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Double layer capacitance on a rough metal surface: Surface roughness measured by “Debye ruler”
1997 Elecrrochimica Acra. Vol. 42. No. 19, pp. 2853-2860. 1997 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 00134686/97 $17.00 + 0.00 s00134586@7)00106-0 0
Pergamon PII:
Double layer capacitance on a rough metal surface: surface roughness measured by “Debye ruler” L. I. Daikhin,“* A. A. Kornyshevb and M. Urbakh “School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel bI.E.V., Forschungszentrum Jiilich GmbH (KFA), 52425 Jiilich, Germany
(Received 17 August 1996; in revised form 16 January
1997)
Abstract-The properties of the double layer capacity on rough electrodes are discussed in terms of the recently developed linear Poisson-Boltzmann theory [L. I. Daikhin, A. A. Kornyshev, M. Urbakh, Phys. Rev. E 53, 6192 (1996)]. This theory offers a concept of a “Debye-length K dependent roughness factor”, ie roughness function, which determines the deviation of capacitance from the Gouy-Chapman result for a flat interface. Analytical expression for the roughness function is available for the case of weak Euclidean roughness. The two parameters-mean square height and correlation length-appear there. The way how the result changes in the case of moderate and strong roughness is analyzed on the basis of a numerical solution obtained for a model roughness profile; an efficient interpolation formula which covers the limits of weak and strong roughness is suggested. The role of anisotropy of the roughness profile is investigated. The predicted effects could be screened by the crystallographic inhomogeneity of a rough surface not taken into account here. Tentatively ignoring such (often important) complications we discuss the results in the context of a possible method for in situ characterization of surface roughness of metal electrodes, based on the double layer capacity measurements in solutions of variable concentration. 0 1997 Published by Elsevier Science Ltd
1. INTRODUCTION Surface roughness is an important property in electrochemistry of solid electrodes. For many practical applications in electrocatalysis and energy conversion, one needs the maximal development of active surface. The majority of fundamental studies, however, focus on smooth, flat surfaces, in order to obtain the reference data for characteristics of flat fractions of the surface. Often, even in these studies, performed on single crystal or polished polycrystalline electrodes, there is a certain residual “roughness”, and it is necessary to be able to distinguish the effects induced by it. Usually the roughness of electrodes is characterized by a geometrical roughness,factor, R = S,,ljS, ie the ratio of the true surface to the apparent surface (flat cross-section area). The cases of weak, moderate
*Author to whom correspondence
should be addressed.
and strong roughness are distinguished, depending whether R - 1 < 1, R - 1 - 1, R - I 9 1, respectively. Weak roughness is typical for single crystal electrodes; moderate roughness for polycrystalhne electrodes, while the case of strong roughness is met for specially fabricated catalysts. The role of roughness in electrochemical kinetics [I, 21, frequency dependent impedance [3-61, optical [7-91 and quartz microbalance [lO-121 response was studied theoretically and experimentally. The effect of roughness on the double layer capacitance was reported for various electrodes and electrolytes in a number of experimental works (for review see [13]). Note, that the effect of crystalline heterogeneity on polished surfaces has been thoroughly studied, both theoretically and experimentally [14-161. However, the deviations of the surface from the flat geometry were not handled in the double layer theory, until recently. The slope of the Parsons-Zobel plot [17], ie the dependence of the measured inverse capacitance us
2853
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L. I. Daikhin et al.
the inverse Gouy-Chapman capacitance was usually regarded as a measure of l/R. However, such a treatment does not take into account the competition between the Debye length and characteristic sizes of roughness, which, generally, will modify the GouyChapman result for the diffuse layer capacitance, Cot. Obviously, the limiting value of the capacitance at short Debye lengths should follow the GouyChapman formula but with S replaced by &,I = RS. In the limit of long Debye lengths the roughness cannot be mamyested in the capacitance which would obey the native Gouy-Chapman expression. These two statements are based on the simple geometrical arguments, based on the interplay between the scales of roughness and the Debye “yardstick”. In the first case the diffuse layer simply follows every bump or dip of the electrode, the surface of which looks flat at the Debye scales. For the second case, the analogy with the macroscopic capacitor can be used. When the distance between plates (cf the Debye length) is much greater than any scales of microroughness of the plates, the latter cannot affect the capacitance. How does the crossover between these two limits occur? One may expect to recover the whole curve by equation C=
i?(K)&
(1)
where the roughness function, I? (K), varies between w(O) = 1 and &co) = R > 1. The problem for the theory is, then, to find this function. Surface roughness is not the only possible reason for deviations of the Parsons-Zobel slope from unity. The majority of efforts to explain them were concerned with the account for crystalline heterogeneity of the surface [1416]. The surface of a “flat” polished polycrystalline electrode (or single crystal with defects) may represent different faces characterized by the different potentials of zero charge. This would affect the charge distribution along the equipotential surface. The theory of this effect has been developed [ 161and applied for the treatment of various experimental data. Generally, this effect may influence the capacitance of rough electrodes, as well, where different crystal microfacets may be represented on the surface. However, a difference in p.z.c.s of different crystal phases for some metals (eg Pb, but not Ag) is not large. For such metals crystalline heterogeneity only slightly affects the slope of the Parson-Zobel plot. Probably, the first mentioning that the apparent roughness factor depends on the electrolyte concentration and potential is due to Valette [14], who ascribed it to crystalline heterogeneity. The question of the K-dependence of the Parsons-Zobel slope was recently discussed by Foresti, Hamelin and Guidelli [18, 191. They considered a model of asperities (hemispheres or hemicylinders) on single crystal electrodes, assuming that the effect of crystalline heterogeneity is here negligible. They proposed an interpolation equation for the capacitance of the
diffuse layer. The &h-)-dependence that results from this equation gives the correct limit at K-+OO (high electrolyte concentrations), R(K) = R. However, their @rc)-curve approaches this limit from above, but not from below, which may be related to the fact that their interpolation formula cannot be used for small K[~?(K) diverges, instead of approaching I with K-d]. In our two previous reports [20,21] a theory was developed for the calculation of I?(K) for rough surfaces. In [20] the problem has been solved for weak roughness of arbitrary shape by means of a perturbation theory. In [21] an exact numerical solution was obtained for any degree of roughness, but for a given characteristic surface profile (rectangular grating); the solution was compared to a closed form extrapolation formula. The results of both works reproduce the physical limits mentioned and predict the behavior for intermediate K in dependence on surface morphology. In the present communication we discuss these findings and the new approach for the treatment and interpretation of experimental data on the capacitance of rough electrodes.
2. THEORY Consider a rough metal surface in contact with electrolyte. We take z axis pointing towards the electrolyte and describe the interface by the equation z = c(x,y). The plane z = 0 is chosen such that the average value of the function t(x,y) over the surface is equal to zero. In the Gouy-Chapman theory, the distribution of electrostatic potential 4(r) in the electrolyte is described by the nonlinear Poisson-Boltzmann equation. As a first step we restrict our consideration by its linearized version, valid for low electrode potentials, 4 < ksT/e: (v* -
K*)&r)
= 0.
