A schematic picture of the model system with the mobile ions, considered as point charges, in between the uniformly charged walls. Downloaded By: [University ...
MOLECULARPHYSICS,1986, VoL. 57, No. 6, 1105-1137
Ion diffusion at c h a r g e d interfaces A stochastic d y n a m i c s s i m u l a t i o n test of the S m o l u c h o w s k i - P o i s s o n - B o l t z m a n n approximation by TORBJC)RN
,~KESSON
a n d BO J ( ) N S S O N
P h y s i c a l C h e m i s t r y 2, C h e m i c a l C e n t e r , P O B 124, S-221 00 L u n d , S w e d e n
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BERTIL
HALLE
P h y s i c a l C h e m i s t r y 1, C h e m i c a l C e n t e r , P O B 124, S-221 00 L u n d , S w e d e n and DEREK
Y. C. C H A N
D e p a r t m e n t of M a t h e m a t i c s , U n i v e r s i t y of M e l b o u r n e , Parkville, V i c t o r i a 3052, A u s t r a l i a (Received 19 July 1985 ; accepted 4 November 1985) We investigate the statistical-mechanical basis and the numerical accuracy of the Smoluchowski-Poisson-Boltzmann (SPB) approximation for describing ion diffusion in non-uniform electrolytes. The many-particle generalized Smoluchowski equation is formally reduced to a hierarchy of coupled nparticle equations. A closure relation, called the Instantaneous Relaxation Approximation (IRA), is used to decouple the equation for t h e one-particle self-propagator. Introducing also a mean field approximation (MFA), we recover the SPB equation. The accuracy of the IRA and M F A is quantitatively assessed for a model system consisting of two parallel uniformly charged plates with an intervening solution containing point ions in a dielectric medium. This is done by comparing diffusion propagators, survival probabilities and mean first passage times obtained by (1) solving the manyparticle generalized Smoluchowski equation by the stochastic dynamics simulation technique (2) numerically solving the one-particle Smoluchowski equation with the exact (simulated) equilibrium potential of mean force, and (3) analytically solving the SPB equation. The IRA is found to be a useful approximation, whereas the M F A can lead to substantial error for systems with strong Coulomb coupling, as in the case of polyvalent counterions. Provided with a realistic potential of mean force, the one-particle Smoluchowski equation thus yields an accurate description of ion diffusion in nonuniform electrolytes.
1. INTRODUCTION T h e electric d o u b l e layer, c o n s i s t i n g of an ionic s o l u t i o n in c o n t a c t w i t h a c h a r g e d interface, e m e r g e s as a central f e a t u r e o f m a n y p r o b l e m s in c o l l o i d science, e l e c t r o c h e m i s t r y a n d b i o p h y s i c s . S i n c e the p i o n e e r i n g w o r k of G o u y [1] a n d C h a p m a n [2], a vast l i t e r a t u r e has a c c u m u l a t e d c o n c e r n i n g the d e r i v a t i o n ,
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accuracy and application of the Poisson-Boltzmann (PB) equation and other theories of the equilibrium ion distribution in the double layer. For a recent review, see Carnie and Torrie [3]. Although of considerable importance, the dynamic properties of the double layer have received m u c h less attention [4]. In this paper, we focus on the self-diffusion of the counterions making up the diffuse part of the electric double layer. This problem has been approached in two fundamentally different ways. Historically, the first approach was the association theory [5], which postulates the existence of bound and free counterions with zero mobility and bulk mobility, respectively. This phenomenological approach is still widely used to interpret experimental data. A more realistic picture of the dynamic consequences of the long-range Coulomb interaction is provided by the Smoluchowski approach. Here Smoluchowski's diffusion equation [6]
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0
; t I ro)]
Ot L ( r ; t ] ro) = D o V 9 [(V - / ~ K ) L ( r
(1)
for the space-time evolution of the self-propagatorfs(r; t lr0) is extended by substituting for the external force K (a uniform gravitational field in Smoluchowski's original application) the equilibrium solvent-averaged mean force acting on a counterion located at r. When the mean force is evaluated in the PoissonBoltzmann approximation we shall refer to equation (1) as the SmoluchowskiPoisson-Boltzmann (SPB) equation. The SPB approach, which was pioneered in the double layer context by Lifson and Jackson [7], has proven successful in explaining counterion self-diffusion [8-10] and spin relaxation Ell, 12] data from polyion solutions. T h e aim of this paper is to investigate some of the statistical-mechanical approximations inherent in the SPB equation. Starting from a many-particle generalized Smoluchowski equation, we identify the further approximations that are needed to obtain the SPB equation. These are (1) an assumption of instantaneous response of the surrounding counterions to the motion of the tagged counterion and (2) a neglect of the equilibrium pair correlation among the counterions. (In a uniform electrolyte solution, approximation (1) corresponds to the neglect of the relaxation effect [-13].) We then assess the effect of each of these approximations for a model system consisting of two parallel charged plates with an intervening solution containing the counterions, see figure 1. This is accom-
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Figure 1. A schematic picture of the model system with the mobile ions, considered as point charges, in between the uniformly charged walls.
Ion diffusion at charged interfaces
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plished by solving the many-particle generalized Smoluchowski equation by the Stochastic Dynamics simulation technique and comparing the resulting selfpropagator with that obtained (1) by numerically solving the Smoluchowski equation, equation (1), with the exact simulated mean force and (2) by analytically solving the SPB equation. Comparisons are also made of mean first passage times and survival probabilities, formally obtained by integrating the self-propagator over time and/or space.
2. THE GENERALIZEDSMOLUCHOWSKIEQUATION
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Our point of departure will be the N-particle generalized Smoluchowski equation which governs the time evolution of the probability density F(rN; t[ r for the configuration of the N counterions
F(rN; t i r g ) = D 0 ~ , v , " ~t
i=l
V,F(r N;t[r~)
I
N
]
}
+ flVi ul(ri) + ~' u2(ri, rj) F(rN; tlrg) . j=l
(2)
For definitions of the quantities appearing in equation (2) ; see Appendix A. Equation (2) may be derived from the Liouville equation and certain welldefined approximations [14-17]. These are as follows: (1) the relaxation of ionic and solvent momenta and of solvent configuration is fast compared to the timescale for relevant changes in the ionic configuration, (2) the potential energy of interaction, N
ul(ri) + ~ ' u2(rl, rj),
(3)
j=l
and its derivatives vary negligibly over the ionic m o m e n t u m correlation length, Do(mfl) 1/2, and (3) the solvent-mediated dynamic coupling between the ions and with the walls may be neglected. Although these approximations do not appear to have been rigorously justified for aqueous electrolytes, simple numerical estimates tend to be reassuring. In this study, however, we shall not be concerned with the accuracy of these approximations. Rather, we shall examine the further approximations that are needed to derive the SPB equation from equation (2). T h e first step in our formal derivation of the SPB equation is the reduction of the N-particle generalized Smoluchowski equation, equation (2), to a corresponding equation for the one-particle self-propagatorfs(t; t[to). As defined in Appendix A, fs(r; t[ r0) dr is the probability of finding, at time t, the tagged ion to within dr of r, given that it was initially at r 0. Using the time dependent distribution functions defined in Appendix A, we perform this reduction in Appendix B. T h e result, equation (B 11), may be expressed in the physically perspicuous form
~t L(r; t l r o ) = D o V - {IV + flVw(r; tl ro)Jf~(r; tl ro)},
(4)
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with a time dependent potential of mean force w(r; t [ to) given by
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Vw(r; t l ro) = Vul(r) + f dr'[Vu2(r, r')]g(r, r'; t[ro)fd(r'; t l r0).
