KEITHB. OLDHAMAND LEO T. TOPOL
1462 To examine prediction xxiv one must determine whether I ( t ) approaches I ( to ) in accordance with eq 2 or 6. The identical experimental data that were used to construct Tables I11 and IV were therefore subjected to a statistical evaluation to see how well they fitted eq 6, and to determine y. The results, encompassing both step and reversal experiments, are summarized in Tables I X and X. If the correlation coefficients in Table I11 are compared, entry by entry, with those in Table IX, it will be seen that sometimes those in the first table are larger, and sometime those in the second, In other words, the experimental data fit eq 6 about as well as they fit eq 2. However, in one respect Table I X presents an analysis which is less satisfactory than Table 111: whereas Yr and Y, are indistinguishable, yr is systematically somewhat smaller than ye. Similar conclusions will be reached if Tables IV and X are compared, though the standard deviations involved here are larger and hence the equality of yy8and yr is more credible. Average y values from Tables I X and X and the corresponding standard deviations have been entered into Table VI1 and are there used to compute the quantities y X and ?r2y/8X to assess the other predictions of the random-tortuosity model. The assessment is incorporated into Table VIII. The final entry (‘NO?’’in this table reflects the fact that Z and y X for the paper fiber matrix do not differ by much
more than their standard deviations. The equality 2 = y X is unlikely but has not been disproved,
Conclusions All the theoretical predictions which are common to the four models have been confirmed for both the matrices studied. As far as the data collected in these experiments are concerned, the uniform-tortuosity model of a porous matrix amply explains all the results obtained with the filter paper media. If, as there is reason to believe,’ the retardation factor for this medium is close to unity, the effective constriction factor a is close to 3.0 and the tortuosity factor X is about 1.9. As Table VLII shows, the random-tortuosity model fits the filter paper matrix much less well than does the uniform-tortuosity model; the gel and diluent models have no success. On the basis of Table VIII, none of the four models is capable of explaining the results obtained with glass fiber matrices. It is beyond the scope of this article to suggest additional models, but it may be significant that the initial decline of I ( t ) is, for this matrix, much slower than would be predicted from results obtained from the steady-state and the approach to it. This is exactly the behavior expected if the medium has a less open structure in its interior than at its surface. Acknowledgments. We are indebted to Charles Lung and George Race for experimental assistance.
Electrochemical Investigation of Porous Media. 111. Theory of Galvanostatic Methods by Keith B. Oldham and Leo E. Topol Science Center, North American Rockwell Corporation, Thousand Oaks, Cal.lforn.la 91360 (Recetoed October 10, 1968)
The measurement of potential following the application of a current step or a current reversal to a porous material soaked in a solution containing an electroactive species offers an advantage in the elucidation of pore structure over the complementary potentiostatic techniques. Theory is presented for both subcritical currents, which lead to an ultimate steady state, and supercritical currents, which engender a transition time. Systems in the absence of a porous material are first considered and then the effect of the matrix is examined in terms of three simple models.
Introduction The two previous articles in this series1n2 were concerned with the application of a constant potential across a cell consisting of a thin disk of porous material sandwiched between metallic electrodes, the pores being with an containing ions Of the metal. The theory derived in the first article’ was The Journal of Physical Chemistry
verified in the second2 using the Hg-HgP couple. These results yielded information on the structure of the porous media. A difficulty which attends the use of potentiostatic techniques is the presence of ohmic resistance in the cell. (1) K. B. Oldham arid L. E. Topol, J. Phys. Chem.. 71, 3007 (1967). and IC. B. Oldham, i w . , 7 3 , 1455 (196s). L. E.
