Transport in Porous Media 13: 79-95, 1993. 9 1993 KluwerAcademic Publishers. Printed in the Netherlands.

79

Diffusion in Pore Fractals: A Review of Linear Response Models R. M U R A L I D H A R Mobil Research and Development Corporation, Paulsboro, NJ 08066, U.S.A.

and D. R A M K R I S H N A School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Received: 24 March 1993)

Abstract. A major aspect of describing transport in heterogeneous media has been that of relating effective diffusivities to the topological properties of the medium. While such effective transport coefficients may be useful for mass fractals or under steady state conditions, they are not adequate under transient conditions for self-similar pore fractal media. In porous formations without scale, diffusion is anomalous with the mean-squared displacement of a particle proportional to time raised to a fractional exponent less than unity. The objective of this review is to investigate the nature of the laws of diffusion in fractal media using the framework of linear response theory of nonequilibrium statistical mechanics. A Langevin/Fokker-Planck approach reveals that the particle diffusivity depends on its age defined as the time spent by the particle since its entry into the medium. An analysis via generalized hydrodynamics describes fractal diffusion with a frequency and wave number dependent diffusivity. Key words: Fractals, anomalous diffusion, generalized Langevin equation, generalized hydrodynamics. 1. I n t r o d u c t i o n It is n o w recognized that porous m e d i a such as catalyst pellets, rock formulations, etc., are often not h o m o g e n e o u s at small length scales but rather are self-similar fractals (Avnir, 1989; Cushman, 1990). These fractal objects possess no characteristic length scale as a consequence o f which they exhibit similar geometrical features at different levels o f magnification. The amount o f matter in a massfractal of length scale L obeys the scaling law (Mandelbrot, 1983)

M ( L ) ~ L dl,

(1.1)

where d f is in general a non-integral quantity and is referred to as the fractal dimension. The fractal dimension satisfies the inequality

dt < d I < d, where dt is the topological dimension and d is the Euclidean dimension. Physically this m e a n s that a fractal in order to a c c o m m o d a t e structure within structure m u s t

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R. MURALIDHAR AND D. RAMKRISHNA

occupy considerably more space than familiar homogeneous objects. Thus, for example a fractal surface (d~ = 2, df > 2) folds out in space to such an extent that it has infinite area although the volume it occupies may be zero. Concepts from fractal geometry have provided a better characterization of heterogeneous media often encountered in scientific and engineering applications. Computer simulations of the processes similar to those employed in the formation of porous catalyst pellets have shown that the resulting structures exhibit fractal scaling. Thus, for example, diffusion limited aggregation (DLA) in two dimensions, a process in which single particles diffuse towards a growing cluster and get irreversibly bound on contact, yields a cluster of fractal dimension 1.71 (Meakin, 1983). Similarly cluster-cluster aggregation (CCA) which involves the simultaneous diffusion of clusters and their irreversible binding on contact, yields in two dimensions fractal clusters with df approximately 1.78 (Meakin, 1983). Experimental observations of small particulates and clusters via scanning electron microscopy (Krohn and Thompson, 1986), small-angle-X-ray scattering (Schaeffer et al., 1984; Schmidt, 1988), etc., have also shown that these are indeed self-similar fractals. Concomittantly, fractal geometry and percolation theory have been employed by investigators to make effective medium approximations for porous media. This framework assumes the validity of classical transport equations and replaces the molecular diffusivity by an effective diffusion coefficient that depends on the topological characteristics of the medium. Mohanty et al. (1982) and Mo and Wei (1986) have predicted effective diffusivities of porous networks using percolation models of the medium. Rosner and Tessapolous (1990) have computed the effective diffusivity tensors of anisotropic fractal networks obtained by computer simulations of aerosol deposition processes. Sheintuch and Brandon (1989) have also theoretically investigated the effectiveness factors for fractal catalyst pellets. These approaches may not be appropriate for transient diffusion in the so called porefractals (Avnir, 1989). Unlike in the case of mass fractals, the volume of the void space, rather than the cluster mass obeys the scaling law given by Equation (1.1). In other words, a mass fractal becomes a pore fractal on exchanging the roles of the solids and pores. Pore fractals may be formed by acid leaching of one phase after phase separation. These materials are also called controlled pore glasses of which vycor glass is one example (Hohr et al., 1987). While Fick's second law with a constant effective diffusivity predicts mean squared displacement of a diffuser (R2(t)) to be proportional to time, for pore fractals it follows the anomalous law given by (Havlin and Ben Avraham, 1987; Hans and Kerr, 1987; Bouchaud and Georges, 1990) (R2(t)) ~ t 1-7,

