and 
for such electrodes.
Amy E. Larsen, David G. Grier, and Thomas C. Halsey
March 17, 1996
We describe measurements of the complex admittance of the
interface between electrodeposited fractal electrodes and
electrolytic solutions over the frequency range 100Hz to 100kHz.
Scaling with a single size-dependent frequency collapses these data
onto a universal curve.
This scaling collapse provides quantitative support for the Halsey-Leibig
theory for constant phase angle (CPA) behavior and a new
technique for measuring the multifractal descriptors
and for such electrodes.
The electrical impedance of electrode-electrolyte interfaces is often observed to have a component which scales non-trivially with frequency:
over at least some range of frequencies, 
.
This phenomenon, referred
to as a constant phase angle (CPA) element, was
reported first by Wolff [1] in 1926.
The possibility that
interfacial roughness might induce CPA behavior was suggested a quarter
century later [2] but was not emphasized until some years after that
[3, 4, 5].
In the decade since the pioneering
effort of Le Mehaute and Crepy [6] to explain CPA scaling in terms
of the fractal structure of rough electrodes, several competing
theories have been proffered
[7].
The majority of these treat the
fractal surface as a blocking electrode, one at which no
electrochemical reactions occur and which, therefore, admits no
faradaic current.
The theory of Halsey and Leibig (HL) [8]
differs from the earlier studies in asserting that the electrodes'
multifractal nature induces this behavior through the relation
Here is the rough electrode's fractal dimension, and d is the
dimension of the embedding space. The multifractal
exponent is the correlation dimension of the surface's
harmonic measure [9].
Recent simulations on fractal
Koch curves embedded
in two dimensions agree well with this prediction [10] and
are not consistent with other predicted forms for 
.
We describe in situ measurements of the frequency dependent admittance of well-characterized fractal electrodes grown by electrochemical deposition. We observe CPA behavior at high frequencies and are able to collapse a wide range of data onto a universal curve by scaling with a single characteristic frequency which in turn scales with system size. The success of this scaling collapse supports the assumptions underlying the HL theory; the extracted scaling exponents also are consistent with its results summarized in eqn. (2). In contrast to the results of Pajkossy [11], our observations are consistent with a geometric origin for CPA scaling for this system. While our results are reproducible from run to run, the following discussion focuses on a typical data set for clarity.
We prepare fractal electrodes in the manner first described
by Brady and Ball [12]. As shown schematically in Fig. 1,
copper is electrodeposited onto the freshly cut tip of an otherwise
insulated wire 25 
m in diameter from an aqueous solution containing
0.01M CuSO 
. The electrolyte fills a rectangular glass
cell and wets a coil of 22 gauge copper wire 1 cm in diameter with a
pitch of roughly 1 mm. This coil serves as the anode in the
electrochemical cell. Its open
structure permits direct observation of the electrodeposition process.
The fine wire tip, which serves as the cathode,
is centered within the coil using a micropositioner. When a
constant voltage is applied across this system, copper ions in solution
deposit onto the cathode wire and form an aggregate.
We apply
-0.5V to the cathode and hold the anode
at virtual ground. An excess of supporting electrolyte
(0.10M Na 
SO ) screens out electric fields in the solution,
so that transport of copper ions to the growth front is limited by
diffusion. The solution is acidified to pH 4 with added H SO and is
deoxygenated by bubbling with argon before being added to the deposition
cell. These precautions minimize the oxidation rate of freshly deposited
copper. The free length of the cathode wire is kept below 5 mm to minimize
mechanical instabilities. We also
add a chemically inert gel [agarose (0.5g/100ml)] to mechanically stabilize
the electrolyte against convection. Finally, the entire growth cell is
vibrationally isolated, maintained at a temperature constant to
within 0.1 
C, and situated in a water-saturated argon atmosphere.
The growth conditions in this system resemble very closely
the diffusion limited aggregation (DLA) model [13, 12] in
which fractal branched structures grow by the sequential accretion
of random walkers.