(2)
1463
ELECTROCHEMICAL INVESTIGATION OF POROUS MEDIA Since the current is time dependent, the “iR drop” is variable, and this destroys the potentiostatic condition to a greater or lesser extent. This imposes a restriction on usable experimental variables. Under galvanostatic conditions, as here described, the “iR drop” is a constant which causes little or no interference with the phenomena being investigated. It is therefore felt that these new techniques will yield more precise data than potentiostatic studies. As in the first article,‘ it is convenient to develop the theory first for the case in which the porous matrix is absent. We shall then briefly indicate how the relationships are modified for the case in which the electrolyte is contained in a porous matrix. In both potentiostatic and galvanostatic cases, there exists a critical current corresponding to the maximum ion flux which the cell can sustain permanently. The galvanostatic behavior of the cell depends markedly on whether or not this critical current is exceeded. We first discuss two methods in which the applied current is subcritical. In the third and fourth methods the current is supercritical.
Subcritical Current Step Consider two parallel electrodes of metal M, each of area A and separated by the distance L as in Figure 1. The space between the electrodes is wholly filled with a solution containing a small concentration C of ions Mmn+ and large concentrations of other, electrochemically inactive, ions. We assume that the reaction
+
Mmn+(soln) ne- = mM
(1)
occurs perfectly reversibly a t both electrodes and that no other reaction is possible.a-6 Because of the presence of large concentrations of other ions, it may be assumed that transport of Mmn+is solely by diffusion and that the diffusion coefficient D of this ion is constant,
a
- C ( X ,t ) =
at
a2
D - C ( X ,t ) ax2
expressing Fick’s second law. Here C ( x , t ) is the concentration of Mmn+a t time t a t a distance x from the cathode (see Figure 1). Equation 2 applies for all 2 in the domain 0 5 x 5 L. We seek a solution to eq 2 subject to the initial condition
C ( x ,0 ) =
c
(3)
and the boundary condition
a a I D - C(0, t ) = D - C ( L , t ) = ax ax nAF
(4)
which follows for t 2 0 from the requirements of Fick’s and Faraday’s laws. The solution of this set of equations is derived in Appendix A. From it, and the Nernst relation (5)
(where F, R , and T are the Faraday constant, the gas constant, and the absolute temperature) the cell potential is obtained as either of two equivalent series expansions
E(t) = nF
-2RT E(t) = nF arctanh 2 n t j D C
potentiol E
[
1
ate0 A
-Figure 1. Schematic diagram of the apparatus.
At time t = 0 the current i through the cell is suddenly changed from zero to a constant value I and the interelectrode potential E ( t ) is monitored a8 a function of time. Reference to Figure 1 will establish the sign convention adopted for E ( t ) and i. The diffusional situation is described by the equation
Equation 6a converges rapidly for small values of t, whereas 6b is more convenient a t long times. Notice that E ( t ) eventually becomes infinjte unless
2nAFDC I < L
(7)
(3) With only minor modifications, these theories apply to cases of Parallel inert electrodes a t which a redox reaction occurs reversibly, provided that both components of the redox couple are soluble and that both are present initially. Somewhat similar hysical situations have been treated earlier by Bersins and Derahayd and by Bowers, Ward, Wilson, and DeFord.5 (4) T. Berzins and P. Delahay, J. Arner. Chem. Soc., 7 5 , 4205 (1953). ( 5 ) R. C. Bowers, G . Ward, 0.M. Wilson, and D. D. DeFord, J. Phys. Chem., 6 5 , 672 (1961). Volume 75,Number 6 May 1969
1464
KEITHB. OLDHAMAND LEO T. TOPOL
The right-hand side of inequality 7 is, in fact, the critical current I , mentioned above, so that E ( t ) remains finite for a subcritical current step. Figure 2 presents an example of the potential variation following a subcritical current step. Setting t equal to infinity in eq 6b leads to
E(a)
=
-2RT arctanh nF
E}
t
(8)
a simple and important result. This final steady state corresponds to a linear concentration profile having been established across the cell. The series in (6) are so rapidly convergent that the approximations tanh
tanh
[$
[g
E ( t ) } = I,L 41
E ( l ) }N
4
0
Dt
(9a)
5
(9b)
Io
= 2nAFDC/L
(19)
( 6 ) P. Delahay. "New Instrumental Methods in Electrochemistry," Interscience Publishers, New York, N. Y., 1954, Chapter 8.