(1.2)

where 3' is the random-walk exponent and has a value between zero and unity. For classical diffusion 7 is strictly zero. Thus, the laws of diffusion in pore fractal media are different from those of homogeneous media.

DIFFUSION IN PORE FRACTALS

81

The objective of this paper is to review approaches based on linear response theory for deriving diffusion equations for pore fractal media and to investigate the nature of the laws of diffusion in pore fractal media. The models exploit the intimate connection between particle diffusion and velocity auto-correlation function (VACF). The first approach akin to the Langevin/Fokker-Planck (LFP) description of Brownian motion (Kubo et al., 1985) consists in describing the particle dynamics via a Generalized Langevin Equation (GLE) which admits memory effects in particle motion via a friction kernel. This GLE serves as a starting point for deriving a diffusion equation in the same way the Langevin equation (McQuarrie, 1976) is the dynamical foundation of the Fokker-Planck equation. A second approach within the framework of linear response theory is rooted in Generalized Hydrodynamics (GH). In this framework, a GLE is written for the intermediate scattering function (McQuarrie, 1976). Anomalous diffusion properties may be used to identify the form of the diffusivity. The outline of this review is as follows. In Section 2, the LFP approach is elucidated and an age dependent transport equation is derived. Section 3 reviews the GH method of analysis culminating in a wave-number and frequency dependent diffusivity. Section 4 summarizes the status of these microscopic theories.

2. LFP Approach: GLE and its Diffusion Equation 2.1. BACKGROUND The disordered porous medium consists of solid particulate matter fixed at random locations in space. In a percolation model for example, the blocked sites may be viewed as the solid phase and the sites belonging to the infinite cluster may be regarded as constituting the pore space. It is assumed that there is a lower cutoff to the fractal nature of the pore space permitting a velocity to be defined. The walls of the porous matrix serve as scattering centres for a diffusing particle. In the so called Lorentz gas limit, the diffusing particle is anisotropically scattered only by the pore walls. At the other extreme, a tagged particle (solute) suffers collisions with other particles (solvent) as well as anisotropic scattering at the pore walls. In both cases, the long time diffusion is anomalous. An adequate theory for investigating diffusion is nonequilibrium statistical mechanics. At the foundation of the linear response theory of non-equilibrium statistical mechanics is the Onsagerfluctuation regression hypothesis (Chandler, 1987) according to which the laws governing macroscopic transport are identical to those governing the regression of the correlation of the corresponding dynamical variables in a system in thermal equilibrium. We assume that there is an upper cut-off to the fractal geometry of the medium which permits the attainment of equilibrium. Further, application of the regression hypothesis, requires that the medium be fractal over a sufficient range of length scales. For particles interacting with each other and the pore walls, the diffusion law is the same as that describing the decay

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of spacetime density correlation. This space time correlation function is given by C'(T, t) =

(5p(v, t)ap(O, 0)).

(2.1)

In the above, the angular brackets denote a thermal equilibrium average, the particle density at v at time t and

p(v, t) is

5p(r,t) = p ( r , t ) - p, where p is the equilibrium density assumed to be spatially uniform. For a dilute system of weakly interacting particles, it can be shown that (p(v, t)p(O, 0)) is proportional to the probability density that a tagged particle is at v at time t given that it was at the origin at time t = 0. Thus, the diffusion equation may be derived from a Lagrangian analysis of single particle random motion. Basic to a Lagrangian description of diffusion phenomena are kinematics and dynamics. Below we first summarize the salient aspects of the kinematics and dynamics of diffusion leading to the GLE discussed previously elsewhere (Muralidhar et al., 1990; Muralidhar et al., 1991). 2.2.

