for a pure two-dimensional process. When quantitatively comparing with experiments it is important to recall that the A, quantities are the ratios of averaged eigenvalues. The alternative quantities B, = (( Ri)2/ ( Rd)*) are 23 '/o smaller than the Ai values.28 Readjusting Effects in Cluster-Cluster Aggregation An important approximation to the original cluster-cluster model is that the clusters stay rigid along their diffusive motion and do not rearrange themselves after sticking.There is experimental evidence that, at least on a small length scale, restructuring and readjusting may occur in aggregation phenomena. An example is shown in plate 1, where one can see aggregates of wax balls floating on water surface under shear forces.29 These aggregates appear to be quite compact on a short-range scale. It is a delicate problem to simulate simply such restructuring effects. An attempt has been made in two dimension^.^' This calculation considers an off -lattice extension of the hierarchical version of the cluster-cluster model in which some readjusting is allowed after sticking. As in the original hierarchical ~ c h e m e , ' ~ one cluster, say cluster 1, stays at the centre of the coordinates while the other cluster, cluster 2, is released randomly on a large circle centred on the origin.Then cluster 2 undergoes a random walk in space until a first contact occurs. Then, as shown in fig. 3 , cluster 2 is rotated rigidly about the centre of the contacting particle on cluster 1 until a second bond is obtained. Sometimes the new contacting point is again located on the same contacting particle of cluster 2, so that another rotation can then be performed around the centre of the contacting particle on cluster 2 until a true loop is obtained. Different options can be adopted to perform these rotations. Either one considers a pure Brownian-like rotational motion or one systematically chooses the smallest possible angle.The precise choice does not affect the result too much. An important point, however, is that, in any case, one waits until the complete readjustment has been made before considering a new collision. Using this procedure, loops are systematically built at each step of the hierarchical procedure. Typical examples of small clusters ( 5 12 particles) built with and without restructuring are shown in fig. 4 (top). Here two rotations have been considered (when possible) and130 Computer Simulations of Cluster- Cluster Aggregation / W Fig. 3. Sketch of the readjusting mechanism in the cluster-cluster model. After contact, the clusters are rotated about their contacting particle until a true loop is obtained. In case ( a ) only one rotation is possible. In case ( b ) two rotations can be performed.the smallest angle has been chosen. Compactification is clearly visible and the readjusted cluster strongly resembles the clusters of plate 1. The calculation has been pursued to reach larger cluster sizes, and typical examples of larger clusters (16 384 particles) are shown in fig. 4 (bottom). Surprisingly, the restructuring effect is now less visible. This is quantitatively checked when calculating the fractal dimension, which characterizes long-distance correlations. When extrapolating to the infinite size, the fractal dimension is estimated to be D = 1.48 f 0.05, only very slightly larger than the one, D = 1.44 f 0.05, without restructuring, the difference being of the order of the error bar. A spectacular experimental confirmation of this has been recently produced by Skjelt~rp.~' His aggregation experiment with uniformly sized polystyrene microspheres confined to thin layers between solid boundaries is shown in plate 2, at two different scales.On the short scale one clearly sees readjusting and compactifications, while this is less visible on larger length scale. Moreover, the fractal dimension D = 1.49 f 0.05, estimated from the experiment is in perfect agreement with the calculations. One can, however, observe that the aggregates are slightly more compact, on the short length scale, in the experiment than in the simulation. This can only be accounted for, in any calculation, by involving more complex restructuring effects such as diffusion of individual particles on the surface and/or slight modifications of the interparticular distances, which have not been allowed in the calculations presented above.30 Such more complicated effects are hard to simulate simply.The Effect of Cluster Polarizability (Tip-to-tip Model) Some aggregation