J. Phys. Chem. B 1997, 101, 3781-3787
3781
Diffusion-Limited Reaction Rates on Self-Affine Fractals Rama Kant Colle´ ge de France, Physique de la Matie` re Condense´ e, 11, Place Marcelin-Berthelot, 75231 Paris Cedex 05, France ReceiVed: October 14, 1996; In Final Form: February 7, 1997X
The diffusion-limited reaction rate is determined on an approximately self-affine corrugated (random) surface fractal. We obtain the exact result for the low roughness and the asymptotic results (in three time regions) for the arbitrary and large roughness surfaces. These results show the anomalous time dependence for the mean flux and the mean excess flux for the large and small roughness surfaces, respectively. The intermediate time behavior of the reaction flux for the small roughness interface has the form 〈J〉 ∼ t-1/2 + const t-3/2+H, but for the large roughness interfaces it has same form as predicted earlier, 〈J〉 ∼ t-1+H/2, where H is Hurst’s exponent. This nonuniversality and dependence of intermediate time behavior on the strength of fractality of the interface is not conceived by earlier works. We also show the localization of the active zones in the presence of roughness. Finally, these results unravel the connection between the total reaction flux and the crossover times to the roughness characteristics like fractal dimension, lower and upper fractal cutoff lengths, and the amplitude of the fluctuations (strength) of the fractal.
1. Introduction The transport across the complex interface is a basic problem in the study of heterogeneous kinetics of several natural and industrial processes. Recently their importance has been emphasized in various fields like spin relaxation,1 fluorescence quenching,1,2 heterogeneous catalysis,3 enzyme kinetics,4 heat diffusion,5 membrane transport,6,7 and electrochemistry.6-21 Two problems in the electrochemical contexts have drawn a lot of attention: (i) the capacitive behavior of the rough/porous electrode19-24 and (ii) the diffusion-controlled charge transfer on the rough/porous interface.8-21 Sources of complexities in these systems either arise from the geometric disorder or the energetic disorder or both of them. These disorders have varying degrees of randomness and can be fractal or nonfractal in nature. The fractal irregularities are usually understood in terms of self-similar25-27 or, in general, as self-affine26-34 fractals. In this paper we are concerned with the understanding of the diffusion-limited reaction rates to the random corrugated surfaces which exhibit the scale invariance over a limited range of scales. The diffusion-limited transport or reactions represent a physical situation where the molecules (or ions) diffuse from the bulk to the interface and react or are absorbed when encountered. The total reaction flux of the molecules or ions, J(t), at the interface is given by the normal derivative, ∂n ) nˆ ‚∇, of the concentration at the interface
∫ΣdS ∂n δC
J(t) ) -D
(1)
where δC ) C(r b,t) - C° satisfies the diffusion equation
∂δC(b,t)/∂t r ) D∇2 δC(b,t), r for t > 0
(2)
There is a (local) transfer kinetics at the interface, and this is represented by the boundary condition, δC|r∈Σ ) δCs at the interface (Σ). Here D is the bulk molecular diffusion constant, and δCs is the difference between surface and bulk concentration. For the absorbing boundary condition it is equal to bulk concentration (-C°). At initial time and far off from the interface a uniform initial and bulk concentration C° is mainX
Abstract published in AdVance ACS Abstracts, April 1, 1997.
