Frequency Dispersion of Electrode Polarization
127
Theory of the Frequency Dispersion of Electrode Polarization. Topology of Networks with Fractional Power Frequency Dependence Walter Schelder Biophysics Research Division, instityte of Science and Technology, University of Michigan, Ann Arbor, Michigan 48 105 (Received July 14, 1974) Publication costs assisted by the National Institutes of Health
Experiments provide strong evidence that surface adsorption is not involved in the frequency dispersion of small-signal electrode polarization capacitance, and that the dissipative processes which are involved are located entirely in the electrolyte phase. The data show that this polarization is neither solvent- nor ionspecific, and that its dependence on electrolyte properties is through a single parameter, g, the bulk electrolyte conductivity. The polarization capacitance varies as ga, where a is the same fractional power involved in the well-known frequency dependence relation. This information leads to a theory of lateral charge spreading in the diffuse double layer which associates the frequency dependence with microscopic surface roughness, and the fractional power a with branching in the pathways of charge spreading. Branched ladder networks with distributed properties are shown to have input properties with fractional power frequency dependence, and to form a class whose topological arrangement may bear a relation to the topology of electrical processes in polarization. The theory requires no detail surface model, uses no distribution functions, and prevails in the absence of faradaic processes. The analysis may have application to other fractional power dispersion phenomena, such as polarization of biological membranes.
1. Introduction The purpose of this work is to attempt to understand the nature of the electric polarization which occurs at the interface between an ideally polarizing metal electrode and an electrolyte solution. In particular, it concerns the frequency dispersion of the small-signal differential impedance of the interface, near the potential of zero charge. This is a matter of both practical and theoretical interest. Electrodes are widely used as impedance probes and as current terminals, and in many of these applications the effect of polarization impedance is among the least well-defined parameters. The number of methods which have been developed for minimizing and correcting for this type of p~larizationl-~ attests to the practical importance of the subject. Theoretical significance attaches to the fact that the form of the frequency dispersion is a relation of great simplicity (eq 1) which occurs also in a number of other phenomena, yet is poorly understood. Among systems in which it has been observed are biological membranes,8-15 tissue and cell suspensions,l6 in the dielectric dispersion of many condensed systems of polar molecules,17J8as well as in active biological transport,lg and in sensory transduction.20 While the underlying mechanism in each case may be quite distinct, analytical methods applicable to the common mathematical form could be enlightening in several or all of these fields. The descriptive literature on electrode polarization is voluminous and extends to the last c e n t ~ r y . ~ , 2 ~ The - 3 ~electrode interface has been found to behave electrically as if it were a capacitor whose capacitance, C,, varies with the frequency w of the applied signal. Grahame33 properly refers to this as a “pseudo-capacitance,” since by definition a capacitance is frequency invariant. As an electrical circuit element at a particular frequency,
however, the interface is linear at small signals. The limits of this linearity have been studied by S c h ~ a n .Super~~,~~ position of signals of different frequencies has been shown to be linearly additive as well.15 The empirical relation which appears to best describe the data for any given electrode and electrolyte is C,(o)
= constant x o-a
(11
where a is a constant over the major portion of the frequency range in which polarization capacitance is measurable. Depending on the system, a takes on values between 0 and y2. This form of dependence we refer to hereafter as “fractional power frequency dependence (fpfd).” In very concentrated electrolytes, a is sometimes observed to be not constant, but to change gradually over the entire frequency range of observation. Earlier observations of this p h e n ~ m e n o n ~were l , ~ ~confirmed for electrodes in physiological saline and mathematically modeled by Jaron, et a1 .35 Observations in more dilute electrolytes are the subject of this investigation. In these, a for any given system remains constant up to a transition frequency, and then changes abruptly (in less than half a frequency decade) to a larger value, probably Y2. These observations cannot be fitted with Jaron’s model; it is not known whether the two types of a variation are related. The work described in this article has two independent starting points, and its convergence to a consistent set of conclusions is one of its attractive features. One starting point is the phenomenological relation, eq 1. An entirely general formal analysis of some implications of this form is combined with a new experimental result (section 6) to derive physically significant information, and to suggest a class of electrical network topology whose use may extend also to other fpfd phenomena. The other starting point is the hypothesis, already exThe Journai of Physical Chemistry, Val. 79, No. 2, 7975
128
plicitly mentioned in 1957,36 that microscopic roughness of the electrode surface is directly responsible for the frequency dispersion. The first attempt to develop a model based on this hypothesis was by DeLevie in 1 9 6 ~ 5 We .~~ have significantly changed the approach in an attempt to improve the results of such an effort. 2. Premises
Three basic premises were adopted. (1)We postulated that the empirical relation, eq 1, is in fact the one which governs the process, in view of the wide range and excellent fit of this form with the available data. This is not a trivial statement; in fact most previous efforts to understand fpfd in terms of familiar mechanisms have implicitly made the opposite assumption, namely, that fpfd is a fortuitous artifact of the superposition of some set of better understood functional relations. We view our premise as representing a trade-off in the direction of greater fidelity to experimental fact at the cost of sacrificing familiar and well-defined mechanisms. (2) We assumed that a theory of the fractional power dispersion should require few architectural specifics, because of the many and diverse systems which exhibit this dispersion. (3) We considered it important that any theory should be able to give some structural interpretation or heuristic significance to the parameter a of eq 1. 3. Ideal Polarization and Faradaic Currents
Walter Scheider character to systems which, in their pure state, would polarize ideally. Notable in this connection was Grahame’s success in demonstrating frequency invariant polarization capacitance in mercury e l e ~ t r o d e s 3by ~ , meticulous ~~ purification and by using a self-replenishing, moving electrode surface. It was natural, though we believe misleading, to place the emphasis on the purity rather than the smoothness in interpreting the results of that experiment. We feel the experiment did not exclude the possibility that a system could be pure, and ideal, and still have polarization dispersion as a result of lack of smoothness. The possibility that in our electrodes we have contaminant-related nonideality is strongly diminished by the fact that our results are substantially independent of the solvent, of the electrolyte, and of the method of surface preparation and contaminant control. Similar systems have been demonstrated to be ideally polarizing by other criteria as we11.42 We believe, furthermore, that recent focus on the microscopic surface topology in relation to polarization disperion^^ provides a better means of explaining the facts than do theories of faradaic processes. This is reinforced by reports that dispersion is diminished on solid surfaces with intrinsic smoothness, such as those formed from the melt43 and those made of single ~ r y s t a l s . ~ ~ > ~ ~ 4. Previous Theories of Non-Faradaic Polarization
If it is accepted that the polarization in our electrode systems is non-faradaic, there are relatively few mechanisms which have been proposed to explain it, and most of these have either explained the data but fit it poorly, or fit it well without explaining it. One of the earliest was Ferry’s46analysis of equilibration rates in the diffuse double layer, in which he recognized that the time constants were off by orders of magnitude, and that without an unusually broad distribution function, the mechanism predicted lorentzian, not fractional power, frequency dependence. B o c k r i ~proposed ~~ that relaxation rates of the proper magnitude could be expected from Ferry’s model if ice-like water, created by surface tension, were supposed to exist in the double layer. This model has received no independent support, the form is again lorentzian, and it is hard to reconcile with our data on nonaqueous electrolytes (section 6). Considerable effort has gone into decomposing the empirical function eq 1,which is not understood, into components, such as relaxation functions, which have a familiar interpretati0n.l6J~,~~ Here, however, one is left with a decomposition distribution which is not understood, and little more has been achieved than what Cole has called, “an alternative expression of our ignorance of the mechanism.” 49 It was recognized early that fpfd implies that the phase angle of the admittance is independent of f r e q u e n ~ yand ,~~ Cole and Cole18 noted that therefore the ratio of maximum electrical energy stored to the energy dissipated per cycle is constant. This fundamental insight may yet turn out to be central to polarization dispersion, but in 30 years since it was pointed out, no one has developed it further.
