diffusion of ionic penetrants in charged disordered mediagold.cchem.berkeley.edu/pubs/dc129.pdf ·...

Diffusion of ionic penetrants in charged disordered media Arup K. Chakraborty and D. Bratko” Department of Chemical Engineering, Universi@ of California, Berkeley, California 94720

David Chandler Department of Chemistry, University of California, Berkeley, California 94720

(Received 6 August 1993; accepted 3 September 1993)

We consider the diffusion of ionic species in technologically relevant materials such as zeolites. These materials are characterized by a disordered density distribution of charged sites that couple with the diffusing species. We present a model for ion diffusion in a specific form of charged disorder. This is a primitive model for ion diffusion in charged or acidic zeolites. The theory relies on a path integral representation of the propagator, and a Gaussian field theory for the effects of the disorder. We use the Feymnan-Bogoliubov variational method to treat the model, and calculate the diffusion coefficient for ions in a medium characterized by randomly located charges. Numerical solution of our equations, and asymptotic analyses of the same, show that in our theory there is a crossover from diffusive to subdiffusive behavior beyond a threshold value for the average density of the disorder. This threshold coincides with the actual diffusion changing from processes well approximated by Gaussian paths to those involving escapes from deep potential wells and barrier crossings. These results are discussed in the context of recent field-theoretic and renormalization group approaches to the problem of diffusion in random media. Our approach to diffusion in random media appears reasonably general and should be applicable to many technologically relevant problems, and is not compute intensive.

I. INTRODUCTION

In this paper we develop a theory for ionic penetrants diffusing in a medium with randomly distributed charges. The approach we adopt should be applicable to a variety of systems involving diffusers influenced by random media. Our focus is rather more specific, however, as it pertains to the physically relevant problem of ionic motion in charge doped zeolites. Zeolites are used in a host of industrial applications involving separation of mixtures and catalytic conversion. These materials are crystalline aluminosilicates with a well-defined and ordered array of lattice points. L* A very simple example of a zeolite is silicalite, which may be considered to be a purely siliceous framework with Si-O-Si bonds that form a pore structure of molecular dimensions (see Fig. 1). Several other technologically useful zeolites can be derived from silicalite by substituting aluminum for silicon at various lattice points. Since the valencies of silicon and aluminum ditfer by one, this leads to a charge imbalance that necessitates the incorporation of a cation into the framework in order to restore electrical neutrality. Recent electronic structure calculations3 have demon- strated that the electron density distribution in the region surrounding these cations is quite localized, and more im- portantly, that the substitution of aluminum (and the corresponding cation) occurs randomly throughout the material. An issue that is of technological importance is the rate at which products (ions or molecules) diffuse out of the zeolite. To date, the only computational studies of diffusion in zeolites have examined the motion of small neutral molecules (e.g., Xe, CH,, short alkanes) through uncharged

a)Also alfiliated with J. Stefan Institute, University of Ljubljana, Ljubljana, Slovenia.

silicalite (e.g., Refs. 4 and 5). An ion making its way through a charge doped zeolite encounters an ordered array of topological barriers, and a disordered density distribution of charged sites with which it may couple strongly. For this type of system, we predict a disorder induced crossover from weakly scattered diffusive behavior to localized states that may hop from one trapped state to another over longer time scales.

The prediction of localized states above a threshold value of disorder density for this specific physical problem appears to be novel. However, the observation is similar to that of Zhang6 who considered the existence of localized states for strong disorders. In recent years, there has been considerable interest in studying diffusion in random media. The formulations due to Martin, Siggia, and Rose and Dominicus and Peliti7 enabled the analysis of diffusion in random media within the framework of field theory. It is generally believed that for the case of weak disorder, and disorders with short-ranged correlations the upper critical dimensional&y for diffusion is d=2.%” For disorders with long-ranged correlations, however, the possibility of anomalous diffusion in any dimension has been considered by Bouchaud and Georges.* Our prediction is different from but not inconsistent with these views. Along with being of technological relevance, the prediction may therefore be of some theoretical interest.

The specific class of models we examine is specified in Sec. II. We imagine that short time and length scale structure are accounted for through a “bare” ditfusivity, D. The charged disorder affecting the low concentration of penetrants is time independent. That is, the disorder is quenched or adiabatic on the time scales relevant to the penetrant’s motion. The nature of the disorder is characterized by charge density pair correlations with finite cor-

1528 J. Chem. Phys. 100 (2), 15 January 1994 0021-9808/94/100(2)/1528/14/$6.00 @ 1994 American Institute of Physics

Chakraborty, Bratko, and Chandler: Ionic penetrants in disordered media 1529

FIG. 1. A typical zeolite structure. For this example (silicalite) the pores that are shown in the figure are roughly 4.5 d; in diameter.

relation length and bulk charge neutrality enforced. The resulting potential energy fluctuations affecting the pene- trams exhibit long-ranged correlations as a result of Cou- lomb’s law.

The disorder can be integrated out to give a Green’s function for the penetrant diffusion with nonlocal couplings. We analyze this function within the framework of path integrals. We carry out its evaluation approximately by employing a variational calculation based upon a Gaussian reference action functional. The procedure is described in Sec. III, and numerical results based upon the approximate treatment are presented in Sec. IV. We also perform an analytical asymptotic analysis of the treatment. In Sec. V we discuss our primary results from a physical point of view, and in the context of the existing literature on diffusion in random media. In Sec. VI, we offer some concluding remarks, and make connection with experiments. We suggest specific experiments that may be able to test whether the change in mechanism of ion motion that we predict for our primitive model occurs in real charged zeolites. A brief Appendix augments the discussion of one of the approximations employed in our theory.

II. MODEL DEVELOPMENT AND THEORY

Consider the standard situation of diffusion in an external field. As noted earlier, we assume that the behavior is diffusive in the absence of the external field, and a corresponding bare diffusivity exists. For ion motion in zeolites, D corresponds to the effective diffusivity in the pres- ence of the topological barriers only. Let c(r,t) dr be the

number of penetrant particles in a volume element, dr. The temporal and spatial evolution of c( r,t> is described by the following Langevin-type equation:

WY> -=DV2c(r,t) -A v * [c(r,t)F], at f

where D is the bare diffusivity, F is the force due to the external field, and f is the friction coefficient. The conditions under which this equation is valid are described in Ref. 12. These conditions are expected to be true for the motion of penetrants in zeolites wherein the length scale over which the field due to the charged centers varies is relatively large compared to the length scale that characterizes the ordered topological barriers.