S1089-5647(96)03141-0 CCC: $14.00
tained viz. C(r b,t)0) ) C(zf∞,t) ) C°. The direct approach to this problem is to solve this random boundary value problem! The random surfaces for many natural and artificial surfaces are best described as self-affine fractal scaling functions.27-34 These functions remain statistically invariant under the scale transformation: (x,y,z) f (λx, λy, λH z) where H is the roughness exponent. The surfaces with limited scales of such an invariance property can be described by the stationary, Gaussian random processes with a power-law spectrum over a limited range of wavenumbers.34 These surfaces also have a power-law mean square height increment function (or the structure function, SF) over a few decades of (intermediate) length scales and saturate to a constant value for the large length scales which is given by34,35
〈[ζ(x) - ζ(x′)]2〉 ≈ τ2(1-H)X2H, Li < X < Lo ≈ 2h2, X > Lo
(3)
where ζ(x) is the centered surface profile function, the angular brackets designate ensemble averaging, X ) |x - x′|, and Li and Lo are the inner and the outer crossover lengths, respectively.34 Lo can be looked upon as a correlation length above which the height is considered as uncorrelated and h2 is the mean square width of the interface. Topothesy (τ)30,35 is defined as the horizontal distance between two points on which SF is equal to τ2. Topothesy of an “ideal” fractal surface is related to the normalizing factor (strength) of the power-law spectra (T) by the following relation: τ2(1-H) ) (T/π1/2H22H) Γ(1 H)/Γ(1/2 + H). In our earlier paper34 we showed that the scaling surfaces of this type can be approximated by the following power-law spectra
{
〈|ζˆ (K)|2〉 ) TK-2H-1 min
K < Kmin Kmin e K e Kmax (4)
T|K|-2H-1 TK-2H-1 exp[-(K - Kmax)/βKmax] K > Kmax max
where ζˆ (K) is the Fourier transform of surface profile ζ(x), K is the wavenumber, and T is the normalizing factor (related to the amplitude of the fluctuations (strength) of the fractal). Kmin © 1997 American Chemical Society
3782 J. Phys. Chem. B, Vol. 101, No. 19, 1997
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and Kmax are low- and high-wavenumber cutoffs, respectively, and β is the coefficient of correlation scale. H is Hurst’s (or roughness) exponent, and it is the measure of persistence in roughness profile.27,34 The values of H are restricted to the interval 0 and 1. In general for a limited range scaling surface, H in eq 3 has a nonlinear dependence on the H in eq 4 and the range of roughness scales (see eq 13 for F), but this nonlinearity is less important for the intermediate values of H and large F.34 A surface with Hurst’s exponent (H) can be treated as a fractal surface with fractional dimensionality (DH)
DH ) 2 - H
(5)
where the fractional dimension of the curved side of the corrugated surface lies between 1 and 2. The power spectrum of eq 4 approximately represents the power spectrum which shows a gradual flattening for low wavenumbers, an intermediate power-law, and a gradual decrease at high wavenumbers. The analytical approaches to this class of problems are considered to be difficult.9,23 The scaling arguments have helped to some extent to increase our understanding of the diffusionlimited reactions which occur in fractal spaces (porous) or on the fractal boundaries (rough). The scaling arguments and numerical and experimental2,8-13,19-21 studies have shown that the reaction rates follow approximately a power-law relation in time. The scaling form of reaction rates is represented by the following relation:
J(t) ∼ t-β
(6)
De Gennes1 showed that β ) (DH - 1)/2 for the problem of diffusion-controlled nuclear magnetic relaxation in porous media with fractal dimension DH. Later the same result was derived in context of diffusion-controlled adsorption on porous fractal catalyst3 and for the electrochemical rough interfaces which follow the diffusion-controlled current under potentiostatic condition.11 Similarly, the power-law flux transient on a fractal interface in an arbitrary Euclidean dimension (DE) of the embedding space is obtained using heuristic considerations5,8,9,36 and rather simple solvable models,8,9 and these works justify the following equation
β ) (DH + 2 - DE)/2
(7)
where DE ) 2 for the curved side of a corrugated surface fractal. Recently, similar scaling results were obtained for the heat diffusion from the self-affine fractal boundary under two different boundary conditions.5 These studies on fractal interfaces have lead to two important conclusions: (i) Under diffusion-limited transport condition, the scale invariant interface has the scale invariant temporal response. (ii) The scaling exponent of the time response is related to the scaling exponent of the spatial structures. But these studies fail to answer the question of how and how much these conclusions are suitable for a realistic interface with a limited scale invariance. What is the (exact) asymptotic form of eq 6 for an approximate scale invariant interface? What other geometrical information is important to understand this problem? Does the scaling form for eq 6 (is universal) hold for the scale invariant roughness of arbitrary magnitude? To answer these questions we look for results on arbitrary geometries. Other results which are very general in approach in understanding the effect of roughness on the interfacial transport are for the arbitrary geometries.14-18,37,38 One type of these studies is by Phillips-Jansons,37 Oldham,38 and Kant-Rangarajan14,18 where these authors obtained the interfacial reaction flux (current) in terms of local surface curvature; another type is by