(2)
The Debye length, K-I, for a 1 - 1 binary electrolyte solution equals (&keT/8nne2)1j2 where n is the electrolyte concentration (number of charge carriers), E, the dielectric constant of the solvent, e, the charge of electron, T, the temperature, and k*, the Boltzmann constant. The solution of equation (2) must satisfy the boundary condition which fixes the value of the potential at the metal-electrolyte interface $J(X,Y,Z= 5(XJ)) = 40
(3)
relative to the zero level in the bulk of the electrolyte. A. Weak roughness Consider first weakly rough surfaces for which h, the characteristic size of roughness in the z direction, is less than the tangential one, 1. The height h denotes the root mean square departure of the surface from
Surface roughness
measured
flatness, and the correlation length (or a period) I is a measure of the average distance between consecutive peaks and valleys on the rough surface. We will also assume that h < K--I. Solving equation (2), it is convenient to Fourier transform the potential and the surface profile function from tangential coordinates R = (XJ) to the corresponding wave vectors K = (K, ,K,.) according to equation,f(K) = j&f(R) exp( - IKR). Application of the perturbation technique [20] gives the capacitance, C = u/&, in the form of equation (I) with the roughness function W(K)=
sdKg(K)[(ti’ + K’)“’ - K].
1 + Kh'(& -
Here we introduced ,function
height-height
the
(4)
correlation
1 g(K) = s
IUK)I’-
(5)
Equation (4) is valid up to the terms of the second order in h. At h = 0, equation (4) gives R(K) = I, reproducing the Gouy-Chapman result for capacitance of a flat interface. For the further consideration it is useful to exclude h, expressing it through R. For this we use the expression for the mean area of a random surface, s -@ = R = 1 + (h/A)’ + 0(h4) S
by “Debye
equation
(4) can be rewritten
a(K)= 1 +
K(R -
as $f(K(),
where I
I
&x2dtru(t)[(l + t*/x*)“*
,f(x) =
- I].
(1 I)
0
In this case the roughness induced change of the capacitance is proportional to the square of the “roughness slope”, 12/l, and the scaling function, f(x), of a single variable: the ratio of the correlation length (period) of roughness to the Debye length. Consider below three examples of surface morphologies. which represent deterministic and random roughness. 1. Sinusoidal corrugation (Fig. l(a)) t(R) = h sin(2n.y//,). In this case, K = (2rrm/L,O) and i(K) = h S (&,J - 6,._,)/2i, where m = 0, 5 1, +2, ., and 6,,.*, is the Kronecker symbol. This leads to the roughness factor R = I + 27~~h?/l: and to the scaling function of argument til*, f(s)
Z
(6)
= x* ((I
I--
+q-
1).
(12)
electrolyte ((1)
1s
I
Z
as
elecirolyte
(b)
i
d(X) =
1+
2855
(4) can be rewritten
where we introduced the notation for the second moment of the correlation function which has the dimension of [length]-?
Then, equation
ruler“
')Ai(2;)i
jdKg(K)[(K’
+ K2)"2 - ti]. (8) x
Equations (4) and (8) relate the capacitance with the morphological features of the interface--h, R and g(K). They can describe the effect of both random roughness and periodical corrugation. In the latter case the integral sm
(9)
2
‘I
c
kd-
electrolyte
electrolyte
cd)
Fig. 1. Schematic sketch of interfacial geometries (a) sinusoidal corrugation; (b) random Gaussian roughness; (c) periodic system of linear defects; (d) rectangular grating.
2856
L. I. Daikhin et al.
2. Random Gaussian roughness (Fig. l(b)). The most widely used approximation for a random Euclidean roughness is the isotropic Gaussian model. It characterizes the roughness spectrum by two parameters: the r.m.s. height h and the tangential correlation length, 1. In this case g(K) = nl& exp( - l&p/4) (ie u(t) = nexp( - ?/4)), the roughness factor R = 1 + 2hz/l:, and the scaling function of ~lo here takes the form ,f((x) = 2Jnxexp(_x2/4)[l
- @(x/2)]
(13)
where m(z) is the probability function [22]. 3. Periodical system of linear defects (Fig. l(c)). In order to simulate the situation when the width of the defects differs from the typical distance between them, we consider the surface profile given by equation
c(R) = h
For this profile the roughness factor
R=
1
+!$$$xp(-f(T);‘),
and the roughness function is given by d(K) =
l + (R - I)F(icd,l/d),
(14)
where F(Kd,l/d)
-1 2nn 12
xCexp n
= 1 + 2(rcd)*
[
2( d 1
]([l+(%yyz-,>
{T(2nn)2exp(
-i(T>‘j.2)F’.
(15)
Note that the meaning of the characteristic lateral length may be different for different types of roughness. For example, the comparison of the models (I) and (2) shows that at the same r.m.s., h, the same geometrical roughness factor, R, would correspond to Is = nlo. Then, according to equation (IO), one should compare the Gaussian scaling function with the sinusoidal scaling function divided by rr2. Such a comparison shows that there is a difference in the range of intermediate ICI,but it does not exceed 10%. Naturally, both models reproduce the limiting laws (Al) and (A3). The roughness function for the random Gaussian model and the periodical system of linear defects is shown in Fig. 2 for typical values of R. The curves for the periodical system of linear defects are plotted for various values of the I./d ratio for the same geometrical roughness factor. We see that the results for two models are very
k
(om”)
Fig. 2. Roughness function li us the inverse Debye length, K: (I) for a random Gaussian roughness (equations (IO) and (13)) and (2)-(4) for a periodic system of linear defects (equations (14) and (15)). E = 80, R = 1.5, 1~ = IO nm, d = I&, I = (2) d/5, (3) d/IO, (4) d/20.
close for i > d/n. However, for the case of rather narrow pores or bumps at the surface (i 4 d) the roughness function calculated within the model (3) approaches R at higher values of K than the corresponding function found for the mode1 (2). Our calculations demonstrate that the region of Debye lengths where the roughness function changes from 1 to R is determined by the smallest lateral scale of roughness. In the mode1 (3) the width of the linear defects plays the role of the such scale. Strictly speaking, the framework of the perturbation theory does not allow us to consider the region of large K, where oh 2 1. However, as we see from Fig. 2, far before this range the roughness function levels off to the geometrical factor R. Therefore, this condition may not be critical. It gives hopes that equation (8) could be used, as an interpolation formula, also in the case of strong roughness. This will be checked on the basis of a mode1 exact solution, discussed in the next section. B. Exact solution for the rectangular grating Consider the profile shown in Fig. l(d) with arbitrary values of h, A, and d. In this case the geometrical roughness factor equals R = 1 + 2h/d. Even for such a simple profile, the solution of the linearized Poisson-Boltzmann equation, equation (2), cannot be obtained in an analytical form for an arbitrary relation between h, A, d and K. The numerical solution has been recently found [21] by matching the solutions of equation (2) at the planes of z = -h/2 and z = h/2. Figure 3 shows the calculated roughness function for R = 5 and a few given values of A/d. On the same figures, corresponding curves obtained from the analytical expressions (14) and (15) are shown for comparison. Figure 4 displays the dependence of W(K) on R, plotted by varying the height of roughness, h, for a few fixed values of red. This dependence is close to the linear one for all values of the parameters. Looking at Figs 3 and 4 we can conclude that the interpolation formula, equations (14) and (I 5) gives
Surface roughness
measured
by “Debye
ruler”
2857
a fairly good representation of the exact solution. In spite of the fact that this formula has been derived within the model of weak roughness it gives a practically universal description of the possible roughness effect on the double layer capacitance for non-fractal electrode surfaces.