We employ the path integral renresentation for the propagator corresponding to Eq. ( fields, i.e.,

F= -V+,

ijJ3 For conservative

(2)

epresentation is easily ransformation:13”4

where 4 is the potential, such a r obtained by making the following t1

e(r,t)=pWhw[& lF*dr], where k is the Boltzmann constant, and T is the absolute temperature of the system. Consistent with standard notation, we will denote l/kT by /?. Note that the exponential term in Eq. (3) is independent of the path for conservative fields. Combining Eqs. ( 1) and (3), and using the Einstein relationship ( Df = kT) obtains

ap(r,t) -p= -DV*p(r,t) + V(r)p(r,t) at

with

-t v*,j+$ (~4~2 . 1

(4)

(5)

Equation (4) has the form of a Schriidinger equation in imaginary time for the propagator in quantum mechanics.” This isomorphism between classical diffusion in real time and imaginary time quantum mechanics is well known (e.g., Refs. 6,11,13, and 14). We can express the propagator or Green’s function corresponding to Eq. (4) as a path integral

G(r,t;r&,) =jj’Dr(t)exp -’ [ 40 j- (32dt

- s v[r(r)ldt] , (6)

where (r,t) and (r,,,tc) are the final and initial space-time coordinates for all paths of the diffuser summed in the integral. A few things are worth noting regarding Eq. (6). First, the quantity that is exponentiated may be considered to be the analog of the action, and henceforth, we will refer to it as such. Second, the potential that enters into the action is not the real potential that the particle experiences; rather it involves the Laplacian and the square gradient of

J. Chem. Phys., Vol. 100, No. 2, 15 January 1994

the potential [see Eq. (5)]. Third, the action contains all the disorder degrees of freedom, q. This is to say that the Ricatti potential, v[r ( t)], depends on r(t) and q. If fruitful use is to be made of Eq. (6), we must find a way to integrate out the effects of the disorder and follow only the tagged degrees of freedom of the penetrant.

The randomly located charged centers in most zeolites are strongly bonded to the framework when the material is synthesized. As such, we are considering ion diffusion through a quenched disordered distribution of charges. This is to say that in the time scale associated with the motion of the ionic penetrant there are no temporal fluctuations of the disorder, and the disorder does not equili- brate with the penetrant. In general, the treatment of quenched and annealed disorders requires different averaging procedures, as is well known in the spin glass’6’17 and polymer statistics literature.‘s-22 The degrees of freedom associated with quenched disorders are usually integrated out using the famous replica trick.‘6*‘7 However, penetrant motion in quenched disorders and annealed disorders in the adiabatic limit are identical on the average provided the penetrants are uncorrelated and equilibrated to the disorder. These conditions imply that we average over all initial positions of the penetrant, and we consider large samples of the disordered material. The physical argument for the equivalence of quenched and annealed disorder for penetrant motion is analogous to ideas presented previously for polymer chain conformations in disordered media’8p21r22 and electrons in fluids. 23-25 This physical argument is provided in Ref. 26. LeDoussal and Machta2’ have also considered this issue for diffusion and find that the two situations are identical in any probable scenario. We also mention that a thermodynamic argument has recently been published28 in a different context that also leads to the conclusion that quenched and adiabatic annealed disorders are equivalent for situations such as the one we are considering.

where p. is the average density of the charged sites, K-’ is the characteristic decay length of the spatial correlations, and il is a constant determined through the electroneutral- ity condition. Thus A is given by

The equivalence between the two averaging procedures can be directly ascertained by performing a calculation wherein the disorder degrees of freedom are integrated out using the replica trick and then comparing the results to those from an annealed calculation in the adiabatic limit. Specifically, if there is no replica symmetry breaking, if one averages over all initial positions of the penetrant, and provided one uses the same probability distribution function for the spatial fluctuations of the disorder in both cases one finds that all statistical results pertaining to the penetrant are identical. For the sake of brevity, we do not reproduce the straightforward replica calculation here; rather, we integrate out the disorder treating it as if it were infinitely slow (i.e., adiabatic) and annealed.

We study disorders that are such that the probability distribution characterizing the spatial fluctuations of the density of charged sites is Gaussian with a mean equal to the average density, po, and a variance equal to the density4ensity correlation function, x( r,r’ ) = (Sp(r)Sp(r’)). Thus the probability distribution function for the density fluctuations of the disorder may be written as

1530 Chakraborty, Bratko, and Chandler: Ionic penetrants in disordered media

PIp( aexp -z [ ’ s dr I dr’Sp(r)X-‘(r,r’)Sp(r’)],

(7) where Sp(r) =p(r) -po, and x-‘(r,r’) is the functional inverse of the density-density correlation function. The density-density correlation function must be appropriate for a charged disorder with charge conservation enforced. In the case under consideration, the disordered material is modeled as randomly distributed charged sites in a neu- tralizing continuum. Physically, the charged sites correspond to zeolite counterions, while the background re- places the charge of the network. For simplicity, we consider the background charge of the network to be described by a uniform distribution. Thus we may express the density-density correlation function to be

K” /2=-

e-Klr--r’l

xhr’)=poS(r--r’) --p$i ,r-r,, , (8)

The above formulation allows for arbitrary choices of the correlation length, K- ’ . In the physical problem of ion diffusion in charged zeolites, the correlation length will be dictated by several factors including the geometry of the zeolite network and electrostatic screening. For example, in the case of Debye-Hiickel statistics:

47rPo * (9)

1=&

and

&c!?@ E ’ (10)

where 1, denotes the particular choice for 1, e. is the elementary charge and E is the permittivity of the medium. While we use Eq. ( 10) in obtaining numerical solutions to our equations in Sec. IV, the analytical asymptotic analysis is carried out for general K. Arbitrary choices for K do not change the qualitative results. In the following, when we integrate out the disorder, we will denote the density- density correlation function by x to keep the notation simple. Leaving x unspecified also points to the fact that our analysis is quite general, and is not restricted only to the problem that we are presently considering; viz., ion diffusion in charged disorders, with x specified by Eqs. (8) and (9).