(s)
In these equations, r refers to the position of the tagged particle and r to the position of any of the N - 1 untagged particles. T h e self-propagator fs(r; t] r and the distinct propagator fd(r t]r are generalizations to finite nonuniform systems of van H o v e ' s space-time correlation functions [18]. T h e definitions and general properties of these propagators and of the time dependent pair correlation function g(r r t ] r can be found in Appendix A. T a k e n together, equation (4) and the corresponding equation (B 12) for the distinct propagator constitute the lowest level of a hierarchy of coupled n-particle generalized Smoluchowski equations (n = 1, 2, . . . , N), which all can be derived along the lines of Appendix B. In order to obtain the self-propagatorfs(r t] r it is necessary to solve the coupled equations, equations (4) and (B 12), subject to an approximate closure which expresses the pair correlation g(r r t]r in terms of known functions. Such a closure has the effect of decoupling the one-particle level from the two-particle level of the hierarchy. If, in addition, we approximate the distinct propagator in equation (5) by some known function, then also the self and distinct one-particle equations decouple and it suffices to solve equation (4). T h e time dependence in the potential of mean force w(r; t] r as exhibited in equation (5), reflects the finite time required for the N - 1 untagged ions to fill the 'correlation h o l e ' at r0 left by the tagged ion. As a consequence, the tagged ion experiences a retarding force which tends to slow down the evolution of the self-propagator. T h e analogous p h e n o m e n o n in uniform systems is the so-called relaxation effect [13], which is usually pictured as a dynamic a s y m m e t r y in the ion atmosphere around the tagged ion. T h e second step in our derivation of the SPB equation is the introduction in equation (5) of the Instantaneous Relaxation Approximation (IRA), defined by
g(r, r'; t I ro)fd(r'; t l ro) = g~q(r,
r')feq(r'),
(6)
where geq(r, r') and feq(r') are the usual equilibrium pair correlation and singlet distribution functions. It is clear from Appendix A that g(r, r'; t]ro)fd(r'; tit0) dr' is the probability of finding, at time t, any one of the N - 1 untagged ions to within dr' of r' (irrespective of the configuration of the remaining N - - 2 untagged ions), given that the tagged ion is simultaneously at r and with the initial configuration of the untagged ions averaged over the conditional equilibrium probability density with the tagged ion fixed at r0. Consequently, equation (6) is exact in the two limits t = 0 and t---~ o0. It would be numerically accurate at all times if the tagged ion were to diffuse m u c h more slowly than the untagged ions. T h e time taken for the untagged ions to develop an equilibrium correlation with the tagged ion would then be short compared to the time required for significant displacements of the tagged ion.
Ion diffusion at charged interfaces
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When equation (6) is inserted into equation (5), we find that the potential of mean force in the IRA is simply the exact equilibrium potential of mean force, Weq(r), given by
Vweq(r) = Vul(r ) + J dr'[Vu2(r , r')]geq(r, r')f~q(r')
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= -fl-;V In feq(r),
(7)
where the last equality follows from the lowest member of the equilibrium B o g o l i u b o ~ B o r n Green Yvon hierarchy [3]. Consequently, the I R A selfpropagator evolves towards the correct equilibrium distribution. (This is expected, since equation (6) is exact in the limit t--* oc). T h e other statistical-mechanical approximation needed in our derivation of the SPB equation is a mean field approximation (MFA), according to which the equilibrium pair correlations are neglected by setting [3, 19]
geq(r,
r')
= 1
(8)
in equation (7). As a consequence, the potential of mean force becomes identical to the mean potential
7J3eq t MFA(r), MFA(r) = ul(r ) + f dr,u2(r ' r)feq
(9)
where fMFA(r) satisfies the integral equation (7) with geq(r, r') = 1. In the MFA, the tagged ion is thus considered as diffusing in the mean potential resulting from statistical averaging over'all the N ions. T h e consequences of the M F A for equilibrium properties have been studied in detail in references [3], [19] and [28].
3. THE SMOLUCHOWSKI--PoISSON--BoLTZMANNEQUATIONIN PLANARGEOMETRY T h e model system which we have chosen as testing ground for the SPB equation consists of two parallel uniformly charged plates located at z = __+b with an intervening solution containing point ions of charge Ze imbedded in a homogeneous dielectric of relative permittivity r (see figure 1). T h e charge of the point ions is opposite in sign to that of the plates, so that the system as a whole is electroneutral. In a system such as this, which contains only counterions, it is possible to set the ionic radius to zero. T h e effects of the two statisticalmechanical approximations (IRA and M F A ) in the SPB equation can thus be investigated without interference from finite ion size effects, i.e. all correlations have a purely electrostatic origin. An additional advantage of the absence of colons is the existence of an analytic solution to the SPB equation for this system
[2o].
T. ~,kesson et al.
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Each plate carries a fixed uniform surface charge density a. T h e permittivity is taken to be the same on either side of the plate, so that dielectric image forces do not occur. T h e singlet and pair potentials are then
u l ( z ) = constant; - b < z < b, (Ze) 2 u2(r, r') =
4neo G I r --
(10) (11)
r'l
The mean electrostatic potential
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~(z) = (Ze)- t f dr, u2(r ' r')/~q(r')
(12)
is approximated by the solution [21, 22] of the Poisson-Boltzmann equation subject to the appropriate boundary conditions ~(z) -- ~(0) =
2k n T
Ze
In cos (nz).
(13)
T h e reciprocal Debye length, ~, is obtained by solving the transcendental equation xb tan (xb) -
[Zeal b
(14)
2eo er kB T
on the interval 0 < ~cb < n/2. In the I R A equation (4) reads St fs(r; t l r 0 ) = D 0 V " {IV + f l V w , q ( r ) ] f s ( r ; t[ r0) }.