3. PARSONS-ZOBEL
PLOTS
A common assumption in electrochemistry is that in addition to dz@se layer capacitance, C, one must also consider the contribution of the compact layer, 2
I
3
4
5
6
GEOMETRICAL ROUGHNESS FACTOR R Fig. 4. Roughness function k us the geometrical roughness factor, R, for the rectangular profile. E = 80. A/d = l/2. dh- = (I) IO, (2) 6, (3) 4, (4) 2, (5) 1, (6) 0.6.
CH. Connected capacitance
0.2
0.6
0.4
K
0.2
0.6
0.6
0.8
1
K (nm”)
0.2
0.4
0.6
0.8
1
K (nm-‘)
3. Results
function, rectangular roughness ((14) and A = d/2:
of exact
calculations
of the roughness length, K, for the grating (solid curves); dashed curves show the function obtained from the analytical expressions (15)). E = 80, R = 5, d = 30 nm, I = 0.75A; (a) (b) A = d/5, (c) A = d/IO.
R vs the inverse Debye
they
give for the
total
1
(rim")
0.4
in series,
Following Grahame [23], CH is assumed to be independent of ionic concentration. Grahame theory was first suggested for a liquid mercury electrode, and was systematically used both for liquid and solid electrodes. This equation is based on the assumption that the boundary between the diffuse and the compact layers is equipotential [24]. Would it be changed for rough electrodes? Since different crystal faces can be presented on a rough surface, it may happen that tlle dipole potential drops across the corresponding segments of the compact layer may be different. Thus, this assumption may break down. As a first approximation, however, we will adopt equation (16) (as no big effect was found due to crystalline heterogeneity on flat surfaces) and focus on the roughness-induced modification of the diffuse layer contribution. The validity of the Gouy-Chapman-Grahame theory is checked by drawing the Parsons-Zobel plots: measured inverse capacitance of the interface, concen1/Got, us ~/CCC for different electrolyte trations [l7]. Straight line with the unit slope approves the Gouy-Chapman theory for the diffuse layer, and the corresponding intercept determines the compact layer contribution, 1/CH. Slopes lower than 1 are usually attributed to the geometrical roughness factor [13]. Deviations from the straight line are regarded as indications of specific adsorption of ions or noncomplete dissociation of electrolyte [ 131. In [ 14, 18-201 different interpretation of such deviations was suggested. We discuss them here on the basis of our findings. Equations (1) and (8) suggest that roughness leads to deviations of Parsons-Zobel plots from linearity.
L. I. Daikhin et al.
2858
Figure 5(a) demonstrates it for the different values of the roughness factor without a compact layer contribution: the intercept is kept zero. Adding the
6o a
compact layer one should bear in mind that Cu would be itself proportional to R, ie CH z RF” where c, is the reference value for the flat surface. Thus, the actual intercept will move down _ l/R with the increase of R. Figure 5(b) demonstrates that for a typical value of e, and several R-values. Important conclusions follow from these figures. 1. “Negative” intercept. Extrapolation of the curves in Fig. 3(a) from the small K range to the limit of large K (l/Coc+O) gives an “apparent” negative intercept. Equation (k3) derived- for nonfractal gives the value of the intercept, surfaces -477(R - l)A*/cLS where
The intercept with account for a compact layer is given by 1 1 _=-Cextr RCo,
I
I
10
20
30 l/C,
10
50
60
(;i>
b
60.
/ / t 10
(17)
The calculations performed within the random Gaussian model for h = 50 A, I= 100 A, which correspond to R = 1.5, give 47r(R - l)A*/&L = 6.5 A. Only positive values for the intercept were so far reported. That means that the negative extrapolation value is compensated by the compact layer contribution. In order to get a value of C,,,, 2 20 pF/cm2, which is typically observed, one must have Cu < 10 pF/cm2. Thus, the treatment of the capacitance data for rough surfaces, should be reconsidered: (i) The value of the roughness factor cannot be taken as the reciprocal slope of the Parsons-Zobel plot in the range of small concentrations. (ii) The intercept, obtained from the extrapolation of the plot from the range of small concentration into the high concentration limit does not give l/RP,. In order to get this value, one must treat the whole curve, making a nonlinear regression analysis with the help of equations (14) and (15). 2. Extended region of non-Gouy-Chapman behavior. Considerable curvature of the plot is seen, eg in Fig. 5 in the region of KI - 1. However, generally, the Parsons-Zobel plots are not convenient for the characterization of surface roughness. More convenient would be the plot of
10.
-20
4a(R - l)A* ELS .
30
20
l/Ccc
10
50
60
CL>
Fig. 5. Parsons-Zobel plots: inverse capacitance, I/C,,,, for a periodic system of linear defects vs the inverse Gouy-Chapman capacitance, l/Cot. (a) without a compact layer contribution; (b) with a compact layer contribution, CA = 20 pF/cm*. Curves: (I) Gouy-Chapman plot; (2) and (3) the plots, calculated via equations (14)-(16) for R = I.5 and 3, respectively; (---) the large Debye length (small CGC) asymptotic laws. E = 80. d = 30 nm, i. = d/2.
where Cu is evaluated from the measurements at high concentration. If the accuracy would allow, the limiting laws (Al) and (A3) may be studied, giving the important roughness parameters: mean square curvature of the interface (S2) (see equations (Al) and (A2)) and (R - l)h*/L. Nonlinear regression fit of the whole curve would give the lateral correlation
Surface roughness
measured
lengths of roughness. It should be noted, that there is another source of deviation from the Gouy-Chapman theory, coming into play at large concentrations, which is due to the structure of the solvent; it may partially compensate the deviations dues to surface roughness [25]. 4. CONCLUSION The new approach to the treatment of capacitance data on rough electrode surfaces does not give us, yet, a new method for the study of surface roughness by measuring the diffuse double layer capacitance at varied Debye lengths. Indeed, before speaking about a new method, one must first check the predictions of the present theory for the deviation of the Parsons-Zobel plots from linearity with experimental data. The results are basically discouraging. Indeed, though there are rare cases of negative curvatures of the Parsons-Zobel plots [19], more often the curvatures are positive [I 3, IS, 191. The reason for that could be a combined effect of different crystal phases, represented on the rough surface, or a more complex interplay between the compact and diffuse layers (rather than the ad hoc Grahame-Parsons [23] combination of the two) for the case of strong roughness, etc. Understanding the consequences of the interference of these effects would require further laborious theoretical investigations and numerical simulations. However, a support to the predictions of the present approach comes, unexpectedly, from another side. The above given formulae were derived within the linearized Poisson-Boltzmann approximation, applicable, strictly speaking, only at low charges of the electrode. In this case, the Debye length variation should be provided by the variation of the electrolyte concentration. For an estimate of the effects due to large deviations from p.z.c., it is tempting to replace K by an effective value, K,R =
64~~ + 47r’L&7’le’
(18)
(where 0 is the surface charge density and LB = e2/&kt,T, the Bjerrum length), which scales the “Debye ruler” to the charge of the electrode. The first results of the nonlinear Poisson-Boltzmann variant of the theory [30] show however that the effect of the potential is more complicated and much more interesting: varying the electrode potential one may induce a crossover from the positive to negative curvatures! A detailed analysis of existing data obtained for different potentials, or obtaining such data systematically in the light of the new theory, is, therefore, necessary. The best interaction between the theory and experiments (for a verification and “calibration” of the “Debye-ruler method”) can be achieved in the studies of surfaces where the roughness is characterized by other methods, such as electron microscopy in uhc, in .situ STM. or diffuse light scattering. An ideal test of the theory predictions (not that much
by “Debye
ruler”
2859
unrealistic in view of the developing nano- and micro-technology) would be to work with electrodes of a given, deterministic, pre-fabricated roughness. In such experiments, one may vary not only the Debye length, but also the corrugation parameters in a series of samples. Experiments with vicinal surfaces could be an approach to our one dimensional corrugated profiles.
