14040
2007, 111, 14040-14044 Published on Web 09/05/2007
Theory of Anomalous Diffusive Reaction Rates on Realistic Self-affine Fractals Rama Kant* and Shailendra K. Jha Department of Chemistry, UniVersity of Delhi, Delhi 110007, India ReceiVed: July 15, 2007; In Final Form: August 15, 2007
We developed a theoretical method based on limited scale power law form of the interfacial roughness power spectrum and the solution of diffusion equation under the diffusion-limited boundary conditions on rough interfaces. This theoretical method is useful for the analysis of diffusion-limited flux/current to limited scale self-affine random fractal interfaces and is compared with experimentally measured electrochemical current for nano- and micron scales of roughness. The new result explains experimental findings of the temporal scale invariance as well as deviation from this in transition, short, and long time regions. Our result show flux/current transients in terms of three dominant fractal morphological parameters, that is, fractal dimension, lower cutoff length scale of fractality, and topothesy, for the limited length scales of fractality. More broadly, these results are applicable for all time scales and roughness factors.
I. Introduction Realistic surface roughness has limited length scales of irregularities and is frequently characterized as self-affine fractals.1-4 Diffusion-limited processes on such interfaces show anomalous behavior of the reaction flux. Some of the diverse realizations of diffusion-limited processes in physical phenomena are: spin relaxation,5 fluorescence quenching,5,6 heterogeneous catalysis,7,8 enzyme kinetics,9,10 heat diffusion,11 membrane transport,12,13 and electrochemistry.14-22 In this paper, we are concerned with the understanding of the diffusion-limited reaction rates to the random but statistically isotropic surfaces that exhibit the scale invariance over a limited range of length scales. The diffusion of the reactant from a bulk medium toward an interface where the reactants either loose their activities or are transformed into product is a common problem in diverse areas of science.5-24 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
J(t) ) -D
∫Σ dS ∂nCR
(1)
where the concentration profile, CR(r b, t), satisfies the diffusion equation
∂ C (b, r t) ) DR∇2CR(b, r t) ∂t R
(2)
where R ≡ O, R, representing the oxidized or reduced species, δCR(ζ) is the difference between surface and bulk concentration, DR is the diffusion coefficient (for simplicity we assume in our calculations DO ) DR ) D) and b r is the three-dimensional vector, b r ≡ (x, y, z) . There is local transfer kinetics at the interface, and this is represented by the boundary condition, * To whom correspondence should be addressed. E-mail: rkant@ chemistry.du.ac.in. URL: http://people.du.ac.in/∼rkant.
10.1021/jp075525t CCC: $37.00
δC|r∈Σ ) Cs 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 (-Co). At initial time and far off from the interface, a uniform initial and bulk concentration Co is maintained viz. C(r b, t ) 0) ) C(z f ∞, t) ) Co. The formalism to solve this random boundary value problem is described in refs 17, 18. We wish to use this formalism for a realistic random fractal roughness that is statistically isotropic. These processes are experimentally approximated as power law relation of reaction rates/flux, J(t) , in the intermediate time (t) and is represented by the following relation:
J(t )∼ t-β
(3)
where the exponent, β, depends on interfacial roughness. Theoretical justification for eq 3 was provided by De Gennes scaling result with β ) (DH - 1)/2 5 and generalized form.25,26 De Gennes analyzed it for the problem of diffusion-controlled nuclear magnetic relaxation in porous media with fractal interfacial dimension DH. Later, similar results were discussed for other diffusion-controlled situations, such as adsorption on a porous fractal catalyst8 in the context of flow of energy and mass through a fractal interface14 for rough fractal electrode/ electrolyte interfacial current under potentiostatic conditions15 and a similar result for heat diffusion from a self-affine fractal boundary.11 Because of its simplicity and the lack of a better alternative, eq 3 captured lots of attention and is used extensively for interpreting large quantity of data.20,21 These include cases where the range of roughness is too small to be taken as idealized fractals. It is commonly known that eq 3 is unable to include a complete set of realistic fractal morphological parameters, and because of this reason it does not capture thesubtle aspect of experimental data. Particularly, difficulties arise in the transition region and often mix data of the scaling region with the transition region, and hence there is an inaccurate prediction of fractal dimension. © 2007 American Chemical Society