(15)
Now from the symmetry of our model system, it is clear that a one-point function such as Weq(t) can depend only on the z coordinate. T h e operator on the righthand side of equation (15) can therefore be split into one part which involves only the cylindrical coordinates p and tp and another part which involves only the z coordinate. As a consequence, the diffusion parallel to the plates (the lateral diffusion) is statistically independent from the diffusion perpendicular to the plates (the transverse diffusion). (This is not true when time dependent pair correlations are taken into account.) In the IRA, therefore, equation (15) may be decomposed into two separate diffusion equations; one describing the evolution of the lateral self-propagator &fs(p;t[po)-
P @
P~pfs(p;
tlpo) ,
(16)
where p is the perpendicular distance (parallel to the plates) from the z-axis, and the other describing the evolution of the transverse self-propagator ~t fs(z; t l Zo) = Do -~z
+ flWeq(Z) f~(z t lZo) ,
where the prime signifies differentiation with respect to z.
(17)
Ion diffusion at charged interfaces
1111
Since the system is laterally unbounded, the solution to equation (16) is simply the G r e e n ' s function for two-dimensional free diffusion
(p
f~(p; tlpo) = (47~Dot) -1 exp
-
p0)
4--~ot ] .
(18)
If, in equation (17), we also invoke the M F A , with Wcq(Z) replaced by Ze~b(z) from equation (13), we arrive at the SPB equation for our model system & f~(z; t l Z o ) = D o - ~ z
- 2Ktan (~z) f~(z;
tlz0)
.
(19)
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In Appendix C we show that the solution to equation (19), with impenetrable boundaries at z = • b and with a delta-function initial condition, is
f~(z;
cos (xz0) t I z0) = ~ c o t (tcb) s e c 2 (Kz) + ~c2b - c o s (Kz) [-6n, odd Un(KZO)Un(KZ) -~ (1 - - (~n, odd)'Un(KZ0)Vn(K"~")] n=l •
exp {-- [(n~/2) 2 -- (Kb)Z](Do t/b2)} [(mz/2) 2 - (~b) 2]
(20)
where 6,, oaa = 1 (0) if n is odd (even). F u r t h e r m o r e
(mrz~_ u.(~:z) = tan (Kz) cos \ 2b /
nrr sin (nrrz~
2Kb
\-2-b-/'
(21 a)
v.(~z) =- tan (Kz) sin \ 2b ] + 2~c---bcos k 2b ] ' Since ~b < n/2, the exponentials in equation (20) decay with time and in the limit t-~ ~ only the first term, corresponding to the equilibrium distribution feq(Z), remains. In the limit tcb = 0, equation (20) reduces to the well-known [23] propagator for one-dimensional free diffusion between reflecting boundaries.
4. STOCHASTIC DYNAMICS SIMULATION T h e starting point for the Stochastic D y n a m i c s simulations is the extended Langevin equation for the ionic m o m e n t a Pi(t) d dt pi(t) = - ~ P / ( t )
+ Ri(t) + Ki(rN; t).
(22)
T h e effects of the solvent are contained in the dissipation coefficient ~, in the random force Ri(t ) and in the dielectric screening of the ionic interactions. T h e time dependence of the ionic force KI(I'N; t) is only implicit through the coordinates r n and it is evaluated from the potential model in equations (10) and (11) N
Ki(rN; t) = -- V VI(rN) = - V i E ' u2(rl, r). j=l
(23)
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et al.
E q u a t i o n (22) can be f o r m a l l y i n t e g r a t e d to o b t a i n r(t) -- r o = ( m ( ) - l ( p 0 -- p(t)) + ( m ~ ) - l
~0t dt'[R(t') +
K ( r N, t')],
(24)
w h e r e the i n d e x i has b e e n d r o p p e d . T h i s is still an e q u a t i o n of m o t i o n in p h a s e space. H o w e v e r , for a t i m e t, w h i c h is long c o m p a r e d to t h e m o m e n t u m c o r r e l a tion t i m e 3 - 1 , we can n e g l e c t the first t e r m on the r i g h t - h a n d side of e q u a t i o n (24) a n d o b t a i n [24]
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r(t) -- ro
= (m~)-t fo dt'[R(t') + K(rN; t')].
E q u a t i o n (25) is a s t o c h a s t i c differential e q u a t i o n , w h i c h has a s o l u t i o n f o r m of a p r o b a b i l i t y d e n s i t y f u n c t i o n W(r s; t [ r~). I n fact it r e p r e s e n t s N tions c o u p l e d t h r o u g h the force K(rn;t). I f we r e s t r i c t o u r s e l v e s to s h o r t b u t still large c o m p a r e d to ~ - 1 , t h e n K(rN; t) is a p p r o x i m a t e l y c o n s t a n t a n d e q u a t i o n s d e c o u p l e to give r(t) -- r o -- K(rN;
O)t/m~ = (m~)-i fo dt' R(t').
(25) in the equatimes, the N
(26)
A p p l y i n g C h a n d r a s e k h a r ' s l e m m a [25] to e q u a t i o n (26) leads to the d e s i r e d p r o b ability density function
W(r; t]ro) = (4nDo t ) 3/2 exp [ - 1 r(t) - to -- Kt/m~12/4Do t]
(27)
w i t h D O = kT/m~ a n d K = K(rN; 0). ( T h e s a m e s y m b o l , e.g. F or PV, will be u s e d to d e n o t e different f u n c t i o n s , i d e n t i f i e d b y t h e i r a r g u m e n t s . ) W e will n o w s h o w that W is also a s o l u t i o n to the g e n e r a l i z e d S m o l u c h o w s k i e q u a t i o n , e q u a t i o n (2). L e t us, in a n a l o g y w i t h w h a t we d i d in o r d e r to arrive at e q u a t i o n (26), r e p l a c e the p o t e n t i a l t e r m in e q u a t i o n (2) w i t h a c o n s t a n t K as is l e g i t i m a t e for s h o r t times. E q u a t i o n (2) t h e n factorizes a n d we have 8t F ( r ; t l ro) = Do V 9 {(V - f l K ) F ( r ; t l ro)}
(28)
with
N F(t n; t] rr~) = I-I F(ri; t r0i).
(29)
i=1
It is s i m p l e to s h o w t h a t e q u a t i o n (27) is a s o l u t i o n to t h e S m o l u c h o w s k i e q u a tion. T h u s the two routes, e q u a t i o n s (2) a n d (25), are i d e n t i c a l (PV-= F ) a n d w h i c h w a y to p r o c e e d is p u r e l y b a s e d on n u m e r i c a l c o n s i d e r a t i o n s . It seems, h o w e v e r , that the stochastic a p p r o a c h is c o n c e p t u a l l y s i m p l e r a n d m o r e s u i t a b l e for n u m e r i c a l c o m p u t a t i o n s . T h e r i g h t - h a n d side of e q u a t i o n (26) can be i n t e r p r e t e d as a r a n d o m d i s p l a c e m e n t d u e to collisions w i t h the s o l v e n t m o l e c u l e s a n d we can r e w r i t e e q u a t i o n (26) as r(t + At) - r(t) = Ar(t, At) = ArG(At ) + K(rN;
t) At/m~.