Kant14-16 and Kant-Rangarajan.17,18 In these studies we obtained results in terms of an arbitrary (small) roughness profile and an arbitrary roughness power spectrum. These studies do not address the questions related to the scale invariant interfaces. In the following sections we are concerned with addressing the above mentioned questions related to approximately selfaffine roughness. This is achieved by using the results for the arbitrary geometries. For the first illustration we concentrate only on the problem of an approximate scale invariant corrugated random roughness. The results for an ideal self-affine surface fractal are obtained as a special case of our work. Unlike earlier scaling approaches we clearly assign geometrical meaning to the time-independent factor for the asymptotic intermediate time response, two crossover times, short and long time response. The qualitatively different time response of the low and high roughness is brought out. Results for the small roughness surfaces are reported in section 2. Results for the arbitrary and large roughness are reported in section 3. Conclusions are reported in section 4. 2. Reaction Rates for Surfaces with Low Roughness The expression for the diffusion-limited reaction rates is obtained by using an exact mapping that exists between the response of an electrode under constant potential conditions and the net diffusion across an interface, say a membrane of the same geometry. The (measured net cathodic) current transient, I(t), caused by heterogeneous charge transfer under diffusionlimited conditions (or reversible transfer), for a redox reaction, O (solution) + ne- (surface) h R (solution), and the reaction flux under similar conditions are related as
J(t) ) -I(t)/nF
(8)
where n is the number of electrons transfered in redox reaction and F is the Faraday constant. The time dependence of the flux on a smooth surface is described by a Cottrell equation39
Jp(t) ) A0xDδCs/xπt
(9)
where δCs is difference between the surface and the bulk concentration and A0 is the area of the (smooth) surface. The Cottrell equation (eq 9) is obtained under the initial condition of homogeneous solution (t e 0), and at time t ) 0 the surface becomes completely active (by the application of a potential step on the electrode). δCs is equal to -C0 for the absorbing boundary in which the particles react or are absorbed with certainty when they encounter the interface, but for a reversible reaction it can be different form the bulk value.16 Recently, Imre, Pajkossy, and Nyikos12 qualitatively discussed that the form of the Cottrell equation is preserved if one replaces area (A0) of smooth electrode by the area of diffusion front A(t) ) A0R(t), say a generalized area. The generalization of the Cottrell equation for a rough interface is
J(t) ) Jp(t) R(t)
(10)
where Jp(t) is same as in eq 9 and R(t) is a generalized roughness factor or the dimensionless area of diffusion front. The diffusion front is not uniform along the surface for a rough interface, except in the limit of very short and very long times. The diffusion layer thickness for very short and very long times is much smaller and larger than the smallest and largest irregularities, respectively. Thus, the diffusion front area reduces to the microscopic and macroscopic area for the very short and very long time, respectively.40 R(t) for all times depends on
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J. Phys. Chem. B, Vol. 101, No. 19, 1997 3783
all the geometrical detail of the surface! This can be seen quantitatively.15-17,37,38 The roughness of a surface is best described in terms of a random surface, and the physical quantity of interest for this model is the mean (total) flux, 〈J(t)〉, at the interface. 〈J(t)〉 on a randomly rough (corrugated) surface is described by the following equation:15-17
{
〈J(t)〉 ) Jp(t) 1 +
∫0∞dK (1 - e-K Dt) 〈|ζˆ (K)|2〉}
1 2πDt
2
(11)
This equation is good for a low roughness interface and reveals the relation between the total mean flux (current) for the electroactive molecules, ions, etc. of a fluid and the power spectral density of roughness. As mentioned earlier the reaction flux for the rough surface is the product of the smooth surface flux and the “generalized roughness factor” (normalized area of diffusion front). The first term in the bracket corresponds to the the smooth surface, but the second term corresponds to the excess flux due to roughness. In case of redox reaction or forward and backward transfer at the interface, the diffusion coefficients for the reactant (oxidized) and product (reduced) species in eq 11, are taken to be equal. The statistically averaged flux transient for an approximately self-affine fractal surface of low roughness is obtained by substituting eq 4 in eq 11. The expression for the reaction rate (total diffusion flux) on the interface is obtained as follows