experiments in two dimensions32 or in three dimensions33 lead to a smaller fractal dimension than the one predicted by the original cluster-cluster model. This discrepancy has been recently explained by introducing polarizability effects in the existing r n ~ d e l s . ' ~ In the case of polarizable clusters, two clusters, before they collide, develop opposite electrical charges on their neighbouring tips. Then, an electrostatic attraction biases their relative diffusive motion. In the limiting case of very strongFaraday Discuss.Chem. SOC., 1987, Vol. 83 Plate 1 Plate 1. Aggregation of wax balls floating on the surface of water in the presence of shear forces (experiment carried out by Camoin and B l a n ~ ~ ~ in Marseille, France). Plate 2. Two-dimensional aggregation of 4.7 pm spheres showing compactification on a short length scale ( a ) and ramified clusters on a large length scale ( b ) (experiment carried out by Skjeltorp3' in Kjeller, Norway). R. Jullien, R. Botet and P. M. Mors (Facing p . 130)R. Jullien, R. Botet and P. M. Mors 131 * < * 80 d i a m e t e r s 60 d i a m e t e r s < * * 1100 d i a m e t e r s 800 d i a m e t e r s Fig.4. Effect of readjusting in the cluster-cluster model. Cases (a) and ( b ) correspond to small clusters (5 12 particles) with and without readjusting, respectively.Cases ( c ) and (d) correspond to larger clusters (16 384 particles). polarizability, one can imagine that the Brownian diffusion does not play any role and that these neighbouring tips effectively become the most probable sticking points. This is the spirit of a new model, the tip-to-tip model, which can be simply described in its hierarchical version. The procedure differs from the standard hierarchical procedure only in the way the collision between two clusters is simulated. As sketched in fig. 5 , a random direction in space is first chosen. Then, the two clusters are placed far apart along this direction. The two nearest particles on the first and second clusters are determined. Then the two clusters are translated so that these two particles come into contact, with their centres aligned along the chosen direction.This model gives a fractal dimension equal to 1.26 and 1.42, respectively, in d = 2 and d = 3. In its particle-cluster counterpart, this procedure leads to star-like clusters with a fractal dimension trivially equal to one. Typical two-dimensional and three-dimensional clusters obtained with this model are given in fig. 6.132 Computer Simulations of Cluster- Cluster Aggregation Fig. 5. Sketch of the sticking mechanism in the tip-to-tip model. Fig.6. Typical aggregates obtained with the tip-to-tip model. Cases ( a ) and (c) correspond to the cluster-cluster version in d = 2 and d = 3 (fractal dimensions equal to 1.26 and 1.42), respec- tively.Cases ( b ) and ( d ) correspond to the particle-cluster version.R. Jullien, R. Botet and P. M. Mors 133 I I - 3 I . 1 6 - h v U Y cl aD 5 - 0 4 - 4 - 3 - 2 log (s) Fig. 7. Experimental small angle X-ray scattering functions of aluminium hydroxyde aggregates Al(OH),, with x = 2.5 ( a ) and x = 2.6 ( b ) , plotted as a function of s = q / ( 2 ~ ) . In each case one can show the corresponding theoretical curve obtained directly from the simulation, using the tip-to-tip cluster-cluster model in case ( a ) and the ballistic cluster-cluster model in case ( b ) . The theoretical curves take care of the diffusion by individual particles and have been adjusted to the experimental curves at the inverse of the particle diameter. (-), (- - -) and (. a - - - -.) correspond to clusters with N = 64, 128, 256 particles in case ( a ) and N = 256, 512, 1024 particles in case ( b ) .The cluster-cluster tip-to-tip model has been recently used to explain small-angle X-ray scattering experiments on aluminium hydroxide aggregate^.^^ The experimental scattering functions obtained with samples of Al(OH),, with x = 2.5 and 2.6, both exhibit a linear part in their log-log plots characteristic of fractal structures, but with a smaller slope in the case x = 2.5 than in the case x = 2.6. These curves have been fitted with the theoretical expression: with where I , ( q ) is the scattering function of a single subunit, which is known from other experiments to be a spherical AlI3 small cluster of radius ro = 10 A, and where P ( r ) is a properly normalized distance distribution function, P( r ) d r being proportional to the134 Computer Simulations of' Cluster- Cluster Aggregation number of inter-particle