While the a priori theory of DLA is not yet complete,
extensive numerical simulations reproducibly generate three dimensional
clusters
with multifractal descriptors 
[14]
and 
[15]. The HL theory thus predicts 
for DLA-like electrodes.
We monitor an electrodeposit's geometry as it grows both by direct
observation and also by measuring the deposition current. An aggregate
such as the example inset in Fig. 2
reaches a diameter of 100 m in
about one hour.
Following Brady and Ball [12], we treat the aggregate as having an
effective spherical radius, r, proportional to its radius of gyration.
The diffusion-limited deposition current arriving at the
surface of a sphere is proportional to its radius [16],
, and should be independent of the applied voltage.
Carro et al. [17] have confirmed that the
deposition current for electrodeposited clusters
is proportional to their apparent radii.
The very small response in the deposition current
to low frequency voltage perturbations (see Fig. 3) is consistent
with diffusive transport. The aggregate's mass scales
with the effective radius and is proportional to the total charge
deposited: 
, so that
. Recasting this as a derivative
relation:
avoids complications due to initial conditions. The fractal dimension then emerges from the slope of the log-log plot of eqn. (3), as shown in the inset to Fig. 2.
As in previous studies [12, 17], the
range of measurable scaling in current, and thus of radius, covers
less than one decade. Ordinarily, such scaling plots should be treated
with extreme skepticism. However, the lower limit of the
scaling domain is set by the 25 m diameter of the cathode and not by the
smallest feature size of the electrodeposit. Scanning electron
micrographs reveal structures as small as 100nm.
While such features are
masked by the dimensions of the wire in Fig. 2, their
appearance means that structure in the disordered branches extends over
three decades in linear
dimension. The value 
extracted for a series of aggregates grown under
similar conditions is consistent with results of the DLA model and
suggests that self-similarity also might extend over three decades.
The frequency-dependent contribution of a fractal blocking
electrode to the overall system impedance arises from the effective
capacitance of the electrode-electrolyte interface. For the geometry
of our experiment, this contribution appears in series with
the resistance of the electrolyte and in parallel with the comparatively
small capacitance of the rest of the cell.
Due to stray reactance in series with the electrochemical cell, it is most
convenient to study the complex admittance, 
.
The range of frequencies over which we expect to see CPA scaling
is limited by the size of the aggregate at low frequencies and by the smallest
feature size at high frequencies. For an isolated 100 m diameter
aggregate with 100 nm features, HL
[8] suggest that CPA scaling should be observed
between 10 Hz and 1 MHz.
Cao et al. [10]
point out that this range will be restricted by the cell
capacitance to between 500 Hz and 500 kHz for our system.
Following the HL theory, we expect to see CPA scaling
above the absorption peak in 
.
This is confirmed by our results below.
We measure the frequency dependence of the system's complex
admittance at regular intervals during an aggregate's growth by
superimposing a 1.6 mV sinusoidal perturbation over the 0.5V
deposition voltage. The sequence of perturbing signals in a single spectrum
includes 13 frequencies ranging between 100Hz and 100kHz.
The duration of each spectral measurement is indicated by the gaps in the
current trace in Fig. 2.
The system's response
is measured with a precision wide-bandwidth current-to-voltage converter
whose output is buffered before measurement with a lock-in amplifier
referenced to the perturbation signal. The overall performance of
this system is calibrated with networks of resistors
and capacitors chosen to mimic the characteristics of the electrochemical
cell. At the highest frequency of this study, 100kHz, the phase
accuracy of the measurement system is found to be better than 1
while the amplitude resolution is better than 50nA. At lower
frequencies, the performance is considerably better.
The main advantage of our approach is that while the response of the cell is
diffusion-limited at the low frequencies on which the electrodeposition
takes place, the high frequency behavior is that of a linear
electrical system.
The real and imaginary parts of the admittance at several stages
in a typical aggregate's growth appear in Fig. 3 with plot symbols
corresponding to those in Fig. 2. To compare our results with theoretical
predictions we define the dimensionless complex admittance
, where 
is the high frequency limit of the real
admittance which arises from the conductance of the electrolyte
bounding the aggregate.