VACF FOR ANOMALOUS DIFFUSION

At the crux of the kinematical description of random motion is the velocity autocorrelation function (VACF) which expresses the content of an ensemble of stochastic particle trajectories in a useful form. In one dimension, the normalized VACF is defined by x(t)-

(V(0)V(t)) (V2) ,

(2.2)

where V(t) is the velocity of the particle at time t and the braces denote an average over a stationary ensemble. The VACF measures the extent to which the velocity fluctuations have coherence. Over very short time intervals, a diffuser rarely interacts with other particles or the medium and as such its velocity changes very little. Thus the VACF starts out at unity (t = 0 limit). With increasing time, the particle suffers many collisions at random and eventually its velocity becomes uncorrelated to its initial value. This implies that the VACF decreases in absolute value to zero (t = oo limit). Now the mean squared displacement of diffusion is intimately related to the VACF by

(x2(t)) ~

- fo X(s) ds -

fo sX(s) ds.

(2.3)

We have in the past (Muralidhar et al., 1990) related the properties of the VACF to that of diffusion. Classical or Fickian diffusion necessarily emerges whenever the VACF decays faster than t -2 for large t and the zeroth moment defined by D = (V 2)

/7

X(S) d ,

(2.4)

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DIFFUSION IN PORE FRACTALS

is positive and finite. This quantity is then the classical diffusion coefficient. A good example of such a situation is an exponentially decaying VACF

x(t) = exp(-/3t).

(2.5)

This exponentially decaying VACF, when substituted in Equation (2.4), yields for the diffusivity D-

(v2) fi

From a physical viewpoint, successive displacements of the particle over time intervals much larger than/3-1 may be regarded as independent and, hence, classical random walk or diffusion arises. Anomalous diffusion can emerge whenever the VACF decays slower than t -2 at large times. A necessary condition for anomalous diffusion seen on fractals is that the classical diffusivity be zero, i.e.

f0

X(8)

(2.6)

-- 0.

In addition to the above, a sufficient condition for the mean squared displacement to obey the power law given by Equation (1.2) is that

x(t) ~ - a t -(1+'y),

a > 0, t --+ ~ .

(2.7)

This VACF is long-tailed and negative because it decays slowly to zero at large times. Such a long-tailed VACF has been observed in the so-called blind-ant random walks on two dimensional percolation clusters (Jacobs and Nakanishi, 1990). The VACF of anomalous diffusion is thus long tailed in sharp contrast to an exponentially decaying function for classical diffusion governed by Langevin dynamics. The characteristic relaxation time for a Langevin particle given by Equation (2.5) is/3-1. Thus correlations are short ranged and displacements over non overlapping time intervals much greater than/3-1 may be regarded as independent. However, in the anomalous case, the VACF has a slow power law decay so that the decay time (~ f~o tx(t) dt) is infinite. We thus have long-ranged correlations. Secondly, the long negative tail of anomalous diffusion implies anti-persistence of motion. In otherwords, if the particle moves in the positive x direction in the current step, it is more likely to move in the opposite direction in the subsequent steps. Indeed in a fractal medium, the particle cannot progress to a significant extent in any direction because it encounters dead ends. In classical diffusion, any persistence or anti-persistence is exhibited only over a short time interval of order/3-1. 2.3.

GENERALIZED LANGEVIN DYNAMICS

We now consider the dynamical aspects of diffusion in pore fractal media. In order to contrast the nature of the dynamical law for anomalous diffusion, we first review

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R. MURALIDHAR AND D. RAMKRISHNA

the classical Langevin equation (CLE) of Brownian motion (Chandrasekhar, 1943; Uhlenbeck and Ornstein, 1930). The Langevin equation describes the motion of a brownian particle which is mesoscopic in scale. While a Stokes sphere perceives the medium as a continuum, a molecule only 'sees' the medium as other molecules. A brownian particle is of intermediate scale and its motion is influenced by continuum as well as molecular features of the medium. Thus, the spirit of the Langevin approach is to decompose the forces acting on the particle into systematic (dissipative) and random (fluctuating) components. This CLE is given by

dV d~ -

flV + F ( t ) / m ,

(2.8)

where V is the particle velocity, fl is the friction coefficient per unit mass and F(t) is the rapidly fluctuating random force. The random force is assumed to have a vanishing mean expressed by

(F(t)) = 0.