ACKNOWLEDGEMENT M. U. is thankful to Deutsche Akademischer Austauschdienst (DAAD) for a Senior Scientist Scholarship and to Professor U. Stimmung for his hospitality.
REFERENCES 1. R. de Levie. J. Electroanal. Chem. 261, I (1989); 281 (1990). 2. R. Kant and S. K. Rangarajan, J. Electroanal. Chem. 368, 1 (1994); 396, 285 (1995). 1 A. Le Mehaute and G. Crepy, Solid Stnre Ionics 9/10, _ 17 (1983); A. Le Mehaute, The Fractal Approach to Heterogeneous Chemistry, p. 311. Wiley. New York (1989). 4. S. Liu. in Condensed Matter Physics Aspects qf Electrochemistry (Edited by M. P. Tosi and A. A. Kornyshev), p. 329, World Scientific, Singapore (1991). 5. B. Sapoval. J.-N. Chazalviel and J. Peyriire, Phys. Ret,. A 38, 867 (1988); T. Pajkossy and L. Nykos, Electrochim. Acfa 34, 171 (1989). 6. T. C. Halsey and M. Leibig, Ann. Pl7y.r. (N. Y.) 219, 109 (1992). 7. H. Raether. Surf&e Plasmons on Smooth and Rough Surf&es and on Gratings, Springer Tracts in Modern Physics Vol. III, Springer, Berlin (1988). 8. M. I. Urbakh, Electrochim. Acta 34, 1777 (1989); in Condensed Matter Physics Aspects qf Electrochemistry (Edited by M. P. TOSI and A. A. Kornyshev). p. 295, World Scientific, Singapore (1991). 9. A. M. Brodsky and M. I. Urbakh, Prog. in Surf. Sci. 33, 991 (1990). IO. M. Yang and M. Thompson, Langmuir 9, 1990 (1993). II. M. Urbakh and L. Daikhin, Phys. Ret>. B 49, 4866 (1994). 12. M. Urbakh and L. Daikhin, Langmuir IO, 2839 (1994). 13. M. A. Vorotyntsev, in Modern Aspects of Electrochemistry (Edited-by J. O’M. Bockris, b. E. donway and R. E. White). Vol. 17. D. 131. Plenum. New York (1986). 14. G. Valetie. J. Elecr;oanal. Chem. &Xl, 425 (19sb). IS. I. A. Bagotskaya. B. B. Damaskin and M. D. Levi, J. Electroanal. Chem. 115, 189 (1980). 16. M. A. Vorotyntsev, J. Electroanal. Chem. 123, 379 (1981). 17. R. Parsons and F. Zobel. J. Electraanal. Chem. 9, 333 (1965). 18. M. L. Foresti. R. Guidelli and A. Hamelin, J. Elecrroanal. Chrm. 346, 73 (1993). 19. A. Hamelin, M. L. Foresti and R. Guidelli, J. Elecrroanal. Chrm. 346, 251 (1993). 20. L. Daikhin, A. A. Kornyshev and M. Urbakh, Phys.
Rw. E 53, 6192 (1996). 21. L. Daikhin.
A. A. Kornyshev
and
M. Urbakh.
in
preparation.
22. M. Abramovitz and 1. Stegan. eds. Handbook Mathema/ical Funcrions. Dover, New York (1965). 23. D. C. Grahame. C/tern. Ret. 41, 441 (1947).
of
L. 1. Daikhin et al.
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24. The whole concept, based on the division of the compact and diffuse layer parts. was always a matter of concern in view of molecular dimensions of the compact layer [26]. However the contribution to capacitance, independent of electrolyte concentration, ie the finite value of the intercept of the Parsons-Zobel plot, is an experimental fact. Furthermore, molecular theory of electrolyte near a charged hard wall [27] and the phenomenological nonlocal electrostatic theory [28] both predict such a contribution without an artifical introduction of any “compact layers”. This results as an effect of the short range structure of the solvent [26-291. Experimental data for a set of simple metals and polar solvents give typical values of the compact layer capacitance [l3, 261, CH/&al = 0.1-0.8 A. 25. The interference between the solvent structure and Debye length rounds the Parsons-Zobel plot down from the straight line in the high concentration region [27]. The roughness effect does the opposite. The two effects may compensate each other for I of molecular dimensions, and the Parsons-Zobel plot will appear “more straight” than it should be for an ideally flat surface. 26. A. A. Kornyshev, in The Chemical Physics of Salvation (Edited by R. R. Dogonadze, E. Kalman, A. A. Kornyshev and J. Ulstrup), Part C, p. 355, Elsevier, Amsterdam (1988). 21. L. Blum, D. Henderson and R. Parsons, J. Electroanal. Chem. 161, 389 (1984). 28. A. A. Kornyshev and J. Ulstrup, Chem. Scripta 25, 58 (1985).
29. A. A. Kornyshev, W. Schmickler and M. A. Vorotyntsev, Phys. Rev. B 25, 5244 (1982). 30. L. Daikhin, A. A. Kornyshev and M. Urbakh, Submitted to J. Chem. Phys.
concept of characteristic correlation length breaks down for fractal surfaces, which have been studied in
WI. For K-’ Q Imi, one may expand (K’ + f?)“2 in the integrand of equation (4) using the smallness of (K/K)', equation (4) then reduces to a(K)-R{l
-$#
(Al)
Here we used equation (6) and the definition for the mean square curvature
Equation (Al) shows, as expected, that the roughness function R(K) approaches the geometrical roughness factor R for small Debye length K-I (large n). With the increase of K-I (the decrease of n) it decreases with respect to R, the correction being proportional to the square of the Debye length, ie it is inversely proportional to the charge carriers concentration. In the range of large Debye lengths (low concentrations), K-’ % I,,,,,, one may expand the term (K~ + f?)"' in the integrand of equation (4) in (~/lu)~ to obtain a(K)
N
1+ q
-
K2h2.
Here the length APPENDIX Limiting Debye
behavior
in the cases
qf
small
and
large
lengths
Consider the behavior of expression (4) for the two extreme cases: (a) the Debye length K-’ is shorter than the smallest characteristic correlation length of roughness Ime, and (b) K-’ is greater than the maximal correlation length I,,,,,. These two limiting cases can be realized experimentally, changing, for instance, the electrolyte concentration. Note that the
is of the order of I,,,. As expected, at very large Debye lengths the roughness of the surface is not “seen” in the capacitance. The first correction to the flat surface result is linear in K. Approving our expectations, equations (Al) and (A3) specify how the roughness function approaches the two obvious limits.
