16690
J. Phys. Chem. 1996, 100, 16690-16699
Adsorption Kinetics on Fractal Surfaces Massimiliano Giona† and Manuela Giustiniani*,‡ Centro InteruniVersitario sui Sistemi Disordinati e sui Frattali nell’ Ingegneria Chimica, Dipartimento di Ingegneria Chimica, UniVersita´ di Cagliari, Piazza d’Armi, 09123 Cagliari, Italy, and Dipartimento di Ingegneria Chimica, UniVersita´ di Roma “La Sapienza”, Via Eudossiana 18, 00184 Roma, Italy ReceiVed: May 23, 1996X
A model is proposed for the description of diffusion-controlled adsorption kinetics on fractal surfaces. This model is based on a constitutive equation between the mass flux and the concentration gradient of the adsorbing species expressed in terms of a Riemann-Liouville (fractional) operator of noninteger order ν. The order ν depends on the fractal dimension df of the adsorbent surface, ν ) df - dT, dT being its topological dimension. The model is compared with Monte Carlo simulations and with the approach proposed by Seri-Levy and Avnir and displays a good level of agreement with Monte Carlo data over all time scales.
1. Introduction In a recent paper, Seri-Levy and Avnir1 proposed a model for adsorption kinetics in diffusion-limited conditions. This model is based on an extension of Delahay’s analysis,2-4 which is valid for flat surfaces, so as to include the effects of fractality. In the article by Seri-Levy and Avnir, the resulting expression for the fractional coverage Γ(t)/Γe vs time t is a modification of the equation derived by Delahay and Trachtenberg2 in order to include the anomalous scaling of the adsorbed volume with the fractal dimension. The model proposed in ref 1 constitutes a significant initial contribution toward the development of an approximate macroscopic model for the kinetics of adsorption on fractal interfaces. The derivation of approximate mean-field models taking advantage of phenomenological and experimental studies and of the results of scaling theories for fractal structures has great importance, especially as regards practical engineering problems, and contributes toward a better understanding of the physics of fractals. The vast literature on mass-transfer kinetics across fractal interfaces and membranes is primarily oriented toward the derivation of anomalous scaling laws and interpretation of the structural properties of the interfaces (see for a review refs 5-7). Most of the literature on this topic is based on the analogy between membrane and interfacial kinetics and the electric response of rough electrodes, for which an anomalous impedance scaling with frequency ω (referred to as constant phase angle behavior, or CPA), Z(ω) ) R0 + R(iω)-η, 0 < η < 1, i ) x-1, has been observed.8-12 The exponent η depends on the roughness of the electrode and ultimately on the fractal dimension df. See ref 6 for a brief analysis of the different models proposed for the relationship between η and df. An initial attempt to derive a macroscopic theory from phenomenological scaling relations on the properties of rough electrodes was made by Le Mehaute,13 who introduced a convolutional relation expressed in terms of Riemann-Liouville integrals (fractional operators)14,15 between fluxes and driving forces (concentration gradients). Macroscopic models for diffusion on fractal media involving fractional operators have been proposed by Giona and Roman16,17 and by Nigmatullin18,19 and are widely applied in modeling the rheological properties of viscoelastic materials20 and the relaxation properties of dielectrics.21-23 †
Universita´ di Cagliari. Universita´ di Roma “La Sapienza”. X Abstract published in AdVance ACS Abstracts, September 1, 1996. ‡
S0022-3654(96)01518-3 CCC: $12.00
This article develops a macroscopic (approximate) model for diffusion-limited adsorption kinetics on fractal structures based on a constitutive equation between fluxes and concentration gradients expressed in terms of Riemann-Liouville operators. The primary concept involved in the application of RiemannLiouville operators in this case is the fact that the presence of surface roughness locally modifies the diffusion properties of the adsorbing species. Fick’s law cannot therefore be directly applied in the neighborhood of the interface, and correlation properties arise as a consequence of diffusion and trapping (adsorption) in particle motion. These correlations can be macroscopically modeled as memory effects in particle motion and can therefore be expressed by means of a constitutive equation of convolutional nature. The article is organized as follows. First we present an extensive Monte Carlo simulation quite close to saturation (Γ(t)/Γe = 0.8) on a self-similar fractal structure, following the technique discussed in ref 1. The approach adopted by SeriLevy and Avnir is reviewed. We then propose a macroscopic model of adsorption kinetics by introducing a fractional constitutive equation between fluxes and concentration gradients. Two cases are considered: the idealized case of an infinitely extended fractal layer and the more realistic case of a fractal layer of finite width. The exact meaning of these two cases is extensively discussed below. The theory presented is subjected to critical analysis and compared both with Monte Carlo data and with the equation proposed by Seri-Levy and Avnir. 2. Monte Carlo Simulations Two-dimensional simulations of diffusion-controlled adsorption kinetics on a fractal interface were performed by means of the Monte Carlo method on a square lattice according to the following rules.1 Nmol adsorptive molecules are randomly distributed in a periodic reservoir. The reservoir size is such that bulk conditions can be approached. The adsorbent surface is placed in the middle of the reservoir. At each step (a) a molecule is chosen at random; (b) if it is adjacent to an unoccupied surface site, it is adsorbed with probability pa; (c) if the molecule is already adsorbed, it can desorbe with probability pd. Steps a-c are repeated Nmol times for each physical time. Adparticles are noninteracting unless through excluded-volume effects. To keep the bulk concentration constant, the Ne particles adsorbed at each time are counted and Ne unadsorbed molecules are added and randomly distributed in the bulk region. © 1996 American Chemical Society
Adsorption Kinetics on Fractal Surfaces
J. Phys. Chem., Vol. 100, No. 41, 1996 16691
Figure 1. Two-dimensional Koch curve used in Monte Carlo simulations. The fractal dimension is df ) 1.5.