〈J(t)〉 ) Jp(t)[1 + RF(t) + RL(t) + RU(t)]
(12)
RF(t) ) m0F/2Dt - (T/4π(Dt)1-H) Γ(-H,Dt/L2max,Dt/L2min) RL(t) ) (m0L/2Dt)[1 - (xπLmax/2xDt) erf(xDt/Lmax)] RU(t) ) (m0U/2Dt)[1 - (xπLmin/2βxDt) exp(-Dt/L2min + g2(t)) erfc(g(t))]
The functions RF(t), RL(t), and RU(t) in the generalized roughness factor are the time-dependent measures of roughness contributions to reaction flux from the intermediate (scaling), lower, and upper nonfractal part of the roughness power spectrum. RU(t)/RF(t) decreases and RL(t)/RF(t) increases with the increase in time. The contributions from RL(t) and RU(t) are important only for the long and short time, respectively. Hence these functions (for large F interface) do not contribute much to the intermediate time reaction flux which mainly depends on the power-law part of the roughness power spectrum. The contribution of RU(t) is unimportant for the surfaces with a short high-wavenumber tail (small β) in the power spectrum. The three-time behavior in diffusion problem can be seen by studying short, intermediate, and long time flux transients. The approximate form of eq 12 can be written as
{
〈J(t)〉 ≈ Jp(t) 1 + m2/2 - m4Dt/4
t < ti -2(1-H) TL TL2H min max ti < t < to (15) 1+ + 1-H 4πHDt 4π(1-H) 4πH(Dt) TΓ(1-H)
1 + m0/2Dt
t > to
where m0 is defined in eq 14, and m2 (see eq 16) and m4 (see eq 17) are the second and fourth moments of the power spectra, respectively. ti and to are the inner and outer fractal crossover times, respectively. The first row in eq 15 is a truncated short time expansion which is good for t , L2min/D. The second row in eq 15 is a truncated intermediate time expansion, i.e. L2max/D . t . L2min/D. The intermediate time expansion is obtained by substituting appropriate expansions (see Appendix) for the incomplete gamma function in eq 12, and we have ignored the contributions from the RL(t) and RU(t) functions which are not important in this region. The third row in eq 15 is a truncated long time expansion, i.e. t . L2max/D. The expression for m2 in eq 15 is given by34
m2 ) m2F + m2L + m2U
g(t) ) Lmin/2βxDt + xDt/Lmin
m2F ) [T(1 - F-2(1-H))/2π(1 - H)]L-2(1-H) min
where Γ(R, x0, x1) ) Γ(R, x0) - Γ(R, x1), Γ(R, x0) and Γ(R, x0) are the incomplete gamma functions,41 and the minimum length scale (Lmin) and maximum length scale (Lmax) are related to the cutoff wavenumber as 1/Kmax and 1/Kmin, respectively. The range of (spatial) scales of roughness is defined as
m2L ) TL2(H-1) max /3π
F ) Lmax/Lmin
(13)
It is an important characteristic in determining the relative importance of lower and upper cutoff scales in the surface. m0F, m0L, and m0U are components of the zeroth moments (m0) of the power spectra of the surface. The mean square height is a measure of the width (m0 or h2) of fluctuation in surface profile, and for a band-limited power-law spectra it is represented by the following form:18]:
m0 ) m0F + m0L + m0U T(1 - F 2πH
-2H
m0F )
) 2H Lmax
m0L ) TL2H max/π m0U ) TβL2H min/π
(14)
(16)
m2U ) Tβ3e1/βΓ(3,1/β)L2(H-1) /π min m2 is the mean square (MS) slope of the interface. We call surfaces with small m2 (small T) as “low roughness surfaces” and surfaces with large m2 (large T) as “large roughness surfaces”. The low roughness surfaces have small R* (microscopic area/macroscopic area) and large roughness surfaces have large R*. m4 in eq 15 is the MS second derivative of the surface and is given by34
m4 ) m4F + m4L + m4U
(17)
m4F ) [T(1 - F-2(2-H))/2π(2 - H)] L-2(2-H) min m4L ) TL2(H-2) max /5π m4U ) Tβ5e1/βΓ(5,1/β)L2(H-2) /π min The positive sign in the first leading term after 1 in eq 15 in all rows means that there is all time enhancement in the reaction rates. The short time behavior (for the large F surfaces) is controlled by the Lmin as this gives the dominant contribution
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to the value of m2 (see eq 16) and m4 (see eq 17). Similarly, the long time behavior is controlled by Lmax as this gives the dominant contribution to the value of m0 (see eq 14). This suggests that the interface with large MS width and MS slope have large excess reaction flux. This for an approximately selfaffine interface means that the interface with large F and T will have a large excess reaction flux. The leading term during intermediate time behavior is independent of both Lmax and Lmin. The corrections from Lmax and Lmin are important only for the small Lmax and the large Lmin roughness. Equation 15 which is an approximate form of eq 12 may not be suitable for the crossover times, i.e. close to ti and to. As we mentioned earlier, qualitatively the mean flux transients can be classified in three categories: very short time domain (t , ti, 〈J〉 ∼ 1/t1/2), very long time domain (t . to, 〈J〉 ∼ 1/t1/2), and an intermediate time domain (to . t . ti) which has the 1/t1/2 term with an added anomalous excess flux term (∼t-3/2+H). The excess flux term has a time exponent which contains a roughness exponent in it. This short and long time 1/t1/2 behavior and a transition region between them were first observed by Vetter and de Levie in their electrochemical experiments in the early sixties.40 Another surprising aspect of this result is that the intermediate time mean reaction flux does not have an usual form of eq 6 for the fractals! But it has an anomalous excess reaction flux (δJ(t)) at the interface which is written as