Equation (6) is our starting point. As we have already noted, V[r,t] in Eq. (6) also includes all the disorder degrees of freedom. For annealed disorders, we obtain the effect of the disorder on the tagged degrees of freedom (the penetrant) by averaging the propagator. Following Feyn- man’s seminal and general work,15,29 and its applications to



static disorders due to Chandler,22-2s and others (e.g., Refs. 18 and 30) we define an influence functional as follows:

gIr(t)l =.QDp(rkxp[ -S Ur(WlPb(r) 1 BWr)P[p(r) 1

and

(G)duodm=fl&(t)exp[ --& s (g)2dt]d[r(‘)1. (12)

The path integrals in Eq. ( 11) are over all possible density distributions of the disorder. Note that v[r(t);qj is given by Eq. (5)) and the disorder enters the problem via 4. We can define Q rather generally as follows:

d(r) =kT s

c(r-r’)p(r’)dr’, (13)

where c( r - r’) is the coupling between the penetrant and the sites that comprise the disorder. For ion diffusion in charged disorders, c(r-r’) is the Coulomb potential. Thus the coupling between points in time on the penetrant’s paths and the sites that comprise the disorder may be written as

Zei 1 c[r(t) -r] =- 47&T jr(t)-r] ’ (14)

where Z is the number of charges on the penetrant. For convenience, in the following we will consider the charge of the penetrant and the dopant to be unity (i.e., Z= 1). Further, we will denote the Coulomb couplings by c[r( t) -r]. Using Eq. (13) we may write

e’=~Dp(r;pIp(r) ] LmP(r)P[P(r) 1

Xev[f Jdt( J dr’V2c[r(t) -r’]p(r’)

1 -- 2 s s

dr’ dr” Vc[r(t) -r’]

XVc[r(t) -r”]p(r’)p(r”) )I . (15) For notational convenience, we will denote ( D/2) V2c[r ( t) -r] by a[r(t) -r] and (dD/2)Vc[r(t) -r] by b[r(t) -r]. We now calculate the influence functional using Eq. (7) for the probability distribution of density fluctuations. This involves evaluating Gaussian path integrals. Specifically, the influence functional is given by

d=exp[poJdtJdr(a[r(t)-rl-pob[r(t)-r] Idr’b[r(t)-r’])]lfDp(r)

Xexp[ -i ldrldr’6p(r)M[r,r’;r(t)]Gp(f)]exp[ - sdt( s dr s dr’ pob[r(t) -r]b[r(t) -r’]6p(r)

+ dr c r dr’ pob[r(t) -r]b[r(t) -r’]Sp(r’) - [ dr a[r(t) -r]8p(r) )I , J J J

where the matrix, M[r,r’;r( t)], represents

Wr,r’;r(t) 1

=f’(r,r’) +2 s

dt b[r(t) -r]b[r(t) -r’]. (17)

Equation ( 16) can be further simplified to read

~=exp[po~dt~dr(a[r(tj-rl-po~dr’ b[r(t)-r]

Xb[r(t)-r’] j’j’Dp(r)exp -A 1 [ 2 JTP

XSp(r)Wr,r’;r(t) ISpW a, 1 a=exp

(18)

(16)

--2po~(W[r(t) -r] )Sp(r) I

(19)

and i; (0) is the k=O component of the Fourier transform of b. The path integral part of Eq. ( 18) is simply the average of a over a Gaussian distribution of density fluctuations with zero mean and variance equal to M-‘[r,r’;r( t)]. This average is easily performed to yield

exp[f Jdtldt’Jdrldr’(a[r(t)-r]

-2pob[r(t) -r]b(O>)M-‘(r,r’;r(t))(a[r(t’) -r’]

-2pob[r(t’)-r’]&O)). (20)

Thus integrating out the *disorder leads to the following expression for ( G)disorder :



FIG. 2. Diagrammatic representation of the coupling between time slices of the Wiener paths.

(c)~i~r~er=.Ufi(r)exp[ - J dt( & ( $)2-p$itOj

+p;b-(0) -; s dt’ v(r(t>;r(t’>) )I ,

(21) where u( r( t);r( t’) ) is short-hand notation for

v(r(t);r(t’))= s J-

dr dr’(a[r(t)-r]

-2poWrO) -r]b(O)) XM-‘(r,r’,r(t)) (a[r(t’) -r’]

-2pob[r(t’) -r’]i;(O)). (22) Note that integrating out the disorder in this manner has led to couplings between various time slices of the Wiener paths. In other words, the effect of interactions with the spatial disorder is to couple different time slices of the diffusive paths. In diagrammatic notation, the coupling, u, may be represented as shown in Fig. 2. The matrix, A?, represents the response of the disorder. Note that M depends on the spatial fluctuations of the disorder and the path fluctuations. We will return to discussing the path dependence of M in Sec. III. Following Feynman,‘5P29 we will refer to the couplings between the time slices of the Wiener paths as influence functional bonds. Figure 3 sche- matically depicts these bonds. Unlike the traditional polaron problem, the influence functional bonds in our case are long ranged in time. This feature is a result of the disorder being static. Its consequence on the dynamics is significant, as we shall discuss shortly.

We have now reduced our problem to that of evaluating the path integrals in Eq. (21). Note that these path

FTG. 3. Schematic representation of a diffusive path (solid line) with influence functional bonds (dashed lines).

integrals are not constrained to any given origin; i.e., we must integrate over all possible locations of the penetrant at time zero, since experimental measurements sample over all possible initial positions of the penetrant species. Fur- ther, this is one of the conditions for the equivalence between quenched and adiabatic annealed averages. Exami- nation of Eqs. (21) and (22) makes it clear that we cannot evaluate these path integrals analytically. In this paper, we employ a variational principle to calculate the path integrals that appear in Eq. (2 1) , This method of obtaining a variationally bounded solution is the subject of the next section.

III. METHODS

In solving Eqs. (21) and (22), we follow Feynman’s method of treating polaron mobility.‘593’ Feynman’s variational method has been previously employed to study problems involving static disorders in a variety of fields such as electron transport in fluids (Chandle?3-25), and polymer statistics ( Muthukumar,‘9P20 Edwards and Singh,32 Honeycutt and Thirumalai,18 des Cloizeaux,33734 Bratko and Dawson,35 etc.).

