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. 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