It is important to make the distinction between faradaic polarization, which results from current crossing the interface by virtue of an electrochemical reaction, and non-faradaic, or ideal, polarization, in which the interface is blocking to charge transfer.33 Thirty years ago, Grahame believed he had shown that ideal polarization is frequency in~ariant.3~ For this and other, historical, reasons, the belief is still widespread that the frequency dispersion of polarization is necessarily associated with faradaic current. The first explanation of electrode polarization due to Warburg30was in terms of faradaic processes, even though it was realized that this explanation yielded a half-power frequency relation, while data even then showed that the exponent in eq 1 is in some cases clearly not j/2.31,32 The recent theory of faradaic processes is due to Grahame.3.1,39He found that they could be classified into those which give rise to frequency-invariant polarization capacitance, those with lorentzian dispersion, and those with half-power dependence. The impedance element associated with the last type he named “Warburg impedance.” This theory thus offers no way to account for fractional power dispersion (other than half-power) through faradaic mechanisms, except by way of a distribution function (see next section). We believe our electrode systems to be non-faradaic (ideal), and we believe the same about the electrodes of other ~ 0 r k e r ~ 2 J ~ , ~upon 5 , ~ whose ~ , ~ 8 data the empirical relation (eq 1) is based. In part we believe this because we know of no reaction involving the known constituents by which charge transfer can occur from the electrolytes we 5. Experimental Section. I. Purpose and Method have used to a precious metal surface at the low potentials The purpose in our experiments was: (1) to study syswe have applied. tematically the dependence of polarization admittance on Small amounts of contaminant, however, can greatly affect the polarizability characteristics of an e l e ~ t r o d e , ~ ~the , ~ ~electrolyte constituent variables (hereafter referred to as the ec variables), specifically, ion type, ion concentraand it has been suggested that impurities can give faradaic The Journal of Physical Chemistry, Vol. 79, No. 2 , 1975
Frequency Dispersion of Electrode Polarization
129
tion, and solvent type, and on electrolyte properties secondary to the ec variables, primarily the bulk electrical properties; and (2) to extend the available data on polarization properties to electrolytes of lower concentration than have previously been reported, in part because of our interest in studying the limit of validity of eq 1as the thickness of the diffuse layer increases. For such data the high precision of the impedance bridge used was essential. In the course of achieving the first listed purpose, we would be also testing Bockris’ hypothesis, as well as any other possible special role of water, and perhaps clearing up the somewhat confused state of knowledge about ion specificity. Our data were obtained using two measurement cells. In each cell, electrodes were cylindrical and concentric, and included a guard electrode to minimize fringe effects; the diameter of the outer electrode was about 1 cm, that of the inner electrode about 1 mm; the surface area of the inner electrode was taken as the “apparent” electrode area, A In cell GP2, the electrode surfaces were of electrodeposited gold, the cell constant was 5.92 cm, A , was 0.50 cm2; cell 66G1 was of burnished machined solid gold, its cell constant was 6.99 cm, A Bwas 0.98 cm2,Insulation was in all cases made of Teflon. Electrode surfaces, in addition to initial preparation by chromic acid purification, were precleaned prior to each series of experiments, by one of three distinct processes: (1) 10% nitric acid, followed by 6-12 rinses with conductivity water; (2) the same, followed by several rinses each, in succession, with absolute methanol and ethyl ether, followed by drying with CO2-free air; (3) hydrochloric acid followed by concentrated ammonium hydroxide and 6-12 rinses with conductivity water. In 1 and 3, water rinses were generally continued until conductivity water placed in the cell retained a conductivity below 3 X mho/cm at 20’ for a t least 1hr. No detectable differences in results have been observed due to differences in surface preparation. Measurements of cell impedance were made using a substitution bridge designed by O n ~ l e yThe . ~ ~value of the series polarization capacitance, C,(w), was derived in the usual way from the measured cell impedance Z,(w), on the assumption that these quantities are related to each other and to the electrolyte conductance in the cell, G, and the high frequency asymptotic value of the capacitance Co as in the circuit of Figure 1. The polarization resistance, R J w ) , had a negligible effect on Z,(w) in the systems we studied.
,.
6. Experimental Section. 11. Results In Figure 2 polarization capacitance is plotted as a function of frequency for a representative sampling of 63 sets of such data. The straight line segments in logarithmic coordinates confirm relation eq 1, with the value of a m 0.2. Though the possibility of sizeable systematic error increases rapidly toward the high frequency end of each curve, we believe the steeper segments, of slope =-Y2, at the high frequency end of several curves, to be real, in part because such sharply bending two-slope compound curves have been reported by others.2128 In section 8 we suggest a saturation mechanism which may account for this high frequency departure. Of all 63 runs, Run 31a shown is the only one in which all the points fall along a steep-segment slope; in this run the electrolyte conductivity was extremely low (less than 1 ymho/cm) and
w G
Figure 1. Equivalent circuit used to derive series polarization capacitance C,(w) from measured cell impedance Z,(w) and high frequency asymptotic values CO and G.