Letters
J. Phys. Chem. C, Vol. 111, No. 38, 2007 14041
We present an analytical model that allows one to probe the complete time behavior of diffusive flux of reacting interface with roughness and to evaluate the consequences of limited length scales of roughness on the anomalous behavior. This theory explains the several experimental findings20,21 of the temporal scale invariance of flux/current in terms of morphological parameters for the limited length scales of roughness. II. Theoretical Model for the Random Fractal Surface Roughness The information about surface roughness enters in this theory through the power spectrum of roughness. The power spectrum of a realistic surface (also called “approximate self-affine fractal”) is described in term of limited scales of wave-numbers (K) power law function3,27
〈|ζˆ (K B)|2〉 ) µ|K|2DH-7,
for 1/L e |K| e 1/l
(4)
There are four morphological parameters of roughness in this framework, namely DH, l, L, and µ. DH is the fractal dimension, a global property that describe scale invariance property of the roughness and anomalous behavior in flux, and its time exponent is usually assumed to be function of this parameter; l and L are lower and upper cutoff length scales of fractality, respectively; and µ is the strength of fractal and is related to topothesy of fractals3,27,28 in which its units are cm2DH-3 and µ f 0 implies no roughness. The lower roughness scale is the length above which surface shows fractal behavior. Also, such a power law spectrum may represent band-limited form of Ausloos-Berman’s generalization29 for multivariate Weierstrass-Mendelbrot function for an isotropic (statistically) rough surface. The moments of power spectrum are related to various morphological features of rough surface viz. root mean square (rms) width (xm0), rms gradient (xm2), rms curvature (xm4), etc. The general moments of power spectrum (i.e., 2k-th moments, m2k) are easily obtained for the above-mentioned power spectrum and are important morphological characteristic of surface roughness. The general formula is
m2k ) µ(l-2δk-L-2δk)/4πδk
(5)
where δk ) δ + k and δ ) DH - 5/2 . Our general formalism17,18 shows that the diffusive flux/ current at random rough surfaces can be described in term of its power spectral density of roughness, 〈|ζˆ (K B)|2〉. The main approximation involved in derivation is the truncation of solution at second order in the surface roughness profile. The total (averaged) flux/current at the stationary, Gaussian random surface is given by17,18
J(t) )
xD A0Cs xπt
1 1+ 4πDt
[
∫0
∞
dKK(1 - e
-K2Dt
)〈|ζˆ (K B)| 〉 2
]
(6)
where A0 is the area of surface around which the rough surface fluctuates. The diffusion-controlled reaction flux is related to potentiostatic current transients of an electrode undergoing fast charge transfer. The reaction flux, J(t), is related to the electrode current (I(t)) as: I(t) ) -nFJ(t), where n is the number of electron transfer in the redox reaction (OSolution + ne- h RSolution), and F is Faraday constant. The present theory of anomalous diffusive reaction rates on realistic self-affine fractals is obtained using the exact solution for the dynamic diffusive flux on an approximate self-affine
Figure 1. This illustrates the effect of the three dominant fractal morphological parameters, (a) DH, (b) l, and (c) µ, on the anomalous scaling behavior of reaction rates/current. The theoretical result (eq 7) is plotted as graphs to show an approximate power law dependence of the flux/current on time. The insets show the relative flux/current (R(t) ) I(t)xπt/nFxDA0Cs) as a function of time in semilogarithmic scales. (a) The solid line is generated using l ) 0.6 µm , µ ) 1.3 × 10-6(arbitrary units), and DH ) 2.45, 2.40, 2.35, and 2.30 from above. Ratio of L/l is kept fixed ()100) in all calculations. (b) The solid line is generated using DH ) 2.4, µ ) 1.3 × 10-6(arbitrary units) and l ) 0.2, 0.4, 0.6, and 0.8 µm from above. (c) The solid line is generated using DH ) 2.4, l ) 0.4 µm and µ ) µ0 ) 1.3 × 10-6(arbitrary units), 1.5 × µ0, 2 × µ0, and 2.5 × µ0 from above. Other fixed quantities in calculations are macroscopic areas (A0 ) 0.1 cm2), diffusion coefficient (D ) 5 × 10-6 cm2/s), and concentration (CO ) CR ) 5 mM).