(2.9)

This requirement ensures that the average dynamics of an ensemble of particles with identical non-zero initial velocity is dissipative. The random force is also assumed to be uncorrelated to the initial velocity of the particle, i.e.

(V(O)F(t)) = 0,

t > 0.

(2.10)

This requirement follows from Onsager's fluctuation regression hypothesis that requires the time correlation function (VACF in this case) to decay in the same manner as a macroscopic variable (the velocity of a Stokes sphere in this case). Further, it is assumed to be have no coherence or correlation, i.e.

(F(O)F(t)) ,,~ 5(t).

(2.11)

In the above equations, the angular braces denote an average over an ensemble of particles. The CLE in addition to giving a good physical insight into the nature of diffusion has the following uses. First it describes the nature of the diffusion for short times during which inertial effects are important and secondly it is a starting point for deriving the partial differential equations of diffusion. From the CLE and the assumed properties of the random force, it follows that xCLE(t) = exp(--fit).

(2.12)

This exponentially decaying correlation is in complete contrast to the negative tail and power law decay exhibited by the VACF on fractals. It is thus evident that the CLE does not describe the kinematics of random motion seen on fractals and a generalized Langevin equation (GLE) is needed. In the GLE framework (Kubo et al., 1985), the dynamics of the particle is given by

dX dt

-

V

(2.13)

DIFFUSIONIN POREFRACTALS

85

together with the GLE dV dt

f o t a ( t - s ) V ( s ) ds + F ( t ) / m .

(2.14)

The first term on the right-hand side is the slowly varying systematic force or friction and the second term is the random force. The function a(t) reflects the extent of memory of the past velocities of the particle and is referred to as the memory function or friction kernel. The random force is again assumed to possess the statistical features expressed by Equations (2.9) and (2.10). For an aged particle in thermal equilibrium, the velocity fluctuations are stationary and this implies that (Kubo et al., 1985)

a(t)-

(F(O)F(t)) ~'b2( g 2 )

(2.15)

This relation between the fluctuating force auto-correlation and the friction kernel is referred to as the fluctuation dissipation theorem and is a generalization of Equation (2.11). Although the GLE was initially a phenomenological approach, it was provided a rigorous statistical mechanical foundation by Mori (1965). While the spirit of decomposing the forces on a particle into systematic and random components is common to the CLE and the GLE, the latter is far more general. The memory term in the GLE, o~(t), enables one to describe random motion with memory or non-Markovian diffusion. Indeed the CLE emerges as a special case of the GLE and corresponds to the case of complete absence of any memory of the systematic force represented by

a(t) = /35(t). By solving Equations (2.13) and (2.24) using Laplace transforms, the velocity and position at any time are determined as v(t)

= v(o)x(t)

+ m -1

x(t -

ds

(2.16)

and X ( t ) = X ( O ) -~- V(O)~)(t) -~- 11t-1 ~0 t ~)(t - 8)F(8) ds,

(2.17)

where ~(t) is the displacement-initial velocity correlation function (DIVCF) defined by

( V ( O ) ( X ( t ) - X(0))} =

(2.18)

(v2)

From the GLE and the assumed properties of the random force, we obtain

x(t) = -

c~(t - s)x(s) ds,

X(0) = 1

(2.19)

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R. MURALIDHAR AND D. RAMKRISHNA

and

x(t) = - ~ .