Low coverages are required in simulations in order to describe the adsorption equilibrium isotherm between adsorbed phase and fluid phase by means of the linear Henry’s law. This practically corresponds to the condition Nmol , Nsites, where Nsites is the number of sites of the adsorbent surface and is equivalent to a low bulk concentration c0. The problem is symmetrically schematized by making diffusion and adsorption possible from either side of the adsorbent interface. This is an abstraction that does not change the resulting adsorption curves in time and reduces the reservoir size required for small coverages on a rough adsorbent surface. This symmetry implies that diffusion occurs in the range x ∈ (-∞, ∞), x being the spatial coordinate in the direction of transport, and adsorption occurs on an interface placed in the middle of the reservoir x ) 0. In this kind of simulation, the diffusion coefficient D is 1 au instead of D ) 1/4 au (see ref 1), when we refer to a model defined for x ∈ (0, ∞), because of the spatial symmetry adopted (mirror symmetry around x ) 0). An adsorption probability pa ) 1 and a desorption probability pd ) 0.05 were chosen, as in ref 1. The equilibrium Henry constant is therefore K ) pa/pd ) 20. Simulations were performed until a coverage Γ(t)/Γe of about 0.8 was reached, which corresponds to about 106 time steps. It should be observed that the unit time in our simulations is 4 times greater than the unit time in the Seri-Levy/Avnir simulations, performed in such a way as to obtain a diffusivity of 0.25 au. A Koch fractal curve with fractal dimension df ) log 8/log 4 ) 1.5 was taken as a model surface in the simulations, Figure 1. The curves obtained at the third and fourth iterations of the generation process of the Koch fractal curve are considered in order to analyze the role played by the thickness of the fractal curve as regards the adsorption kinetics. 3. The Seri-Levy/Avnir Approach A brief analysis is given of diffusion-controlled adsorption, which has already been analyzed by Delahay on flat surfaces and by Seri-Levy and Avnir on fractal surfaces. As reported by Delahay and co-workers,2-4 in the presence of diffusion-controlled adsorption on an initially empty flat surface, the surface concentration of the adsorbed molecules at time t can be defined as
Γ(t) ) -∫0 J(x, τ)|x)0 dτ t
(1)
where x is the spatial coordinate in the direction of transport, x ∈ [0, ∞), the adsorbent interface is localized at x ) 0, and J is the component of the flux of the adsorptive species in the direction orthogonal to the surface,
∂c(x, t) ∂x
J(x, t) ) -D
(2)
Figure 2. Γ(t)/Γe vs t. Prediction of the Seri-Levy/Avnir model (solid line) and Monte Carlo simulations (dots). The simulation parameters are Nmol ) 30 000, Nsites ) 819 200, reservoir size 51 200 × 1200 lu, c0 ) 4.88 × 10-4, K ) 20, and D ) 1. The model parameters, eq 7, are D ) 1 and k ) 20.
where c(x, t) is the concentration of the adsorbing species in the fluid phase. The problem is described by means of the diffusion equation
∂c ∂ 2c )D 2 ∂t ∂x
(3)
equipped with the boundary and initial conditions c(∞, t) ) c0, for t g 0, c(x, 0) ) c0 for x > 0 with c(0, 0) ) 0. The other boundary condition derives from the condition that diffusion is controlling. At the surface, x ) 0, the condition of equilibrium between the bulk fluid phase with concentration c(0, t) and the adsorbed phase with concentration Γ(t) is assumed. The boundary condition at x ) 0 therefore reads as
Γ(t) ) f(c(0, t))
(4)
f(c) being a generic function representing the adsorption isotherm. The linear Henry’s adsorption isotherm is assumed in the hypothesis of low coverage. Equation 4 can therefore be rewritten as
Γ(t) ) Kc(0, t) The analytic solution of this problem is expressed
(5) by2
[ ] [ ]
Γ(t) xDt Dt ) 1 - exp 2 erfc Γe K K
(6)
where Γe is the equilibrium surface concentration for the bulk concentration c0, i.e. Γe ) Kc0. As observed by Delahay and Trachtenberg,2 the ratio Γ(t)/Γe is zero for t ) 0 and approaches unity as t tends to infinity. Moreover, the adsorption kinetics, eq 6, is independent of the bulk concentration c0 as a consequence of the linear adsorption isotherm. Together with scaling analysis, eq 6 was taken as a starting point by Seri-Levy and Avnir1 in order to derive an extension for fractal adsorbents:
[
] [
]
(Dt)dT+1-df (Dt)(dT+1-df)/2 Γ(t) ) 1 - exp erfc Γe k k2
(7)
where dT is the topological dimension (dT ) 1 or 2 for interfaces embedded respectively in two- and three-dimensional spaces), and k is a constant differing (also dimensionally) from K. Equation 7 was compared by Seri-Levy and Avnir with simulation data obtained on different adsorbent surfaces in the range t ∈ [0, 50 000]. Figure 2 shows the model prediction, eq 7, compared with Monte Carlo data of adsorption kinetics on
16692 J. Phys. Chem., Vol. 100, No. 41, 1996
Giona and Giustiniani
Figure 3. The same as Figure 2 observed at longer time scales.
the fractal interface with fractal dimension df ) 1.5, obtained under the simulation conditions Nmol ) 30 000, Nsites ) 819 200, reservoir size 51 200 × 1200 lattice units, corresponding to a bulk concentration of c0 ) 4.88 × 10-4, K ) 20, and diffusion coefficient D ) 1. The model, eq 7, with D ) 1 and k ) pa/pd ) 20, is represented by the solid line. Identity between k and K was analyzed by Seri-Levy and Avnir. A good level of agreement is observed up to coverages of about Γ(t)/Γe ) 0.45. This result is qualitatively the same as that reported by SeriLevy and Avnir up to coverages of Γ(t)/Γe ) 0.45. Direct quantitative comparison is not possible because the normalizing value Γe is different. This was calculated by Seri-Levy and Avnir by means of a Monte Carlo simulation of the equilibrium conditions at a bulk concentration c0, which is higher than Kc0. We assume that the adsorption isotherm, eq 5, is independent of the surface roughness, i.e. that the Henry constant K in eq 5 is independent of the surface fractal dimension. This is in line with experimental observations and with several theoretical works24,25 which point out that the influence of geometric (and energetic) heterogeneities is negligible at low coverages. The maximum coverage is therefore the same as in the flat case, i.e. Γe ) Kc0, and at low coverages the surface roughness is assumed to influence only the diffusional phenomena adjacent to it. The good level of agreement between simulation data and model prediction shown in Figure 2 is no longer observed at large time scales, as is evident in Figure 3, which shows the same data as in Figure 2 but in the broader range t ∈ [0, 200 000]. From Figure 3, it is evident that agreement between model and simulation data is restricted to short timee scales and that the model predicts slower adsorption kinetics than those observed from Monte Carlo simulations. 4. Fractional Diffusion Equation The theory for adsorption kinetics on fractal interfaces based on Riemann-Liouville (fractional) constitutive equations is developed over this section and the next. To develop the theory without “discontinuities” in the analysis, comparison with Monte Carlo experiments is postponed to section 6. Let us start with a phenomenological consideration. In their studies on the electric response of fractal electrodes, Pajkossy and Nyikos26,27 found that the flux of the transported entity J on a fractal interface behaves as follows: -γ