Consider the general situation, wherein

G=flWthp( --S[rW I> (23)

and S[r(t)] is the action that is of concern to us. Suppose we have another action (a reference action), such that

Go=flMt)exp( -Sdr(t)l>. (24) Noting that the exponential function is real and convex, it follows that15P36

G>exp[ - (S-SO>OI GO (25)

with

(s--so)o= LTMt) (S--So)exp[ -SOI

flDr(t)exp[ -So] * (26)

Equation (25) allows us to obtain a variational bound for G by choosing a reference action for which the right-hand side of the equation can be calculated analytically, and then tiding its extremum with respect to parameters that define So[r(t)]. Once the optimum reference action has been obtained in this manner, we may calculate the average value of any fluctuating quantity by averaging with respect to this variationally bounded reference action.

Equation (2 1) may be rewritten as

(‘3 tir~er=flMtbw[ - JOT dt( & ( g)2 -Wr(dl , )I (27)

where

U[r(t)] =po dr a[r(t)-r] -pg dr dr’ b[r(t)-r] s ss

Xb[r(t) -r’] +i s

dt’ v(r(t);r(t’)). (28)



As we have noted earlier, the path integrals in Eq. (27) cannot be evaluated analytically. As such, we will use the Feynman-Bogoliubov bound that we have just established to obtain a variationally bounded estimate for these path integrals. We choose the reference action to have the following general Gaussian form:

So=-& fdt(;)‘-; s,’ ; s,’ +,t’)

X Ir(t>-r(t’> 12. (29) In Eq. (29) T is the total time over which we follow the motion of the penetrant (not to be confused with absolute temperature). Our reference action (or trial action) is such that different time slices of the Wiener paths are coupled via harmonic springs, with r( t,t’) being the matrix of force constants. Our trial action attempts to mimic the complicated form of the influence functional bonds with harmonic oscillators. In our variational treatment we will

maximize the right-hand side of Eq. (25) with respect to the matrix of force constants, I’( t,t’ ). Since Eq. (29) represents a Gaussian reference, the terms on the right-hand side of Eq. (25) can be evaluated analytically.

We now proceed with the variational treatment of the influence functional. We are concerned only with the behavior of the ionic penetrant in the limit of long times, T. In the limit of long times we may consider the different Wiener paths to be periodic (with period T). This is to say that we will consider cyclic paths, and calculate ( 1 r(t) - r( t’) I 2, over shorter portions of the path. Using periodic paths makes the following analysis a bit easier, and in the limit of long times, should not affect our results in any important way. Since the paths are periodic, the matrix of force constants depends only upon the time difference between time slices, and will thus be denoted by l?( t- t’).

We must now calculate Go and (S--So)o. These quantities are given by

Go=flDr(t)exp --!- [ 4. Jo’dt(z)‘+f s,’ $ s,’ $r(t-t’)lr(t)-r(t’~12]

and

(30)

(31)

Evaluating Eqs. (30) and (3 1) involves performing Gauss- ian path integrals. In order to do this we introduce the following Fourier components:

1 = rn=-

I T o dt e’*&(t), (32)

where

r(t) = B,rg-‘nnf (33)

and

R,=2nn/T. (34)

Let us calculate Go first. In terms of the Fourier components that we have introduced, we may express the two integrals that constitute So as follows:

1 r dr2 -40 0 J 0

dt ;i; =--&Z,lr,,12C$

and 1 z r(t-ty [r(t)-r(t’) I2

(35)

where

Yn=xr,--ro), (37)

rn=f dx r(x)e”ti, (38)

In obtaining Eq. (36) we have invoked the fact that the matrix, I’, is cyclic. Note that the set of Fourier components, {r,), contain as much information as the matrix of force constants, l?. As such, in the variational treatment to follow the set, CT,}, will be our variational parameters. The optimal set of these quantities will enable us to obtain the variational bound.

In view of Eqs. (35) and (36), we may write So in Fourier space to be

(4)

We now transform the path integrals in Eq. (35) from the space of paths, r(t), to the space of Fourier components, {r,). This obtains

~~(t).g-SOtr(t)l= ~drl~dr2***~drn*..

XexP[ -i L41r,/2], (41)

where



A,= (& #+u.)- (42)

Equation (41) involves Gaussian integrals that are easily evaluated. A little care needs to be exercised, however, because of the subtlety associated with the fact that r, and rTn are not statistically uncorrelated. We will calculate Eq. (41) during the course of the variational treatment. Now we turn to the evaluation of (S--S,-,),.

Evaluating Eq. (31) involves the averaging of three terms over a Gaussian reference action. Only one of these presents any difhculty; viz., the average over a Gaussian trial of

1z = a=- I s

dt 20 0

dt’ v(r(t);r(t’)). (43)

In essence, we must work out the average of u( r (t);r( t’ ) ) over a Gaussian reference. The first difficulty that we en- counter in calculating this average can be seen by considering Eqs. (17) and (22). Calculating v(r(t);r(t’)) involves the matrix, M, which depends on the path fluctuations. This dependence of M on the path fluctuations precludes a simple analytical treatment. As such, we develop an approximation for isotropic disorders where we average over all initial positions of the penetrant. We develop the approximation at two different levels, both of which lead to the same results. In essence, our approximation for M ignores terms that are of fourth or higher order in the fluctuations of the density of the disorder (see the Appendix).

Consider first a relatively simple approximation. Sup- pose we average the path-dependent term in Eq. ( 17) over all possible paths. In other words,

+ 2 dt b(r(t)-r)b(r(t)-r’) (s >

. paths

(4.4) Noting that we are concerned with an isotropic disordered material this average is easily performed, and in the thermodynamic limit (large sample) yields

(M=‘(Wpaths=iW (45) The physical meaning of this result is simple. The averaging procedure that we have adopted for isotropic disorders in the thermodynamic limit samples the points on different Wiener paths with equal probability. Most paths that are included in calculating the average are far away from r and r’ and, since b[r(t) -r] is short ranged, their contribution to the average is vanishingly small.

In the Appendix, we carry out a more detailed analysis. This analysis shows that the error embodied in Eq. (45) is of fourth and higher order in the density fluctuations of the disorder. Thus for density fluctuations that are not very large, Eq. (45) represents an acceptable approximation that should lead to only small quantitative discrep- ancies. Replacing M-‘(k) with z(k), we proceed with the averaging of a.