(30)
Ion diffusion at charged interfaces
l 113
T h e r a n d o m d i s p l a c e m e n t Ar G has, in the l i m i t of infinite d i l u t i o n , a gaussian d i s t r i b u t i o n w i t h t h e first a n d s e c o n d m o m e n t s e q u a l to ( A r G ) ----0
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( I A r GI 2) = 6 D 0 At.
(31 a) (31b)
W e n o w a s s u m e t h a t Ar G has the s a m e statistical p r o p e r t i e s at finite c o n c e n t r a tions as in an infinitely d i l u t e s y s t e m , w h i c h is c o n s i s t e n t w i t h the n e g l e c t of the inertial t e r m in e q u a t i o n (24), see also [24]. T h i s m e a n s t h a t n e i t h e r the friction coefficient n o r the r a n d o m force d i s t r i b u t i o n are affected b y the e x t r a force K ( t ~ ; t). I t is t h e n s t r a i g h t f o r w a r d to solve e q u a t i o n (30) w i t h a t i m e step At small e n o u g h so t h a t K(rN; t + At) ~ K(rN; t), b u t still At >> ~ 1. T h e l a t t e r r e q u i r e m e n t has a f o r m a l c h a r a c t e r , since it is r e l a t e d to the m a s s of the B r o w n i a n particle, w h i c h is w i t h o u t significance in the diffusion limit. S i m u l a t i o n s w i t h t i m e steps of 0-1 a n d 0 - 2 p s w e r e w i t h i n the statistical fluctuations i d e n t i c a l a n d we u s e d the larger value in all s i m u l a t i o n s p r e s e n t e d here. P e r i o d i c b o u n d a r y c o n d i t i o n s w e r e a p p l i e d in t h e two d i r e c t i o n s p a r a l l e l to the c h a r g e d walls. T o g u a r a n t e e t h a t the flux across the walls is zero (perfect reflection), an ion was a l l o w e d to m o v e p e r p e n d i c u l a r l y o n l y if its n e w p o s i t i o n was c o n f i n e d b e t w e e n t h e walls. T h i s c o n d i t i o n leads to the c o r r e c t e q u i l i b r i u m d i s t r i b u t i o n , as can be verified f r o m the p r i n c i p l e of d e t a i l e d balance. T h e electric force a c t i n g on an ion was e v a l u a t e d u s i n g the m i n i m u m i m a g e c o n v e n t i o n , that is, it was a l l o w e d to i n t e r a c t w i t h all ions w i t h i n a p a r a l l e l e p i p e d c e n t r e d on itself. D u e to the long r a n g e c h a r a c t e r of the e l e c t r o s t a t i c force a n d the l i m i t e d n u m b e r of p a r t i c l e s e x p l i c i t l y t r e a t e d ( u s u a l l y 50 c o u n t e r i o n s ) , one has to i n c l u d e a c o r r e c t i o n f r o m the c h a r g e s o u t s i d e the s i m u l a t i o n box. T h i s can in p r i n c i p l e be d o n e s e l f - c o n s i s t e n t l y , t h a t is the o r i g i n a l l y u n k n o w n d i s t r i b u t i o n is a p p r o x i m a t e d b y the d i s t r i b u t i o n o b t a i n e d f r o m an initial s i m u l a t i o n . A s e c o n d s i m u l a t i o n will t h e n give an i m p r o v e d d i s t r i b u t i o n , w h i c h in t u r n can b e u s e d to a p p r o x i m a t e the exact one a n d so on. T h i s a p p r o a c h has s o m e d r a w b a c k s : it is for e x a m p l e n o t o b v i o u s t h a t it will c o n v e r g e a n d it is r a t h e r t i m e - c o n s u m i n g . I n t h e p r e s e n t s i m u l a t i o n s we have a p p r o x i m a t e d the d i s t r i b u t i o n o u t s i d e the b o x w i t h the c o r r e s p o n d i n g P o i s s o n - B o l t z m a n n d i s t r i b u t i o n . T h i s a p p r o a c h has b e e n e x t e n s i v e l y t e s t e d in [19], w h e r e also a m o r e d e t a i l e d a c c o u n t is given.
5. CALCULATIONS I n this section we p r e s e n t a q u a n t i t a t i v e a s s e s s m e n t of t h e a c c u r a c y of t h e S P B a p p r o x i m a t i o n in d e s c r i b i n g c o u n t e r i o n diffusion in the m o d e l s y s t e m of figure 1. T h e effects of the two s t a t i s t i c a l - m e c h a n i c a l a p p r o x i m a t i o n s , i n s t a n t a n e o u s r e s p o n s e ( I R A ) a n d n e g l e c t of e q u i l i b r i u m spatial c o r r e l a t i o n ( M F A ) , i n h e r e n t in the S P B p r o p a g a t o r , will be e x a m i n e d s e p a r a t e l y . T h i s can be d o n e b y c o m p a r i n g (1) t h e exact ( w i t h i n the m o d e l ) S D p r o p a g a t o r , o b t a i n e d f r o m stochastic d y n a m i c s s i m u l a t i o n s as d e s c r i b e d in w (2) the I R A p r o p a g a t o r , o b t a i n e d b y n u m e r i c a l l y solving e q u a t i o n (17) w i t h the exact ( s i m u l a t e d ) m e a n force, as d e s c r i b e d in A p p e n d i x D, a n d (3) the a n a l y t i c S P B p r o p a g a t o r , as given in w w h i c h i n v o l v e s b o t h t h e I R A a n d the M F A . T h e s i m u l a t i o n s , as well as o t h e r calculations, were d o n e w i t h the f o l l o w i n g p a r a m e t e r v a l u e s : T = 293 K, 8r = 80"36, b = 10"5.~ ( e x c e p t figure 9), D O = 2 x 1 0 - 9 m 2 s -1 a n d [a[ = 0 ' 2 2 4 C m - 2
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( c o r r e s p o n d i n g to one e l e m e n t a r y c h a r g e p e r 71"4,~2). T h e c o u n t e r i o n v a l e n c y was Z = 1 or 2 as i n d i c a t e d , c o r r e s p o n d i n g to ~b = 1'3655 a n d ~b = 1"4599, respectively. T h e s e l f - p r o p a g a t o r fs(P, tp, z; tJpo, tPo, Zo) o b t a i n e d f r o m the s i m u l a t i o n m a y be r e g a r d e d as t h e s o l u t i o n to e q u a t i o n (4). B e c a u s e o f t h e t r a n s l a t i o n a l s y m m e t r y of t h e m o d e l s y s t e m , this p r o p a g a t o r can d e p e n d o n l y on t h e lateral d i s p l a c e m e n t Ap = [(x -- Xo) 2 + (y --y0)2] 1/2, a n d on t h e t r a n s v e r s e c o o r d i n a t e s z a n d z 0 . I n g e n e r a l the t r a n s v e r s e a n d lateral diffusion p r o c e s s e s are c o u p l e d b u t one m a y define a p u r e l y t r a n s v e r s e p r o p a g a t o r b y a v e r a g i n g o v e r the lateral d i s p l a c e m e n t . F o r o u r m o d e l s y s t e m , then,
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f~(z ; t J z0) =
d(Ap) f~(Ap, z ; ~0~176
t J Zo).