J∝t
df - dT + 1 2
∂c(x,t) ) DF(t) * ∇2c(x, t) ∂t
(10)
where x is the vector of the spatial coordinates, t is the time variable, c(x, t) is the concentration of the adsorbing species in the bulk phase, and the convolutional operator * is applied to a diffusiVity memory kernel DF(t). It should be pointed out that DF is not, strictly speaking, a diffusivity, but rather a diffusivitydependent parameter. Equation 10 is equivalent to the assumption of the generalized flux-concentration constitutive equation
J ) -DF(t) * ∇c(x, t)
(11)
The functional form of the diffusivity kernel DF(t) is derived below and is physically justified. In the following, we refer always to a one-dimensional transport model; that is, eqs 10 and 11 simplify as follows:
∂c(x, t) ∂2c(x, t) ) DF(t) * ∂t ∂x2
(8)
where the exponent γ is given by
γ)
Equations 8 and 9 correspond to a situation in which the concentration of the transported entity can be assumed to be equal to zero at the interface and the bulk phase concentration is initially constant. In this case, the classical diffusion equation predicts an exponent γ ) 1/2, which is recovered for df ) dT, i.e. in the limit of a flat interface (surface). It is evident that eqs 8 and 9 cannot be predicted by means of the classical Fick’s law, i.e. in terms of a flux that is proportional to the concentration gradient through a constant. A generalized approach to the problem of transport through fractal interfaces should therefore be attempted in terms of temporal correlations in the constitutive equation between the flux and the concentration gradient of the transported species. Memory effects in diffusion through non-homogeneous media have been theoretically studied by Goddard,28 who shows that this phenomenon can be represented by introducing memory kernels and convolution integrals in time to describe the history effects arising from microscopic diffusional relaxation. Fractional diffusion equations have been applied by Giona and Roman16,17 and by Nigmatullin18,19 to model diffusion in fractal media macroscopically and find application in the study of the relaxation of dielectric materials,21-23 in the rheology of viscoelasticity,29-33 and in general in connection with correlated random walks.34-37 In terms of the macroscopic approximate balance equations, the constitutive equation for the flux-concentration gradient dependence underlying memory effects can be expressed by means of a Riemann-Liouville integral operator with a characteristic noninteger order depending on the adsorbent fractal dimension. This model applies to the typical phenomenology occurring in electrochemistry and adsorption, i.e. diffusion in a bulk phase with adsorption on a surface, which was first investigated by Delahay et al.2 and further studied for fractal structures by SeriLevy and Avnir.1 To describe a diffusion-controlled phenomenon in the presence of fractal interfaces, a generalized diffusion equation is proposed of the following form:
(12)
or equivalently
(9)
J(x, t) ) -DF *
∂c(x, t) ∂x
(13)
Adsorption Kinetics on Fractal Surfaces
J. Phys. Chem., Vol. 100, No. 41, 1996 16693
where x is the coordinate in the direction of transport (roughly speaking, “macroscopically” orthogonal to the interface). It should be observed that in the flat case flux and concentration are rigorously calculated at x ) 0, the adsorbent surface being really identified at x ) 0. On the contrary, for a fractal adsorbent surface, a virtual line at x ) 0, acting as a planar projection of the rough interface, must be considered in order to impose the boundary conditions. This is made necessary by the fact that a fractal curve is nondifferentiable almost everywhere, and the direction orthogonal to the surface is therefore not defined at each point. This schematization represents the intrinsic approximation of any macroscopic model (balance equation) involving fractal (nondifferentiable) structures. Before applying eq 12 to diffusion-controlled adsorption kinetics, we shall derive the functional form of DF(t) by making use of the results expressed by eqs 8 and 9. The boundary and initial conditions associated with the model of Pajkossy and Nyikos26 are c(0, t) ) 0, c(∞, t) ) c0 ) constant for t > 0 and c(x, 0) ) c0 for x > 0. With these conditions, eq 12 can be solved in the Laplace domain to yield
c˜ (x, s) )
[ ( ) ]]
[
s c0 1 - exp s D ˜ F(s)
1/2
x
(14)
where the tilde symbol identifies the Laplace transforms of the functions of time c(x, t), DF(t), and s is the Laplace variable. The flux at the interface is therefore given in the Laplace domain by
()
˜ F(s) L[J]x)0 ) D
∂c˜ ∂x
x)0
) c0
( ) D ˜ F(s) s
1/2
(15)
where L[ ] indicates the Laplace transform. Equation 15 can be compared with the anomalous diffusion scaling law obtained for fractal interfaces, eq 8, in order to derive a functional form for the diffusivity kernel D ˜ F(s) as a function of s. In the Laplace domain, eq 8 becomes
L[J] ) Csγ-1
(17)
where the exponent ν is given by
ν ) df - dT
(18)
The constant DF0 has the dimensions [m] 2[s]df-dT-1 and is, strictly speaking, a diffusivity only if df ) dT, i.e. in the nonfractal case. Since eq 17 is derived from a scaling relation, eq 8, its validity is of course restricted to a given range of s values. We shall return to this topic below. Equation 17 can be regarded as an intrinsic property of the model, independently of the particular equilibrium condition assumed at the interface, thus making it possible to describe the general power law behavior observed for diffusion through fractal interfaces. It should be observed that eq 17 has a singularity, corresponding to the condition df g dT, i.e. ν g 0. The inverse of the Laplace transform of the flux can be obtained in terms of fractional derivatives.15 By substituting eq 17 into the expression for the Laplace transform of the flux, we obtain
(∂x∂c˜)
(19)
x)0
The inverse Laplace transform of eq 19 can be expressed in terms of Riemann-Liouville integral operator and is given by (see Appendix I)
(
)
DF0 ∂ t ∂c(x, τ) J|x)0 ) ∫ (t - τ)-ν ∂x Γ(1 - ν)∂t 0
x)0
dτ (20)
where Γ(1 - ν) is the gamma function of argument 1 - ν. Equation 20 represents the explicit form of the constitutive equation assumed for the dependence of the flux on the concentration gradient, i.e.
