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The Transient Potentials Produced by the Creation or Annihilation of the Diffuse Double Layer in Dilute Solutions
by Fred C. Anson Contribution ,Vo. $51 6 f r o m the Gates and Crellin Laboratories of Chemistry, California Institute of Technology, Pasadena, California 91109 (Received A p r i l 19, 1967)
Changes in the electronic charge density on electrode surfaces are accompanied by changes in the ionic concentrations next to the electrode even in the absence of faradaic electrode reactions. I n dilute solutions of electrolyte these concentration changes produce appreciable transient potentials arising from both the ionic strength dependence of the diffuse double-layer capacity and the concentration cell created by the concentration changes. A derivation of the equations describing this behavior is given and the magnitudes of the transient potentials resulting from galvanostatic and coulostatic charge injection are calculated.
The properties and behavior of the diffuse double layer at charged electrode-electrolyte interfaces under nonequilibrium conditions have received considerable recent attention.‘s2 The problem was also considered much earlier by Ferry3 and by Grahame,4 who concluded that for most accessible experimental situations “the effects of concentration polarization on the (double-layer) capacity are inappreciable.” Recent experiments in these laboratories5 involving the rapid injection of charge into a mercury electrode contained in dilute solutions (0.5-1 mM) of electrolytes led to a calculation of the magnitude of the transient effects to be expected when the charge on the electrode is suddenly altered and the electrolyte concentration just outside the diffuse double layer differs from that in the bulk of the solution. The results of these calculations for the two cases of galvanostatic (current-step) and coulostatic (charge-step) perturbation are quite similar to the equation given by Barker2for the case of alternating current perturbation. However, since the derivation given here is more detailed and conventional, it seemed worth presenting it to facilitate understanding of the physical situation giving rise to the calculated effects. The problem is the following. When the charge density (or potential) of an electrode is abruptly altered, the ions that are attracted toward (or repelled from) the electrode surface in order to create the diffuse double
layer corresponding to the new charge density must come from (or be expelled into) the initially homogeneous solution just outside the diffuse layer. I n solutions containing a high concentration of electrolyte (0.1-1 144) the resulting concentration perturbations are extremely minute and the diffuse layer can be safely assumed to remain in equilibrium with a solution having the composition of the bulk (at least in the absence of ionic specific adsorption). However, in more dilute solutions the concentration changes produced just outside the diffuse layer can be far from negligible. In such cases the diffuse layer resulting initially after the perturbation will be in equilibrium with a concentration of electrolyte different from that in the bulk, and its properties (e.q., capacity) will vary in time as the concentration just outside the diffuse layer equilibrates with the bulk concentration. To calculate the magnitude of this transient effect, we may consider an experiment in which we change the charge density of an ideally polarized mercury electrode in a solution of a single, uni-univalent salt and calculate (1) P. Delahay, J . P h y s . Chem., 70, 2067, 2373, 3150 (1966); 71, 779 (1967). (2) G. C . Barker, J . Electroanal. Chem., 12, 495 (1966). (3) J. D. Ferry, J . Chem. P h y s . , 16, 737 (1948). (4) D. C. Grahame, J . Am. Chem. Sac., 68, 301 (1946). (5) F. C. Anson, A n a l . Chem., 38, 1924 (1966).