δJ(t) ) 〈J(t)〉 - Jp(t) ) [A0DδCsTΓ(1 - H)/4π3/2H] (Dt)H-3/2 (18) Equation 18 is an exact form for the low roughness (small T) ideal self-affine surface fractals. An ideal fractal with low roughness is an extreme form of the surface with small T and large F (large Lmax and small Lmin). It is important to note that the ideal fractals are characterized by their roughness exponent H and the amplitude factor T but no cutoff length scales as F f ∞. The total mean flux is the summation of smooth surface flux and an anomalous excess flux due to ideal fractal roughness. The mean reaction flux on a real surface with finite F has contributions from the cutoff scales but is more important when close to crossover times and regions away from the intermediate time scales. The crossover time scales, within which the excess current on the surface shows approximately anomalous excess reaction flux, can roughly be estimated by the limiting behavior. The inner crossover time (ti) is obtained by equating the leading terms in small and intermediate time behavior, and ti is obtained as
ti ≈ (Γ(2 - H)/H)1/1-H L2min/D
(19)
ti is not a monotonic function of H. It has a minimum at H ≈ 1/ and t ≈ 3.5L2 /D. 2 i min The outer crossover time (to) is obtained by equating the leading terms in large and intermediate time behavior. to is obtained as
to ≈ (Γ(1 - H))-1/H L2max/D
(20)
The value of to decreases with an increase in the roughness exponent. The value of to < L2max/D for H > 0.53 and to > L2max/D for H < 0.53. The ratio of the outer and inner crossover time scales is the range (Ft) of scaling region of the excess reaction flux. Ft is proportional to the square of F,
Ft ≈ [(Γ(1 - H))-1/H/(Γ(2 - H)/H)1/1-H]F2
(21)
Ft is a monotonically decreasing function of H and satisfies the condition: Ft e F2. This suggests that scaling (time) behavior of the excess flux may have a scaling region on (less than or equal to) double the decades of space scaling behavior of the roughness of the interface! In the preceding discussion we analyzed the problem of diffusion to the self-affine rough surfaces by using results obtained from second-order perturbation theory, and these results are valid for the whole time domain. As the results are obtained by perturbation analysis, their applicability is limited to surfaces with small roughness (small T). Results for surfaces with an arbitrary or large roughness (large T) can be understood in a limited sense and are discussed in the following section. 3. Reaction Rates for a Surface with an Arbitrary Roughness In this section we show that the results obtained by PhillipsJanson37 and Oldham38 for an arbitrary surface geometry in terms of its surface area (contour length of the curved side) and local surface curvatures can be used to understand statistically rough surfaces as well as the self-affine surface fractal too! Assuming that the interface is dominantly made up of points of small curvatures, the total reaction flux under diffusionlimited conditions on such interfaces in two dimensions is given by37,38
J(t) ≈ DδCs
[
()]
κ κ2 Dt + πDt 2 4 π
∫Σdx dy x 1
1/2
(22)
Here κ is taken positive if the center of curvature lies on the electrode (membrane) side of the solution/electrode (membrane) interface. The first integral in eq 22 is the surface area of a corrugated surface and is defined as
A)
∫Σ dx dy ) ∫Σ
[ ( )] ∂ζ(x) ∂x
1+
0
2 1/2
dx dy
(23)
where Σ0 is the average plane about which the surface Σ is fluctuating. Similarly, the second surface integral of the curvature is defined as:
κm ) (1/A0)
∫Σdx dy κ
(24)
The last term in eq 22 is a surface integral of square curvature of the rough surface, and it is defined as
∫Σdx dy κ2 ) (1/A0)∫Σ dx dy (1 - ζ′2)1/2 κ2
κ2m ) (1/A0)
(25)
0
The ratio of flux density of rough and planar interfaces can be used to point out local flux inhomogeneity. This expression up to first order in local surface curvature is written as (see eq 22)
J/Jp ≈ 1 + κxπDt/2
(26)
As time progresses interface zones with κ > 0 and κ < 0 will have higher and lower reaction flux, respectively. For time t J 4/πDκ2, the reaction zones with κ < 0 will have zero reaction flux, i.e. time of formation of inert zones is inversely proportional to the square of the local curvature. This means that the reaction activity will be localized to zones with κ > 0. The roughness causes the fluctuation of κ along the surface which
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leads to fluctuation in local reaction flux or the localization of active zones on the interface. This conclusion is more general and consistent with the earlier studies on a specific type of geometry under steady state conditions.6 Our main interest is to obtain the Phillips-Jansons and Oldham equation for a random surface geometry where the random fluctutions of an arbitrary magnitude are taken about a reference plane, i.e. z ) 0. The ensemble averaged form of eq 22 for the total reaction rate for the random surface is obtained as