Pergamon PII:
Double layer capacitance on a rough metal surface: surface roughness measured by “Debye ruler” L. I. Daikhin,“* A. A. Kornyshevb and M. Urbakh “School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel bI.E.V., Forschungszentrum Jiilich GmbH (KFA), 52425 Jiilich, Germany
(Received 17 August 1996; in revised form 16 January
1997)
Abstract-The properties of the double layer capacity on rough electrodes are discussed in terms of the recently developed linear Poisson-Boltzmann theory [L. I. Daikhin, A. A. Kornyshev, M. Urbakh, Phys. Rev. E 53, 6192 (1996)]. This theory offers a concept of a “Debye-length K dependent roughness factor”, ie roughness function, which determines the deviation of capacitance from the Gouy-Chapman result for a flat interface. Analytical expression for the roughness function is available for the case of weak Euclidean roughness. The two parameters-mean square height and correlation length-appear there. The way how the result changes in the case of moderate and strong roughness is analyzed on the basis of a numerical solution obtained for a model roughness profile; an efficient interpolation formula which covers the limits of weak and strong roughness is suggested. The role of anisotropy of the roughness profile is investigated. The predicted effects could be screened by the crystallographic inhomogeneity of a rough surface not taken into account here. Tentatively ignoring such (often important) complications we discuss the results in the context of a possible method for in situ characterization of surface roughness of metal electrodes, based on the double layer capacity measurements in solutions of variable concentration. 0 1997 Published by Elsevier Science Ltd
1. INTRODUCTION Surface roughness is an important property in electrochemistry of solid electrodes. For many practical applications in electrocatalysis and energy conversion, one needs the maximal development of active surface. The majority of fundamental studies, however, focus on smooth, flat surfaces, in order to obtain the reference data for characteristics of flat fractions of the surface. Often, even in these studies, performed on single crystal or polished polycrystalline electrodes, there is a certain residual “roughness”, and it is necessary to be able to distinguish the effects induced by it. Usually the roughness of electrodes is characterized by a geometrical roughness,factor, R = S,,ljS, ie the ratio of the true surface to the apparent surface (flat cross-section area). The cases of weak, moderate
*Author to whom correspondence
should be addressed.
and strong roughness are distinguished, depending whether R - 1 < 1, R - 1 - 1, R - I 9 1, respectively. Weak roughness is typical for single crystal electrodes; moderate roughness for polycrystalhne electrodes, while the case of strong roughness is met for specially fabricated catalysts. The role of roughness in electrochemical kinetics [I, 21, frequency dependent impedance [3-61, optical [7-91 and quartz microbalance [lO-121 response was studied theoretically and experimentally. The effect of roughness on the double layer capacitance was reported for various electrodes and electrolytes in a number of experimental works (for review see [13]). Note, that the effect of crystalline heterogeneity on polished surfaces has been thoroughly studied, both theoretically and experimentally [14-161. However, the deviations of the surface from the flat geometry were not handled in the double layer theory, until recently. The slope of the Parsons-Zobel plot [17], ie the dependence of the measured inverse capacitance us
2853
2854
L. I. Daikhin et al.
the inverse Gouy-Chapman capacitance was usually regarded as a measure of l/R. However, such a treatment does not take into account the competition between the Debye length and characteristic sizes of roughness, which, generally, will modify the GouyChapman result for the diffuse layer capacitance, Cot. Obviously, the limiting value of the capacitance at short Debye lengths should follow the GouyChapman formula but with S replaced by &,I = RS. In the limit of long Debye lengths the roughness cannot be mamyested in the capacitance which would obey the native Gouy-Chapman expression. These two statements are based on the simple geometrical arguments, based on the interplay between the scales of roughness and the Debye “yardstick”. In the first case the diffuse layer simply follows every bump or dip of the electrode, the surface of which looks flat at the Debye scales. For the second case, the analogy with the macroscopic capacitor can be used. When the distance between plates (cf the Debye length) is much greater than any scales of microroughness of the plates, the latter cannot affect the capacitance. How does the crossover between these two limits occur? One may expect to recover the whole curve by equation C=
i?(K)&
(1)
where the roughness function, I? (K), varies between w(O) = 1 and &co) = R > 1. The problem for the theory is, then, to find this function. Surface roughness is not the only possible reason for deviations of the Parsons-Zobel slope from unity. The majority of efforts to explain them were concerned with the account for crystalline heterogeneity of the surface [1416]. The surface of a “flat” polished polycrystalline electrode (or single crystal with defects) may represent different faces characterized by the different potentials of zero charge. This would affect the charge distribution along the equipotential surface. The theory of this effect has been developed [ 161and applied for the treatment of various experimental data. Generally, this effect may influence the capacitance of rough electrodes, as well, where different crystal microfacets may be represented on the surface. However, a difference in p.z.c.s of different crystal phases for some metals (eg Pb, but not Ag) is not large. For such metals crystalline heterogeneity only slightly affects the slope of the Parson-Zobel plot. Probably, the first mentioning that the apparent roughness factor depends on the electrolyte concentration and potential is due to Valette [14], who ascribed it to crystalline heterogeneity. The question of the K-dependence of the Parsons-Zobel slope was recently discussed by Foresti, Hamelin and Guidelli [18, 191. They considered a model of asperities (hemispheres or hemicylinders) on single crystal electrodes, assuming that the effect of crystalline heterogeneity is here negligible. They proposed an interpolation equation for the capacitance of the
diffuse layer. The &h-)-dependence that results from this equation gives the correct limit at K-+OO (high electrolyte concentrations), R(K) = R. However, their @rc)-curve approaches this limit from above, but not from below, which may be related to the fact that their interpolation formula cannot be used for small K[~?(K) diverges, instead of approaching I with K-d]. In our two previous reports [20,21] a theory was developed for the calculation of I?(K) for rough surfaces. In [20] the problem has been solved for weak roughness of arbitrary shape by means of a perturbation theory. In [21] an exact numerical solution was obtained for any degree of roughness, but for a given characteristic surface profile (rectangular grating); the solution was compared to a closed form extrapolation formula. The results of both works reproduce the physical limits mentioned and predict the behavior for intermediate K in dependence on surface morphology. In the present communication we discuss these findings and the new approach for the treatment and interpretation of experimental data on the capacitance of rough electrodes.
2. THEORY Consider a rough metal surface in contact with electrolyte. We take z axis pointing towards the electrolyte and describe the interface by the equation z = c(x,y). The plane z = 0 is chosen such that the average value of the function t(x,y) over the surface is equal to zero. In the Gouy-Chapman theory, the distribution of electrostatic potential 4(r) in the electrolyte is described by the nonlinear Poisson-Boltzmann equation. As a first step we restrict our consideration by its linearized version, valid for low electrode potentials, 4 < ksT/e: (v* -
K*)&r)
= 0.