FREQUENCY
KHz
Figure 2. Series polarization capacitance as a function of frequency; representative sampling of 63 sets of data. Cell 66G1: (0)run 7.27, KCI, 18.6 ymholcm; (e)run 8.24, KCI in CH3CN, 9.1 1 ymho/ cm; cell GP2: ( + ) run l l a , KCI, 786 wmho/cm; ( 0 )run 14a, KCI, 59.7 pmho/cm; (B)run 35c, HCI, 1.89 wmholcm; V run 31a, HCI, 0.95 ymholcm; (A)run H14, human serum albumin, 5.95 ymho/cm (1.7 ymho/cm due to hydrogen ion plus 4.25 ymho/cm due to ion-
ized protein; see footnote 52).Arrows indicate calculatgd values of branch saturation frequency marking beginning of steeper slope segment for high frequency behavior (see section 8).
it is credible that all points are in the “high frequency” region. In Figure 3a each point is from a different run. Here the frequency of the applied signal is fixed (at 1 kHz) and the independent variable is the electrolyte conductivity, g. The fact that all the points (with the exception noted below) fall along a single line suggests that the dependence on all the electrolyte constituent (ec) variables is primarily through their effect on the conductivity. Furthermore, the slope of the line in Figure 3a is, within experimental error, in absolute value the same as the slope of the frequency dependence, Figure 2. This functional relation is expressed in the equation [c,(ec)lw = constant x [g(ec)]”
(2 )
in which a is the same parameter appearing in eq 1,and the subscript w indicates that measurements are taken at fixed frequency. It had been evident from a number of earlier investigat i o n that ~ ~the ~polarization ~ ~ ~ capacitance ~ ~ ~ tends to increase with electrolyte conductivity, but that the increase is by no means proportional. In the data of Smiley and Smith24 two pairs of points are directly comparable with our data and these are plotted on Figure 3a. Data of The Journal of Physical Chemistry, Vol. 79, No. 2, 1975
Walter Scheider
i312
a
0 L-
I
1
'10
'
'
CONDUCTIVITY
q/ 8
oai~
I
lOlN
'
" " ' I
I
100
'
$
'
pmho/cm
I 'IN
'
'
14,
CONCENTRATION
Figure 3. Series polarization capacitance at 1.0 kHz as a function of electrolyte conductivity. (a) Cell 66G1: (0)KCI; ($) KCI in CHsCN; (0) HCI; cell GP2: (0)KCI: (a)HCI; (V)serum albumin, high concentration, ionized protein is major charge carrier; (A)serum albumin, low concentration, hydrogen ion is major charge carrier; the solid line is drawn through all data points of this work except those in which hydrogen ion was the principal charge carrier: the dotted line is through points for HCI in cell 66G1; data for HCI in cell GP2 are not sufficient to determine which line fits best; data of Smiley and Smith:24(34) NaCI; (%) H2S04. The point at 800 pmho/cm was extrapolated to 1 kHr in the plot for run l l a of Figure 2. (b) Data of Wolff2' in a cell with unreported electrode dimensions: (e)H2SO4; (W)HCI. WolfP' are more extensive and more convincing, but electrode dimensions are not available, so these are plotted separately in Figure 3b. These earlier data are presented because we draw strong conclusions from the empirical relation eq 2, and we find it encouraging that, though this dependence has not previously been intentionally studied, data from at least two other experimenters show that our results are not unique. The apparent down-turn to steeper segments below about 3 @mho/cmmay be related to the fact that for these conductivities the measurement at 1 kHz is in the region of the "high frequency" departure in which the frequency dependence is correspondingly steeper. An upward displacement of the curves in which hydrogen ion is the principal charge carrier appears both in our data for cell 66G1 and in Smiley's. Our hydrogen ion data in cell GP2 is limited to low conductivity samples; from the available data it appears this displacement is less pronounced than in the other cell; the reason is not apparent. We speculate that the increased polarization capacitance for hydrogen ion is related to closeness of approach a t the interface, because the charge-free layer consisting of a single layer of water5I may be accessible in a different way to hydrogen ion than to other ions. Though displaced, the data for hydrogen ion are still described by eq 2, suggesting that hydrogen ion represents an exception in a quantitative, and not in a qualitative sense. We suggest that the case of hydrogen ion is sufficiently understandable in these terms to permit of the generalization that there is virtually no discernible ion specificity, even between small singly charged ions (potassium) and large multiply charged ions (albumin52).No difference is The Journal of Physical Chemistry, Vol. 79, No. 2 , 1975
noted between anions and cations as principal charge carriers. The change resulting from substitution of acetonitrile (CH3CN) for water as a solvent for potassium chloride is small, suggesting that polarization does not depend in any essential way on properties unique to water. In combination, the absence of either ion or solvent specificity suggests, in itself, that the polarization characteristics are determined in the bulk electrolyte phase rather than in adsorption or interaction of ions at the electrode surface. Stronger evidence for this conclusion is contained in the quantitative relation eq 2, but this requires the theory of the next section to demonstrate.