The averaging of a over a Gaussian reference involves averaging u(r (t);r (t’ ) ) . Thus we must calculate

= -r) -2poj;(0)

[a(r(t’) -r’) -2pob(0) xb(r(t) -r) IxW)

XW(f’)-r’)l)ref, (46)

Introducing Fourier components, we can perform the average easily to obtain

1 WW;r(t’) ))mf.=m I dk 3k)

Xexp -#C 1 I-cos qt-t’)

n [u7W~2,+y,l ’ 1 (47)

where

F(k)= - [Z(k) -2poi;(0)6(k)]2T(k). (48)

Note that the quantity that is exponentiated in Eq. (47) corresponds to the product of k2 and the squared and averaged displacement of the penetrant. Evaluating Eq. (3 1) also involves calculating the average of the reference action, which is easily obtained to be

(f JUST I l?(t-t’) r(t)-r(f) I’), (49)

with A,,, having been defined in Eq. (42). The calculation of Eq. (3 1) is now complete, and we may write

(S--so)o= - dt[p$(O) --p;[b(0)]2] -; ; p m

1 dt’ o3

s dk E(k)

Xexp --PC 1 1-cos i-l,(t-t’) m 1 Am *

(50)

Equation (50) is further simplified by noting that for isotropic disorders, 6 (0) is zero. We have completed the calculation of Go exp[ - (S--So) J, and may now proceed directly to the variational calculation of the Fourier components of the matrix of force constants, {r,}. In order to do this we must calculate

ShG Sh Go S(S--so)o sy,=sy,- w?l -

(51)

Combining Eqs. (41), (50), and (51), and setting the derivative in Eq. (5 1) equal to zero obtains

J. Chem. Phys., Vol. 100, No. 2,15 January 1994

1 -cos n,r (T/2D) n2,+y, * 1 (52)

We have used polar coordinates to simplify the integral over k in order to obtain Eq. (52). Equation (52) consti- tutes a set of coupled transcedental equations that must be solved for the set {r,}. The set of Fourier components of the matrix of force constants obtained in this manner yields the Feynman bound. Our ultimate objective is to calculate the squared and averaged displacement of the penetrant. This is easily obtained in terms of the set, {r,}, as follows:

(Ir(t)-r(t’) 12>= flDr(t)e-So~rcr)ljr(t)-r(t’) I2

JJDr( t)e+Jr(‘)l

6(1-cos n,(t-t’) (53)

Equations (52) and (53) allow us to calculate the effective diffusivity of the ionic penetrant in our primitive model. Thus the theory for ion motion in the field of randomly located quenched charges (with charge conservation enforced) is complete. We now proceed to numerically calculate the effective diffusivity as a function of the parameters in our model. The results of these calculations and asymptotic analysis of our equations that predict the existence of localized states as the strength of the disorder is increased are presented in the following section.

IV. RESULTS

We have solved our model for ion diffusion in charged disorders numerically. Equation (52) is solved by direct iteration. For Coulomb couplings and ~(r,r’) given by Eqs. (8)-( lo), the integral over the wave vector in Eq. (52) can be determined analytically. The initial guess, ~~yn=O for all n, is used to obtain a second estimate for the set of variational parameters, and the process is repeated to self-consistency. Depending upon the number of points (N) used to discretize the path, and the total loop time (T) this iterative process typically entails 15-500 cycles. We have monitored the effect of discretizing the paths by varying the step size. The number of points N needed to avoid any step size dependence varied between lo2 and 104, being larger for longer loop times and stronger penetrant disorder interactions. The calculations should be carried out using loop times (T) that correspond to path dimensions, ([r(t) -r(t’) 12)1’2, which sufficiently exceed the range of correlations between the elements of the path. In our calculations, the loop times were chosen to be large enough to eliminate any discemable dependence of the effective “diffusivity” on the cycle time T.

We have calculated the squared and averaged displacement of the ionic penetrant for different choices of the permittivity of the medium and the average density of the disorder. As noted earlier, for the numerical calculations 2 is specified according to Eq. ( 10). We present our results

Chakraborty, Bratko, and Chandler: Ionic penetrants in disordered media

FIG. 4. D&D as a function of the disorder strength as measured by the dimensionless parameter, ,I #.

in terms of the ratio of the effective diffusivity to the bare diffusivity in the absence of the field due to the disorder. Prior to doing so, we note the following. In the limit of very long sampling times ( T), by nondimensionalizing Eq. (52) we obtain

=-&[ d? Joa dE& (+)

x l-cos!% (

Xexp -2 1 J +m dii (l-cosdr3 2n n2+f(b) ’ 1 (527 -cc

where ?=~DI?, E= k/K, d =R 02, and we treat fi as a continuous variable. Similarly, Eq. (53) can be nondimen- sionalized as follows:

+(t)-r(tt)12)=--$ s_‘” dfi (l-co?!?. co fi2+U~2)

(53’)

According to Eq. (52’) the strength of the disorder is measured by just one nondimensional variable involving po, E, and K. This single measure of the strength of the disorder is ( p. @e$/4&K), which also equals ilfl. Figure 4 depicts the results of our numerical calculations for D&D as a function of this variable. As is evident, as the average density is increased the effective diffusivity decreases smoothly until it acquires a value that is roughly half the bare diffusivity. More interestingly, there exists a threshold value of the strength of the disorder beyond which the diffusivity abruptly goes to zero. In other words, these numerical results suggest that there is a crossover from the weakly scattered diffusive behavior to a localized state when the strength of the disorder, 1, K, exceeds a value close to unity. In order to shed further light on these results, we now embark upon an analytical treatment of the pertinent equations.


WGPO s qd/1-cos5anT1+

0 9 I Tcd~[l-cosfi T] n

9


Consider Eqs. (42) and (52). If there were no fields imposed by the disorder, y,, would equal zero for all n. The quantity, A,, , would then simply equal the ideal piece ( ={T/2D)Ri). Under these circumstances, A,, would scale as n2, and the squared and averaged displacement would exhibit diffusive behavior. Let us now consider the situation when the disorder does impose fields and the set, {y,), are not zero. Specifically, we examine how y,, scales with n in the limit of long times. If a transition from diffusive behavior to a localized state does exist, then y,, must scale with n2 under certain circumstances and with some other n-dependence when localization occurs. In the following, we ask whether yn can scale with different n-dependences under different circumstances, and if so, we aim to trace the condition for the transition.