(2)
The Debye length, K-I, for a 1 - 1 binary electrolyte solution equals (&keT/8nne2)1j2 where n is the electrolyte concentration (number of charge carriers), E, the dielectric constant of the solvent, e, the charge of electron, T, the temperature, and k*, the Boltzmann constant. The solution of equation (2) must satisfy the boundary condition which fixes the value of the potential at the metal-electrolyte interface $J(X,Y,Z= 5(XJ)) = 40
(3)
relative to the zero level in the bulk of the electrolyte. A. Weak roughness Consider first weakly rough surfaces for which h, the characteristic size of roughness in the z direction, is less than the tangential one, 1. The height h denotes the root mean square departure of the surface from
Surface roughness
measured
flatness, and the correlation length (or a period) I is a measure of the average distance between consecutive peaks and valleys on the rough surface. We will also assume that h < K--I. Solving equation (2), it is convenient to Fourier transform the potential and the surface profile function from tangential coordinates R = (XJ) to the corresponding wave vectors K = (K, ,K,.) according to equation,f(K) = j&f(R) exp( - IKR). Application of the perturbation technique [20] gives the capacitance, C = u/&, in the form of equation (I) with the roughness function W(K)=
sdKg(K)[(ti’ + K’)“’ - K].
1 + Kh'(& -
Here we introduced ,function
height-height
the
(4)
correlation
1 g(K) = s
IUK)I’-
(5)
Equation (4) is valid up to the terms of the second order in h. At h = 0, equation (4) gives R(K) = I, reproducing the Gouy-Chapman result for capacitance of a flat interface. For the further consideration it is useful to exclude h, expressing it through R. For this we use the expression for the mean area of a random surface, s -@ = R = 1 + (h/A)’ + 0(h4) S
by “Debye
equation
(4) can be rewritten
a(K)= 1 +
K(R -
as $f(K(),
where I
I
&x2dtru(t)[(l + t*/x*)“*
,f(x) =
- I].
(1 I)
0
In this case the roughness induced change of the capacitance is proportional to the square of the “roughness slope”, 12/l, and the scaling function, f(x), of a single variable: the ratio of the correlation length (period) of roughness to the Debye length. Consider below three examples of surface morphologies. which represent deterministic and random roughness. 1. Sinusoidal corrugation (Fig. l(a)) t(R) = h sin(2n.y//,). In this case, K = (2rrm/L,O) and i(K) = h S (&,J - 6,._,)/2i, where m = 0, 5 1, +2, ., and 6,,.*, is the Kronecker symbol. This leads to the roughness factor R = I + 27~~h?/l: and to the scaling function of argument til*, f(s)
Z
(6)
= x* ((I
I--
+q-
1).
(12)
electrolyte ((1)
1s
I
Z
as
elecirolyte
(b)
i
d(X) =
1+
2855
(4) can be rewritten
where we introduced the notation for the second moment of the correlation function which has the dimension of [length]-?
Then, equation
ruler“
')Ai(2;)i
jdKg(K)[(K’
+ K2)"2 - ti]. (8) x
Equations (4) and (8) relate the capacitance with the morphological features of the interface--h, R and g(K). They can describe the effect of both random roughness and periodical corrugation. In the latter case the integral sm
(9)
2
‘I
c
kd-
electrolyte
electrolyte
cd)
Fig. 1. Schematic sketch of interfacial geometries (a) sinusoidal corrugation; (b) random Gaussian roughness; (c) periodic system of linear defects; (d) rectangular grating.
2856
L. I. Daikhin et al.
2. Random Gaussian roughness (Fig. l(b)). The most widely used approximation for a random Euclidean roughness is the isotropic Gaussian model. It characterizes the roughness spectrum by two parameters: the r.m.s. height h and the tangential correlation length, 1. In this case g(K) = nl& exp( - l&p/4) (ie u(t) = nexp( - ?/4)), the roughness factor R = 1 + 2hz/l:, and the scaling function of ~lo here takes the form ,f((x) = 2Jnxexp(_x2/4)[l
- @(x/2)]
(13)
where m(z) is the probability function [22]. 3. Periodical system of linear defects (Fig. l(c)). In order to simulate the situation when the width of the defects differs from the typical distance between them, we consider the surface profile given by equation
c(R) = h
For this profile the roughness factor
R=
1
+!$$$xp(-f(T);‘),
and the roughness function is given by d(K) =
l + (R - I)F(icd,l/d),
(14)
where F(Kd,l/d)
-1 2nn 12
xCexp n
= 1 + 2(rcd)*
[
2( d 1
]([l+(%yyz-,>
{T(2nn)2exp(
-i(T>‘j.2)F’.
(15)
Note that the meaning of the characteristic lateral length may be different for different types of roughness. For example, the comparison of the models (I) and (2) shows that at the same r.m.s., h, the same geometrical roughness factor, R, would correspond to Is = nlo. Then, according to equation (IO), one should compare the Gaussian scaling function with the sinusoidal scaling function divided by rr2. Such a comparison shows that there is a difference in the range of intermediate ICI,but it does not exceed 10%. Naturally, both models reproduce the limiting laws (Al) and (A3). The roughness function for the random Gaussian model and the periodical system of linear defects is shown in Fig. 2 for typical values of R. The curves for the periodical system of linear defects are plotted for various values of the I./d ratio for the same geometrical roughness factor. We see that the results for two models are very
k
(om”)
Fig. 2. Roughness function li us the inverse Debye length, K: (I) for a random Gaussian roughness (equations (IO) and (13)) and (2)-(4) for a periodic system of linear defects (equations (14) and (15)). E = 80, R = 1.5, 1~ = IO nm, d = I&, I = (2) d/5, (3) d/IO, (4) d/20.
close for i > d/n. However, for the case of rather narrow pores or bumps at the surface (i 4 d) the roughness function calculated within the model (3) approaches R at higher values of K than the corresponding function found for the mode1 (2). Our calculations demonstrate that the region of Debye lengths where the roughness function changes from 1 to R is determined by the smallest lateral scale of roughness. In the mode1 (3) the width of the linear defects plays the role of the such scale. Strictly speaking, the framework of the perturbation theory does not allow us to consider the region of large K, where oh 2 1. However, as we see from Fig. 2, far before this range the roughness function levels off to the geometrical factor R. Therefore, this condition may not be critical. It gives hopes that equation (8) could be used, as an interpolation formula, also in the case of strong roughness. This will be checked on the basis of a mode1 exact solution, discussed in the next section. B. Exact solution for the rectangular grating Consider the profile shown in Fig. l(d) with arbitrary values of h, A, and d. In this case the geometrical roughness factor equals R = 1 + 2h/d. Even for such a simple profile, the solution of the linearized Poisson-Boltzmann equation, equation (2), cannot be obtained in an analytical form for an arbitrary relation between h, A, d and K. The numerical solution has been recently found [21] by matching the solutions of equation (2) at the planes of z = -h/2 and z = h/2. Figure 3 shows the calculated roughness function for R = 5 and a few given values of A/d. On the same figures, corresponding curves obtained from the analytical expressions (14) and (15) are shown for comparison. Figure 4 displays the dependence of W(K) on R, plotted by varying the height of roughness, h, for a few fixed values of red. This dependence is close to the linear one for all values of the parameters. Looking at Figs 3 and 4 we can conclude that the interpolation formula, equations (14) and (I 5) gives
Surface roughness
measured
by “Debye
ruler”
2857
a fairly good representation of the exact solution. In spite of the fact that this formula has been derived within the model of weak roughness it gives a practically universal description of the possible roughness effect on the double layer capacitance for non-fractal electrode surfaces.