surface. Substituting eq 4 for the band-limited power law spectrum in eq 6 and solving resultant integral, we obtained the following equation
J(t) )
xD A0Cs xπt
( ( 1+
µ l-2δ - L-2δ + 8π δDt
))
Γ(δ, Dt/l2, Dt/L2) (Dt)1+δ
(7)
: where δ ) DH - 5/2 , Γ(R, x0, x1) ) Γ(R, x0) - Γ(R, x1) ) γ(R, x1) - γ(R, x0), and Γ(R, xi) and γ(R, xi) are the incomplete Gamma functions.33 Equation 7 is graphically analyzed in Figure 1, which shows dependence of the scaling region on (a) DH, (b) l, and (c) µ. The scaling region has a very weak dependence on L so it is not shown in Figure 1. These calculations break the earlier beliefs based on idealized fractal models that the exponent of the anomalous diffusion region purely depends on the fractal dimension of roughness. These figures clearly demonstrate that the exponent of the anomalous region is dependent on all three dominant fractal morphological parameters. It is important to note that eq 7 is valid in all time regimes. Most of experimental data recorded are for the intermediate and long time regime, that is, t J l2/D. The expression for this time regime is obtained by expanding two incomplete gamma
14042 J. Phys. Chem. C, Vol. 111, No. 38, 2007
Letters
functions in eq 7 viz., one for small Dt/L2 and another for large Dt/l2 33 (see appendix for formulas for these expansions) andretaining only leading orders. The final equation has a simple and elegant form as:
J(t) ≈
xD A0Cs xπt
(
1+
(
Γ(δ) µ l-2δ 8π δDt (Dt)1+δ
))
(8)
that captures the anomalous behavior of the reaction current. Γ(x) in eq 8 is the gamma function.33 The small length scale of fractal roughness commands the flux transient for roughness of limited scales, that is, L/l ∼ 10. Under this approximation, there is no dependence on the upper length scale (L) for fractality in eq 8. This equation shows good agreement with eq 7 for time t > ti. Equations 7 and 8 extends the conventional representation of the Cottrell current transient (1/xt) on the planar electrode in electrochemistry34 to the fractally rough electrode. These equations achieve a more realistic characterization of limited scale rough surface diffusive flux as it includes the fractal dimension dependent power law as well as the contribution from the length scale and strength of fractality (µ). The total mean flux is the summation of smooth surface flux and an anomalous excess flux due to fractal roughness, or it can also be looked upon as the product of the 1/xt current and dynamic roughness factor (term inside parenthesis of eqs 7 and 8). III. Results and Discussion Nature of the plots in Figure 1 elucidates anomalous scaling behavior in the intermediate time regime. The current transients increased with a decrease in small scale of roughness in the early time domain, which is similar but larger in the magnitude of slope as compared to the planar 1/xt response. It follows a power law behavior in the intermediate time regime that merges with large time 1/xt behavior. The insets show similar behavior for the relative flux/current (R(t) ) I(t)xπt/nFxDA0Cs) plots. As the range of roughness increases, the roughness factor and total current output also increases simultaneously on small time scales. So one can say that the total current output at small times is dependent upon the lower cutoff length scale l. No such impact of l has been observed at the large time scale as this regime is controlled by the width of interface m0. The width of interface is a strong function of L and a weak function of l. Another important feature that can be verified from these graphs is that there is no such sharp outer cutoff time but the inner cutoff time decreases with the decrease in the lower length scale of roughness or with the increase in roughness factor. What is remarkable about eqs 7 and 8 is that they govern the diffusive flux of the diverse set of roughness features. Its validity is not only tested in this paper for various ranges of timescales but also for the magnitude of roughness factor. The conventional scaling eq 3 cannot explain deviation and regime of timescales from linear behavior in a log-log plot that is often seen in the transition region of data20,21 and is easily captured by this theory. The major drawback in the result of eq 3 is that it is claimed to be valid only for the intermediate time regime, and there is no proper prescription for characterization of this region. Several authors20,21 ended up fitting eq 3 to the long time transition data (see Figure 2 for dashed lines) due to improper characterization of intermediate time data. Our theory is valid for all time regimes, and hence they do not suffer from such deficiency. Figure 2 has several current transient curves and experimental
Figure 2. Comparison of model predictions from eq 7 (solid lines), eq 3 (dashed lines), and current transients data of (a) Pajkossy20 for micron scale roughness and (b) and (c) of Ocon et al.21 for nanoscale roughness. Morphological parameters used for calculations are listed in Table 1.
