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
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s00134586@7)00106-0
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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</(27c)* should be replaced by the
sum S-l &. Approving
our expectations
equations
(4) and (8) reproduce two obvious limits: !?+I for
n-+0 and &+R for ~--+a. In the Appendix
we
specify how the roughness function approaches these
limits.
Examples of dlflerent surface morphologies. The
asymptotic laws for the capacitance at small and large
Debye lengths [given by equation (1) and equation
(Al) or equation
(A3)] are universal
for weak
nonfractal roughness. However, the specific form of
the roughness function depends on the morphology
of the interface.
When the roughness
may be
described by a one-scale correlation
function,
g(K) = 12u(H),
(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.
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2. R. Kant and S. K. Rangarajan,
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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
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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.