Volume 75, Number 6 M a y 1069
1466
KEITHB. OLDHAMAND LEO T. TOPOL
.B ..-
I
I
2.0
I/I,
-
I
3.0
Figure 5. Dependence of the transition time 71 for a supercritical current step on the ratio of current Z to the critical current I,. curve &PO: exactly; curve RPO: according to approximation 21a; curve QPS: according t o approximation 21b.
t
restricted range of current magnitudes, we shall concentrate on (21a). By thus restricting ourselves to I 2 1.851,, we satisfy also the condition (see Figure 5) 15 7 1 5 L2/20D, so that eq 9a applies. Combining (9a) and (21a) we arrive at the remarkably simple relationship
-50
-100
-
E(t) =
-2RT
-arctanh
nF
di
-150
(22)
for a supercritical current step with I 2 1.851,. Figure 6 shows an example of this potential-time relationship. The point of inflection lies always at t = r1/3 and (at 25') at -33.8/n mV as is readily shown by differentiating eq 22 twice. It is to be noted that the potential changes extremely rapidly as r1 is approached; this implies that a measurement of r1 may be made with great precision. There is no steady state accompanying supercritical galvanostatic experiments, and therefore no quantity analogous to W , or W,. Moreover, it is not fruitful to define any analog of V,. The quantity Us, as defined in eq 12, may, however, be associated with a supercritical current step, though (for I 2 1.851,) there is no longer any voltage limitation attaching to eq 12. A new dimensionless parameter 8 -
21
may also be defined; eq 21a shows it to equal unity in the absence of a porous matrix.
Supercritical Current Reversal In this section we treat the situation which ensues if, a t the transition time of a supercritical current step, the direction of current flow is suddenly reversed. Qualitatively, the result is intuitive: on reversal, the potential will cease to shift negatively and commence a drift toward positive potentials, passing through zero a t an instant we may denote by tl* and continuing The Journal of Physical Chemistry
figure 6. Potential-time plot for a supercritical current step. Parameters used: n = 2 equiv mol-'; T = 298°K; Z 1 1.851,; 71 = 5.00 sec. Note the fivefold change in ordinate scale interval at E ( t ) = 50.0 mV.
toward a second transition time a t 1 = r2. We may generalize this situation by considering that the current is again reversed a t r2, and a t every subsequent transition time, to give a behavior typified by Figure 7. We shall here restrict consideration to I 2 1.851,, but a similar treatment can be used if I , < I 5 1.851,. Appendix C presents an outline of the proof that under the conditions specified above -
tanh { -nF s E ( t ) ]=
$+ 2 2 71
$2
(-)k
k=1,2
(24)
71
where Tk is the kth transition time characterized by E(Tb)
(25)
= (-)'"a
+
When t lies between the kth and the (k 1)th transition time, there are (k 1) real terms in eq 24. The imaginary terms are to be ignored. Equation 24 may be solved to find all Tk and all tk*. For example
+
r-
I
1467
ELECTROCHEMICAL INVESTIGATION OF POROUS MEDIA Table I k
60’
b
b
Ib
-
Ib f (seos)
io
1 2 3 4 5 6 7 8 9 10 11 12
5;
7. Potential-time plot for supercritical current reversal. The same parameters were used as for Figure 6.
Figure
m
and
+
whence tl* = 4r1/3, r2 = 25r1/9, and tz* = 8(17 2/m) 4 3 5 . Other values of these critical times are listed in Table I. Eventually, as the table shows, the potential becomes a periodic function. The period is
where q(--+) is a constant (0.38010) related to the Figure 7 shows a typical potenRiemann p f ~ n c t i o n . ~ tial-time relation during the first few supercritical reversals. The analog of S,
tk’h
7k/Ti
1 * 3333 3 . 1017 4.8176 6.5564 8.2812 10.0156 11 .7429 13.4757 16,2040 16.9361 18.6650 20.3966 0.3259 (&I)
1.0000 2.7778 4.4907 6.2311 7.9548 9.6900 11.4167 13.1500 14.8779 16.6103 18.3389 20.0708 1.7304 ( T ~ - I / T I )
+
+
of potential-time curves will be most serious where 1 dE/dt I is large, i.e., a t the beginning of current steps and current reversals, and as transition times are approached. (b) This distortion will reduce the magnitude of dE/dt. Clearly, 1 dE/dt cannot exceed I / c A , for this is the rate of potential change when all the applied current is used to charge the double layer. (c) All r and t” time intervals will be somewhat lengthened in comparison with the expectation based on wholly faradaic current. It might be expected that the extra time required would be of order cAA,E/I, where AE is the potential change occurring during the timed interval. (d) The effect of double-layer distortion will be minor, provided that cAAE/I is small compared with the electrolysis time, Consider the case of the supercritical current step shown in Figure 6. Using eq 21a, we expect a negligible double-layer effect if
I
is shown by eq 29 to be unity for large k.