3. PARSONS-ZOBEL
PLOTS
A common assumption in electrochemistry is that in addition to dz@se layer capacitance, C, one must also consider the contribution of the compact layer, 2
I
3
4
5
6
GEOMETRICAL ROUGHNESS FACTOR R Fig. 4. Roughness function k us the geometrical roughness factor, R, for the rectangular profile. E = 80. A/d = l/2. dh- = (I) IO, (2) 6, (3) 4, (4) 2, (5) 1, (6) 0.6.
CH. Connected capacitance
0.2
0.6
0.4
K
0.2
0.6
0.6
0.8
1
K (nm”)
0.2
0.4
0.6
0.8
1
K (nm-‘)
3. Results
function, rectangular roughness ((14) and A = d/2:
of exact
calculations
of the roughness length, K, for the grating (solid curves); dashed curves show the function obtained from the analytical expressions (15)). E = 80, R = 5, d = 30 nm, I = 0.75A; (a) (b) A = d/5, (c) A = d/IO.
R vs the inverse Debye
they
give for the
total
1
(rim")
0.4
in series,
Following Grahame [23], CH is assumed to be independent of ionic concentration. Grahame theory was first suggested for a liquid mercury electrode, and was systematically used both for liquid and solid electrodes. This equation is based on the assumption that the boundary between the diffuse and the compact layers is equipotential [24]. Would it be changed for rough electrodes? Since different crystal faces can be presented on a rough surface, it may happen that tlle dipole potential drops across the corresponding segments of the compact layer may be different. Thus, this assumption may break down. As a first approximation, however, we will adopt equation (16) (as no big effect was found due to crystalline heterogeneity on flat surfaces) and focus on the roughness-induced modification of the diffuse layer contribution. The validity of the Gouy-Chapman-Grahame theory is checked by drawing the Parsons-Zobel plots: measured inverse capacitance of the interface, concen1/Got, us ~/CCC for different electrolyte trations [l7]. Straight line with the unit slope approves the Gouy-Chapman theory for the diffuse layer, and the corresponding intercept determines the compact layer contribution, 1/CH. Slopes lower than 1 are usually attributed to the geometrical roughness factor [13]. Deviations from the straight line are regarded as indications of specific adsorption of ions or noncomplete dissociation of electrolyte [ 131. In [ 14, 18-201 different interpretation of such deviations was suggested. We discuss them here on the basis of our findings. Equations (1) and (8) suggest that roughness leads to deviations of Parsons-Zobel plots from linearity.
L. I. Daikhin et al.
2858
Figure 5(a) demonstrates it for the different values of the roughness factor without a compact layer contribution: the intercept is kept zero. Adding the
6o a
compact layer one should bear in mind that Cu would be itself proportional to R, ie CH z RF” where c, is the reference value for the flat surface. Thus, the actual intercept will move down _ l/R with the increase of R. Figure 5(b) demonstrates that for a typical value of e, and several R-values. Important conclusions follow from these figures. 1. “Negative” intercept. Extrapolation of the curves in Fig. 3(a) from the small K range to the limit of large K (l/Coc+O) gives an “apparent” negative intercept. Equation (k3) derived- for nonfractal gives the value of the intercept, surfaces -477(R - l)A*/cLS where
The intercept with account for a compact layer is given by 1 1 _=-Cextr RCo,
I
I
10
20
30 l/C,
10
50
60
(;i>
b
60.
/ / t 10
(17)
The calculations performed within the random Gaussian model for h = 50 A, I= 100 A, which correspond to R = 1.5, give 47r(R - l)A*/&L = 6.5 A. Only positive values for the intercept were so far reported. That means that the negative extrapolation value is compensated by the compact layer contribution. In order to get a value of C,,,, 2 20 pF/cm2, which is typically observed, one must have Cu < 10 pF/cm2. Thus, the treatment of the capacitance data for rough surfaces, should be reconsidered: (i) The value of the roughness factor cannot be taken as the reciprocal slope of the Parsons-Zobel plot in the range of small concentrations. (ii) The intercept, obtained from the extrapolation of the plot from the range of small concentration into the high concentration limit does not give l/RP,. In order to get this value, one must treat the whole curve, making a nonlinear regression analysis with the help of equations (14) and (15). 2. Extended region of non-Gouy-Chapman behavior. Considerable curvature of the plot is seen, eg in Fig. 5 in the region of KI - 1. However, generally, the Parsons-Zobel plots are not convenient for the characterization of surface roughness. More convenient would be the plot of
10.
-20
4a(R - l)A* ELS .
30
20
l/Ccc
10
50
60
CL>
Fig. 5. Parsons-Zobel plots: inverse capacitance, I/C,,,, for a periodic system of linear defects vs the inverse Gouy-Chapman capacitance, l/Cot. (a) without a compact layer contribution; (b) with a compact layer contribution, CA = 20 pF/cm*. Curves: (I) Gouy-Chapman plot; (2) and (3) the plots, calculated via equations (14)-(16) for R = I.5 and 3, respectively; (---) the large Debye length (small CGC) asymptotic laws. E = 80. d = 30 nm, i. = d/2.
where Cu is evaluated from the measurements at high concentration. If the accuracy would allow, the limiting laws (Al) and (A3) may be studied, giving the important roughness parameters: mean square curvature of the interface (S2) (see equations (Al) and (A2)) and (R - l)h*/L. Nonlinear regression fit of the whole curve would give the lateral correlation
Surface roughness
measured
lengths of roughness. It should be noted, that there is another source of deviation from the Gouy-Chapman theory, coming into play at large concentrations, which is due to the structure of the solvent; it may partially compensate the deviations dues to surface roughness [25]. 4. CONCLUSION The new approach to the treatment of capacitance data on rough electrode surfaces does not give us, yet, a new method for the study of surface roughness by measuring the diffuse double layer capacitance at varied Debye lengths. Indeed, before speaking about a new method, one must first check the predictions of the present theory for the deviation of the Parsons-Zobel plots from linearity with experimental data. The results are basically discouraging. Indeed, though there are rare cases of negative curvatures of the Parsons-Zobel plots [19], more often the curvatures are positive [I 3, IS, 191. The reason for that could be a combined effect of different crystal phases, represented on the rough surface, or a more complex interplay between the compact and diffuse layers (rather than the ad hoc Grahame-Parsons [23] combination of the two) for the case of strong roughness, etc. Understanding the consequences of the interference of these effects would require further laborious theoretical investigations and numerical simulations. However, a support to the predictions of the present approach comes, unexpectedly, from another side. The above given formulae were derived within the linearized Poisson-Boltzmann approximation, applicable, strictly speaking, only at low charges of the electrode. In this case, the Debye length variation should be provided by the variation of the electrolyte concentration. For an estimate of the effects due to large deviations from p.z.c., it is tempting to replace K by an effective value, K,R =
64~~ + 47r’L&7’le’
(18)
(where 0 is the surface charge density and LB = e2/&kt,T, the Bjerrum length), which scales the “Debye ruler” to the charge of the electrode. The first results of the nonlinear Poisson-Boltzmann variant of the theory [30] show however that the effect of the potential is more complicated and much more interesting: varying the electrode potential one may induce a crossover from the positive to negative curvatures! A detailed analysis of existing data obtained for different potentials, or obtaining such data systematically in the light of the new theory, is, therefore, necessary. The best interaction between the theory and experiments (for a verification and “calibration” of the “Debye-ruler method”) can be achieved in the studies of surfaces where the roughness is characterized by other methods, such as electron microscopy in uhc, in .situ STM. or diffuse light scattering. An ideal test of the theory predictions (not that much
by “Debye
ruler”
2859
unrealistic in view of the developing nano- and micro-technology) would be to work with electrodes of a given, deterministic, pre-fabricated roughness. In such experiments, one may vary not only the Debye length, but also the corrugation parameters in a series of samples. Experiments with vicinal surfaces could be an approach to our one dimensional corrugated profiles.