(2.20)

Thus, the VACF and the DIVCF are determined from a knowledge of the friction kernel, a(t) which characterizes the dissipative interaction between the diffusing particle and the surrounding medium. The problem of describing diffusion in the GLE framework then reduces to that of modelling the friction kernel. For example, if we assume a power-law memory kernel

c~(t ) = C~ot-(1-'y)

(2.21)

which reflects the absence of a time scale characterizing the dissipation process, then the large time behavior of x(t) is given by (Muralidhar et al., 1991) lim x(t) ~ t--4oo

7sinTrTt-(l+'Y),

(2.22)

O~oTr

As discussed earlier, the negative tail of the VACF implies antipersistence of motion. The DIVCF also decays as a power law for large times. This implies that the characteristic time for the displacement to forget the initial velocity (,,~ f~o t~b(t) dt) is infinite. In other words the random walker forever remembers his initial velocity in complete contrast to classical diffusion. One of the uses of the GLE is that it serves as a starting point for deriving the partial differential equations (PDEs) of diffusion. Later in this section we derive the PDE for the joint probability density of position and velocity of the diffuser on a fractal. From this equation, the diffusion equation for the probability density of position may be derived. 2.4. AGE DEPENDENT DIFFUSION EQUATION We assume that a probability density p(r, t) may be defined such that p(r, t)dV is the probability of finding the particle in an Euclidean volume element dV about 7,. This density is defined as an ensemble average over several realizations of the porous network and dV is large enough to make the density function continuous and differentiable. We now derive evolutionary equations for the probability density p(r, t). Whenever the velocity autocorrelation time is finite, the successive displacements of the particle over time intervals larger than this correlation time may be regarded as independent. The total displacement then being a sum of such independent random variables is Gaussian as a consequence of the CLT (Feller, 1957). An important consequence of the long-time correlations of velocity fluctuations, typical of anomalous diffusion, is the breakdown of the CLT. In other words, the probability density for the position of the diffuser is no longer Gaussian at large times in general. The non-Markovian nature of the diffusion process as well as the break down of the Gaussian displacement law make the derivation of diffusion laws difficult.

DIFFUSION IN PORE FRACTALS

87

We assume that the stochastic force in the GLE is Gaussian and derive a diffusion equation following the approach of Mazo (1978). For this Gaussian case, the linearity of Equations (2.16) and (2.17) implies that the velocity and displacement are jointly Gaussian random processes and are, hence, completely characterized by the correlation matrix, Q whose components are given by Q l l = ((V - V(0)X) 2) = (V2)(1 - X2(t)),

(2.23)

Q12 = Q21 = ( ( v - v ( o ) x ) ( X - x ( o ) - V(0)~b)) = (V2)~b(1 - x),(2.24) Q22 = ( ( X - X ( 0 ) - V ( 0 ) ~ b ) 2) = (V 2) (2fot~b(s)ds-~b2(t)) .

(2.25)

In other words, on defining

Y1 = V - V(O)x

and Y2 = X -

X ( O ) - V(O)~b,

the probability density P(Yl, Y2; t) is given by

P(Yl, Y2; t) = ~:-1 ]det Q[-1/2exp(-yt Q - l y ) . Using the method of characteristic functions (Mazo, 1978), one can show that this probability density p( x , v, t) satisfies

Op Op O( 2 O) 2 02p 0-7 + u-~z = ~(t) v + (V )-~v p + (V )~(t)O-~v ,

(2.26)

where ((t) = _ 2 X

(2.27)

~r(t) = ~b - 1 + X.

(2.28)

and

The above is the analog of the Fokker-Planck equation (FPE) of classical diffusion theory. The classical FPE results for the Markovian diffusion described by the friction kernel .~(t) = ~(t). For this choice of memory function, we have x(t) = exp ( - - ~ ) and

88

R. MURALIDHARAND D. RAMKRISHNA

so that o- = 0 and ((t) = fl/ra. On using the above in Equation (2.26) and using the classical equipartition result (ra(V z) = kT) we recover the FDE, o-7 +

-

m 0v

v + kT/m

p.

One may be only interested in the position of the particle described by the density function

F(x, t) = f f p(x, v, t) dv.

(2.29)

It can be shown (Mazo, 1978) that this distribution function satisfies the equation

. OZF OF OFOt = (vZ)~b(1 - X)-~x2 - V(O)x Ox .