For the specific problem that we are examining, the integral over k can be performed analytically. This is so because F(k) has a particularly simple form. Recall that F(k)=[Z(k)12*f(k), and a[r( t) -r] is the Laplacian of c[r(t) -r]. Since the coupling is Coulombic [Eq. (14)], a[r(t) -r] is

where 7 is a time such that ~5%’ remains small, and T, is a time above which K-L?? is sufficiently large for Eq. (57) to hold. Now consider the n dependence of each of the terms in Eq. (59) for small values of n (or long times). For the first two terms, we can always choose total loop times (T) that are large enough for a,~ to remain small throughout the range of integration. Thus these terms will scale as n2 in the limit of small values of n. Now consider the third term. In this case, 0,~ cannot remain small throughout the range of integration under all circumstances. However, if l/33” decays rapidly with time (e.g., the behavior is diffusive) then the upper limit of integration becomes irrelevant. In this circumstance 0,~ will be small over the range of integration wherein the integrand contributes appreciably. Ex- actly the same argument that we made for the first two terms would then hold, and the third term would also scale as n2 for small values of n. On the other hand, if l/B” does not decay rapidly with time the third term does not scale with n2. For example, consider the extreme case when 9’ scales with the zeroth power of 7; in this case the third term scales as sin(bn)/n, where b is a constant.

2

a[r(t) -r] =Tz 8(r(t) -r).

C(k) is obtained by taking the Fourier transform of Eqs. (8) and (54), and forming the product. Thus F(k) is

F(k)=(%)lp,[&]. (55)

Using Eq. (55), the integral over k in Eq. (52) can be performed analytically. The result is a function of the square root of the squared and averaged displacement, and is given by

3,6 2 K~& -- 2 is+ 29

-$ erfc( $)*tP~6], (56)

where 9 is the square root of the squared and averaged distance, and A is ( 27-rD;1J2po. The complementary error function in Eq. (56) can be expanded in the limits of large and small values of ~3%’ This procedure yields

for large ~9, and

(57)

(58)

for small ~9%‘. In Eqs. (57) and (58), C and E are pure constants that do not depend upon the average density of the disorder. Given these expansions for large and small values of KS?‘, and retaining only the leading order terms, Eq. (52) becomes

(59)

The above analysis demonstrates that, at long times, yn may scale with n2 under certain circumstances, while under other conditions it does not have a n2 dependence. n2 scaling implies diffusive behavior, while other kinds of n-dependence lead to anomalous behavior. Our approximate variational equations therefore suggest that both diffusive and nondiffusive behavior are possible in the problem that we are considering. Further analysis of Eq. (59) also shows two other features that support this conclusion. Firstly, we can show that diffusive behavior is consistent with the equations that describe our theory for this model problem. This is to say that if the behavior is diffusive, yn decays with time at least as fast as the ideal term in the limit of long times. Second, we can also show that our equations are consistent with two possible ways in which (W2) scales with time; viz., 1 and 0. This result also implies that the behavior can be both diffusive and subdiffusive. Analysis of Eq. (59) leading to these conclusions is straightforward, and so we do not reproduce this scaling analysis here for the sake of brevity. We will discuss these issues in Sec. V; but first, let us trace the condition for crossover to a localized state.

Our analysis aimed toward finding a condition for the transition begins with an observation from our numerical calculations. Consider the variation of the squared and averaged displacement at half the loop time (T/2) with loop time, T. Of course, all values of the effective diffusivity reported in Fig. 4 are *for values of the loop time that are sufficiently large to eliminate any dependence on the loop time. However, we observe that in cases wherein localization occurs, for short loop times a’( T/2) increases with loop time; in the long time limit, we find that 9’(T/2)


vanishes when localization occurs. Note that L%“( T/2) is zero because we sample only cyclic paths. Since z??‘( T/2) increases with loop time for short T, and is zero for large values of T when localization occurs, a necessary condition for the localization transition to occur is that .@‘( T/2) exhibits a maximum with respect to loop time, T. If the condition for the maximum to occur is not satisfied, then s2( T/2) continues to grow and reaches an asymptotic slope for large loop times. Under these circumstances the behavior is diffusive. If the condition for the maximum in s2( T/2) with loop time is satisfied then a localization transition (signaled by D,,/D=O) occurs.

Let us find the condition that leads to a maximum in s2( T/2) with respect to loop time, T. Thus we set

d9P2( T/2)

f3T =o (60)

s2( T/2) is given by Eq. (53), and is strongly dominated by small values of n. Differentiating Eq. (53) with this in mind leads to the conclusion that Eq. (60) will be satisfied when

ay, UrnI ar=2* (61)

The derivative of y,, with respect to loop time, T, is easily obtained by differentiating Eq. (52) with respect to T. Performing this derivative while noting that we are search- ing for the condition for i192/ilT=0, yields

G92(11) d, (64) where f is a number larger than unity. This implies that the upper limit of integration, q, takes a value close to f/( 02). Assuming that the behavior is diffusive up to the maximum in g2( T/2), and invoking Eq. (63) obtains the following condition for the average density of the disorder required for localization to occur:

poz 2K@e;. (65)

The numerical results have been obtained with the correlation length, K, chosen according to Eq. ( 10). In this case, Eq. (65) reduces to

Poa$. (66)

Equation (65) corresponds to saying that the nondimensional parameter that measures the strength of the disorder is of order unity when our equations predict localization. This is borne out by the numerical results shown in Fig. 4. Our variational calculations, and the asymptotic analysis of the equations corresponding to a Gaussian reference action predict that for the case at hand there is a crossover from weakly scattered diffusive behavior to a localized state when the strength of the disorder exceeds a threshold value. In the following section, we discuss this further in the context of previous field theoretic studies. This discussion suggests that a localized state does not live forever. Rather, the localized state may move over longer time scales; a feature not captured by our calculation.

(62) V. DISCUSSION


Combining Fqs. (61) and (62), and invoking the defini- tion of a,, obtains the following condition for a maximum in 5p2( T/2) with respect to T.