Deviation from Ideal Behavior As stated in the Introduction, ohmic resistance introduces no serious effect in galvanostatic studies. If p is the resistance, a constant Ip is to be added to or subtracted from the predicted E(t) to account for ohmic effects. More serious is the effect of double-layer capacitance. The theory above ignores the current needed to charge the double layer a t each electrode-solution interface, whenever the interelectrode potential is changing. This current, the magnitude of which is cAdE/dt, where c is the electrode capacitance per unit area, does not contribute to reaction 1. Therefore, some inconstancy in faradaic current is introduced and the theory is accordingly inappropriate to a greater or lesser extent. Quantitative correction for double-layer charging has proved a stumbling block to semiinfinite chronopotentiometry,8 and the problem would be equally intractible in the present problem. Qualitatively, however, the following predictions seem reasonable. (a) Distortion
Using the typical values A E = 100 mV (see Figure 6), = 2 equiv mol-’, andDc = 2 X 10-6 farad we obtain Cz/, >> 4 X mol concentration of not less sec1‘2. Hence for an Mmn+ than mol ~ m - and - ~ a current selected to give a transition time of not less than 1sec, overall interference from double-layer charging should be negligible.
D = 10“ cmz sec-‘, n
Application to Porous Matrices The first part of this series1 detailed four models of a porous matrix. Of these, three will be considered here. The fourth, the random-tortuosity model, is incompatible with a galvanostatic treatment. (7) M. Abramowite and I. A. Stegun, Ed., “Handbook of Mathematical Functions,” National Bureau of Standards, Washington, D . C., 1964, Chapter 23. (8) D . G. Davis, “Electroanalytical Chemistry,” A. J . Bard, Ed., Vol. I, Marcel Dekker Inc.. New York, N. Y., 1966, p 157 ff. (9) P. Delahay, “Double Layer and Electrode Kinetics.” Interscience Publishers, New York, N. Y., 1965, Chapter 3. Volume 73, Number 6 Mag 1969
1468
KEITHB. OLDHAMAND LEO T. TOPOL and
Table I1
Exptl quantity
Homogeneous medium
Theoretical prediction-Model of porous matrix-
P(X, T) =
Gel
Diluent
tortuosity
T,
4s
1
dS
1
e
1 a
l/a
Ad6
ads x/ff
The gel model assumes that the sole effect of the porous medium is to retard the motion of the M,n+ ions by a retardation factor 6. All the equations derived for the absence of a porous matrix apply to a matrix obeying the gel model except that each D in those equations should be replaced by the ratio D/6. The effect of this on the various dimensionless quantities is shown in Table 11. In the diluent model, the matrix serves merely to dilute the solution by a factor a, the constriction fact0r.l The effect of replacing C in the definitions of We,V,, . . . X,by C / a is shown in Table 11. According to the uniform-tortuosity model, the porous matrix is equivalent to a bundle of equally sized tubules. The effect of such a matrix is to replace D by D/6, A by A/a, and L by XL, where X is a tortuosity factor. The net result of these replacements is also shown in Table 11. In addition the relationships’ between the porosity 0 and the factors a and X are listed in Table I1 for the diluent and uniform-tortuosity models. Note that 0 is necessarily less than unity and that a>X>1 1 = V = T for a particular matrix would strongly suggest that the diluent model was applicable to that matrix. The success of this electrochemical approach to matrix structure was established by potentiostatic methods.2
(-43)