(2.30)

The above represents the diffusion characteristics of an ensemble of particles starting with a fixed initial velocity V(0). The last term represents the influence of memory of the initial velocity. For the case of the CLE, we have

x(t) -+ O,

~(t) --+ constant,

t --->

and the equation for the distribution function goes over to the familiar diffusion equation (Fick's second law). On averaging over all possible initial velocities V(0) and noting that the average initial velocity is zero, Equation (2.30) reduces to

OF ot

-

(v2)

OZF (1 - x) 0x2"

(2.31)

The above is the generalization of Fick's second law for non-Markovian diffusion. On taking the long time limit of the above, we obtain

OF

z

O2F

~-~ - ( V ) ~ b ( t ) ~ x 2 .

(2.32)

The above equation is expected to predict the asymptotic regime of diffusion. For the case of anomalous diffusion, we have from Equations (2.6) and (2.7)

~b(t) ~ Dt -~,

t -+

and Equation (2.32) becomes

OF OzF Ot = (V2)Dt-'~ Ox 2'

(2.33)

which is what one would also get by simply replacing the diffusion constant in the classical diffusion equation by a time dependent one noting that (Rz(t)) ,,~ t 1-'y. The solution to the above from the initial condition

F(x, 0) = 5(x)

(2.34)

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DIFFUSION IN PORE FRACTALS

may be readily obtained on making the transformation U = t 1-'r

which reduces Equation (2.33) to the classical form OF

D ( V 2) 0 2 F -

-

Ou

~

1 -- 7 0 x 2

The solution to the above for the delta function initial condition is

(DI(V;)-l/z F(z,u) = 47r 2) u

( exp

(1--7)z2~

47rD(VZ)uj ,

or

F(:c,t)

=

47r

t 1-'y

exp

-47rD(V2)tl_. r

(2.35)

Thus we find that at large times the diffusion is anomalous and (X2(t)) ~ t ~-'Y.

(2.36)

Thus, on making a Gaussian assumption for the random force, we arrive at a diffusion equation describing Gaussian anomalous diffusion. This process has been termed fractional Brownian motion by Mandelbrot (1968). Thus, a Gaussian random force is tantamount to assuming that the porous medium induces a fractional Brownian motion. 2.5.

MACROSCOPIC DIFFUSION EQUATIONS FEATURING AGE

A notable feature of the Equation (2.33) is the time dependent diffusivity, (V2)Dt-'L This time dependence restricts its utility to the special case of tracer diffusion in free space. One runs into difficulties for other situations because two particles entering the diffusion medium at different times, have different diffusivities. Thus, in order to know the diffusivity of a particle, knowledge of its age r, defined as the duration for which the random walker has been in the medium, is necessary. The age dependence of the diffusivity arises due to coarse graining the description of the medium of transport. Consider a one dimensional description of solute transport in a porous network. In considering the transport of a particle from one end to the another, only the pore connecting the two boundaries (the backbone) effectively transfers particles. The pores branching from the backbone, form deadends from which the particle has to extricate itself. These blocked ends may be, hence, viewed as traps. In a self-similar random medium, traps of different length scales exist, the smaller ones predominating the larger in number. After entering

90

R. MURALIDHARAND D. RAMKRISHNA

the medium, a random walker initially samples mostly the ubiquitous smaller traps. With increasing time, the particle samples bigger traps and escapes to the backbone with increasing difficulty, thus decreasing its effective motion along the x direction. In a coarse-grained description of the medium, the mere location (in this case the x coordinate) does not account for the nature of the trap in which the particle is present. The age dependent diffusivity accounts for the possible nature of the traps sampled by an average particle during its motion in the medium. Hence, the appropriate coordinates needed to describe the particle state are its location and age. The state of the system is then specified by the bivariate density function c(x, T, t) which is the concentration of particles of age x and age T at time t. The bivariate function satisfies the conservation equation

Ot

+

Or

- D(r)

ff~z2

(2.37)

The solution of Equation (2.37) for appropriate boundary conditions specifying the geometry and the nature of the interaction between the medium and the surrounding, completes the analysis of diffusion in pore fractals. The experimentally measurable concentration C(x, t) is readily obtained from the bivariate density from

C(x,t) =

~0 ~

c(x,'r,t)dT.