[
()]
〈κm〉 Dt R* 〈J(t)〉 ≈ DA0δCs 4 π xπDt 2
1/2
(27)
where R* is the roughness factor () average area/projected area) of the surface and 〈κ2m〉 is the ensemble and surface averaged square (MIS) curvature. The exact expression for the roughness factor of a random surface in eq 27 is given by18
R* )
1
(
)
1 1 U , 2, 2 2m 2 x2m2
(28)
where U(a, b, z) is a confluent hypergeometric function.41 The mean area increases with the second moment of power spectra (m2) of the surface. The average value of the surface integral of square curvature per unit projected area (MIS curvature) is obtained by taking two averages over the Gaussian joint probability density. The MIS curvature is obtained as
〈κ2m〉
m4
(
)
1 1 ) U , -1, 2 2m 2 x2m2
The arbitrary length (L) in a problem of diffusion-limited reaction rates is related to the diffusion length. We assume that L in eq 31 is proportional to the diffusion length
L ) xDt
(33)
where 1/2 is the proportionality parameter (see eq 36). The (asymptotic) intermediate time expression for the reaction rates on the interface with an arbitrary roughness can be obtained by substituting eqs 31 and 33 in eq 27 and represented as
[
〈J(t)〉 ≈ Jp(t) R*(L) -
]
〈κ2m(L)〉 Dt 4
(34)
where the square bracket is the dimensionless area of diffusion front (see eq 10). R*(L) and 〈κ2m(L)〉 are the exact scaling forms of the roughness factor and MIS curvature, respectively. The area of diffusion front during intermediate time scales depends on an effective diffusion length (L). The value of in eq 33 can be obtained by comparing the small roughness expansion of eq 34 and the intermediate time behavior in eqs 15 and 18. The small roughness expansion for eq 34 is obtained as
[
〈J(t)〉 ) Jp(t) 1 +
T [1 4π(1 - H)1-H
]
(29)
(30)
m2 and m4 are given by eq 16 and eq 17, respectively. Equation 30 is the same result as that for short time behavior in eq 15. To understand the intermediate time scaling behavior of the reaction flux we use the scaling form of m2 and m4 for a surface with a broad range of roughness length scales. m2 and m4 for the intermediate length scales are given by their scaling form and are given by the following equations18
m2(L) ≈ (T/2π(1 - H))(1/L)2(1-H) m4(L) ≈ (T/2π(2 - H))(1/L)2(2-H)
(32)
(1 - H)/2(2 - H) ](Dt)-(1-H) (35)
The MIS curvature increases with the increase in the magnitude of the fourth moment of the power spectra of the surface and it decreases with increase in second moment of power spectra. The second moment dependent function, viz. 1/(2m2)1/2 U(1/2, -1, 1/2m2), varies between 1 and 0. Equation 27 suggests that the reaction flux increases with an increase in R* and a decrease in the MIS curvature of the interface. Equation 27 can be used for two purposes: (i) to obtain the early nonscaling behavior and (ii) as an asymptotic expression for the intermediate scaling time behavior of the interfacial reaction flux. The early nonscaling behavior for the surface with arbitrary roughness is obtained by substituting eq 16 for m2 and eq 17 for m4 in eq 27. The small T expansion yields the result for the small roughness surface as
〈J(t)〉 ≈ (DA0δCs/xπDt) [1 + m2/2 - m4Dt/4]
L e 5-1/1-H Lmax
(31)
where L is an arbitrary (intermediate) length scale and is restricted between two fractal crossover length scales.34 The scaling form in eq 31 is a good approximation if L satisfies the following condition:34
The parameter in diffusion length satisfies the following equation
2-H - [H(2 - H)/Γ(3 - H)] + H(1 - H)/2Γ(3 - H) ) 0 (36) The parameter in diffusion length depends only on the roughness exponent. for the large H is approximately given by ≈ (H/Γ(2 - H))1/1-H. The three time regions for the large roughness surfaces are obtained by taking the large T limit in eqs 27 and 34 and are written as
{
〈J(t)〉 ≈ (DA0δCs/xπ)
x2m2/πDt - m4xDt/3x2πm2 t < til til < t < tol (37) fxT (Dt)H/2-1 t > tol 1/xDt + h2/2(Dt)3/2
where f is given by the following equation:
f ) f(H) )
x(1 - H)
1 πx(1 - H)
(1-H)/2
-
6π(2 - H)(3-H)/2
(38)
f for the large H is approximately given by f ≈ (Γ(1 - H)/ π2H)1/2. m0 (or h2) is defined in eq (14), m2 (see eq 16) and m4 (see eq 17) are the second and fourth moments of the power spectra, respectively. til and tol are the inner and outer fractal crossover times, respectively. The first row in eq 37 is the short time expansion and is obtained from eq 27 for the large T. The second row in eq 37 is an approximate expansion for the intermediate time domain which is obtained by expanding eq 34 for the large T. The third row in eq 37 is the long time behavior. The derivation of an exact long time series for the
3786 J. Phys. Chem. B, Vol. 101, No. 19, 1997