In addition, it is useful to employ the abbreviation
LI 2nAFDC
Equations 2 through 4 become simplified on introduction of the new variable set and they readily Laplace transform to
and
a F(,tl,
ax
1
p) =P
where and F(x, p ) are the transforms of T and F(x, T). The solution of the differential eq A5 subject to eq A6 is obtained straightforwardly as
F(x, p )
,= ~
p + ‘sech ~
z/i sinh{xl/p]
(A7)
Though eq A7 may not be inverse transformed directly, its expansion as either an infinite series or an infinite product permits Laplace inversion. Thus
whence by using standard formulaslO and the Heaviside theorem,lO respectively, we derive a
qx,T) = 2
4
(-)i
+oJ
x (ierfc
{2 j 2z/F +1-
= LX - 8L
C
?r2 +1,2
X]
(-)j
- ierfc {2 j 2+f il + x }) -
sin { s ( 2 j 1)x/2) (2j- 1 p
Appendix A Before solving eq 2 through 4, we find it convenient t o define the following set of dimensionless variables to replace the experimentally more significant terms x, t, and C(z, t )
-c
C
Uniform
L -
s., sr
C(z, t )
x exp[
--a2(2jq -
(A9)
where ierfc denotes the first integral of the error function complement.ll Finally, on reverting to the original, (10) M . Abramowitz and I. A. Stegun, Ed.. “Handbook of Mathematical Functions.” National Bureau of Standards, Washington, D. C., 1964, Ohapter 29. (11) Reference 10, Chapter 7.
The Journal of Physical Chemietry
1469
ELECTROCHEMICAL INVESTIGATION OF POROUS MEDIA experimentally significant, variables, the alternative results
riz])
- ierfc -
=
C--
4L
(A1O) are obtained from eq A9.
Appendix B The final steady state after a subcritical current step is obtained by setting infinite in eq '10This result then represents the initial condition
C ( x , 0) = c -
Appendix C ~The complete equation set for the supercritical current-reversal situation comprises eq 2, 3, 5, 25, and
D-C(O, ax t)
We define dimensionless distance and concentration variables somewhat differently than in Appendix A; thus
L
where the summations are those of eq 6. Equations 13 represent a near-perfect approximation to (Be).
a
I ( L - 2x) 2nAFD
for current reversal. We seek to solve Fick's eq 2 subject to ( B l ) and to the new boundary condition
x'
The variation of potential with time follows straightforwardly by application of the Nernst equation (5)
a
=
i
D -ax C ( L , t ) = nAF -
i ( t < 0) = 0 i(0 < t
< TI)
=
I
> 1.8510
(C1) (C2) (C3)
The superposition principle12leads one to expect by analogy with eq A10 that the exact solution of the set might be
- 22
=-
L
and
C ( x ,1) 2C
T) = ___
rl(Xr,
- 2-1'
( L - 2x)I 4nAFCD
(B4)
The undimensionalized time and current, T and c, are retained without redefinition. When x, t, C ( x , t ) , and I are replaced in eq 2, B1, and B2 by x', T, r'(x', T), and L, expressions are derived which on Laplace transformation yield equations which are identical (apart from the presence of primes) with (A5) and (A6). The solution of'the subcritical current-reversal problem then proceeds along the course mapped out in Appendix A. A final reversion to the original variables gives
C(x,t) 8L
c ...)
. ~.
h1.a
(B5)
where the summations are identical with those in (ANI).
- ierfc { 2 4jDL(-tx- r k ) })
(C5)
where U ( t - a) is the unit step function.l8 This expectation proves to be correct irrespective of the magnitude of the I / I . ratio. If the latter exceeds 1.85, we may employ approximations which lead ultimately to ea 24. (12) K.B. Oldham, Anal. Chem., 40,1799 (1968). (13) W. T. Thomas "Laplace Transformation," Prentice-Hall Inc., Englewood cliffs, N. J., 1960, Chapter 2.
Volume 79, Number 6 May 1969