distances located between r and r+dr.The normalization is such that P P ( r ) d r = N - 1 J where N is the number of subunits in an aggregate. P ( r ) has been directly obtained from the simulations of an N-particle aggregate, using the hierarchical procedure. The quantative fit to the experimental curve for x = 2.5, using the three-dimensional tip-to-tip model, is shown in fig. 7(a). The only adjustable parameter is the intensity at q = l / r o . The fit of the low-q part of the curve (the so-called Guinier regime) allows one to estimate the number of subunits to be N ~ 6 4 . The same kind of fit has been obtained in the case x=2.6 [fig.7 ( b ) ] , but using the usual three-dimensional cluster-cluster model (for simplicity linear trajectories have been considered but the results would not have been so different with brownian trajectories). Here the number of particles is estimated to be N = 512. Since in these fits one does not take care of any size distribution of clusters, these deduced numbers of particles must be considered as very rough averages. There are some qualitative arguments to explain why the tip-to-tip model is better adapted to the x = 2.5 case. The Al,, subunits are positively charged, and water molecules are oriented near their surfaces, producing an electrical field whose intensity is directly linked with the intensity of the charge on the subunits.For x = 2.5, this charge is quite large and one can imagine that the polarization of water molecules on the surface of the aggregates might induce a sticking mechanism like in the tip-to-tip model. However, when x increases, the charge decreases, leading back to the regular cluster-cluster mechanism for x = 2.6. Flocculation of Colloids in the Presence of Polymers The aggregation of charged colloids can be induced by the adjonction of polymers of opposite charges into the colloidal solution. This is a quite common procedure which has a lot of industrial applications. The colloidal particles, attracted by polymers, can cover their surface with a limited number of polymeric units. If the polymeric chain is sufficiently long, the same chain can be attached to several colloidal particles and, since a given particle can be covered with units coming from distinct polymeric chains, a random set of polymeric connections is progressively built, leading to an efficient aggregation mechanism between particles.A first approach to simulating this mechanism could be to extend the existing cluster-cluster model to the case of two diffusing species A (particles) and B (polymers), in which A-B connections are allowed while A-A and B-B connections are forbidden. Such a model has recently been introd~ced,'~ but with another motivation. Very interesting results, such as a dependence of the fractal dimension on the ratio of concentrations of A and B species, or saturation effects occurring when one species is in excess, are observed.However, this model, which treats the A and B species equally, as though they were both diffusing colloidal particles, is too crude a model to describe aggregation through polymers, since it does not take into account the specific properties of polymers. A more adequate simulation is in progre~s.'~ This is an iterative method in which a collection of aggregates of balls connected by polymers is built in a finite box, using a chemical-type mechanism. The first version is restricted to two dimensions. The general principles are the following. The aggregates are assumed to be rigid (the interparticle distances remain constant), while all the polymeric chains have a different random configuration at each step. Once a polymer is attached to a ball, however, this sticking remains permanent and stays at the same distance (counted along the chain)R.Jullien, R. Botet and P. M. Mom 135 Fig. 8. A typical two-dimensional aggregate of 190 particles built with a model of aggregation of balls with polymers (see text). of the polymer end. One assumes that a ball cannot be attached by more thanf contacts to the same or different polymers. The iterative process starts with a given number P of polymers and a given number B of balls. Then at a given iteration one has a collection of aggregates together with remaining single balls and/or polymers. An iteration step is performed as follows. All these objects are randomly disposed in the box under the constraint that balls cannot overlap (many trials are performed if necessary).