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reaction rates on an arbitrary roughness interface is a difficult task but the leading correction will have the same form as for the low roughness surfaces. The long time response of the rough surface will have the t-1/2 term as of a smooth interface and first correction to it comes from the MS width (h2) scaled with the diffusion length. The reaction flux for a large roughness surface has an intermediate (tol . t . til) anomalous time domain. The scaling exponent in eq 37 for the intermediate time is the same as in eq 6 for DE ) 2. Equation 37 not only has a scaling form of eq 6 for DE ) 2 but also unravels its prefactor which remains as an unidentified geometry-dependent parameter in earlier calculations. We can also claim that the scaling form of eq 6 is valid only for a large roughness surface or large T. Qualitatively, the leading behavior of the mean flux can be classified in three categories: very short time domain (t , til, 〈J〉 ∼ 1/t1/2), very long time domain (t . tol, 〈J〉 ∼ 1/t1/2), and an additional term in them which shows the deviation from t-1/2 behavior. These three behaviors are observed experimentally in electrochemistry for the diffusion-controlled interfacial charge transfer.12,40 Comparing equations 15 and 37 we see that the short time expansion has the same time dependence but with different coefficients. The intermediate time behavior is a qualitatively different form for small and large T surfaces. The leading long time dependence is the same for two cases. Finally, we conclude that the fractal dimension is not sufficient to characterize a self-affine fractal. The knowledge of T, Lmin, and Lmax is essential for a proper characterization. The two crossover times between the three time regions can be estimated by the following procedure. The inner crossover time (til) is obtained by equating the leading terms in small and intermediate time behavior, and an approximate value for til is obtained as
til ≈ π x1 - Hf2/1-H L2min/D
(39)
The inner crossover time depends on the Hurst’s exponent and the lower cutoff length of the surface. The outer crossover time (tol) is obtained by equating the leading terms in large, and intermediate time behavior will overestimate it. Another way of estimating tol is by equating eq 32 and eq 33. An approximate value for tol is obtained as
tol ≈ L2max/52/1-H D
(40)
(iv) These results show three time regimes: a short time region, and long time region, and a (anomalous) transition region between them. This is consistent with and explains the qualitative behavior of the measured electrochemical current transients on a solid electrode under diffusion-limited charge transfer condition. (v) We show that the fractal dimension of a surface is not sufficient in itself to understand the problem of diffusion-limited reaction rates. It appears from all earlier studies that the exponent β in eq 6 is universal; it depends only on the scaling property of the interface. As it has been found to be the same for the self-similar porous and self-affine rough interface by earlier authors. Our work shows that this temporal scaling property indeed depends on the extent of the roughness of a self-affine interface, i.e. the crossover from anomalous flux (large roughness) to anomalous excess flux (small roughness) behavior with reduction in the strength of fractal (T). (vi) The mean flux on the small roughness interfaces follows the form: 〈J〉 ∼ t-1/2 + const t-3/2+H. The large roughness interface has the form: 〈J〉 ∼ t-1+H/2, and for an intermediate (arbitrary) roughness it has a more complicated form. (vii) Any process in which transport across a rough interface is diffusion limited can be enhanced by increasing surface area, MS width, and decreasing MS curvature. The enhancement for a scale invariant surface of large roughness increases with the increase in the strength and the range of the fractality. These interfaces show an anomolous diffusion behavior for the intermediate time scales. (viii) Finally we suggest that one should use the small roughness result for the surfaces with R* j 1.6, the large roughness result for R* J 2 and an arbitrary roughness result for 1.6 j R* j 2. It will be intresting to verify these results using simulations13 on two-dimensional rough surfaces and experiments on directionally polished surfaces in electrochemistry. The results related to an isotropic self-affine fractal and the effect of finite interfacial transfer will be disscused elsewhere. Acknowledgment. I gratefully acknowledge fruitful suggestions from P. G. De Gennes. Appendix Useful Expansions. The small z expansion for the incomplete gamma functions41 is ∞
4. Conclusions The aim of this paper is to analyze the mean reaction rates to an approximately self-affine scaling surface with a band limited power-law spectra and to correlate known information concerning the diffusion flux of the reacting particles. Other important conclusions are the following. (i) It is shown that the general results obtained for the diffusion-limited reaction rates for arbitrary geometry interfaces by Phillips-Jansons, Oldham, and Kant-Rangarajan can be used for the self-affine surface fractals when a finite range of fractality is taken into account. (ii) This work unravels the connection between the gross roughness features such as the roughness exponent, the inner and outer cutoff lengths, the magnitude of roughness, and the mean reaction flux transient. (iii) The presence of roughness leads to localization of the active zones. The time of formation of such localized zones is inversely proportional to the square of local curvature of the surface.