7. Theory. I. Implications of the Fractional Power Relations Any empirisal relation can be formally decomposed in a number of equally valid ways. Such a decomposition, however, is useful only if it does something more than restate the initial information. We directed our attention to studying the implications of the separability of the admittance function into dissipative and energy-storing elements, in the belief that these processes may be physically identifiable, and perhaps subject to experimental characterization. First, we require the expression for the complex polarization admittance of which the capacitance we measure is one part. According to the relations of Kronig and Kramers53 the admittance Y,(w) is completely determined when, as in eq 1,the imaginary component is known at every frequency. It is given by (31
where .r' = 6 1 , and YPois a constant which can be determined from the value of C, at any reference frequency, W O , by the relation yPo= o,C,(w,)/sin
lr
-
2 (I
(Y)
(34
From the separability of the term containing G w ) in this equation it follows immediately that the admittance is separable into two types of function, fl and gl
where all the frequency dependence is contained in the functions fi, which are functions of capacitive parameters y k , ~as follows fl
= fl(jwYk,I)
k = 1,2,
. . .K ,
(5 )
and where gl are frequency-independent functions of dissipative parameters CJ;J g, =
41(Or,z)
i = 1,2,
.. .I,
(6 )
By definition, parameters y are of dimensions (ohm-l sec), and parameters CJ are of dimensions (ohm-l). The particular form of eq 3, furthermore, determines that for each term in the sum, 1 = 1,2, . . . L, separately, there exists a quantity such that ff(jwyk,I)=
L+zl'-a
(7)
and consequently, for dimensional reasons, there exists a quantity fi for each 1 such that
Frequency Dispersion of Electrode Polarization
131
Thus far these statements are formal, and contain no information concerning the relation of any of the parameters to physical properties. These relations are now used to prove a theorem with physical consequences. Theorem o n Dependence o f [C, 1, o n Electrolyte Properties. If (1)an admittance of the form eq 3 is composed of dissipative elements d and capacitive elements y as in eq 4-6; and if (2) the conductance of each element CTQ depends on the electrolyte constituent (ec) variables only through, and in direct proportion to, the bulk electrolyte conductivity g (ec), that is, if ai,l(ec) =
x g(ec)
b
(94
and if (3) the capacitive elements are (ec) invariant Yk,l
(ec) = (constant),,
(9b)
-L
0
Then, by eq 4-6 and 8 [Y,(ec)lw = (complex constant) x [g(ec)]" and consequently [C,(ec)lW= (real constant) x [g(ec)J"
(10)
(11) This is the same as the empirical relation eq 2. It should be emphasized that eq 11 was derived from data which is independent of that upon which eq 2 is based. Equation 2 is an empirical generalization of the data of Figure 3, in which conductivity is the independent variable; eq 11 was derived from eq 1, which derives from the data of Figure 2, in which frequency is the independent variable. This theorem shows that these two sets of data are related to each other by way of the conditions of the theorem, expressed in eq 9a and 9b. The conditions 9a,b are important physical statements. The theorem proves them sufficient to explain the concurrence of the two sets of data, Figures 2 and 3. Though they are not proven necessary, they are an attractive and by far the simplest way to understand the observed facts. In addition, they are supported by the direct observation (section 6) that polarization properties appear to depend on the electrolyte constituent variables almost entirely through the single parameter g, the bulk conductivity. An important conclusion is that dissipative processes which occur in proportion to the concentration of ions, as one would expect in connection with adsorption, are ruled out, because these do not conform to condition 9a, even apart from ion specificity. Conductivity is related to concentration principally by the mobility. The widely different mobilities of ions used in our experiments permit of a sufficiently strong experimental distinction between conductivity and concentration dependence. Network Topology. The view which emerges is that the interface polarization admittance can be mathematically separated into (1) capacitive elements which are invariant not only with frequency but also with electrolyte variation, and (2) true conductance elements which are directly proportional to the bulk electrolyte conductivity, and hence act as if they were spatially fixed and entirely in the electrolyte phase. This view gives legitimacy to the question, can the admittance be represented by a pure-element network? It has been pointed that no finite combination of linear circuit elements has the admittance of the form eq 3. A number of recent effort^^^-^^ to synthesize practical circuitry approximating fractional power frequency depen-
1 .
TCO
T
T
C Figure 4. (a)RC ladder network; (b) first-order parallel-branched ladder network; (c) first-order series-branched network with CR branches. dence for engineering purposes has yielded no relations which we have found useful in this context. Since the polarization process takes place in a continuum, it is not unreasonable to consider infinite networks, provided we restrict consideration to those which can be defined by a small finite number of parameters, corresponding to the number of parameters which determine the relevant polarization properties. While there is an infinity of networks whose admittance is arbitrarily close to that in eq 3, the permutations are limited if the parameters are few. We can synthesize a network having the required properties by writing eq 4-6
and then, according to eq 7 and 8
and
E%,,= I
0
In the simplest case, a = Yz,eq 4a reduces to
This is the familiar expression for the admittance of an infinite RC ladder (Figure 4a) in the limit of distributed properties, y being parallel capacitance c per unit length, and 0-l being series resistance r per unit length. With appropriate interpretation of the symbols, it also represents the input characteristic for diffusion down an infinite linear region. The well-known mathematical isomorphism between these networks and diffusion processes makes these networks natural candidates for description of charge movement in solution. Because of strong damping in RC networks, these ladder networks need not be long to be effectively infinite equivalent. A straightforward calculation shows that at a length of $A,, the magnitude of the input admittance is within 0.4% and its phase within 0.07O of the admittance of the infinite The Journal of Physical Chemistry, V d , 79, N o . 2, 1975
Walter Scheider
132
TABLE I: Input Admittance of RC and CR Branched Ladder Networks up to Second-Order; RCo and s l Onlya Branching type
Admittance
RC 0 ( j w ) ’ (C, 1‘ /’ (Go) q 1 / 2 RC,; CRsl RCo; RCSl RCo; CRs1; RCP2 RC,; CRs1; RC*2 RCo; CRs1; CRP2 RCo;CR,1;CRs2 RCo; RCSf; R C R 2 RC,; RCs1; RCS2 RC,; RCS1;CRR:! RCO; RC,1;CRs2 a The admittance expressions given are those calculated for infinite networks in the limit of infinitely distributed properties. These expressions are excellent approiimations for finite networks in which the length of the branches of each order exceed the appropriate half-propagation wavelength (eq 12 and 13), and in which the properties are distributed over at least 20 equal segments per half wavelength. The exponent of (jw) corresponds to (1 a) (see eq 3). Symbols used: s, series branching; p, parallel branching; e, capacitance; r, resistance; g, conductance; n, number of branches per unit length; subscript numbers refer to orders of branching; overline indicates that the quantity is a distributed property, per unit length. As indicated in the text, 6 is used in CR branches to indicate the total branch length, aAd ca to denote the lengthwise total branch capacitance; the product (&) is a constant independent of branch length, and = (l/e), the inverse-capacitance per unit branch length,
-