DF0 ∂ t ∂c(x, τ) J(x, t) ) ∫ (t - τ)-ν ∂x dτ Γ(1 - ν)∂t 0
(21)
Having derived the functional form of the diffusivity kernel, we shall go on to analyze the kinetics of adsorption in the presence of diffusional control, i.e. in the Delahay hypothesis for a fractal interface. As in the flat case discussed in section 3, two phases are assumed: a fluid phase with concentration c(x, t) and an adsorbed phase with a surface concentration Γ(t), defined by eq 1. The boundary conditions are the same as in the flat case, bearing in mind that in eq 1 the expression of the interfacial flux is given by eq 20 and that the equation describing diffusion is given by eq 12. By solving the balance equations with the given initial and boundary conditions, one obtains for the Laplace transforms Γ ˜ (s) the expression
Γ ˜ (s) xD˜ F(s)/K ) Γe s(xs + D ˜ (s)/K)
x
(22)
F
where Γe, as previously defined, is Γe ) Kc0. Substitution of eq 17 into eq 22 leads to the model solution in the Laplace domain:
(16)
C being a generic constant. To satisfy eqs 8 and 9, the Laplace transform of the kernel D ˜ F(s) must therefore satisfy the equation
D ˜ F(s) ) DF0s2γ-1 ) DF0sν
L[J]x)0 ) -DF0sν
Γ ˜ (s) xDF0/K ) (1-ν)/2 Γe s(s + xDF0/K)
(23)
The limiting case of a flat adsorbent surface is recovered for df ) dT, i.e. ν ) 0, and DF0 ) D. Equation 23 can be analytically inverted by means of the complex inversion formula, as shown in Appendix II, thus obtaining
Γ(t) ) Γe 1-
xDF0 Kβπ
∫0
∞
exp[-ty1/β] sin(βπ)
(y cos(βπ) + xDF0/K)2 + y2 sin2(βπ)
dy (24)
where the notation has been simplified by replacing β ) (1 ν)/2, and y is a real variable. It can be observed that the limiting flat case is satisfied for β ) 1/2 (i.e. ν ) 0), and eq 6 is recovered for DF0 ) D. Moreover, eq 24 satisfies the conditions Γ(0) ) 0 and Γ(∞) ) Γe ) Kc0. Here we anticipate an important point discussed extensively in section 6. The results obtained from the model expressed by eq 17, or equivalently by eq 24, are not as satisfactory at long time scales
16694 J. Phys. Chem., Vol. 100, No. 41, 1996
Giona and Giustiniani
as at short time scales. A good level of agreement between model and simulation data is observed only at short time scales. To some extent, this can be accounted for by pointing out that the effect of adsorbent roughness is expected to exercise a strong influence upon short-term behavior. This is a consequence of the fact that the spatial region in which diffusive motion is dominated by anomalies is restricted to the neighborhood of the fractal interface. At a distance from the interface, diffusion is still regular since the influence of fractality is negligible for x . 0. This effect can be accounted for either by describing it in terms of two temporal length scales or by considering the fact that the spatial region over which diffusion is anomalous has a finite width. The former approach is described in this section, and the latter, which leads to the concept of the fractal layer, in the next. Let us consider the first model formulation. Two different temporal behaviors can be identified and distinguished. At short time scales, the fractality of the adsorbent surface strongly affects the resulting adsorption kinetics, whereas interface roughness is be a negligible factor at long time scales. Starting from this observation, the model, eq 11, can be modified in order to describe the existence of these two characteristic temporal regions. The constitutive equation can be regarded as a superposition of the two characteristic behaviors:
J(x, t) ) -D0
∂c(x, t) ∂c(x,t) - DF(t) * ∂x ∂x
(25)
where the first term on the right-hand side is the Fickian contribution significant for large time scales (i.e. far from the interface) and the second one is the anomalous constitutive equation previously defined. The governing equation is obtained by substituting this expression into the balance equation ∂c/∂t ) -∂J/∂x (or in vector form ∂c/∂t ) -∇‚J). Equation 25 corresponds in the Laplace domain to the assumption of an equiValent diffusiVity kernel of the form
˜ F(s) ) D0 + DF0sν D ˜ eq(s) ) D0 + D
(26)
where eq 17 still gives the diffusivity kernel D ˜ F(s). It should be observed that eq 26 is still consistent with eq 8 since the latter is strictly a scaling relation. The resulting solution in the Laplace domain can therefore be obtained in the form of eq 22, where the equivalent diffusivity ˜ F(s) appears, i.e. D ˜ eq(s) rather than D
Γ ˜ (s) xD˜ eq(s)/K ) Γe s(xs + D ˜ (s)/K)
x
(27)
eq
Together with eq 26, eq 27 can be analytically inverted back to the time domain. The details can be found in Appendix II. In the case of the Koch curve with fractal dimension df ) 1.5 considered in the simulations, the solution attains the following form in the time domain:
[
Γ(t) Az2 2 ∞ ) 1 + ∫0 exp(-z2t) 2 Γe π (B + z )2 + A2z2 (B2 + A2z2)1/4[(B + z2) cos(θ/2) + Az sin(θ/2)] (B + z ) + A z 2 2
2 2
]
The parameters DF0 and D0 must of course depend on the bulk diffusion D in the following way:
D0 ) q0D
DF0 ) q1Dν
q0 being a dimensionless constant and q1 a constant with dimension [m]2(1-ν). As will be discussed in the following sections, very good results are obtained by correlating eq 28 with the simulation data. It should be pointed out that the proposed model does not take into account the influence of the fractal nature of the adsorbent structure on the local adsorption equilibrium. The same linear adsorption isotherm has therefore been applied as in the flat case. This assumption is theoretically justified on the grounds that the concentrations in the bulk and adsorbed phases are very low. In these conditions, the effect of surface roughness is expected to be negligible, as suggested in other works and as discussed in section 2. 5. Fractal Layer The macroscopic adsorption kinetic model expressed by eqs 27 and 28 represents the role of the adsorbent roughness in terms of a memory effect by means of the characteristic fractal dimension dependent exponent ν. This constitutes a global modeling of different effects due to roughness. In general, roughness gives a higher adsorbent site number than the planar projection of the interface, different site accessibility, and the existence of a nonzero thickness influencing diffusion close to the interface. The model of eq 27 neglects the fact that a fractal interface has a finite thickness. A step toward the description of this point is given by an extended version of the proposed model characterized by three phases: (i) a bulk region, far from the fractal interface, in which the diffusional phenomena are regular, i.e. not influenced by the fractality of the solid surface, and Fick’s law applies; (ii) a fractal region (as referred to the fractal layer), in which anomalous diffusion occurs, described by means of model eq 10; (iii) an adsorbed surface phase. A finite thickness of the fractal interface LF is thus introduced, delimiting the region in which anomalous diffusion occurs. The resulting balance equations should therefore read as follows. 1. In the bulk phase (z ∈ [0, ∞))
∂ 2c ∂c )D 2 ∂t ∂z
where A ) DF0/K2, B ) D0/K2, z is a real variable, and tan(θ) ) Az/B.