a
Γ(a,z) ) Γ(a) - z
∑
(-z)n
n)0(a
(41) + n)n!
The large z asymptotic expansion for the incomplete gamma function41 is
[
Γ(a,z) ≈ za-1e-z 1 +
a - 1 (a - 1)(a - 2) + + ... z z2
]
(42)
References and Notes (1) De Gennes, P. G. C. R. Acad. Sci. (Paris) 1982, 295, 1061. (2) Kopelman, R. J. Stat. Phys. 1986, 42, 185; Science 1988, 241, 1620. (3) Pfeifer, P.; Avnir, D.; Farin, D. J. Stat. Phys. 1984, 36, 699. (4) Pfeifer, P.; Welz, U.; Wippermann, H. Chem. Phys. Lett. 1985, 113, 535. (5) Vandembroucq, D.; Boccaro, A. C.; Roux, S. Europhys. Lett. 1995, 30, 209. (6) Gutfraind, R.; Sapoval, B. J. Phys. I (France) 1993, 3, 1801. (7) Sapoval, B.; Fractal electrodes, fractal membranes and fractal catalyst. In Fractals and disordered systems; Bunde, A., Havlin, S., Eds.; Springer-Verlag: Heidelberg, Germany, 1991.
Reaction Rates on Self-Affine Fractals (8) Sapoval, B.; Chazalviel, J. N.; Peyriere, J. Phys. ReV. A 1988, 38, 5867. (9) Nyikos, L.; Pajkossy, T. Elecrochim Acta 1986, 31, 1347. (10) Nyikos, L.; Pajkossy, T. Elecrochim Acta 1989, 34, 171. (11) Pajkossy, T.; Borosy, A. P.; Imre, A.; Martemyanov, S. A.; Schiller, R.; Nyikos; L. J. Electroanal. Chem. 1994, 366, 69 and references therein. (12) Imre, A.; Pajkossy, T.; Nyikos, L. Acta Metall. 1992, 40, 1819. (13) Borosy, A. P.; Nyikos, L.; Pajkossy, T. Elecrochim Acta 1991, 36, 163. (14) Kant, R. Electrochemistry at Complex Interfacial Geometries; Indian Institute of Science: Bangalore, India, 1993. (15) Kant, R. Phys. ReV. Lett. 1993, 70, 4094. (16) Kant, R. J. Phys. Chem. 1994, 98, 1663. (17) Kant, R.; Rangarajan, S. K. J. Electroanal. Chem. 1994, 368, 1. (18) Kant, R.; Rangarajan, S. K. J. Electroanal. Chem. 1995, 396, 285. (19) Nyikos, L.; Pajkossy, T. Elecrochim Acta 1990, 35, 1567 and references therein. (20) Pajkossy, T. J. Electroanal. Chem. 1991, 300, 1 and references therein. (21) De Levie, R. J. Electroanal. Chem. 1990, 281, 1 and references therein. (22) Halsey, T. C.; Leibig, M. Ann. Phys. (NY) 1992, 219, 109; Europhys. Lett. 1991, 14, 815 and references therein. (23) Mulder, W. H.; Sluyters, J. H. Electrochim. Acta 1988, 33, 303. (24) Ball, R.; Blunt, M. J. Phys. A: Math. Gen. 1988, 21, 197 and references therein. (25) Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983, 79, 3558. Avnir, D.; Farin, D.; Pfeifer, P. J. Chem. Phys. 1983, 1983, 3566.
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