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All p r o p a g a t o r s will be p r e s e n t e d in d i s c r e t i z e d f o r m . T h u s fs(z; t Jzo) is r e p l a c e d b y P(k; tJ ko), w h i c h gives the p r o b a b i l i t y of f i n d i n g the t a g g e d ion in t h e k t h z - i n t e r v a l at t i m e t, given that it was l o c a t e d in k0 initially. W i t h an i n t e r l a m e l l a r s p a c i n g of 21 ,~ a n d an i n t e r v a l w i d t h of 1 ,~, k r u n s f r o m 1 to 21 (k = 1 a n d 21 b e i n g a d j a c e n t to the c h a r g e d walls). T h e t r a n s v e r s e I R A p r o p a g a t o r was o b t a i n e d b y first solving e q u a t i o n (17) w i t h a g r i d s p a c i n g of 0"1 A a n d t h e n c o n v e r t i n g the r e s u l t i n g p r o p a g a t o r to the c o a r s e r 1 A g r i d u s e d in the figures. T h e d i s c r e t i z a t i o n e r r o r was f o u n d to be n e g l i g i b l e . T h e d i s c r e t i z a t i o n of the t r a n s v e r s e S P B p r o p a g a t o r was p e r f o r m e d a n a l y t i c a l l y as d e s c r i b e d in A p p e n dix C. T h e lateral diffusion is c h a r a c t e r i z e d b y t h e d i s c r e t i z e d p r o p a g a t o r P(l, m; t Jmo), w h i c h gives t h e p r o b a b i l i t y t h a t t h e t a g g e d ion at t i m e t has suff e r e d a lateral d i s p l a c e m e n t o f 1 radial units, while its initial a n d final t r a n s v e r s e c o o r d i n a t e s were in the i n t e r v a l s m o a n d m, r e s p e c t i v e l y . T h e radial i n t e r v a l w i d t h is t a k e n to be 1 .~ (l = 1,2 . . . . ), w h i l e the z - i n t e r v a l is 3"5 A so t h a t m r u n s f r o m 1 to 6 (m = 1) a n d 6 b e i n g a d j a c e n t to the c h a r g e d walls).
5.1. Lateral propagators As n o t e d in w3, the I R A a n d S P B lateral p r o p a g a t o r s are b o t h i n d e p e n d e n t of the t r a n s v e r s e c o o r d i n a t e a n d given b y t h e f r e e - d i f f u s i o n G r e e n ' s f u n c t i o n , e q u a tion (18). T h i s is a c o n s e q u e n c e of the s y m m e t r y of t h e m o d e l s y s t e m , w h i c h e n s u r e s t h a t the (exact) m e a n force has no lateral c o m p o n e n t . H o w e v e r , t h e e v o l u t i o n of the s i m u l a t e d lateral p r o p a g a t o r is e x p e c t e d to d e v i a t e f r o m e q u a t i o n (18) d u e to the d y n a m i c c o r r e l a t i o n s a s s o c i a t e d w i t h the t i m e - d e p e n d e n t m e a n force in e q u a t i o n (4). F i g u r e 2 s h o w s t h e s i m u l a t e d p r o p a g a t o r P(l, m; tim) for m o n o v a l e n t c o u n t e r i o n s as a f u n c t i o n o f t h e s q u a r e of t h e lateral d i s p l a c e m e n t . F i r s t l y , we n o t e the a l m o s t p e r f e c t G a u s s i a n b e h a v i o u r ; the p r o p a g a t o r s are well d e s c r i b e d b y e q u a t i o n (18) b u t w i t h an effective diffusion coefficient w h i c h is l o w e r t h a n D O b y u p to 15 p e r cent. T h i s r e t a r d a t i o n is a n a l o g o u s to the w e l l - k n o w n r e l a x a t i o n effect in u n i f o r m s y s t e m s . T h e s e c o n d p o i n t to n o t e a b o u t figure 2 is t h a t t h e effective diffusion coefficient is t i m e - d e p e n d e n t , i.e. a finite t i m e , o f the o r d e r 1 0 0 p s , is r e q u i r e d for t h e d y n a m i c c o r r e l a t i o n s to be fully m a n i f e s t e d . T h i s t i m e
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(c) Figure 2. T h e logarithm of the simulated lateral p r o p a g a t o r P(l, m; tim) for m o n o v a l e n t c o u n t e r i o n s as a f u n c t i o n of ( l - 89 T h e s t r a i g h t lines d r a w n c o r r e s p o n d to the l o g a r i t h m of t h e G r e e n ' s f u n c t i o n , e q u a t i o n (18), w i t h different values of the diffusion coefficient, D (given b e l o w in u n i t s of 1 0 - 9 m 2 s-1). T o separate the g r a p h s a c o n s t a n t c has b e e n added. (a) t = 3 2 p s : Q , m = 1, c = 8 9 D = 1.84; + , m = 2 , c=O, D = l ' 9 1 ; x , m = 3 , c = - - 8 9 D = 1"94. (b) t = 6 4 p s : Q, m=l, c=l, D = 1.74; + , average for m = 2 a n d m = 3, c = 0, D = 1-87. (c) t = 1 2 8 p s , D = 1"70 a n d 1.77 respectively. O t h e r w i s e as in (b).
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m a y be t h o u g h t of as a r e l a x a t i o n t i m e for t h e ionic a t m o s p h e r e ; it s h o u l d be closely r e l a t e d to the t i m e taken for an ion to diffuse a d i s t a n c e e q u a l to t h e r a d i u s of its ' c o r r e l a t i o n h o l e ' . S i n c e t h e c o u n t e r i o n s a c c u m u l a t e n e a r the walls, this c o r r e l a t i o n l e n g t h s h o u l d b e a b o u t 6 - 8 , ~ (the c h a r g e d e n s i t y on each wall c o r r e s p o n d s to one e l e m e n t a r y c h a r g e p e r (8.4,&)z). W i t h a diffusion coefficient of 2 x 1 0 - 9 m Z s - 1 , this y i e l d s a r e l a x a t i o n t i m e of a b o u t 100-150 ps, in r e a s o n a b l e agreement with the simulations. I n a n a l o g y w i t h u n i f o r m e l e c t r o l y t e solutions, w h e r e the r e l a x a t i o n effect increases w i t h c o n c e n t r a t i o n , one e x p e c t s the lateral diffusion to be m a x i m a l l y r e t a r d e d n e a r the c h a r g e d walls w h e r e the local c o u n t e r i o n c o n c e n t r a t i o n is highest. T h i s e x p e c t a t i o n is b o r n e o u t b y the s i m u l a t i o n d a t a in figure 2. F r o m the slopes we t h u s e s t i m a t e a diffusivity r e d u c t i o n of ca 15 p e r cent w i t h i n 3 - 5 A of t h e c h a r g e d walls ( w h e r e a c o u n t e r i o n resides a b o u t 75 p e r c e n t of the time), c o m p a r e d to ca 8 p e r cent in the r e m a i n d e r of the s y s t e m . I n view of the large c o n c e n t r a t i o n difference b e t w e e n t h e s e regions, t h e o b s e r v e d z - d e p e n d e n c e in t h e lateral diffusion m a y s e e m s u r p r i s i n g l y weak. H o w e v e r , since the i n t e r - l a m e l l a r s e p a r a t i o n is of the s a m e o r d e r of m a g n i t u d e as the c o r r e l a t i o n length, also c o u n t e r i o n s m i d w a y b e t w e e n the walls are d y n a m i c a l l y c o r r e l a t e d w i t h t h o s e r e s i d i n g n e a r the walls.