(30)
where D is the diffusion coefficient. 2. In the fractal layer with thickness LF, q(x, t) being the concentration in this phase,
∂2q ∂q ) DF * 2 ∂t ∂x
(31)
where x ∈ [0, LF]. The spatial coordinates x and z are of course related by z ) x - LF. 3. In the adsorbed phase, characterized by a surface concentration,
Γ(t) ) ∫0 DF * t
dz (28)
(29)
(∂q∂x)
x)0
dτ
(32)
The boundary and initial conditions are, as in the case of flat adsorbents, given by c(∞, t) ) c0, Γ(t) ) Kq(0, t) for t > 0, c(z, 0) ) c0 for z ∈ (0, ∞), q(x, 0) ) c0 for x ∈ (0, LF). The
Adsorption Kinetics on Fractal Surfaces
J. Phys. Chem., Vol. 100, No. 41, 1996 16695
continuity between the fractal layer and the bulk phase implies that
(∂c∂z)
c(0, t) ) q(LF, t)
D
z)0
) DF *
(∂q∂x)
x)LF
(33)
It is evident that the model discussed in the previous section corresponds to the particular case for LF f ∞, and the planar adsorbent surface is recovered if LF ) 0. The solution of the model in the Laplace domain is then obtained:
Γ ˜ (s) 1 ) + Γe s (J + 1) - (J - 1) exp[-2LF/M] K (34) s exp[-2LF/M](J - 1)(M - K) - (J + 1)(M + K) where M and J are defined as
M)
x
D ˜ F(s) s
J)
x
D ˜ F(s) D
Figure 4. D ˜ eff(s) vs s. The proposed models are compared with Monte Carlo simulations (]). Solid lines represent (a) eq 23 with DF0 ) 0.71, (b) eq 26 with D0 ) 3.5 × 10-3 and DF0 ) 0.71, and (c) the fractal-layer model, eq 34, with DF0 ) 0.73, LF ) 11. The simulation parameters are Nmol ) 30 000, Nsites ) 819 200, reservoir size 51 200 × 1200 lu, c0 ) 4.88 × 10-4, K ) 20, and D ) 1.
(35)
It should be observed that the Laplace transform of the solution of the model, eq 34, satisfies the particular cases for LF ) 0 and LF f ∞. In the case LF f ∞, eq 23 is recovered from eq 34; in the case LF ) 0, eq 34 yields
Γ ˜ (s) xD/K ) Γe x s( s + xD/K)
(36)
which is the Laplace transform of the solution in the flat interface case, eq 6. Moreover, eq 34 satisfies the condition Γ(0) ) 0. The description of simulation data by means of eq 34 proves quite satisfactory, as will be shown in the next section. 6. Results A convenient way to compare the proposed models with Monte Carlo results is to use eqs 22, 27, and 34 directly in the Laplace domain. We therefore calculated the Laplace transform of an interpolating function of the adsorption curves vs time. The interpolating function for the Monte Carlo data obtained on the Koch fractal interface (df ) 1.5) was chosen in the form f(t) ) 1 - exp[-at0.25 - bt0.5]. This interpolation satisfies the characteristic short-time behavior for the interface with fractal dimension df ) 1.5, i.e. J ∝ df(t)/dt ∝ t0.25. From eq 22, an effective diffusivity D ˜ eff(s) can be defined as
˜ (s)K sΓ 3 2
D ˜ eff(s) )
2
(Γe - sΓ ˜ (s))2
(37)
This representation allows direct comparison of the models discussed in the previous sections and Monte Carlo simulations. Figure 4 shows the simulation data (]) already reported in Figures 2 and 3 (simulation conditions: Nmol ) 30 000, Nsites ) 819 200, reservoir size 51 200 × 1200 lattice units, c0 ) 4.88 × 10-4, K ) 20) together with the model, eq 23, with ˜ eff(s) ) D ˜ F(s). This DF0 ) 0.71 (solid line a). In this case, D level of agreement is fairly poor at long time scales, i.e. at s f 0, and prediction in time domain is comparable with the results obtained by means of the Seri-Levy/Avnir model.
Figure 5. Γ(t)/Γe vs τ. Comparison of Monte Carlo simulations (dots) and the model, eq 28 (solid line), with D0 ) 3.5 × 10-3 and DF0 ) 0.71. The simulation parameters are Nmol ) 30 000, Nsites ) 819 200, reservoir size 51 200 × 1200 lu, c0 ) 4.88 × 10-4, K ) 20, and D ) 1. (a, top) Up to 600 000 time steps; (b, bottom) up to 50 000 time steps.
Figure 4 also shows the improvement in the description of the same simulation data achieved by means of the equivalentdiffusivity model eq 26 (solid line b) and the fractal-layer model (solid line c), where the effective diffusivity defined in eq 37 is plotted against s. The values of the parameter obtained from curve fitting are respectively D0 ) 3.5 × 10-3 and DF0 ) 0.71 for the equivalent-diffusivity model and DF0 ) 0.73 and LF ) 11 for the fractal-layer model. Since D ) 1, eq 29 yields the values q0 ) 3.5 × 10-3 and q1 ) 0.71 for the parameters qi. The level of agreement is satisfactory for both the models, as is evident in Figure 5a, where the inverse of the Laplace transform of the model, eq 28 (solid line), is shown together with simulation data (dots) up to 600 000 time steps. In Figure 5a, the normalized variables Γ(τ)/Γe and τ ) tD/L2, with L ) 1 lu are reported respectively on the y and x axes. The average
16696 J. Phys. Chem., Vol. 100, No. 41, 1996
Figure 6. D ˜ eff(s) vs s. The proposed models are compared with Monte Carlo simulations (]). The simulation conditions are Nmol ) 20 000, Nsites ) 409 600, reservoir size 51 200 × 1200 lu, c0 ) 3.25 × 10-4, K ) 20, and D ) 1. The adsorbent surface is the Koch fractal curve (df ) 1.5) at the third iteration. Solid lines represent (a) the fractal layer model and (b) the equivalent diffusivity model, eq 26. Fitting parameters: (a) DF0 ) 0.95 and LF ) 7; (b) DF0 ) 0.9 and D0 ) 0.013. N error, calculated as 1/N∑i)1 |ytheor - ysim|/ytheor, is 1.2% over 751 points. In Figure 5b, the good level of agreement between the model and Monte Carlo simulations is shown in the restricted range τ ∈ [0, 50 000]. Analogous results were obtained for the same fractal interface at the third iteration. In this case, the thickness of the fractal region is, of course, smaller than at the fourth iteration (42 as against 170 lattice units). The simulation conditions are Nmol ) 20 000, reservoir size 51 200 × 1200, Nsites ) 409 600, c0 ) 3.25 × 10-4, K ) 20, and D ) 1. Figure 6 shows the comparison of the fractal-layer (solid line a) and equivalent-diffusivity (solid line b) models with simulations in the Laplace domain in terms of effective diffusivity, eq 37. For the equivalent-diffusivity model, the values of the parameters obtained are DF0 ) 0.9 and D0 ) 0.013. It is interesting to observe that D0 is strictly related to the thickness of the fractal interface. As the iteration in the generation of the structure increases, and the thickness of the interface increases correspondingly, the parameter D0 becomes smaller. This is in agreement with the observation that the case LF f ∞ of the fractal-layer model, eq 34, corresponds directly to the particular case D0 ) 0 in eq 26, i.e. to eq 23. For the fractal-layer model, eq 34 was used to find the best fit for LF ) 7, D0 ) 0.95. From Figure 6, it is evident that both models provide a good description of the Monte Carlo simulation. Figure 7 shows the inverse Laplace transform of the equivalent-diffusivity model, eq 28, with DF0 ) 0.9 and D0 ) 0.013 (solid line), together with simulation data (dots). The dimensionless variables Γ(τ)/Γe and τ, where τ ) Dt/L2, L ) 1 lu, are reported as the x and y axes. The average error is about 2% over 2000 points. The higher fluctuations in Monte Carlo data are due to the smaller number of particles Nmol in the simulation. The model analysis can be completed by examining the behavior for different Henry’s adsorption equilibrium constants K and for different diffusivities in the bulk fluid phase D. Figure 8a,b presents a comparison in the time domain of Monte Carlo data with the equivalent-diffusivity model, eq 28, for different values of the Henry’s constant K. In Figure 8a, dots are used to represent the Monte Carlo results obtained with the following simulation conditions: Nmol ) 100 000, reservoir size 51 200 × 1200 (resulting concentration c0 ) 1.63 × 10-3),
Giona and Giustiniani
Figure 7. Γ(τ)/Γe vs τ. Comparison of Monte Carlo simulations (dots) and the model, eq 28 (solid line), with D0 ) 0.013 and DF0 ) 0.9. The simulation parameters are Nmol ) 20 000, Nsites ) 409 600, reservoir size 51 200 × 1200 lu, c0 ) 3.25 × 10-4, K ) 20, and D ) 1. The adsorbent surface is the Koch fractal curve at the third iteration.