(2.38)

3. Generalized Hydrodynamics Description 3.1. BACKGROUND Generalized hydrodynamics (GH) may be employed to model diffusion in a selfsimilar porous medium. The GH theory has been disscussed in detail by McQuarrie (1976) and we here summarize the salient aspects. The framework is ideally suited for incorporating spatial effects in transport processes such as for example nonlocal diffusion. As the name suggests it is applicable for long-range (so-called hydrodynamic) fluctuations characterized by small wave-numbers k. The application of this theory to fractal diffusion in porous media has been made by Cushman and coworkers (Cushman, 1990; Cushman et al., 1991). The probability density for a particle to suffer a displacement a~ in time t is given by t) =

- (,,(t)-

,-(o))]/

where r represents the location of a tagged particle and the angular brackets denotes an average over an equilibrium ensemble. The space Fourier transform of the above, known as the self-part of the intermediate scattering function is given by G(k,t) =/R3eik'~G(w,t)dze.

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DIFFUSION IN PORE FRACTALS

This function may be determined experimentally (McQuarrie, 1976). From the above, one can write G in the form of a correlation function G(k, t) = (exp(ik. r ( t ) ) e x p ( - i k , r(0))).

(3.1)

Since r(t) - r(0) =

v(t')

Equation (3.1) may be written as G(k, t) = (exp(ik 9

v(t') dt')).

(3.2)

Since G is a time-correlation function, it satisfies a GLE (McQuarrie, 1976)

OG(k,t) Ot -

fo ~7(k, t)G(k, t - t') dt',

(3.3)

where 7 is the friction or memory kernel. Particle conservation along under the assumption of validity of continuum is expressed as

OG = -V. j Ot

(3.4)

where j is the probability or particle current. On assuming Zwanzig's constitutive relationship (Zwanzig, 1964) j = - D ( k , w ) . ikG(k,w)

(3.5)

we obtain a transport law. In Equation (3.5), the tilde represents a Laplace transform with respect to time. Equation (3.5) is the generalization of Fick's first law of diffusion. Classical Fickian diffusion is readily recovered from the above by making the assumption that the diffusion tensor is isotropic and independent of frequency and wave-number, i.e., D ( k , ~ ) = DI.

(3.6)

Using Equations (3.6) and (3.5) in Equation (3.4), one can show that

OG(k,t) = _Dk2~(k,t)" Ot Since G(k, 0) = 1 from Equation (3.1), we obtain from the above

G(k, t) = e -Dk2t,

(3.7)

which is the self-part of the intermediate scattering function for classical diffusion (McQuarrie, 1976). Thus, on inverting the above, one obtains a (

,t) -

1

-

(47rDt)3/2 exp which is the probability distribution function for classical diffusion starting at the origin at time t = 0.

92 3.2.

R. MURALIDHAR AND D. RAMKRISHNA DESCRIPTION OF ANOMALOUS DIFFUSION

In the previous section, we recovered classical diffusion by making the diffusivity wave-number and frequency independent. In this section we review the approach of Cushman and coworkers (Cushman, 1990; Cushman et al., 1991) of identifying the form of the diffusivity from the properties of fractal diffusion. Anomalous transport can emerge whenever the diffusion tensor is wave-number and frequency dependent. From Equations (3.5), (3.4) and (3.3), one can show that the friction kernel and diffusivity are related by "~(k, w) = k . D(k, co). k.

(3.8)

Thus, models for the friction kernel can serve to quantify the diffusion tensor. In this case, information on VACF and mean-squared displacement can be exploited to arrive at the diffusivity tensor. By expanding Equation (3.2) to second order in k and assuming stationarity of the VACF, Cushman et al. (1991) have shown that

k,co( )

-

a)

1

co

where C is the velocity correlation tensor C = (~,(t)~,(0)). Using Equation (3.3) and using Equation (3.8), they obtain

k. k. D(k,w). k ~

(co)k

1 - k . ~(co)co-1. k

.