5.2. Transverse propagators I n the case of the t r a n s v e r s e c o u n t e r i o n diffusion, all t h r e e c a l c u l a t e d p r o p a g a tors (the exact S D p r o p a g a t o r a n d the a p p r o x i m a t e I R A a n d S P B p r o p a g a t o r s ) will, in general, be different. A n y d e v i a t i o n b e t w e e n the S D a n d I R A p r o p a g a t o r s can be t r a c e d to d y n a m i c c o r r e l a t i o n s , as in t h e case of lateral diffusion, w h e r e a s a difference b e t w e e n t h e I R A a n d S P B p r o p a g a t o r s reflects static c o r r e l a t i o n s via the e q u i l i b r i u m m e a n force in e q u a t i o n (17). T h e d i s c r e t i z e d t r a n s v e r s e p r o p a g a tor, P ( k ; t]k0), p r e s e n t e d in this s u b s e c t i o n refers to reflecting b o u n d a r y c o n d i tions at the c h a r g e d walls. T h i s p r o p a g a t o r evolves t o w a r d s t h e e q u i l i b r i u m d i s t r i b u t i o n . I n t h e l i m i t t---~ o0, the S D a n d I R A p r o p a g a t o r s s h o u l d t h u s c o i n cide, w h e r e a s the S P B p r o p a g a t o r s h o u l d e x h i b i t the k n o w n [19] deficiencies of the PB a p p r o x i m a t i o n for this s y s t e m , viz. a too h i g h c o u n t e r i o n c o n c e n t r a t i o n in the m i d p l a n e (z = 0) region. F i g u r e s 3-5 illustrate the t r a n s v e r s e s e l f - d i f f u s i o n of m o n o v a l e n t c o u n t e r i o n s ; in figures 3 a n d 4 we have p l o t t e d d i s c r e t i z e d p r o p a g a t o r s P ( k ; t ] k0) as f u n c t i o n s of k at fixed time, w h e r e a s in figure 5 we p l o t P ( k ; t ] k0) as a f u n c t i o n of t i m e for given k a n d k 0 . It is seen that the S D , I R A a n d S P B p r o p a g a t o r s agree r e m a r k a b l y well for m o n o v a l e n t ions. W h e r e d e v i a t i o n s do o c c u r , t h e y are d u e m a i n l y to static c o r r e l a t i o n s . T h i s is m o s t clearly seen in the z - i n t e r v a l a d j a c e n t to t h e c h a r g e d wall (k = 1) a n d at long times. S i n c e the I R A is exact in the two l i m i t s t = 0 a n d t--~ oo (cf. w we can e x p e c t to see effects of d y n a m i c c o r r e l a t i o n s o n l y at i n t e r m e d i a t e t i m e s . E x c e p t for the small differences b e t w e e n the S D a n d I R A p r o p a g a t o r s in figures 4 (b) a n d 5 (b), t h e r e is no i n d i c a t i o n in figures 3-5 of significant d y n a m i c c o r r e l a t i o n s . I n figure 5 ( a ) the S D a n d I R A P ( 1 ; t ] 1) p r o p a g a t o r s are seen to c o i n c i d e at all times. T h i s m a y be u n d e r s t o o d on the basis that, at long a n d i n t e r m e d i a t e t i m e s , d y n a m i c c o r r e l a t i o n s r e t a r d to r o u g h l y e q u a l e x t e n t the c o u n t e r i o n fluxes a w a y f r o m a n d t o w a r d s the c h a r g e d wall. O n t h e o t h e r h a n d , at s h o r t t i m e s , the ion has
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F i g u r e 3. T h e t r a n s v e r s e p r o p a g a t o r P(k; t i t ) for m o n o v a l e n t c o u n t e r i o n s as a f u n c t i o n o f k at t h r e e d i f f e r e n t t i m e s : (a) t = 2 ps, (b) t = 32 ps, a n d (c) t = 1 0 2 4 p s . S y m b o l s : SD (x), IRA (- --), SPB ( ). T h e s t a n d a r d d e v i a t i o n s are a b o u t 1 p e r c e n t ( S D ) a n d 0"2 p e r c e n t ( I R A ) .
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(c) F i g u r e 5. T h e t r a n s v e r s e p r o p a g a t o r P ( I ; t [ k0) for m o n o v a l e n t ions as a f u n c t i o n o f t i m e for t h r e e d i f f e r e n t initial p o s i t i o n s k o . S y m b o l s as in figure 3. T h e b a r s s h o w o n e s t a n d a r d d e v i a t i o n for S D .
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m o v e d a s h o rt d i s t a n c e c o m p a r e d to its c o r r e l a t i o n length, w h i c h results in m i n o r effects of d y n a m i c c o r r e la ti o n s . H o w e v e r , e v e n the p r o p a g a t o r P ( 2 1 ; t ] l ) reveals no significant relaxation effect. ( T h i s p o i n t is f u r t h e r d i s c u s s e d in w5.3.) F i g u r e s 6 and 7 illustrate the t r a n s v e r s e self-diffusion of d i v a l e n t c o u n t e r i o n s . As e x p e c t e d , static c o r r e l a t i o n s are m o r e i m p o r t a n t t h a n for m o n o v a l e n t ions.
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Figure 6.
k . . . .
T
11 (c)
. . . . . . . . . .
2'1
T h e transverse propagator P(k; t ] 1) for divalent counterions as a function of k at (a) t = 10ps, (b) t = 160ps, and (c) t = 5000ps. Symbols as in figure 3.
Ion diffusion at charged interfaces
1121
H o w e v e r , d y n a m i c c o r r e l a t i o n s are still n e g l i g i b l e ; the S D a n d I R A p r o p a g a t o r s are v i r t u a l l y identical. A n a l t e r n a t i v e to the plots in figures 5 (a) a n d 7 is s h o w n in figure 8, w h e r e we have p l o t t e d the q u a n t i t y [1 - P ( 1 ; t l l ) ] / [ 1 - Peq(1)] as f u n c t i o n of time. T h i s
P(1;tll)
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.5-
2log t/5ps The time dependence of the transverse propagator P ( 1 ; t I 1) for divalent counterions. Symbols as in figure 3.