data21 on rough electrodes with limited length scales of fractality. Electrodes in these diffusion-limited current experiments used were nanoscale electrodispersed gold wire with surface layer of thin columnar gold.21 Rough surface of a wire electrode is imagined to be a randomly fluctuating surface around a macroscopic plane as the curvature contribution of macroscopic wire geometry is insignificant. Data are analyzed keeping in mind the experimental information about roughness factor that is available about their roughness. Surprisingly, limited order perturbation analysis is able to capture the features of large roughness also, although one expects that the scaling results would be seen only in the large roughness form of theory to match with the assumptions of eq 3.19 Most important of all, this work shows an intermediate anomalous power law form for time above inner transition time (ti). This suggests that this theory has an extended range of validity much beyond expectations, which are also seen in comparison with experimental results in Figure (2). Another
Letters
J. Phys. Chem. C, Vol. 111, No. 38, 2007 14043
TABLE 1: Morphological Quantities Used in Figure 2 and tia data
D1H
D3H
R/e
h (µ m)
xm2
rc (nm)
ti (µs)
µ × 106
l (nm)
A0 (cm2)
Figure 2a (b) Figure 2b (2) Figure 2b (1) Figure 2b ([) Figure 2c (4) Figure 2c (3) Figure 2c (])
2.46 2.52 2.32 2.24 2.4 2.20 2.00
2.46 2.516 2.381 2.375 2.378 2.230 2.020
4.27 100 20 20 50 3 1.1
19.64 4.89 4.31 4.39 2.01 2.27 0.67
3.40 79.79 15.96 15.96 39.89 2.39 0.45
1045 0.35 7.87 7.97 0.40 250.49 2524.07
16927 0.21 20.11 20.56 0.33 364.77 742.45
17.11 0.214 3.354 3.910 0.60 15.56 87.80
2500 20 86 87 11 400 750
0.0164 0.25 0.085 0.060 0.85 0.067 0.066
D1H and D3H are calculated using eq 1 and 8, respectively. Roughness factor R/e is estimated for Figure 2a, but it is experimentally measured from voltammetric experiment21 for remaining the figure and is used to constrain the value of µ parameter in our calculations. The predicted width of interface is h, root-mean-square gradient is xm2, average radius of curvature is rc, and inner crossover time is ti. Upper cutoff length (L) is kept constant in all calculations, that is, L ) 3 µm. a
feature of this theory is that ti decreases with a decrease in l. Inner transition time is evaluated equating small time expansion
J(t) )
xD A0Cs xπt
(
1+
m2 m4 - Dt + ‚‚‚ 2 4
)
(9)
and eq 8 at t ) ti. The results developed in eqs 7 and 8 are based on statistical models for the current on random surface fractals. The advantage of such formulation is that it is based on four morphological parameters: DH, µ, l, and L. In most cases, we do not have independent information about these morphological parameters as experimental studies rarely measure roughness power spectrum. In some cases, small and large length scales of roughness are characterized from a scanning electron microscopy or a scanning tunneling microscopy image of the surface. Similarly, the knowledge of the surface roughness factor or width of roughness or both can help to fix these statistical parameters. The remaining unknown parameters can be obtained by minimizing the variance of experimental data for current from theoretical values. With the knowledge of these four parameters, one can predict the various roughness features of the roughness profile, such as mean roughness factor (R*), rms width of roughness (h or xm0), rms gradient (xm2), and inverse rms curvature (rc ) 1/xm4). The roughness factor is a function of mean square gradient m2, that is, R* ≈ xπm2/2 for large roughness, which in turn is a strong function of l and a weak function of L. Figure 2 compares our theory for current transient with rough gold deposit on wires. Equations 7 and 8 are also obeyed by other experiments on surfaces like replica or gold masking of surfaces like fractured steel, dental surface and liquid-liquid interface.30-32 This theory predicts various roughness and morphological features that are listed in Table 1 along with ti. We calculate the variance between experimental data and theoretical curves, and in all cases it is below 0.0036. IV. Conclusions The central results, eqs 7 and 8, constitute an elegant and simple test that any data must pass to be called limited scale self-affine fractals. This model gives a very good description of the large quantity of data that have the scaling region as well as transition region. It does not suffer from the limitation of eq 3, which claimed to be valid only for the intermediate time regime, and there is no proper result for characterization of this region. It is important to note that the slope of the scaling region does not purely depend on fractal dimension alone but also on lower cutoff length scale of roughness and strength of fractality also. Another important observation is that the simplified eq 8
is excellent for the current transient in intermediate and long time regimes (i.e., t > ti). This formulation opens new avenues toward an understanding of diffusion-limited reaction rates at realistic rough surfaces with limited scales of fractality. This work unravels the connection between the anomalous intermediate power law regime exponent and the morphological parameters of limited scales of fractality, that is, DH, l, and µ. We believe that this will provide insights into many other diffusion-limited processes on random interfaces and the generality of this approach will provide a logical extension to several complex situations. As more data becomes available with clear power spectral characterization of roughness, quantitative analysis should provide an improved way to understand the diffusive processes on them. Acknowledgment. The authors thank Dr. T. R. Seshadri for his useful suggestions, and University of Delhi for financial support. S.K.J. gratefully acknowledge the financial assistance from the CSIR, New Delhi. Appendix: Useful Expansions The small z expansion for the incomplete gamma functions33 is ∞
Γ(a, z) ) Γ(a) - za
∑
(-z)n
n)0(a
(10)
+ n)n!