ACKNOWLEDGEMENT M. U. is thankful to Deutsche Akademischer Austauschdienst (DAAD) for a Senior Scientist Scholarship and to Professor U. Stimmung for his hospitality.
REFERENCES 1. R. de Levie. J. Electroanal. Chem. 261, I (1989); 281 (1990). 2. R. Kant and S. K. Rangarajan, J. Electroanal. Chem. 368, 1 (1994); 396, 285 (1995). 1 A. Le Mehaute and G. Crepy, Solid Stnre Ionics 9/10, _ 17 (1983); A. Le Mehaute, The Fractal Approach to Heterogeneous Chemistry, p. 311. Wiley. New York (1989). 4. S. Liu. in Condensed Matter Physics Aspects qf Electrochemistry (Edited by M. P. Tosi and A. A. Kornyshev), p. 329, World Scientific, Singapore (1991). 5. B. Sapoval. J.-N. Chazalviel and J. Peyriire, Phys. Ret,. A 38, 867 (1988); T. Pajkossy and L. Nykos, Electrochim. Acfa 34, 171 (1989). 6. T. C. Halsey and M. Leibig, Ann. Pl7y.r. (N. Y.) 219, 109 (1992). 7. H. Raether. Surf&e Plasmons on Smooth and Rough Surf&es and on Gratings, Springer Tracts in Modern Physics Vol. III, Springer, Berlin (1988). 8. M. I. Urbakh, Electrochim. Acta 34, 1777 (1989); in Condensed Matter Physics Aspects qf Electrochemistry (Edited by M. P. TOSI and A. A. Kornyshev). p. 295, World Scientific, Singapore (1991). 9. A. M. Brodsky and M. I. Urbakh, Prog. in Surf. Sci. 33, 991 (1990). IO. M. Yang and M. Thompson, Langmuir 9, 1990 (1993). II. M. Urbakh and L. Daikhin, Phys. Ret>. B 49, 4866 (1994). 12. M. Urbakh and L. Daikhin, Langmuir IO, 2839 (1994). 13. M. A. Vorotyntsev, in Modern Aspects of Electrochemistry (Edited-by J. O’M. Bockris, b. E. donway and R. E. White). Vol. 17. D. 131. Plenum. New York (1986). 14. G. Valetie. J. Elecr;oanal. Chem. &Xl, 425 (19sb). IS. I. A. Bagotskaya. B. B. Damaskin and M. D. Levi, J. Electroanal. Chem. 115, 189 (1980). 16. M. A. Vorotyntsev, J. Electroanal. Chem. 123, 379 (1981). 17. R. Parsons and F. Zobel. J. Electraanal. Chem. 9, 333 (1965). 18. M. L. Foresti. R. Guidelli and A. Hamelin, J. Elecrroanal. Chrm. 346, 73 (1993). 19. A. Hamelin, M. L. Foresti and R. Guidelli, J. Elecrroanal. Chrm. 346, 251 (1993). 20. L. Daikhin, A. A. Kornyshev and M. Urbakh, Phys.
Rw. E 53, 6192 (1996). 21. L. Daikhin.
A. A. Kornyshev
and
M. Urbakh.
in
preparation.
22. M. Abramovitz and 1. Stegan. eds. Handbook Mathema/ical Funcrions. Dover, New York (1965). 23. D. C. Grahame. C/tern. Ret. 41, 441 (1947).
of
L. 1. Daikhin et al.
2860
24. The whole concept, based on the division of the compact and diffuse layer parts. was always a matter of concern in view of molecular dimensions of the compact layer [26]. However the contribution to capacitance, independent of electrolyte concentration, ie the finite value of the intercept of the Parsons-Zobel plot, is an experimental fact. Furthermore, molecular theory of electrolyte near a charged hard wall [27] and the phenomenological nonlocal electrostatic theory [28] both predict such a contribution without an artifical introduction of any “compact layers”. This results as an effect of the short range structure of the solvent [26-291. Experimental data for a set of simple metals and polar solvents give typical values of the compact layer capacitance [l3, 261, CH/&al = 0.1-0.8 A. 25. The interference between the solvent structure and Debye length rounds the Parsons-Zobel plot down from the straight line in the high concentration region [27]. The roughness effect does the opposite. The two effects may compensate each other for I of molecular dimensions, and the Parsons-Zobel plot will appear “more straight” than it should be for an ideally flat surface. 26. A. A. Kornyshev, in The Chemical Physics of Salvation (Edited by R. R. Dogonadze, E. Kalman, A. A. Kornyshev and J. Ulstrup), Part C, p. 355, Elsevier, Amsterdam (1988). 21. L. Blum, D. Henderson and R. Parsons, J. Electroanal. Chem. 161, 389 (1984). 28. A. A. Kornyshev and J. Ulstrup, Chem. Scripta 25, 58 (1985).
29. A. A. Kornyshev, W. Schmickler and M. A. Vorotyntsev, Phys. Rev. B 25, 5244 (1982). 30. L. Daikhin, A. A. Kornyshev and M. Urbakh, Submitted to J. Chem. Phys.
concept of characteristic correlation length breaks down for fractal surfaces, which have been studied in
WI. For K-’ Q Imi, one may expand (K’ + f?)“2 in the integrand of equation (4) using the smallness of (K/K)', equation (4) then reduces to a(K)-R{l
-$#
(Al)
Here we used equation (6) and the definition for the mean square curvature
Equation (Al) shows, as expected, that the roughness function R(K) approaches the geometrical roughness factor R for small Debye length K-I (large n). With the increase of K-I (the decrease of n) it decreases with respect to R, the correction being proportional to the square of the Debye length, ie it is inversely proportional to the charge carriers concentration. In the range of large Debye lengths (low concentrations), K-’ % I,,,,,, one may expand the term (K~ + f?)"' in the integrand of equation (4) in (~/lu)~ to obtain a(K)
N
1+ q
-
K2h2.
Here the length APPENDIX Limiting Debye
behavior
in the cases
qf
small
and
large
lengths
Consider the behavior of expression (4) for the two extreme cases: (a) the Debye length K-’ is shorter than the smallest characteristic correlation length of roughness Ime, and (b) K-’ is greater than the maximal correlation length I,,,,,. These two limiting cases can be realized experimentally, changing, for instance, the electrolyte concentration. Note that the
is of the order of I,,,. As expected, at very large Debye lengths the roughness of the surface is not “seen” in the capacitance. The first correction to the flat surface result is linear in K. Approving our expectations, equations (Al) and (A3) specify how the roughness function approaches the two obvious limits.