Figure 7.
I
1-P(1;t11) I-P~I)
2log t/ps (a)
P(1;tll)
x
.x. . . . . . . . .x. . .
.--~
2log t/5ps 5
10
(b) Figure 8.
Transverse counterion diffusion as illustrated by the time dependence of P(1 ; t] I)]/[1 -- Peq(1)] for (a) monovalent and (b) divalent ions. Symbols as in figure 3. [1
--
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et al.
quantity is zero initially and evolves towards unity. At very short times, where t[1) is virtually unaffected by electrostatic interactions, the SD and I R A curves exceed the SPB curve in figure 8. This is simply due to the larger value of 1 - P,q(1) in the MFA. However, at longer times there is a cross-over and the SD and I R A curves evolve more slowly than the SPB curve, due to the deeper electrostatic potential well in the presence of static correlations.
P(1,
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5.3.
Absorption statistics
In some applications one is primarily interested in the statistics of counterion diffusion from one spatial region to another rather than in the, hitherto discussed, decay of an initial delta-function towards the equilibrium counterion distribution. T h e former process may be studied theoretically by introducing an absorbing boundary condition at some point where the tagged counterion loses its label. For the transverse counterion diffusion in our model system, for example, we can define the survival probability, Q(t[Zo, zA), that a counterion, initially located at z0, has not yet reached (and become absorbed at) z a at time t. Choosing the left wall (at z = - b ) as reflecting and the absorbing boundary at - b < z a < b, we can express Q(t[Zo, za) as an integral over the transverse self-propagator, fs(z; t l z0), subject to the corresponding boundary conditions
Q(tlZo,ZA)=ffidzfs(z;tlZo).
(33)
The mean first passage time ( M F P T ) , z(za[z0) , is the average time taken for a counterion, initially located at z0, to reach z a for the first time. This quantity is obtained by a further integration as
Z(ZA IZ0) =
dt Q(tlZo, za).
(34)
T h e discretized versions of the absorption probability and the M F P T can be obtained directly from a simulation (SD). In the IRA, when the transverse propagator obeys equation (17), the propagator and the absorption probability can be calculated numerically as described in Appendix D. T h e M F P T , however, can be calculated without the need to solve equation (17). By combining equations (17), (33), (34) and the boundary conditions and then performing the integrations in equations (33) and (34) in a formal way, one obtains the simple formula [26, 27] z(zA [z~ = Doo 1 ; zA dz exp [~Wcq(Z)] f f 0
dz'
exp [--flWcq(Z')].
(35)
b
Using this result and the simulated (exact) potential of mean force, W~q(Z), we calculated the I R A M F P T by summation over 0"1A z-intervals. T h e SPB absorption probability and M F P T , finally, were obtained from the general analytical results of [20]. Figure 9 shows the transverse MFPT z(zl -b) as a function of - b < z < 0 for monovalent and divalent counterions. T h e initial position is within 0"5 A of the left wall (z = --b). (In contrast to the other data presented in this paper, figure 9 refers to a system with b = 13 ,~.) T h e close agreement between the SD and I R A
1123
Ion diffusion at charged interfaces Tins
,/
.Y
i T / . x///
....
/A
6
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(a)
Tins
/ 10
,,
/" // 7 _
e
/"
_ ~ "'/" ~ "
zlA 6
(b) Figure 9. The M F P T x(Z[Zo) with --b < z o < --b + 0.5,~, for (a) monovalent and (b) divalent counterions. Symbols as in figure 3. System parameters as elsewhere, except b = 13 A.
results implies that d y n a m i c correlations are u n i m p o r t a n t for this M F P T . Static c o r r e l a t i o n s are of m i n o r i m p o r t a n c e for m o n o v a l e n t ions, for which the SPB result deviates by less than 20 percent. F o r divalent counterions, however, the reduced well d e p t h in t h e PB potential causes the SPB a p p r o x i m a t i o n to z(01 - b ) to fall short by a factor of 4. F i g u r e 10 shows the t e m p o r a l decay of the survival p r o b a b i l i t y Q(tl Zo, ZA), and of its logarithm, for m o n o v a l e n t counterions. T h e initial position is within 0"5 ,~ of the left wall and the a b s o r b i n g b o u n d a r y is at the m i d p l a n e (z n = 0). One sees again that t h e r e is virtually no effect of d y n a m i c correlations on the counterion diffusion out of the potential well. T h e decay of the survival p r o b a b i l i t y is seen to be exponential, except d u r i n g an initial phase of ca. 200 ps. T h i s fact m a y be exploited in simulations of the M F P T . R a t h e r than extending the simulation to times that are long c o m p a r e d to the M F P T (which m a y mean up to 1 0 - 7 s for divalent ions), one can terminate the simulation as soon as the exponential decay in figure 10 is established and then evaluate the r e m a i n d e r of the integral in equation (34) analytically. T h i s p r o c e d u r e m a y reduce the c o m p u t e r r e q u i r e m e n t s by more than an order of magnitude.
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1-
O(t ]-b,0)
...........T g
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(a)
InQ(t I-b,O) 0-~.~.,~...
-5-
/ns (b)
Figure 10. Time dependence of (a) the survival probability Q(t]Zo, ZA) and (b) its logar!thm for monovalent counterions, z A = 0 and -- b < z 0 < -- b + 0"5 A with b = 10.5 A. Symbols: S D ( . . . . . . ), IRA (. . . . ), SPB ( ).