Figure 8. Γ(τ)/Γe vs τ at different values of Henry’s constant. Simulation data: Nmol ) 100 000, Nsites ) 819 200, reservoir size 51 200 × 1200 lu, c0 ) 1.63 × 10-3, D ) 1, and (a, top) K ) 2, (b, bottom) K ) 10. The solid line is the prediction obtained by the model, eq 28, with the same parameters as Figure 5, D0 ) 3.5 × 10-3 and DF0 ) 0.71.
D ) 1, and K ) 2 on the Koch fractal interface with df ) 1.5 (fourth iteration). The model predictions (solid line) correspond to the values of the parameters previously obtained, i.e. D0 ) 3.5 × 10-3 and DF0 ) 0.71, K ) 2, and can be considered fairly satisfactory. Figure 8b refers to the same simulation conditions as in Figure 8a with a different value of the Henry’s constant, K ) 10. The level of agreement between model (solid line is eq 28) and simulations (dots) is very good. In Figure 8a,b the characteristic model parameters appearing in eq 28 are the same as in Figure 5. This confirms that the assumed model of the equivalent diffusivity, eq 28, simply describes anomalous diffusion due to the fractal nature of the adsorbent surface, the parameters D0 and DF0 being independent of the equilibrium Henry’s constant. The same conclusion can be drawn by analyzing the adsorp-
Adsorption Kinetics on Fractal Surfaces
Figure 9. Γ(t)/Γe vs t at bulk diffusivity D ) 10. The Monte Carlo simulation (dots) is compared with the model prediction, eq 28 (solid line). The simulation conditions are Nmol ) 100 000, c0 ) 1.63 × 10-4, Nsites ) 819 200, and K ) 10, and the model parameters q0 ) 3.5 × 10-3 and q1 ) 0.71.
J. Phys. Chem., Vol. 100, No. 41, 1996 16697 Giona et al.38 recently analyzed the sorption properties of volume fractals by applying exact Green function renormalization. More specifically, a fractal structure, in which the concentration of the diffusing species is initially zero, is placed in contact with a reservoir (of infinity capacity) at constant concentration c0. This situation corresponds to sorption experiments customarily performed in membrane science, in the study of microporous materials,43 and in the study of the controlled release of swollen polymeric matrices.42 Solute motion is assumed to be purely diffusive within the fractal structure. The quantity of interest as providing a macroscopic description of sorption kinetics is the ratio between the quantity of solute entering the structure up to time t and the total quantity entering the structure asymptotically (i.e. for t f ∞). In the case of bulk fractals (e.g. the Sierpinski gasket), the results obtained by Giona et al. can be summarized by the scaling relation
Mt/M∞ ∝ t(df-dfb)/dw
Figure 10. Domain of the complex plane used to obtain the Laplace inverse, eq 23 and eq 27.
tion curves obtained at different values of diffusivity in the bulk phase. Figure 9 shows simulation (Nmol ) 100 000, reservoir size 51 200 ×1200 lu) and model predictions for D ) 10 and K ) 10. The model is fully predictive since the parameters D0 and DF0 are obtained by applying eq 29 with the values of q0 and q1 derived from Figure 4, i.e. D0 ) 3.5 × 10-2 and DF0 ) 2.25. As in the case of a flat adsorbent surface, the adsorption curve vs the dimensionless time τ ) tD/L2, with L ) 1 lu, is independent of the bulk diffusion coefficient D. This can be directly obtained from the functional form of the model, eq 28, which depends on D only through the dimensionless group τ ) tD/L2. This is also confirmed by the simulations, by observing that the adsorption curves obtained for D ) 1 and D ) 10 coincide for the same values of the other simulation parameters. To sum up, both the equivalent-diffusivity model and the fractal-layer model furnish satisfactory predictions of kinetic data. In the case of the equivalent-diffusivity model, a complete analysis of the functional dependence of the model parameters on K and D has been carried out and is summarized in eq 29. The only topic requiring further investigation is the behavior of the fractal-layer width LF with respect to the structural (geometric width of the interface) and dynamic parameters, i.e. D. Unfortunately, such analysis requires extremely large lattice (reservoir) size and is beyond our present capacity. This important topic will hopefully be analyzed elsewhere. 7. Sorption Kinetics: Scaling Observations This section focuses on an interesting analogy between the sorption behavior of volume fractals and the adsorption properties of fractal interfaces.