network, where A,, is the propagation wavelength58 A,, = 2n[Ycw/2]-”2 (12) When a is a fraction not equal to l/2, eq 4a can be shown to represent the class of branched ladder networks. No literature on such networks has been found, probably because no easy methods of constructing them for engineering applications are known. In concept such branched networks are an elementary extension of ladder network theory, and in multidimensional continua their realizability is not out of the question. It should be noted, however, that these are not simply multidimensional networks. As an example, one obtains what we call a first-order parallel branched ladder (Figure 4b) if, in the RC ladder of Figure 4a, one replaces each capacitor with the input to another ladder network, called a “branch.” Branch parameters may differ from those of the zero-order ladder. In the expression for the network admittance, the admittance of the parallel capacitor is replaced by the input admittance of the branch by which it has been replaced. A first-order branch may itself be branched, and this is branching of order 2. If series elements are replaced by branches, we have series branching. Figure 4c shows a firstorder series branched network. In Table I are listed some of the branch configurations up to order 2, with an equation for the input admittance of each. The values of the frequency dependence exponent (1 - a ) are discrete, but limited only by the extent of branching. The intervals between values of a which can be Synthesized by such networks is [l/Zls+l, where 0 is the order of branching. A thorough discussion of branched ladder networks is not attempted here. Our main purpose is to define the class, to demonstrate that one is led to its consideration by eq 4a, 7a, and 8a, and to propose that it contains the topology required for an exact synthesis of fpfd circuits. The parameters n, the number of branches per unit length, are retained explicitly in the expressions for admittance in Table I, rather than going to the limit as n m, largely for simplicity and ease of visualization. In fact, of course, branching must be physically discrete, and there is
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The Journal of Physical Chemistry, Vol. 79, No. 2, 1975
an acceptable level of discreteness which has no substantial effect on the result, just as there is an acceptable level of finiteness in ladder length (eq 12). Included among the branches are CR ladder networks (as distinguished from RC) in which capacitors are series elements and conductances are parallel elements. This concept is useful in continuous media, though one must define a series element whose measure is inverse capacitance per unit length. With a view both to simplicity and to ultimate application, this measure has been expressed as the inderse of the product of an overall capacitance cg with an overall branch length 6, these quantities being, for example, measures of capacitance and transverse dimension of the diffuse layer. The propagation wavelength for a CR ladder, in these terms, is A,,
= 2?7[2w6~~/2]~’~
(13)
One of the key properties of these branched ladder networks is that the frequency dependence is completely determined by the branching type, and is not affected by the magnitude of any of the circuit elements. Thus, many networks, each with circuit elements of widely different magnitude, may be connected with their inputs in parallel, and if the branch type of all networks is the same, the frequency dependence of the entire system will be homogeneous. In terms of eq 4a, each network thus connected in parallel is represented by one term in the sum over the index 1. In terms of eq 3, in such a case, all the variability in the magnitudes of the elements y and n is absorbed in the term Y This property of these networks is important in relation to premise 2 (section 2), since it provides a way in which a system of great architectural heterogeneity can be assembled to exhibit external characteristics of the great simplicity of eq 1. 8. Theory. 11. Charging t h e Double Layer on a Rough, Ideally Polarizing Surface We turn now to a physical description of the electrodeelectrolyte interface, beginning with the most general and obvious truth that the lines of electric force do not converge evenly onto an uneven surface.
Frequency Dispersion of Electrode Polarization
133
Figure 5. Scanning electron micrograph of a partian of a gakl surface. 24K gold rod from Engalhard Industries. Newark, N.J.. polished lo a matte appearance. as used in the electrodes of cell 66G1. In Figure 5, a smooth-appearing gold surface of the type used in our cell 66G1 is shown magnified by the scanning electron microscope. Irregularities of various shapes, cliffing, cratering, and channeling are all apparent, with dimensions down to the submicron range. Regardless of the details of the structure, it follows that the surface will be charged unevenly, and that the admittance will be time dependent. The supposition that surface roughness, per se, results in frequency dispersion is thus a truism, though of course the real question is in relation to its form and magnitude. T o proceed with that question, we consider some general characteristics of the surface. We assume that the Helmholtz, or compact, part of the double layer follows every detail of the electrode surface, for the thickness of this layer is of molecular dimensions."~sg~@' In the case of ideally polarizing electrodes, this layer is totally blocking to charge transfer, and hence purely capacitive. Outward from the Helmholtz boundary is the diffuse part of the double layer6' extending (in dilute solutions) for a characteristic distance 6 = 1/(4w)%, where x is the Debye-Hiickel parameter. Beyond the diffuse layer, in the bulk electrolyte, electrical neutrality is the defining condition. The polarization characteristics are assumed to be determined by the manner in which ions are accumulated in the double layer. The equation which governs the flux J; of ion species i involves the combined effect of the gradients Vnj of concentration and Vqof electric potential J, = - p # I k T V n ,
+
II~Z~CVU]
(14)
where pi is the mobility, e the electronic charge, z; the valence. and kT the thermal energy term. We will show that space charge effects, contained in the first term in the parentheses. while confined to the double layer, are of primary importance. In attempting to proceed
further to constructing a model, one faces the fact that even for an extremely simple geometric approximation, such as the V-groove model used hy delevie,:'? the solution of eq 14 is quite intractable. In neglecting the space charge term, d e b v i e was able to reduce the problem to one which he could solve exactly, but in doing so we believe he seriously overapprozimated his model, resulting in the rather poor quantitative agreement with Schmid's experiment,"* which was done to test the model. We retain delevie's idea that in microscopic surface topology lies the explanation for frequency dispersion, and we will attempt to show that, with space charge effects retained, although we are unable to give an exact solution, a model with the desired properties can be devised, with the help of some approximations and the use of appropriate network concepts. Diffusion within the Double Layer. The idea that tangential components of surface admittance can be related to frequency dispersion of polarization is not new. It has been applied, for example, to spherical particles in suspension.6"d5 We find this concept useful, and apply it here in modified form to the irregularities of a metal surface. For the purposes of our calculations, the equilibration of the ion distribution across the thickness of the diffuse layer will be assumed to be instantaneous.'" This is justified by the fact that.the time constant for this transverse equilibration re = 1 / 4 n ~ ~ D
(15)
is approximately 5 X 10-9 sec in a 1mM solution of sodium chloride. D is the (n;zj*)-weighted average diffusion constant for the ions in solution. A t equilibrium, a transverse concentration gradient opposes and compensates the potential gradient. ThG field strengths in the double layer are related to the rapid equilibration rate, and can be very highM for parallel plane electrodes, the ratio of field strength E , at the Helmholtz plane to that in the bulk electrolyte E. is
At the frequencies at which polarization is observed, this ratio is large, - . and in conseauence large diffusion potentials also exist. For olane electrodes. these ootentials are entirelv transverse, and are opposed by the electric field. However, on a rough surface, to the extent that roughness causes uneven charging of the double layer, sizeable tangential diffusion potential gradients are expected to occur; the force of these diffusion gradients can be the principal means by which remote and shielded portions of the surface are charged. A tangential propagation wavelength can be calculated (eq 18). and provided distances are of the order of at least half such a wavelength, equilibration will not be instantaneous, as in the transverse direction, hut will be determined by a complex admittance function derived from ladder network equations. The relative magnitude of lateral charge spreading along diffusion gradients, compared with that due to charge flow along electric field lines from the bulk electrolyte, depends on the radius of curvature of the roughness contour and on the shielding effect of the configuration. An extreme example serves to illustrate the point. A pocket in the electrode surface (Figure 6a), with an opening small compared with 6, is almost completely shielded from The Journal olPhys;calCnern;slry. Vol. 79. NO. 2. 1975
Walter Scheider
134
0
~*
O .
