Generalized diffusion equation of interacting brownian particles

GENERALIZED

DIFFUSION

Toshiya OHTSUKI Lkpartmentof -4pplied Received 26 October

17 June 1983

CHEhlICAL PHYSICS LETTERS

Volume 98. number 2

EQUATION

OF INTERACTING

Physics, Faculty of Engineering,

Universi@

BROWNIAN

of Tokyo, BunLyo-hu,

PARTICLES

Tokyo 113. Japan

1982: in fmal form 6 April 1983

Macroscopic behavior of a system of brownian particles interacting with each other through potential forces is described by a generalized diffusion equation (GDE) for the density of particles. The diffusion coefficient in the GDE is given by the generalized Stokes-Einstein relation and generally depends on the density. In the presence of long-range interactions, the GDE becomes non-local in space. When a Coulomb interaction exists, the GDE corresponds to an improvement of the Poisson-Boltzmann equation.

1_ Introduction

2. The GDE in the presence of short-range

interactions

Diffusion is observed in almost every field of science and engineering and many mass transfer phenomena are governed by a diffusion equation. In an ordinary inves-

We consider a system of N spherical brownian particles interacting with each other through a short-range pair potential qk. The frictional damping of particles

tigation of diffusion, diffusing particles are regarded as coupled only with a heat bath and interaction between particles are neglected. In a real system, however, various interparticle interactions frequently play an essential role. Recently the study of interacting brownian

caused by a medium is assumed to be large enough. The temporal evolution of such a system is described by a Smoluchowski-type diffusion equation for the ?&particle distribution function PN(RI .R2,..R,,;r) inconfiguration space [4],

particles has received attention and numerous theoretical and experimental works have been reported [ 1,2] _ But it is usually hard to take into account many-body effects due to interactions at a microscopic level [3] _ On the other hand, in order to solve many practical problems, detailed knowledge at a microscopic level is not necessary and only coarse-grained information at a macroscopic level is required. Then it might be significant to develop a systematic method for a macroscopic analysis of systems of interacting brownian particles,

where D,-, is a bare diffusion coefficient and F is an external force. Integrating eq. (1) over the coordinates we have an equation for the one-particle R,,R,,..R,,

which is just the purpose of this letter. In section 2, we derive a generalized diffusion equation (CDE) which describes the macroscopic behavior of brownian particle systems in the presence of short-range potential interactions. In this letter, hydrodynamic interactions are not taken into consideration and effects of long-range interactions are investigated in section 3. In section 4, we discuss our results.

distribution function P, (R 1 ;f), cles p(R r ;r), as follows [5] I

0 009-2614/83/0000-0000/S

03.00

0 1983 North-Holland

g=D,V,

-

i.e. the density of parti-

V,P - k&F,~

(2)

121

\‘olume

9s. number

17 June 1983

CHEMICAL PHYSICS LETTERS

2

Here we treat only macroscopic behavior of the system B em p varies very dowry over the microscopic mnge of the system such as an average interparticle distance. and 3 range of interactions. Then in the righthand side of eq. (2). we expand Pz in a series of density gradients and retain the temls up to linear order because the integrand is not zero only when the interparticle drstance R 12 = IR2 - R 1I is of the order of the range of the interactions. In a Slightly non-uniform system, a 13d131distribution functiong(R1 ,Rz) is expanded as [for details, see the 3~pe~ldix~

the macroscopic change ofp is and they can describe non-linear behavior far from equilibrium. They are also valid at high densities in the presence of strong interactions. The expression of the concentration diffusion coefficient has already been derived by various analyses [L&7,8], but we show here that eq. (6) is applicable not only to the concentration diffusion coefficient defined only near equilibrium but also to the diffusion coeffcient appearing in the GDE which is meaningful even far from equilibrium, as mentioned above.

3. The GDE in the presence of long-range

>: (VP)- &RI + RI11 - Ro] + O(‘c”p)+

(3)

wberegu is the rsdial distribution function of the system with 3 uniform density pu = p(Ro)_ Then P2 is grven by

X

(Vph(RI +Rz -7R,)+6(V3p).

(4)

Puttmg R, = RI and substituting eq. (4) into eq. (21, we get 3 gener3lized diffusion equation (GDE) for p. apiat = VW[D(pWp

D@l=D,,

- (D&T)Fp].

(5)

I -

In the presence of a long-range interaction like a Coulomb interaction, the macroscopic behavior of the system differs essentiahy front that in the absence of a long-range interaction and cnnnot be described by eqs. (5) and (6). That is, when \k decreases more slowly than Rm3 in the limit R + m. the integrand in eq. (6) diverges. A macroscopic change of the density influences the microscopic circumstance around particles and the screening effect. In this case. we separate \k into two parts, Ik=Jrs+*=,

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where @ is the short-range part decreasing more rapidly than R-3 in the limit R + m and \kL is the long-range part which decreases more slowly than RD3 (includes the Rm3 term). Then we rewrite the integral term of eq. (2)as

i

= ~+~(lik7-)

interactions

aiI/ap.

(6)

I\ here p = g(R .p) ts the radial distributiol~ function of the system \\ith d uniform density p and D is the (osmotic) plcssure. In arriving at these equations. we assume that the system is 3t local equilibrium and time dependence IS introduced only through p and F. This assumption is reasonable for macroscopic behavior \\ here p tmd F v,.~ryvery slowIy on the microscopic time scale given by r = L.‘/D,. Eq. (6) represents the generalized Stokes-Einstein relation and corresponds to the expression of the concentr&ion diffusion coefficient [6] _ It should be emphasized that 3s Iong 3s p and Fare slowly varying functions of space 3rd time over the microscopic range L xnd r, eqs. (5) and {6) hold good no matter how large

‘(V,\k~#-$ - PIP21+ @,q,)P,P,l

m,=

(8)

pi = p@:f)_ Since the first and second terms in the right-hand side of eq. (8) have no singularity, they are treated in the same way 3s in section 2, while the third term is left as it is. Substitution of eqs. (4 and (8) into eq. (2) leads to a genemhzed diffusion equation (GDE) in the presence of a long-range interaction,

where

aPiat = v. [D(p)vp

+ fD,fkT)(vU

- F)p],

(9)

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98. number

CHEhlICAL

2

U= U(R;t) = j’l’=(R

- R’)p(R’;t)

dR’.