(38)
where dw is the walk dimension (dw ) 2 for Euclidean media, dw > 2 for fractals), df the fractal dimension of the structure, and dfb the dimension of the transfer surface exposed to the external reservoir concentration c0. In particular, if the exchange area reduces to a single point, dfb ) 0, and eq 38 becomes Mt/ M∞ ∝ tds/2, where ds is the spectral (fracton) dimension, since 2/dw ) ds/df. In the case dfb ) df - 1, Mt/M∞ ∝ t1/dw. It is interesting to observe that eq 38, which was obtained by applying exact renormalization to bulk fractals, also describes the sorption properties of fractal interfaces. Indeed, from eqs 8 and 9 one obtains
Mt/M∞ ∝ ∫0 J(τ) dτ ∝ t(1+dT-df)/2 t
s
(39)
where dfs is used to indicate the fractal dimension of the interface in order to avoid confusion with the fractal dimension df of the bulk entering into eq 38. Let us compare eq 39 with eq 38. In the case of adsorption on a fractal surface, diffusion takes place on a regular Euclidean medium with fractal dimension df ) dT + 1 and with walk dimension dw ) 2. In this case, the transfer area dimension dfb coincides with the fractal dimension of the interface dfs. This indicates that eq 38 is a unifying relation that describes the sorption properties of both bulk fractals and fractal interfaces. The result expressed by eq 38 shows that the quantity Mt/ M∞, which is customarily measured in sorption experiments, depends strongly on the dimension of the exposed surface dfb. Different scaling behaviors may therefore be expected as a consequence of different distributions of adsorbing centers, e.g. in the case of a multifractal distribution, as discussed by Gutfraind et al.41 in connection with catalytic properties. 8. Concluding Remarks This article proposes a macroscopic theory for the kinetics of adsorption on fractal structures. Although the analysis is focused exclusively on the diffusion-controlled regime, it should be borne in mind that the theory applies to a broader phenomenology since different regimes are mathematically described by the boundary condition at x ) 0. Indeed, in the case of linear transfer kinetics at the interface, the boundary conditions become
16698 J. Phys. Chem., Vol. 100, No. 41, 1996
∂Γ ∂c ∂c ) - DF * |x)0 -DF * |x)0 ) h(Kc - Γ)|x)0 ∂t ∂x ∂x
Giona and Giustiniani
(40)
where h is the mass-transfer coefficient and eq 12 can be solved with these conditions. The diffusion-controlled regime can be obtained from eq 40 for h f ∞. By analogy, the representation of the anomalies in diffusion/ adsorption kinetics of fractal interfaces by means of fractional operators finds confirmation in the exact results obtained in the case of diffusion on fractal media by applying Green function renormalization.38 In the case of sorption kinetics on fractals, a power-law behavior in s was observed for the Laplace transforms of the Green functions. Both the phenomenologies, i.e. (1) adsorption across a fractal interface and (2) sorption kinetics of bulk fractals, can be unified by means of a single scaling relation, eq 38, in which the exponent depends on the dimension of the exchange area dfb, on the fractal dimension of the bulk structure df, and on the walk dimension dw. To achieve a complete understanding of diffusion across fractal interfaces and to overcome the intrinsic limitations of Monte Carlo simulations, it would of course be highly advisable to extend to this case the Green-function renormalization successfully applied in the case of diffusion and adsorption on fractal media. This article introduces the concept of the fractal layer, which corresponds to the physical region in which correlation properties in the motion of the adsorbing particles are modified by the presence of the fractal interface. While the width of the fractal layer may not coincide with the geometric width of the interface in the direction of transport, a relationship between these two quantities does exist. Diffusion within the fractal layer is described by means of the constitutive eq 13. In conclusion, we should just like to mention another approach to model diffusional dynamics within the fractal layer that is currently under investigation. This approach is based on the introduction of a multifractal field of diffusivities.39 The physical explanation of this approach is grounded on the properties of projected densities of fractal interfaces which display a multifractal nature.41 The fractional approach discussed in this article and the multifractal description discussed by Giona and Adrover39 are not antithetical but complementary to each other. Details of the latter approach are presented elsewhere.39 Acknowledgment. The authors thank R. R. Nigmatullin for sending useful material. Appendix I. Derivation of Eq 20 The inverse Laplace transform of eq 19 can be expressed in terms of fractional derivatives15 by recalling that a basic property of the Laplace transform of a ν-order fractional derivative (ν > 0) of a continuous function f(t) with vanishing initial conditions is15
L[Dνf(t)] ) sνL [f(t)]
(41)
where Dν represents the fractional derivative of noninteger order ν, ν > 0, of the function f(t) with respect to time. By applying eq 41 to eq 19, one obtains
[ (∂x∂c) ]
L[J]x)0 ) -DF0L Dν
x)0
The explicit expression of the flux, eq 19, is therefore given by
[ (∂x∂c) ]
J|x)0 ) -DF0 Dν
x)0
(42)
with a noninteger positive order ν depending on the adsorbent fractal dimension and on the topological dimension, ν ) df d T. The fractional derivative, eq 42, can also be expressed in terms of the Riemann-Liouville integral operator. To do this, we recall that the integral form of the fractional derivative D-qf(t) of a function f(t), given that q > 0, is defined as
D-qf(t) )
t 1 (t - τ)q-1f(τ) dτ ∫ 0 Γ(q)
(43)
where Γ(q) is the gamma function of argument q. In the case of positive ν, the Laplace transform of Dνf(t) can be expressed as
L[Dνf(t)] ) L [Dm[D-qf(t)]]
(44)
where q ) m - ν > 0 and m is the smallest integer greater than ν. Given that 0 e ν e 1, by virtue of the definition ν ) df dT, it follows that m ) 1, and therefore q ) 1 - ν. By applying eqs 43 and 44 to eq 42, it follows that
DF0 ∂ t ∂c dτ J|x)0 ) ∫ (t - τ)-ν ∂x x)0 Γ(1 - ν)∂t 0
( )
(45)
Equation 45 is the explicit representation in the time domain of the flux-concentration constitutive equation at the interface, x ) 0. Appendix II. Laplace Inversion of Eqs 23 and 27 The inverse Laplace transform of eq 23 can be obtained analytically by means of the complex inversion formula. To simplify the notation, let us indicate a ) xDF0/K > 0, so that eq 23 becomes
Γ ˜ (s) a ) β Γe s(s + a)
(46)
where β ) (1 - ν)/2 ) (1 - D + dT)/2 > 0. If β-1 is a multiple of 2, then eq 46 has only one singularity (a branching point) at s ) 0. In the case of the Koch curve considered, ν ) 1/2, so that β ) 1/4. In this case, it can easily be seen that the term sβ + a * 0 for every value of s in the complex plane. In fact, let us assume that there exists a value s* satisfying the equation sβ* + a ) 0. Then its βth power must be a real negative number as a is a real positive number. Let us write s* ) r exp(iθ). Its power is sβ* ) rβ exp(βiθ) ) rβ (cos(βθ) + i sin(βθ)). Since sβ* is a real number, sin(βθ) ) 0. Therefore, if 1/β is a multiple of 2, it follows that sin(θ) ) 0. In this case, it follows that Γ ˜ /Γe admits no singularities other than s ) 0, the square root of a real number always being non-negative. By using the complex inversion formula, the inverse of the Laplace transform of eq 46 can be expressed by the contour integral in the complex plane
Γ(t) a x0+i∞ ) ∫ exp(ts)s(sβ 1+ a) ds Γe 2πi x0-i∞ where x0 is a generic real abscissa in the half-plane of convergency, Re[s] > 0. The residual of Γ ˜ /Γe at s ) 0 is given by limsf0 a/[(sβ + a)] ) 1.