(ii) The single polymers (if there are any) and all the polymeric chains connecting the balls in aggregates are reconstructed in a random order, using a pure random walk for the single polymers and the dangling ends, and an adequate approximately biased random walk for the segments connecting two balls. (iii) Once a polymer sticks a non-saturated ball, a contact is permanently obtained and a new cluster is built. The parameters are as follows: the number P of polymers, the number B of balls, the functionalityf of the balls, the length N of the polymers (i.e. number of monomers), and the size L of the box. A big difference from the usual cluster-cluster process is that, if there is a sufficient number of polymers, all the balls within a cluster can be saturated and the overall process can stop with more than one cluster in the box. Fig.8 gives a typical example of a final cluster (the largest) obtained in a simulation with B = 200, P = 100, f = 4, L = 100, N = 100. This cluster contains 190 balls. The theoretical Fourier transform of the positions of particles in this cluster, S ( q ) , has been calculated and reported in fig. 9. We have used the three-dimensional formula given above as if this two-dimensional cluster would be randomly disposed in space. One observes a characteristic linear qPD regime whose slope gives D = 1.7, in agreement with a direct estimation of D in real space. This is a preliminary result obtained on a unique and quite small aggregate which must be checked by averages over many samples. One can, however, already notice that this fractal dimension is larger than the one of the chemical model in two dimensions ( D = 1.55), as if the effect of polymers would be to increase the density compared to a pure chemical process.This must be understood as a preliminary conclusion, and a systematic study of the fractality and other characteris- tics of such an aggregate as a function of the parameters of the model is in progre~s.”~ This model is quite crude in its early stages, and some improvements are looked for in the future. First, it must be extended to three dimensions. Then, instead of randomly disposing the objects (which corresponds to a chemical mechanism where the objects touch many times before sticking) one can build a true diffusion. Also, instead of randomly constructing the polymers before they are irreversibly attached to the balls136 Computer Simulations of Cluster- Cluster Aggregation 0 -2 - 4 Fig.9. Theoretical curve for S ( q ) , calculated as if the cluster of fig. 8 would be randomly disposed in three-dimensional space. (which corresponds in assuming a strong short-range attraction between balls and polymers), one can envisage a biased random walk to take care of long-range attractions. Finally, one could take care of the excluded volume for polymers, and other effects, such as repulsion between balls and a possible variation of the distances between balls, can also be considered. Conclusion In conclusion, the cluster-cluster model appears to be quite appropriate to describe aggregation of colloids and aerosols. Several realistic extensions are able to reproduce quantitatively specific experimental results.In many cases a direct comparison between computer simulations and experiment has been very useful in providing a better under- standing of the underlying mechanisms of real aggregation processes. Such studies will be developed in the future. We thank M. A. V. Axelos, B. Cabane, J. P. Chevalier, M. Kolb, P. Meakin, A. T. Skjeltorp, D. Tchoubar and M. Ten& for stimulating discussions. This work has been supported by an A.T.P. and an A.R.C. from the C.N.R.S. Computer simulations have been performed at C.I.R.C.E. (Centre Inter-Rkgional de Calcul Electronique), Orsay, France. P.M.M. acknowledges financial support from C.N.Pq, Brazil. References 1 S.Forrest and T. Witten, J. Phys. A, 1979, 12, L109. 2 B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1982). 3 R. Jullien and R. Botet, Aggregation and Fructal Aggregates (World Scientific, Singapore, 1986). 4 T. Witten and L. Sander, Phys. Rev. Lett., 1981, 47, 1400. 5 D. Houi and R. Lenormand, in Kinetics of Aggregation and Gelation, ed. D. Landau and F. Family 6 M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo and Y . Sawada, Phys. Rev. Lett., 1985, 53, 286. 7 L. Niemeyer, L. Pietronero and H . Wiesmann, Phys. Rev. Lett., 1984, 52, 1033. 8 J. Nittmann, G. Daccord and H . 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ISSN:0301-7249    

DOI:10.1039/DC9878300125

出版商:RSC    

年代:1987

数据来源: RSC