.. .. . . ... . . .*..... .... . '
EQ4:
. *
.m.o.
0
IBl
b
a
b
Figure 6. Visualization of a "pocket" in the elenrode surface. (a) charge accumulates in lhe double layer at t h e outer surface. where lhe electric fiela vector E is opposed by m e concentration gradient vector N. and these are equal at equilibrium.Charges are forced into the pocket opening by the concentration gradient oetween the outer Surface and the pocret interior, and continue to flow in until Ib) t h e eleclr c field resulting hom t h e accumulation 01 charge .n the poc*et is great enough at tne opening to oppose t x l h e r influx of cnarge. In a spherical pocket. the distribution of charges and the flela are everywnere in the radial directnon.
Figure 7. V i s u a l i i l b n of a "mound" type of surface *regularity. (a) lneially electric field lines E converge toward the peak of the mound while the valley is. in effect. partially shielded. (b) Wm accumulation of charge, following the electric field lines to the peak of the mound, a concentration gradient is established and the line of constant e b s tric potential moves closer and folds around the peak; at the peak, the concentration gradient vector N opposes the electric field. while at the side the concentration gradient has a tangenial component which acts in the Same direction as the elenric field. (c)An artk ficial boundary B is drawn at the edge of the mound to help visualize the process by which accumulated charge in the peak area A spreads to the valley side C under the force of the tangential grad+ ent in the eienrodiffusion potential. This movement along the surface C is described mathematically by the branched network shown, whose frequencyresponse corresponds to a = 'i4.
unit input perimeter length a t B y = .ijwep-/4n
external fields. The ionic excess at the opening causes ions to diffuse into the pocket. This continues until the pocket is lined with a charge layer whose electric potential gradient at the opening is equal and opposite to the diffusion gradient there (Figure 6h). In this equilibration, the role of driving force and opposing force is reversed from what these are in the plane diffuse layer. For visualization of what is probably a more nearly typical roughness contour, we examine diffusion over the surface of a "mound," Figure 7. This process is most easily visualized in the time domain. One considers the movement of charges subsequent to the beginning of a step function in applied current a t time t = 0. What one should expect to find is determined by translating the empirical relation eq 1 into the time domain by Fourier transformation; this yields a pseudo-capacitance which increases with time as t a (except near t = 0, where for physical reasons it must become constant). At t = 0 no concentration gradients are present, and ionic flow follows the lines of electric force (Figure 7a) to the peak of the mound. As charge accumulates in the donble layer at the peak, the folding down of the equipotential lines (Figure 7h) combines with lateral concentration gradielits to cause tangential spreading of ions to the remoter portions of the surface. As more surface becomes accessible to charge with the elapse of time, the effective capacitance increases. To estimate the orders of magnitude involved in this process, it is supposed that an artificial boundary B may he drawn (Figure 7c) dividing the surface into a peak side, A, assumed to charge uniformly as if it were a segment of a perfectly smooth parallel plane electmde, and a valley side, C, assumed t o be completely shielded from the bulk electrolyte. The double layer in the valley is thus charged exclusively from the input a t B by ions moving under the lateral electrodiffusion potential gradient. An expression, involving rather gross approximation~,6~ suffices nonetheless to calculate quantities correct to an order of magnitude for the lateral input admittance y per The Journal of Physical Chemistry, Vol. 79,No. 2, 1975
(17)
and for the propagation wavelength A, along side C
where c and g are the dielectric constant and the conductivity, respectively, of the bulk electrolyte, and x is the Debye-Hiickel parameter. In order to visualize the manner in which the model may account for a frequency dispersion in which the fractional power, a, is %, a first-order series branched network is drawn in the region C in Figure 7c, with its input a t B. We suppose the branching to come about through some mechanism which causes discontinuities or discreteness in the transverse pathways within the diffusion layer. The slope of the polarization capacitance data constitutes the only evidence for such branching. However, the result in section 7 requires that, whatever mechanism may he responsible for the branching, the dissipative elements of the branches must be elements of bulk electrolyte, whose conductivity varies directly with the bulk conductivity. One may speculate that the branches extend away from the electrode surface (more precisely, from the Helmholtz boundary) for an effective distance approximately equal to the thickness, 6, of the diffuse layer, and that they possess a total lengthwise capacitance, car equal to that of the diffuse layer (adjusted for branch width). Such a speculation is based on no substantial evidence, but is presented because of an interesting consequence of such a picture: If one calculates, based on the finite lengths so defined for these branches, a t what frequency the branch length, 6, hecomes short compared with the corresponding half propagation wavelength (eq 13), one finds that these frequencies (indicated by arrows on Figure 2) coincide roughly with the frequencies a t which the polarization curves change slope from about -% to about -%. This is precisely the slope change which a RCo; CR,I network exhibits a t the saturation frequency of its branches. In Figure 8, the dashed line represents the input capaci-
Frequency Dispersion of Electrode Polarization
..