PHYSICS

(11)

In the presence of long-range interactions, the GDE becomes non-local in space. In particular, when qL is the Coulomb interaction given by (.ze)2/&, eq. (11) becomes Poisson’s equation for the electric potential V = Vjke, AV = 447rze/e)p,

(12)

where ze is the charge of a particle and E the dielectric constant. If we neglect a short-range interaction and correlation between particles, viz. put es = 0 and g = 1, D(p) equals Do and at equilibrium, eq. (9) gives the Boltzmann distribution p = p,, exp(--U/M’). Thus, eqs. (9) and (12) reduce to the Poisson-Boltzmann equation, AV = -(477ze/e)p,,

exp(-zeV/W).

LETTERS

17 June 1983

dependent diffusion coefficients which are frequently used for various systems and given rather phenomenologically [lo]. In the field of mathematical ecology, a density-dependent diffusion coefficient is popular as a result of a population pressure [ 111. Although the microscopic law is different, the concept of the GDE is also applicable to such cases. In this work, hydrodynamic interactions are not brought into consideration and the theoretical investigation of effects of hydrodynamic interactions remains as one of the most significant for future work. However, it seems that even in the presence of hydrodynamic interactions, macroscopic behavior of a system of interacting brownian particles is also governed by the diffusion equation with a density-dependent “concentration” diffusion coefficient_

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Therefore, the GDE in the presence of a Coulomb interaction can be regarded as the improvement of the Poisson-Boltzmann equation where a short-range interaction and interparticle correlation are taken into consideration and is applicable to the study of non-equilibrium properties_

Appendii In this appendix, we calculate the density gradient expansion of a radial distribution function of a slightly inhomogeneous system. In a classical system, the s-particle distribution function P, is defied by

4. Discussion In this letter, we derive a generalized diffusion equation (GDE) for a system of interacting brownian particles where the diffusion coefficient is given by the generalized Stokes-Einstein relation and generally depends on the density. As long as a system varies slowly over the microscopic range, the macroscopic change of the density of particles is described by the GDE even for a large change. The crucial effects of a long-range interaction are clarified and the GDE in the presence of a Coulomb interaction turns out to correspond to the improvement of the Poisson-Boltzmann equation. The present work is thought to be useful in many cases where potential interactions play an important role. First, the GDE gives a systematic method, which is tractable and widely applicable, to analyze macroscopic behavior of such systems. Second, eqs. (Q)-(1 1) are one approach to improvement of the PoissonBoltzmann equation (see, e.g. ref. [Q]) which is the most fundamental equation in ionic systems. Third, eq. (6) or (10) provides the microscopic basis of density-

XJ___Je-P”hr

dR,,

1

___dR,,

with

fl=

l/?cT,

where E is the grand partition function. v is the fugacity and QJV is the potential energy of the system. Here we divide \kN into two parts as qAr = *;

+ e*k

with *~(R,&-..RN)=

(E < l),

(15)

N c

i=l

I,

where *_iT is the potential energy of the uniform reference system and e\k;V is the weak perturbation. Substi123

VoIumc

98. number 2

CHEMICAL PHYSICS LETTERS

tution of eq. ( f 5) into eq. (14) and expansion order in E lead to

to linear

17 June 1983

Differentiating eq. (20) and substituting it into eq. (21), we get the density gradient expansion of the radial distribution function,

Acknowledgement

--go~“Ij)-SOt~lj)+~IclR3+15(~7) f. where

pg.gO

and k,

are the density.

071

The author wishes to thank Professor K, Okano of the University of Tokyo for his valuable comments and advice.

the radial distribu-

tion ~~11ctio~l2nd the three-particle correlation function of the reference system. In order to introduce a dennty d(R) = m-4-(R - Ro). and substitute tinn [ 121

gradient.

we put (1%

it into eqs. (I 6) and (I 7). Using the rela-

[ I] W. Dieterich, P. Fulde and 1. Peschel, Advan. Phys. 29 (1980) 517. [Z] W.B. Russcl, Ann. Rev. fluid Bfech. 13 (1981) 125. [3] T. Ohtsuki. Physica 110A (1982) 606. [4] T-J. Murphy and J.L. Aguirre, 1. Chem. Phys. 57 (1972)

‘098. [5] A.R. Altenbeger .md J.M. Deutch, J. Chem. Phys. 59 (1973) 894. 161 T. Olttsuki and K. Ok;mo, J. them. Phys. 77 (1982) 1443. f7] C. Tanfurd, Physical chemistry of macromolecules (Wiky, New York, 1961). [8] C.D.J. Phillies, J. Chem. Phys. 60 (1974) 976. [ 9 j L.B. Bhuiyan, C-IV. Outhwaite and S. Levine, Mol. Phys. 42 (1981) 1271, and references therein. [IO] 1V.f. Ames, Noniincas partial differential equations in engineerin (Academic Press, New York, 1965). [ 111 A. Okubo, Diffusion and ecofogical problems: mathemat1ca.lmodels (Springer, Berlin, 1980). [ 121 P. Schofield, Proc. Phys. Sot. (London) 88 (1966) 149.

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