Adsorption Kinetics on Fractal Surfaces
J. Phys. Chem., Vol. 100, No. 41, 1996 16699
By using the Cauchy theorem on a contour of the complex plane excluding the singularity s ) 0, Figure 10, and letting the radius R of the arc of circumference CR tend to infinity and , the radius of circumference surrounding s ) 0, tend to zero, it follows that
Γ(t) a 1 ds )1exp(ts) β ∫ C 1 Γe 2πi s(s + a) a ∫ exp(ts)s(sβ 1+ a) ds 2πi CII where on the branch CI we can replace s ) u2 exp(iπ) ) -u2, -∞ < u e 0, and on the branch CII, s ) u2 exp(-iπ) ) -u2, 0 < u e ∞, and on the branch CII, s ) u2 exp(-iπ) ) -u2, 0 < u e ∞. These substitutions lead to the following expression:
exp(-tu2) Γ(t) a ∞ du ) 1 + ∫0 Γe πi u[u2β(cos(βπ) + i sin(βπ)) + a] exp(-tu2) a ∞ du (47) ∫ πi 0 u[u2β(cos(βπ) - i sin(βπ)) + a] which can be expressed, after some algebraic manipulation, in the form of eq 24. The inverse Laplace transform of eq 27, can be derived in the same way. By substituting eq 26 into eq 27, we have
Γ ˜ (s) xAsν + B ) Γe s(xs + xAsν + B)
(48)
where A ) DF0/K2 and B ) D0/K2. Since ν ) 1/4, there is only one singularity at s ) 0, as discussed above. The residual at s ) 0 is 1. We thus have, as above,
Γ(t) xAsν + B a ds )1exp(ts) ∫ Γe 2πi C1 s(xs + xAsν + B)
xAsν + B a ds (49) exp(ts) ∫ 2πi CII s(xs + xAsν + B) By replacing on CI s ) u2 exp(πi) ) -u2, with -∞ < u e 0, and on the CII, s ) u2 exp(-iπ) ) -u2, with 0 < u e ∞, eq 49 becomes 2 Γ(t) 1 ∞exp(-tu ) xAui + B ) 1 + ∫0 du Γe πi u (iu + xAui + B) 2 x-Aui + B 1 ∞exp(-tu ) du (50) ∫ πi 0 u (-iu + x-Aui + B) from which eq 28 follows after some algebra.
References and Notes (1) Seri-Levy, A.; Avnir, D. J. Phys. Chem. 1993, 97, 10380. (2) Delahay, P.; Trachtenberg, I. J. Am. Chem. Soc. 1957, 79, 2355. (3) Delahay, P.; Fike, C. T. J. Am. Chem. Soc. 1958, 80, 2628. (4) Delahay, P.; Mohilner, D. M. J. Am. Chem. Soc. 1962, 84, 4247. (5) Sapoval, B. In Fractals and Disordered Systems; Bunde, A., Havlin, S., Eds.; Springer Verlag: Berlin, 1991. (6) Larsen, A. E.; Grier, D. G.; Halsey, T. C. Fractals 1994, 2, 191. (7) Pajkossy, T. J. Electroanal. Chem. 1991, 300, 1. (8) Nyikos, L.; Pajkossy, T. Electrochim. Acta 1985, 30, 1533. (9) Bates, J. B.; Chu, Y. T.; Stribling, W. T. Phys. ReV. Lett. 1988, 60, 627. (10) Kaplan, T.; Gray, L. J.; Liu, S. H. Phys. ReV. B 1987, 35, 5379. (11) Pajkossy, T.; Nyikos, L. Electrochim. Acta 1989, 34, 181. (12) Chassaing, E.; Sapoval, B.; Daccord, G.; Lenormand, R. J. Electroanal. Chem. 1990, 279, 67. (13) Le Mehaute, A. J. Stat. Phys. 1984, 36, 665. (14) Oldham, K. B.; Spanier, J. The Fractional Calculus; Academic Press: New York, 1974. (15) Miller, K. S.; Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equations; J. Wiley & Sons: New York, 1993. (16) Giona, M.; Roman, H. E. J. Phys. A 1992, 25, 2093. (17) Roman, H. E.; Giona, M. J. Phys. A 1992, 25, 2107. (18) Nigmatullin, R. R. Ph.D. Thesis, Kazan State University, Kazan, 1992 (in Russian). (19) Nigmatullin, R. R. Theor. Math. Phys. 1992, 90, 354 (in Russian). (20) Rabotnov, Yu. N. Elements of Hereditary Solid Mechanics; Mir Publ.: Moscow, 1980. (21) Nigmatullin, R. R. Phys. Status Solidi 1984, 123, 739. (22) Nigmatullin, R. R. Phys. Status Solidi 1984, 124, 389. (23) Nigmatullin, R. R. Phys. Status Solidi 1986, 133, 425. (24) Pfeifer, P.; Obert, M.; Cole, M. W. Proc. R. Soc. London A 1989, 423, 169. (25) Fripiat, J. J.; Gatineau, L.; Van Damme, H. Langmuir 1986, 2, 562. (26) Pajkossy, T.; Nyikos, L. Electrochim. Acta 1989, 34, 171. (27) Borosy, A. P.; Nyikos, L.; Pajkossy, T. Electrochim. Acta 1991, 36, 163. (28) Goddard, J. D. Ind. Chem. Res. 1992, 31, 713. (29) Nonnenmacher, T. F. J. Phys. A 1990, 23, L697. (30) Giona, M.; Cerbelli, S.; Roman, H. E. Physica A 1992, 191, 449. (31) Smit, W.; de Vries, H. Rheol. Acta 1970, 9, 525. (32) Bagley, R. L. J. Rheol. 1983, 27, 201. (33) Rogers, L. J. Rheol. 1983, 27, 351. (34) Nonnenmacher, T. F.; Metzler, R. Fractals 1995, 3, 557. (35) Hilfer, R.; Anton, L. Phys. ReV. E 1995, 51, R848. (36) Tatom, F. B. Fractals 1995, 3, 217. (37) Hilfer, R. Fractals 1995, 3, 549. (38) Giona, M.; Schwalm, W. A.; Adrover, A.; Schwalm, M. K. Chem. Eng. J., in press. (39) Giona, M.; Adrover, A. First International Conference on Chaos and Fractals in Chemical Engineering CFIC96, Rome, 2-5 September, 1996. (40) Giona, M.; Giustiniani, A.; Patierno, O. First International Conference on Chaos and Fractals in Chemical Engineering CFIC96, Rome, 2-5 September, 1996. (41) Gutfraind, R.; Sheintuch, M.; Avnir, D. J. Chem. Phys. 1991, 95, 6100. (42) Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, 1986. (43) Ruthven, D. Principles of Adsorption and Adsorption Processes; J. Wiley & Sons: New York, 1984.
JP961518L