FREQUENCY
KHz
Figure 8. Input capacitance of a type RCo;CR,l network with finite
branch length resulting in the two-slope curve (dashed line). Network parameters were chosen to simulate polarization capacitance of run 14a (solid dots, see also Figure 2). See text for discussion of network parameters.
tance of such a network in which the parameters have been adjusted so that the curve falls near the points of run 14a (Figure 2). The network elements are represented in the schematic of Figure 4c. The physical parameters which could be independently assigned were chosen to correspond to an electrolyte solution similar to that of run 14a. The bulk conductivity was taken as 50 pmholcm; the diffuse layer thickness, 6, as 5 X cm; the diffuse layer capacitance as 2 pF/cm2; the Helmholtz layer capacitance as 50 pF/cm2. Thus, with the CR branch length 61 = 6 (see nomenclature of Table I), requiring the branch saturation frequency to coincide with the slope transition frequency of run 14a, at about 50 kHz, determines that the branch thickness, l / n l = 5 X cm; the branch inverse capacitance per unit branA1ength per unit input perimeter length (at boundary B), l / c l = ( 6 1 c ~ l ) -=~ (4 X F/cm perimeter)-l/cm; the branch conductance f i = 10 mho cm-l (cm perimeter)-l; the main network capacitance, = 50 X F cm-l (cm perimeter)-l. If the network of the schematic Figure 4c is regarded as having a “width” in a dimension perpendicular to the plane of the schematic, then this width corresponds to the B boundary input perimeter length, and in order that the input capacitance correspond in magnitude with that of run 14a, this perimeter length must be approximately 4 X lo4 cm per square centimeter of apparent electrode area. These calculated quantities, as well as Figure 8, are presented with a caution to the reader not to over-interpret them. The microscopic heterogeneity of the surface guarantees that at best this model is an approximation representing some average properties. The calculations are intended to demonstrate that with not entirely unreasonable physical parameters, the model we have suggested can account for the magnitude, the slope of both segments, and the transition frequency of a typical set of polarization data. 9. Membranes and Other Systems
The similarity in the appearance of electrode polarization impedance data and a variety of impedance measurements on biological membranes, when these are plotted according to the method of Cole and Cole,18 suggests that two aspects of this work may be worth considering in the context of biological impedances. (1) The concept of lateral, time dependent, spreading of charge from centers of initial accumulation can provide one possible explanation of biological impedance measurements which have been described in terms of frequency-
135
dependent impedances of constant phase angle greater than zero and less than ne.^-^^ Surface roughness, or infolding, has been observed in some biological membranes,68though in others with similar dispersion properties the evidence appears to be against extensive folding.69 In the case of membranes, one ought to consider that the centers of initial charge accumulation may not be primarily geometrical features, but may be points of chemical heterogeneity (“hot spots”), or functional substructures. For example, the effect of substructure in muscle fibers, written in electrical analog form by Fatt? bears a resemblance to branched ladder networks as we have defined them. Biologists often emphasize a fundamental difference between electrodes and membranes, established firmly by elegant experiments on single nerve a x o n ~ , ~namely, O that the capacitance of membranes involves charge separation across a physical dielectric, whereas this is not so a t the electrode surface. Nevertheless, at each face of the membrane dielectric there is an interface with an electrolyte at which double layer processes may contribute to polarization properties, particularly its frequency dispersion. (2) The concept of branched networks has provided for the first time a mechanism for producing an electrical admittance with constant fractional power frequency dependence (other than 1h power) without the intermediary of an unexplained distribution function. It is not necessary that one conceive of such a network as existing in the membrane, or other structure, in a literal sense. The usefulness of the branched network concept is broader than that, for it represents most fundamentally a set of relations in time and space among dissipative and energy-storing processes. 10. Conclusions
We have shown that a simple, quantitative explanation for our experimental results is contained in the hypothesis that the dissipative processes giving rise to the frequency dispersion of polarization capacitance in ideal electrodes occur in the diffuse double layer, entirely within the electrolyte phase. This conclusion is consistent with other evidence linking this dispersion to microscopic surface roughness. We have associated the frequency-dependent processes of polarization with tangential charge spreading in the double layer, in which diffusion potential gradients play a significant role. The fractional exponent, a , which we find in the empirical relations for the dependence of electrode polarization capacitance both on frequency and on electrolyte conductivity, is interpreted as a branching factor in the pathways of surface charge movement. The branching concept arises out of our characterization of the class of branched ladder networks, which we have shown to be a conceptually simple arrangement of dissipative and capacitive elements capable of exactly representing the electrical admittance of phenomena with fractional power frequency dispersion. Acknowledgments. I am indebted to Dr. J. L. Oncley for‘ my introduction to this problem and for having pioneered some of the experimental techniques. A considerable portion of the data presented here is the work of Mr. John Keto. This work was supported in part by NIH grants #HL09739 and #GM1355. The Journalof Physical Chemistry, Vol. 79, No. 2 , 7975