The large z asymptotic expansion for the incomplete gamma function33 is
[
Γ(a, z) ∼ za-1e-z 1 +
a - 1 (a - 1)(a - 2) + + ‚‚‚ z z2
]
(11)
References and Notes (1) Feder, J. Fractals; Plenum: New York, 1988. (2) Mandelbrot, B. B.; Passoja, D. E.; Paullay, A. J. Nature 1984, 308, 721. (3) Sayles, R. S.; Thomas, T. R. Nature 1978, 271, 431. (4) Zaiser, M.; Grasset, F. M.; Koutsos, V.; Aifantis, E. C. Phys. ReV. Lett. 2004, 93, 195507. (5) De Gennes, P. G. C.R. Acad. Sci., Paris 1982, 295, 1061. (6) Kopelman, R. Science 1988, 241, 1620. (7) Chaudhari, A.; Yan, C.-C. S.; Lee, S.-L. Chem. Phys. Lett. 2002, 351, 341. (8) Pfeifer, P.; Avnir, D.; Farin, D. J. Stat. Phys. 1984, 36, 699. (9) Pfeifer, P.; Welz, U.; Wippermann, H. Chem. Phys. Lett. 1985, 113, 535. (10) Dewey, T. G. Proc. Natl. Acad. Sci. U.S.A. 1994, 91, 12101. (11) Vandembroucq, D.; Boccara, A. C.; Roux, S. Europhys. Lett. 1995, 30, 209. (12) Sapoval, B. C.R. Acad. Sci., Paris 1991, 312, 599.
14044 J. Phys. Chem. C, Vol. 111, No. 38, 2007 (13) Sapoval, B. Fractal electrodes, fractal membranes, and fractal catalyst in Fractals and disordered systems. Bunde, A.; Havlin, S., ed.; Springer-Verlag: Heidelberg, 1996. (14) Le Mehaute, A.; Crepy, G. Solid State Ionics 1983, 9-10, 17. (15) Pajkossy, T.; Borosy, A. P.; Imre, A.; Martemyanov, S. A.; Nagy, G.; Schiller, R.; Nyikos, L. J. J. Electroanal. Chem. 1994, 366, 69. (16) Halsey, T. C.; Leibig, M. Ann. Phys. (NY) 1992, 219, 109. (17) Kant, R. Phys. ReV. Lett. 1993, 70, 4094. (18) Kant, R.; Rangarajan, S. K. J. Electroanal. Chem. 1994, 368, 1. (19) Kant, R. J. Phys. Chem. 1997, 101, 3781. (20) Pajkossy, T. J. Electroanal. Chem. 1991, 300, 1. (21) Ocon, P.; Herrasti, P.; Va´zquez, L.; Salvarezza, R. C.; Vara, J. M.; Arvia, A. J. J. Electroanal. Chem. 1991, 319, 101. (22) Go, J.-Y.; Pyun, S.-I. J. Solid State Electrochem. 2007, 11, 323 and references therein. (23) Ball, R. C.; Somfai, E. Phys. ReV. Lett. 2002, 89, 135503.
Letters (24) Sapoval, B. Phys. ReV. Lett. 2006, 96, 180601. (25) Maritan, A.; Stella, A. L.; Togio, F. Phys. ReV. B 1989, 40, 9269. (26) Dassas, Y.; Duby, P. J. Electrochem. Soc. 1995, 142, 4175. (27) Yordanov, O. I.; Atanasov, I. S. Eur. Phys. J. B 2002, 29, 211. (28) Berry, M. V. J. Phys. A: Math. Gen. 1979, 12, 781. (29) Ausloos, M.; Berman, D. H. Proc. R. Soc., London, Sect. A 1985, 400, 331. (30) Imre, A.; Pajkossy, T.; Nyikos, L. Acta Metall. Mater. 1992, 40, 1819. (31) Eftekhari, A. Appl. Surf. Sci. 2005, 227, 331. (32) Eftekhari, A. Colloids Surf., B 2003, 32, 375. (33) Handbook of Mathematical Functions; Abramowitz, M., Stegan, A., Eds.; Dover Publications Inc.: New York, 1972. (34) Bard, A. J.; Faulkner, L. R. Electrochemical Methods: Fundamentals and Application; Wiley: New York, 1980.