The crossover from weakly scattered diffusion to a localized state in the problem that we have been considering is not difficult to understand from the point of view of the

(63) formulation that we have employed. There is a competition between the kinetic term in the action and the disorder- induced couplings between different time slices of the Wiener paths. For weak disorders, the kinetic term domi- nates, while the disorder-induced couplings lead to localized state (signaled by D,s=O> for sufficiently strong disorders.

Note that the condition for the maximum in g2( T/2) is very close to the condition for localization, but not exactly the same. This is to say that shortly after the maximum is reached localization occurs. Thus it is not unreasonable to say that Eq. (63) must be roughly true just before the localization transition. Equation (63) states that y,, equals the ideal part of A,. This implies that just before localization the effective diffusivity must be close to one half the bare diffusivity. Examining Fig. 4 shows that this is indeed true. Thus Eq. (63) seems to be fairly accurate in compar- ison with numerical calculations. We now use Eq. (63) to obtain a condition that relates the threshold density above which localization occurs to other relevant parameters.

One may show that ph > pot where p. is given by Eq. (65), coincides with E>kT where E is the typical depth of potential energy wells. Specifically, E2 can be estimated as the mean square fluctuations in potential. This is to say that

E2= (Srj2>, (67)

where

In order to derive such a relationship, we must com- bine Eqs. (59) and (63). Equation (59) is an expression for y,, wherein only the leading order terms have been retained for large and small values of 9’. For a limiting analysis, we concern ourselves only with the first and dominant term in Eq. (59), The upper limit of the integral in Eq. (59) (7) is such that ~5%’ is small. v is therefore chosen according to the condition

^ e’

X4relr’-r”l x( f’,r”‘)

pie4 1 e-Klr-r’I

~~ jr-r’] - (r-r’/ 1 I ’ (68)


The second equality follows from performing the indicated convolution integrals with x( r,r’) given by Eqs. (8) and (9). Equations (67) and (68) give

(69)

Equation (69) implies that the condition for the transition derived in Eq. (65) coincides with E>kT.

It is appropriate to consider our results for a specific system in the context of previous general studies of diffusion in random media. Several author&” have shown that the application of RG methods to a weak-disorder fixed point leads to the conclusion that diffusive behavior is re- covered in dimensions greater than 2. Anomalous behavior has been found for dimensions less than 219-l’ a result that has been confirmed by a Lyapunov exponent analysis.” It has also been found that when the correlations that describe the disorder are short ranged, the behavior is diffusive in dimensions greater than 2.&” Zhang6 and oth- ers37f38 have argued that for sufficiently strong disorders localized states will exist. The localized state, however, does not live forever; rather, the localization center hops discontinuously. Zhang6 predicts that the displacement of the localization center is sub-ballistic. Ebeling et al. 37 argue that for short range correlations for potential energy fluctuations, the average time spent in a localized state grows as exp[P2E2]. So the time between hops can be very long. Further evidence suggesting that the localized state that we find should not persist forever is provided by the work of De Masi et uL3* who predict that the diffusion coefficient should have a tinite bound for the kind of problem that we consider. When our calculation predicts a localized state, this bound is finite. We believe that the following physical picture probably describes the situation. For strong enough disorders transition to a localized state occurs. However, over longer times the localized state can hop to another localized state. In Feynman path integral language, the paths connecting the localized states are instantonlike paths. Thus, the mechanism of diffusion changes when localization occurs. Our Gaussian approximation for the influence functional bonds cannot capture the discrete hops between localized states. It is worth noting, however, that as the strength of the disorder is increased (large E) the time spent in a localized state may exceed experimental time scales.

The above picture, however, is subject to some doubt in light of some RG calculations due to Bouchaud and Georges’ who have considered the possibility of anomalous diffusion in any dimension when the disorder exhibits long ranged correlations. As Eq. (68) shows, in our problem the potential energy fluctuations have a long ranged component. It is a manifestation of the Coulomb potential and the correlations that describe the disorder. Bouchaud and Georges’ characterize the length scale of the correlations by a parameter u. This parameter is the exponent that characterizes the long-distance decay of the force-force correlation function. For a < d (dimensionality ) , and a < 2 they find anomalous behavior. Our problem corresponds to u=2. For this marginal case, the diffusion fixed point is


found to be marginally stable, and the diffusion exponent depends upon the strength of the disorder with logarithmic corrections leading to subdiffusion. This implies that for our problem even the very long time behavior may not be diffusive.

In summary, our calculation is consistent with RG studies in that the behavior is diffusive for weak disorders. The crossover to a localized state that we observe is also consistent with previous studies. However, we believe that this disorder induced change in the mechanism of penetrant motion for a specific physical problem has not been described heretofore. Previous studies suggest that this localized state hops from one region to another over longer times. Gaussian calculations cannot capture this behavior. However, the picture is not entirely clear since RG calculations* predict that for the general class to which our problem belongs the behavior should be subdiffusive for strong disorders. It is hoped that in the future more so- phisticated calculations will be able to shed light on this issue.

VI. CONCLUDING REMARKS

We have considered ion motion in a particular form of a spatial charged disorder. The problem may be considered to be a primitive model for the technologically important issue of ion diffusion in charged zeolites. Molecular simu- lation studies are not computationally tractable for the study of these problems. We have developed an approximate theory that provides a coarse-grained description of this problem. Our theory relies on a path integral representation of the propagator, and the Feynman variational bound is used. We believe that this approach is rather general, and can be used and extended to study many practical problems such as diffusion in polymeric membranes and zeolites. Results of such calculations will be presented in a forthcoming sequel.

Analytical considerations and numerical results indi- cate that for the case under consideration there is a crossover from weakly scattered diffusion to a localized state as the strength of the disorder is increased. Specifically, as the strength of the disorder increases the effective diffusivity decreases smoothly until it acquires a value that is roughly half its bare value; if the strength of the disorder is increased further, the effective ditfusivity abruptly goes to zero. These results are not inconsistent with field-theoretic studies of diffusion in random media. However, we believe that such a clear localization transition with increasing strength of the disorder has not been predicted previously for a specific physical problem.