T h e i n s i g n i f i c a n t effect o f d y n a m i c c o r r e l a t i o n s on t h e t r a n s v e r s e c o u n t e r i o n diffusion, as e v i d e n c e d b y figures 9 a n d 10, m a y be c o n t r a s t e d w i t h t h e 15 p e r c e n t r e d u c t i o n of the effective l a t e r a l diffusion coefficient (figure 2). T h i s difference can b e u n d e r s t o o d as follows. T h e M F P T ~(0] - b ) m e a s u r e s t h e t i m e r e q u i r e d for a c o u n t e r i o n to escape f r o m a well d e p t h of several k T in the e q u i l i b r i u m p o t e n t i a l of m e a n force. T h i s p r o c e s s m a y be d e c o m p o s e d into t w o stages: a t r a n s i e n t initial phase, w h e n the p o s i t i o n of the t a g g e d ion b e c o m e s e s s e n t i a l l y equilibrated, followed by a quasi-steady-state phase, during which the tagged p a r t i c l e d e n s i t y s l o w l y leaks o u t o f t h e s y s t e m at t h e a b s o r b i n g b o u n d a r y w i t h o u t significantly d i s t u r b i n g the e q u i l i b r i u m d i s t r i b u t i o n . T h i s q u a s i - s t e a d y - s t a t e p h a s e , w h i c h c o r r e s p o n d s to the e x p o n e n t i a l r e g i o n in figure 10, m a k e s t h e o v e r w h e l m i n g c o n t r i b u t i o n to r(0] --b). As a c o n s e q u e n c e ~(0] --b) b e c o m e s essentially i n d e p e n d e n t o f d y n a m i c c o r r e l a t i o n s , w h i c h o n l y affect the t r a n s i e n t initial phase. T h e M F P T ~ ( - - b ] 0 ) for diffusion into the p o t e n t i a l well, on the o t h e r h a n d , s h o u l d s h o w s o m e effects of d y n a m i c c o r r e l a t i o n s . W h i l e we have n o t c a l c u l a t e d
Ion diffusion at charged interfaces
1125
M F P T s , z(z[b), from simulations compared to M F P T s obtained from equation (34) with O(t { - b, - z0) from IRA. The quotient ZSD/qRA reflects the effect of dynamic correlations. z0(A)
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8 6 4 2 0
"csb(ps)
~IRA(PS)
37.6 169 438 837 1299
32.3 158 426 829 1334
~'SD/~'IRA 1.16 1.07 1.03 1"01 0.97
this M F P T , a r e l a t e d t r e n d can be seen in figures 4(b) a n d 5 (b) w h e r e the S D p r o p a g a t o r evolves m o r e s l o w l y t h a n the I R A p r o p a g a t o r . R e l a x a t i o n effects s h o u l d also b e a p p a r e n t in z(z[ --b) for z in the n e i g h b o u r h o o d of the c h a r g e d wall; the c o n t r i b u t i o n f r o m the t r a n s i e n t initial p h a s e is t h e n n o t negligible. S u c h a b e h a v i o u r is i l l u s t r a t e d b y the ratio of t h e S D a n d I R A M F P T ' s in the table.
6.
CONCLUSIONS
I n this w o r k we have i n v e s t i g a t e d the s t a t i s t i c a l - m e c h a n i c a l basis a n d the n u m e r i c a l a c c u r a c y o f t h e S m o l u c h o w s k i - P o i s s o n - B o l t z m a n n a p p r o x i m a t i o n for d e s c r i b i n g ion diffusion in n o n u n i f o r m e l e c t r o l y t e s . O u r m a i n c o n c l u s i o n s are as follows. T h e m a n y - p a r t i c l e g e n e r a l i z e d S m o l u c h o w s k i e q u a t i o n m a y be f o r m a l l y r e d u c e d to a diffusion e q u a t i o n for the o n e - p a r t i c l e s e l f - p r o p a g a t o r . T h i s e q u a tion c o n t a i n s a t i m e - d e p e n d e n t g e n e r a l i z a t i o n of the e q u i l i b r i u m p o t e n t i a l of m e a n force. T w o a p p r o x i m a t i o n s , e x p r e s s i b l e in t e r m s of w e l l - d e f i n e d o n e - a n d t w o - p a r t i c l e d i s t r i b u t i o n f u n c t i o n s , are n e e d e d to d e r i v e the S P B e q u a t i o n . T h e s e are an I n s t a n t a n e o u s R e l a x a t i o n A p p r o x i m a t i o n ( I R A ) , c o r r e s p o n d i n g to t h e n e g l e c t of d y n a m i c c o r r e l a t i o n s (or the r e l a x a t i o n effect), a n d a M e a n F i e l d A p p r o x i m a t i o n ( M F A ) , c o r r e s p o n d i n g to t h e n e g l e c t of static c o r r e l a t i o n s . I n t h e i n v e s t i g a t e d m o d e l s y s t e m , the effective diffusion coefficient for lateral d i s p l a c e m e n t s of m o n o v a l e n t c o u n t e r i o n s is r e d u c e d b y u p to 15 p e r cent d u e to a r e l a x a t i o n effect ( d y n a m i c c o r r e l a t i o n s ) . T h i s r e d u c t i o n d e p e n d s r a t h e r w e a k l y on t h e d i s t a n c e f r o m the c h a r g e d walls. T h e t r a n s v e r s e diffusion of m o n o - a n d d i v a l e n t c o u n t e r i o n s o u t o f t h e well in the e q u i l i b r i u m p o t e n t i a l of m e a n force is v i r t u a l l y u n a f f e c t e d b y d y n a m i c c o r r e l ations b u t is sensitive to the well d e p t h . F o r this m o d e of diffusion the I R A is t h u s an e x c e l l e n t a p p r o x i m a t i o n ; as long as an a c c u r a t e e q u i l i b r i u m p o t e n t i a l of m e a n force is used, the d y n a m i c a l b e h a v i o u r is f a i t h f u l l y d e s c r i b e d in the I R A . T h e M F A (i.e. the P o i s s o n - B o l t z m a n n a p p r o x i m a t i o n ) s h o u l d be useful for m o n o v a l e n t ions, b u t can lead to s u b s t a n t i a l e r r o r in t h e d y n a m i c a l d e s c r i p t i o n of p o l y v a l e n t ions. T h i s w o r k was s u p p o r t e d b y g r a n t s f r o m the S w e d i s h N a t u r a l S c i e n c e R e a search C o u n c i l . O n e of us (Bo JSnsson) also w a n t s to a c k n o w l e d g e a V i s i t i n g F e l l o w s h i p at the A u s t r a l i a n N a t i o n a l U n i v e r s i t y , C a n b e r r a .
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APPENDIX A
Time dependent distribution functions We consider a collection of N identical classical solute particles without internal degrees of freedom. T h e system is in thermal equilibrium at a temperature T. T h e solvent-averaged hamiltonian of the system is
H(r N, ON) =
P? + v(tN), i=1
(A 1)
2m
where cN denotes the 3 N particle coordinates it, t2, . . . , tN and ON denotes the 3 N particle m o m e n t a lal, 02, - - - , las. T h e potential energy of the system in the configuration cN is N
N
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V(CN) = E u,(r i=1
N
E ' u~(r,, r
+ 89 E i=1
(A 2)
j=l
where the prime signifies exclusion of t h e j = i term. T h e singlet potentials ul(r which include the effects of confining boundaries, produce a spatially nonuniform average particle density at equilibrium. Let F(tN; t lr be the specific N-particle probability density in the canonical ensemble. T h e n F(rN; t l t0~) dr N is the probability of finding, at time t, the N distinct particles (labelled 1, 2, . . . , N) to within dr of the configuration iN, given that they were in the configuration t N at t = 0. It follows that
fdtNF(tN; tl toN) = 1.
(A 3)
Furthermore, N
F(r N; 01 rg) = H 6(ri - rol),
(A 4)
i=l
and F(rN; ~ froN) = Feq(r N) =
exp
[--flV(rM)]
,
(A 5)
f dr N exp [ - f l V ( r n ) ] where fl -- (k. 7 ) - 1. T h e reduced specific n-particle probability densities are defined through
F(t n ; t I rN) -- f deN- nF( CN; t I iN) ; n ],.