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Walter Scheider
References and Notes H. P. Schwan, 2.Naturforsch. B, 6, 121 (1951). T. M. Shaw, J. Chem. Phys., 10,609 (1942). J. L. Onciey, Chem. Rev., 30, 433 (1942). J. D. Ferry and J. L. Oncley, J. Amer. Chem. Soc., 63, 272 (1941). J. L. Oncley, J. Phys. Chem., 44, 1103 (1940). J. D. Ferry and J. L. Oncley, J. Amer. Chem. SOC., 60, 1123 (1938). J. L. Oncley, J. Amer. Chem. SOC., 60, 1115 (1938). P. Fatt, Proc. Roy. SOC., Ser. 6,159, 606 (1964). H. J. Curtis and K. S.Cole, J. Gen. Physiol., 21, 757 (1938). E. Bozler and K. S. Cole, J. Cell. Comp. Physiol., 6, 229 (1935). K. S. Cole and H. J. Curtis, Cold Spring Harbor Symp. Quant. Biol., 4, 73 (1936). K. S.Cole, Science, 79, 164 (1934). K. S. Cole, J. Gen. Physiol., 15, 641 (1932). H. Fricke, Cold Spring Harbor Symp. Ouant. Biol., 1, 117 (1933). H. Fricke, Physics (New York), 1, 106 (1931). H. P. Schwan, Advan. Biol. Med. Phys., 5, 147 (1957). R. H. Cole, J. Cell. Comp. Physioi., 68, Suppl. 2, 13 (1965). K. S. Cole and R. H. Cole, J. Chem. Phys., 9, 341 (1941). J. R. Segal, Biophys. J., 12, 1371 (1972). J. Thorson and M. Biederman-Thorson, Science, 183, 161 (1974). H. P. Schwan, Ann. N. Y. Acad. Sci., 148 (I), 191 (1968). J. F. Johnson and R. H. Cole, J. Amer. Chem. SOC., 73, 4536 (1951). D. C. Grahame, J. Amer. Chem. SOC., 63, 1207 (1941). W. G. Smiiey and A. K. Smith, J. Amer. Chem. SOC., 64,624 (1942). C. C. Murdock and E. E. Zimmerman, Physics (New York), 7, 211 (1936). G. Jones and S.M. Christian, J. Amer. Chem. SOC., 57, 272 (1935). I. Wolff, Physics (New Yo&), 7, 203 (1936). i. Wolff, Phys. Rev., 27, 755 (1926). H. Fricke, Phi/. Mag., [7] 14, 310 (1932). E. Warburg, Ann. Phys. Chem., 67, 493 (1899). E. Neumann, Ann. Phys. Chem., 67, 500 (1899). M. Wien, Wed. Ann., 58, 37 (1896). D. C. Grahame, J. Electrochem. SOC., 99, 370C (1952). H. P. Schwan and J. G. Maczuk, Dig. lnf. Conf. Med. Biol. Eng., 6th, 1965, 24 (1965). D. Jaron, H. P. Schwan, and D. B. Geselowitz, Med. Biol. Eng., 6, 579 (1968). J. N. Sarmousakis and M. J. Prager, J. Electrochem. SOC., 104, 454 (1957). R. delevie, Electrochim. Acta, I O , 113 (1965). D. C. Grahame, J. Amer. Chem. SOC., 68, 301 (1946). D. C. Grahame, Annu. Rev. Phys. Chem., 6, 337 (1955). W. D. Robertson, J. Electrochem. SOC., 100, 194 (1953). D. C. Grahame, Chem. Rev., 41,441 (1947). G. M. Schmid and N. Hackerman, J. Electrochem. SOC., 109, 243 (1962). T. i. Borisova, B. Ershler, and A. Frumkin, Zh. Fiz. Khim., 22, 925 (1948); 24, 337 (1950). Chuan-sin Tza and Z. A. iofa, Dokl. Akad. Nauk SSSR, 131, 137 (1959).
The Journal of Physical Chemistry, Vol. 79, No. 2, 1975
D. I. Leikis and B. N. Kabanov, Tr. lnst. Fiz. Khim., Akad. Nauk SSSR, 6 (1957); Chem. Abstr., 53, 918 (1959). J. D. Ferry, J. Chem. Phys., 16, 737 (1948). J. O'M. Bockris and 6. E. Conway, J. Chem. Phys., 28, 707 (1958). R. M. Fuoss and J. G. Kirkwood, J. Amer. Chem. Soc., 63, 385 (1941). K. S. Cole, "Membranes, Ions, and Impulses," University of California Press, Berkeley, Calif., 1968. N. R. S. Hollies and J. L. Oncley, Nat. Acad. Sci., Nat. Res. Counc., Publ. (1950). J. R. McDonald, J. Chem. Phys., 22, 1857 (1954). Albumin, in these experiments, was human and bovine serum mercaptaibumin in iso-ionic solution. Though on the average neutral, these moiecules carry a root-mean-square net valence of l .7, according to calculations in ref 71. In this reference data also show that at low protein concentration the principal charge carrier is hydrogen ion, while at concentrations of several grams per 100 ml the greater part of the conductivity is due to ionized protein. R. deL. Kronig, J. Opt. SOC. Amer., 12, 547 (1926); H. A. Kramers, Atti. Congr. Fisici, Como, 545 (1927). G. E. Carlson and C. A:Halijak, /E€€ Trans. Circuit Theory, CT-11, 210 IIQBAl
R.-M.'Cerner, I€€€ Trans. Circuit Theory, CT-10, 98 (1963). S.C. D. Roy and 6. Shenoi, J. Franklin inst., 282,318 (1966). C. A. Hesselberth, "Synthesis of Some Distributed RC Networks," National Technical Information Service Document No. AD418171, 1963. W. R. LePage and S. Seely, "General Network Analysis," McGraw-Hili, New York, N.Y., 1952, p 301 ff. (59) J. A. V. Butler, Ed., "Electrical Phenomena at Interfaces," MacMillan, New York, N.Y., 1951. (60) 0. Stern, 2.Eiektrochem., 30, 508 (1924). (61) F. Stillinger and J. G. Kirkwood, J. Chem. Phys., 33, 1282 (1960). (62) G. M. Schmid, Electrochim. Acta, 15, 65 (1970). (63) H. Fricke and H. J. Curtis, J. Phys. Chem., 40, 715 (1936). (64) H. Fricke and H. J. Curtis, J. Phys. Chem., 41, 729 (1937). (65) G. Schwarz, J. Phys. Chem., 66, 2636 (1962). (66) D. C. Grahame, J. Chem. Phys., 18, 903 (1950). (67) The following further approximations are involved. The input driving potential is assumed representable by a single value, that being the value of the electric potential at the point where the diffuse layer merges with the bulk electrolyte, where the diffusion potential vanishes. Because of transverse equilibrium across the diffuse layer, this potential is equal to the diffusion potential at the electrode surface where the electric potential vanishes, and is likewise equal to the total electrodiffusion potential everywhere on the boundary "B." Equations 17 and 18 are derived for a simple RC ladder network, In which the resistivity is set equal to that of a sheet of electrolyte equal in thickness to the diffuse layer, and the capacitance is taken to be that of the double layer, calculated by standard methods. These are indeed gross approximations, but are probably as good as the model, and accurate to an order of magnitude. (68) E. H. Eyiar, M. A. Madoff, 0. V. Brody, and J. L. Onciey, J. Biol. Chem., 237, 1992 (1962). (69) G. Falk and P. Fatt, Proc. Roy. SOC., Ser. 6,160, 69 (1965). (70) K. S.Cole and A. L. Hodgkin, J. Gen. Physiol., 22, 671 (1939). (71) W. Scheider, J. Phys. Chem., 76, 349 (1972).