We estimate by extrapolating from curves such as those in
Fig. 3 under the assumption that
. For 
the real and
imaginary parts of 
fall on the single smooth
curve shown as the inset to Fig. 3.
Estimated values of 
appear as
dashed lines in Fig. 3.
The low frequency limit of the real admittance
is a measure of the faradaic contribution to the electrode's transport
properties.
From Figs. 2 and 3, we estimate this conductance
to be smaller than 
mmho which is more than
two orders of magnitude smaller than
the high frequency admittance.
To this extent, our aggregates act as
blocking electrodes in the frequency range of interest.
In addition to this conventional static scaling, which corrects for the size dependence of the overall magnitude of the admittance, we examine size-dependent scaling in the dynamical response. Without reference to any particular model, we assume a scaling form for the dimensionless normalized admittance:
where the function f(x) is, as yet, undetermined.
The strongest assumption in eqn. (4) is that the admittance
scales with a single characteristic frequency, 
.
This
differs diametrically from models which introduce a range of
relaxation rates to account for surface inhomogeneity [11].
If we further assume that the characteristic frequency, ,
scales with the size of the aggregate, 
,
and recall that the aggregate's size is proportional to the deposition
current, we find
where 
is a characteristic current scale [12]
corresponding to
the observed typical crystallite size of 100 nm and
rad/s is a
characteristic frequency [8] for an
electrolyte of resistivity 
cm,
and dielectric constant 
.
If eqns. (4) and (5)
reflect the dynamics adequately,
then we expect to find values of 
for which all the admittance
data for a given aggregate at different sizes collapse onto a single
universal curve.
Fig. 4 shows the real and imaginary parts of
the rescaled admittance data for the aggregate in Fig. 1 at 12
different stages of its growth collapsed according to
eqns. (4) and (5) with
. The success of this scaling collapse
justifies our assumptions that the dynamical response is characterized
by a single frequency which in turn scales with system size
and is the central experimental observation of this Letter.
If we now assume CPA scaling, 
, for
we then obtain a new expression for the
imaginary part of the dimensionless admittance:
from which we can extract the CPA exponent .
Multiplying Im by the dimensional factor
gives a typical scale of
nanofarads for the interfacial capacitance.
The slope of the dashed line in Fig. 4 indicates
.
Comparable CPA exponents are extracted from each of the spectra at
different sizes individually.
These measurements and similar results for other aggregates
(see, for example, the inset to Fig. 4)
are not consistent
with the scaling hypothesis
claimed in other studies on model fractal electrodes
[18, 19].
The same assumptions which are supported by the successful collapse of the admittance data also underlie the HL theory for CPA scaling. Their result for the dimensionless admittance:
where 
,
reflects these assumptions and suggests an overall scaling form
Comparing eqn. (8) and the HL result in eqn. (2) with the experimentally observed scaling form in eqn. (6) gives
in d=3 dimensions.
Eqn. (9) can also be derived from the scaling form given
in eqn. (4) if we assume low frequency capacitive behavior,
with the capacitance proportional to the surface area of the fractal
electrodeposit.
The data collapse in Fig. 4 thus provides an
independent measure of the fractal dimension, 
,
whose agreement both with the value found from
current scaling and also with the
accepted value for DLA provides additional quantitative
support for the HL theory.
Eqns. (2), (8) and (9) enable us to extract
the multifractal descriptor 
from our scaling data.
The result, 
is roughly 10% smaller than the
presently accepted value for DLA.
This small but real discrepancy
may arise from
a systematic error in calculating this numerical value from simulation data or
from subtleties in the electrodeposition process that
lower in the physical system with respect to its
DLA value.
If the latter is the case, then the impedance method allows us
to distinguish electrodeposits from DLA clusters despite their
identical fractal dimensions.
We would like to acknowledge the donors of the Petroleum Research Fund of the American Chemical Society for supporting this research.