It is obvious that we should compare our predictions with experimental results on ion diffusion in zeolites. Un- fortunately, direct comparisons are hard to make because the experimental situation at the current time is not entirely clear. The data39 show that there are dramatic decreases in ion mobility upon changing the charge on the diffusing cation. This is equivalent to making the disorder stronger. However, these studies have all been carried out using zcolites with small pores, and so this effect could be just due to larger steric hindrance for the motion of the



strongly hydrated cations. Experimental data regarding the mobility of an ion in the zeolitic structures with varying aluminum substitution (or average charge density) are still unavailable, because it is difficult to synthesize zeolites wherein the aluminum loading is carefully controlled. We suggest the following experiment that will be able to test the predictions reported in this work. Ion mobility should be measured in large-pore zeolites, such as VPI-5,40 wherein steric barriers to difliusion are weak. The extent of loading can be controlled for VPI-5:’ and so measurements of ion mobility in a series of differently charged VPI-5 type zeolitcs should provide a test of our predictions. It is important that such a test be carried out. This is so because our results have important implications for both scientific and technological reasons. From a fundamental point of view, we find an interesting phase transition in a very simple physical system. From a technological view- point, if our theoretical predictions are correct, and the model is an appropriate primitive model for ion difTusion in charged zeolites, then we have the striking result that product ions (e.g., NIQ ) present in low concentrations will not be able to escape from the zeolite if it has an aluminum loading exceeding a certain threshold value. The experiments that we suggest above should serve to establish whether the simple theory presented in this work and ex- tensions thereof may find application in describing subtle transport phenomena in technologically relevant disordered materials.

ACKNOWLEDGMENTS

This work was supported by the National Science Foundation via a National Young Investigator Award to

I

A.K.C. Fruitful discussions with M. W. Deem, K. Leung, D. Rokhsar, M. Teubner, and D. N. Theodorou are grate- fully acknowledged.

APPENDIX

The evaluation of the influence functional, 2, entails averaging over a Gaussian distribution of density fluctuations, Q(r), with zero mean and variance equal to M- ‘. The resulting expression for the propagator has a form that is familiar from studies of polaron dynamics*5*3’ and the statistics of polymer conformations in Gaussian disorders. The difference is that the density-density correlation function is replaced by a functional, M- ‘[r,r’;r(t)]. This ren- ders the averaging of exp[ -s] as per Eq. (26) a more ditllcult task. However, as we shall demonstrate below, if the density fluctuations are sufficiently small higher order moments can be ignored, and the standard procedures can still be followed. The analysis below shows that Eq. (45) which was obtained by averaging over all paths ignores terms that are of fourth and higher order in the density fluctuations.

First, we note that the interesting interactions embodied in v[r( t)] stem from the density fluctuations, and hence, the deviation of 2 from unity can be treated as a functional expansion in terms of the average moments, @p”(Cr,l)), where (@‘(r))=@p(r)), @p2(rl,r2)) =(6p(rl)6p(r2))=X(rl,r2), and so on. We begin with Eq. ( 15). For simplicity, we will omit all terms that do not explicitly depend on the density fluctuations and on the path fluctuations. We will also assume that the normaliza- tion of the probability, qp( r)] has already been carried out. With these considerations, Eq. ( 15) can be written as

daflDp(r)P[p(r)]exp[ IdtJdrap(r)(a[r(t)-r]- Jdr’b[r(t)-r]*b[r(t)-r’]6p(r’))

Functional expansion of the exponential term in Eq. (Al) in terms of the density fluctuations yields

(Al)

flWr)P[p(r)l(l+ J dtj dra[r(t)-rl6p(r)- J J J dt dr dr’ b[r(t) -r] *b[r(t) -r’]Sp(r)Sp(r’)

+; j-dtj-dt’j- j- dr dr’ a[r(t) -r]a[r(t’> -r’]Sp(r)Sp(r’) - IdtSd~~~drldr’Idr”n[r(r)-r]

Xb[r(t’) -r’] gb[r(t’) -r”]6p(r)8p(rt)i3p(rtf) +i Sdfldf’IdrSdr~Sdr”Idr”b[r(t)-r]*b[r(r)--r’]

Xb[r(t’) -r”] l b[r(t’) -r”]Sp(r)Sp(r’)Gp(r”)Sp(r”) +i . Sd*Sdr’Idr”ldrldr’ldr”a[r(t)-r]

Xa[r(t’) -r’]a[r(t”) -r”]~p(r)~p(r’)f3p(rt~) -i Idtldr’Idr”ldrldr’ldr”J‘dr”a[r(r)-r]

Xa[r(t’) -r’]b[r(t”) -r”] l b[r(t”) -r”‘]c5p(r)~p(r’)6p(r”,)Sp(r”)

+i sd~ld*‘~d~~‘sdrldr’sdr”J‘dr~~~Idr”” a[r(t) -r]b[r(t’) -r’] l b[r(t’) -r”]

Xb[r(t”) -,“‘I l b[r(t”) -r”“]8p(r)fip(r’)6p(r”)Sp(r”“)



-f Sd~~d~‘Id~‘~Sdrldr’Idr”ldr”ldr”“ldr””’ b[r(t) -r] l b[r(t)-r’]b[r(f’) -r”]

Xb[r(t’) -r”‘]b[r(t”) -r”“] *b[r(,“)-r”“‘]sp(r)sp(r’)sp(r”)sp(r”’)~~(r””)Sp(r”“‘) +. . . . (AZ)

Performing the average in Eq. (A2) would merely lead to the replacement of the terms involving density fluctuations by their average moments, (Sp”({r,})). For a Gaussian distribution of density fluctuations with zero mean, all odd moments vanish.42 If we now truncate the expansion at fourth order, we are left with the following second order term

d-l+; Jdtldt’Idrldr’o[r(t)-r]x(r,r’)

xa[r(t’)-r’]+i JdfJdrJdrt b[r(t)-r]

xX(r,r’)b[r(t)--‘l+0((8p4[(r4)l)), (A31 where X(r,r’) is defined as usual. Inspection of Eq. (A3) reveals that only the first term that is different from unity couples different time slices of the Wiener paths. Further- more, by shifting coordinates, the second term can be simplified to a form that does not have any path dependence. Thus the second term in Eq. (A3) will not affect the probability of the path, and further, will be irrelevant during our variational treatment. Thus we conclude that the only relevant contribution to the influence functional, up to fourth order in the density fluctuations, is the second term on the first line of Fq (A3). This indicates that in the present context, the Laplacian of the potential [in Eq. (6)] represents the dominant contribution to the influence functional. Of course, this also implies that Eq. (45) is correct up to fourth order in the density fluctuations.

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