(3.9)

The diffusion tensor may now be identified from a knowledge of the velocity correlation tensor. Cushman et al. (1991) assume that the porous medium induces an isotropic fractional brownian trajectory for which the mean squared displacement is 6 D t d / d , 0 < d < 1. For this trajectory they obtain

1)col-d/d

1 3Dr(d + D(k,co) = 3 [1 - 3DF(d + 1)k2co-d/d] "

(3.10)

Thus we find that the diffusivity to leading order is independent of k for small k and its frequency dependence obeys a power law. From Equations (3.4), (3.5) and (3.8), Cushman et al. (1991) obtain O GOt (x,t) - Vx.

~ot fn dr

3d y D ( y , r ) V x _ y G ( x - y,t-

r).

(3.11)

This integro-differential equation is the replacement of Fick's second law of diffusion. The space and time dependent diffusivity is obtained as the inverse transform of the frequency and wave-number dependent diffusivity.

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4. Summary and Unresolved Problems 4 . 1 . REVIEW OF APPROACHES

In this paper, we have reviewed the application of linear response theory to the problem of anomalous diffusion in self-similar pore fractals. The approaches exploit the intimate connection between particle diffusion and the velocity-autocorrelation function (VACF). In a homogeneous environment, velocity fluctuations of a particle lose coherence quickly and the diffusion is Markovian. In a self-similar pore fractal, a random-walker statistically samples larger and larger blocked ends with increasing time and as such the perceived motion displays memory. Embellished in linear response theory is a Generalized Langevin Equation (GLE) that describes dynamics of fluctuating microscopic variables. This equation provides for incorporating memory effects and serves as a starting point for deriving the laws of diffusion. The first approach presented in this paper is akin to the Langevin/Fokker-Planck description of classical diffusion. A GLE with a power-law friction kernel is used to describe particle dynamics consistent with the observed VACE Modeling the random force in the GLE as a Gaussian process leads to the analog of Fick's second law of diffusion. Generalized hydrodynamics serves as the basis for the second approach discussed in this review. In this theory, a GLE is written for the time-correlation function, which in this case is the self-part of the intermediate scattering function. The theory incorporates a frequency-wavenumber dependent constitutive law relating particle flux to the concentration gradient. Using the VACF characteristics of fractional Brownian motion, a transport law is derived. The transport laws derived by the two approaches are quite different and predict the mean squared displacement of anomalous diffusion. In view of the mathematical differences in the two diffusion equations, the diffusivities in the two approaches are not in the same footing. Although both models agree on the meansquared displacement (second moment for probability distribution of position), higher moments could be different. 4.2.

UNRESOLVED ISSUE~

While the theories described in this paper serve as a basis for understanding fractal diffusion, there are unresolved issues. The VACF for anomalous diffusion may be oscillatory in certain situations (Jacobs and Nakanishi, 1990) and not a simple power-law. The diffusion process may be anisotropic. It has been observed (Muralidhar et al., 1991) that tracer diffusion on random percolation clusters is anisotropic and isotropy is recovered (self-averaging out of anisotropy) only when diffusers are initially dispersed throughout the medium. Of greater concern is the presence of non-Gaussian effects in particle diffusion. Diffusion in a porous formulation may not in general be Gaussian. Indeed, computer simulations (Havlin and Ben Avraham, 1987) of random walks on percolation

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clusters, Sierpinski gaskets show marked deviations from Gaussian diffusion. The approaches presented in this paper can account for non-Gaussian effects. The first requires either making the random force spatially dependent or modeling it as a non-Gaussian process or both. The second requires a proper characterization of the wave-number and frequency dependence of the diffusivity. On a broader scale, coupled transport processes and chemical reaction must be investigated. Another issue is the importance of non-linear effects on transport and reaction.

Acknowledgements The authors are grateful to the Department of Chemical Engineering at Purdue University for a grant that made this research possible. The authors wish to acknowledge H. Nakanishi and D. Jacobs of the Physics Department at Purdue University for fruitful discussions.

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