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October 1967
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the resulting change in the potential of the electrode measured with respect to a nonpolarizable reference electrode (in a cell with liquid junction)
where E is the measured potential of the mercury electrode, qm is the electronic charge density on the mercury surface, and p is the chemical potential of the salt composing the electrolyte. The fiist term on the right-hand side of eq 1 is the reciprocal of the differential double-layer capacity, C. Making this substitution, eq 1 can be rewritten d E = l[dqm C
+
’(Z),dU]
The coefficient (bq”/bp), can be obtained from the electrocapillary equations and the relation 1 2F
(3)
where E- is the potential of the mercury electrode with respect to a reference electrode reversible to the anion of the electrolyte in a cell without liquid junction, and F is the Faraday. The result is
Equation 5 gives the changes in potential that would result from the effect of a charge perturbation on the double layer. There is an additional source of potential change arising from the fact that the concentration perturbation which gives rise to the second term on the right-hand side of eq 5 also constitut’esa concentration cell which will contribute to the measured potential change by an amount
where Tbi and Tb- are the bulk transference numbers of the cation and anion, respectively, and p is the chemical potential of the salt. The total measured potential change is the sum of eq 5 and 6 dEt,t
=
(;)dqrn
f
1 -[(Tdl+ F
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- ‘/z(Tb+ - Tb-)] dp (7)
Equation 7 can be simplified by noting that Tb+ -ITb- = 1 for a single electrolyte dEt,t
where q+ is the charge in the diffuse double layer due to cations. For cases where there is no specific adsorption of either ion of the electrolyte the coefficient (dq+/dqm), can be calculated directly from the Gouy-Chapman theory of the diffuse double layer as described in detail by Parsons.6 Grahame’ termed this coefficient “the transference number of the cation in the double layer” while Barker2calls it “the effective transference number for the cation at the boundary’’ between the diffuse layer and the solution. It measures the fraction of the total charge moving into or out of the diffuse layer that is carried by cations; it mill be abbreviated here as Tdit. For electrolytes in which neither ion is specifically adsorbed at the concentrations employed, an assumption that will be made throughout the following discussion, the value of Tdl+ (and Tdl-) can be readily calculated from diffuse layer theory.6 Thus, Tdl+ will be 0.5 a t the point of zero charge (pzc) and will approach unity as the potential is made more negative, or zero as the potential is made more positive.8 Substitution of eq 4 into eq 2 yields
‘/z)
=
(i)dpm
+ FAT+ dLc
where AT+ is the difference between the cation transference numbers at the outer boundary of the diffuse double layer and in the bulk of the solution, ie., AT+ = Tdl+ - Tb+. Inasmuch as this whole treatment will only be of importance in rather dilute solutions, it seems justifiable to approximate the salt activity by its concentration and write (for uniunivalent electrolytes) dp
=
2RT d In Csalt
(9)
Substitution in eq S then gives dEtot = (b)dqm
+ 2AT+-dR T
In Csaitr=O (10)
F The first term on the right-hand side of eq 10 corre(6) R. Parsons, Proc. Intern. Congr. Surface Actitity, 2?id, London, 1957, 3, 38 (1957). (7) D. G. Grahame, J . Chem. Phys., 16, 1117 (1948). (8). I n the presence of specific adsorption Tdi+ and T d l - may exceed unity or become negative and the qualitative conclusions that are about t o be presented with the assumption of no specific adsorption will not hold. The equations given will still apply, however, and the behavior to be expected can be predicted if appropriate values for Tdl+ and T d l - are available.
DIFFUSEDOUBLE LAYERIN DILUTESOLUTIONS
sponds to the usual equilibrium potential change resulting from charging of the double layer. The second term represents the (usually neglected) transient potential change arising from the concentration changes produced by the movement of ions into and out of the diffuse double layer. To evaluate the magnitude of this term the mass transfer problem must be solved for the particular experimental technique employed to perturb the double layer. It is apparent already, however, that the contribution from the second term will not vanish even a t the point of zero charge for salts with bulk cation (or anion) transference numbers much different from 0.5 (e.g.J strong acids or bases). The Mass-Transfer Problem. In the case of a pure binary electrolyte having equal cation and anion concentrations C+ and C- it is well knowng-" that the Fick diffusion equation in the presence of migration becomes
bC+ = D8-d2C+ dt
dX2
where
D,
=
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layer. Applying the condition of electroneutrality outside the diffuse layer (i.e.j C+ = C-), multiplying (15) by D- and (16) by D+ and summing, the terms in dpj'dx can be eliminated to obtain
Equation 17 is the chief boundary condition to be applied to solve eq 11. The other boundary conditions are the usual C+ = C- = Co for all x when t = 0 and C+ = C- = Cofor all t when x + m . The solution to eq 11 for these conditions is
where C+z=ois the concentration of cation a t the outer boundary of the diffuse layer, Co is the bulk salt concentration, A is the electrode area, and the other terms have been previously defined. Equation 18 can be rearranged and simplified with the use of eq 12 and by noting that
2D+DD+ D-
+
and D+ and D- are the individual ionic diffusion coefficients of the cation and anion, respectively. We are interested in the case where no faradaic reaction occurs a t the electrode but the fluxes of both cations and anions at the outer boundary of the diffuse layer are nevertheless nonzero because they cross this boundary in the formation of the diffuse double layer.I2 These fluxes in the presence of migration areg
where dp/bx is the potential gradient to which the ions are subjected. If we consider first the case of charging the electrode by means of a constant current, i, the boundary conditions are
and
where Tdl+ and Tdl- are the effective cation and anion transference numbers at the boundary of the difluse
The result is
where AT+ is once again the difference in cation transference number between the boundary of the diffuse double layer and the bulk of the solution. Equation 20 leads to several interesting observations. Inasmuch as C+ = C- = CsaltJeq 20 gives the time dependence of the salt concentration at the surface of the electrode. It may be useful to trace the may this concentration changes during the course of a typical experiment. Suppose the electrode potential is initially quite positive of the pzc, e.g., 0 v us. sce, in a solution of salt having equal anion and cation bulk transference numbers Tb+ = Tb- = 0.5, and a cathodic current step is applied @.e., i > 0). Tdl' = (dq+/ dq"), will be essentially zero at this potential (assuming (9) M. D. Morris and J. J. Lingane, J . Electroanul. Chem., 6 , 300 (1963). (10) H. L. Kies, ibid., 4, 156 (1962). (11) V. G. Levich, "Physicochemical Hydrodynamics," PrenticeHall, Inc., Englewood Cliffs, N. J., 1962, p 280 ff. (12) F o r the purposes of the mass-transfer calculations the reasonable initial assumption can be made that the diffuse-layer thickness is vanishingly small compared with the distances moved by the diffusing and migrating ions. The z = 0 plane can then be considered to coincide with the outer boundary of the diffuse layer rather than with the true surface of the electrode.
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no specific adsorption), the concentration of salt at the electrode surface (i.e., just outside the diffuse layer) will increase continuously as the current flows, and the electrode potential moves toward the pzc. When the potential reaches the vicinity of the pzc Tdt+ will have increased to 0.5 and the salt concentration a t the electrode surface will reach a maximum and momentarily stop changing even though the current continues to flow. As the potential moves cathodic of the pzc Tdl+ becomes larger than 0.5 and the salt concentration thus begins to be decreased by the current flow. By the time the potential becomes much more negative than the pzc, Tdl+ will have reached unity and the current will be decreasing the salt concentration a t the maximum rate. For certain combinations of AT+ and i, eq 20 implies that a transition time, T , would ultimately be reached a t which the salt concentration a t the electrode surface would be decreased to zero
The assumptions involved in the derivation of eq 20 are not valid for the extreme conditions implied by eq 21 (i.e., pure, salt-free solvent at the electrode) and no clear significance can be attached to this transition time. It may be important, however, to calculate 7 so that a t the currents and salt concentrations employed in experiments this time is not too closely approached and certainly not exceeded. So long as Tb+ = Tb- it is a general consequence of eq 20 that, regardless of the initial electrode potential, whenever the current is of such a sign that the potential moves away from the pzc the surface salt concentration decreases and vice uema. This conclusion is also valid for nonconstant current techniques. The Transient Potential Change. Equation 20 provides the surface concentration needed for the evaluation of the second term in eq 10
This equation closely resembles the equation given by Barker2 for the case of a small ac perturbation. The fact that AT+ enters t,he equation squared means that the transient potential observed will be of the same sign on either side of the pzc even though (Tdl+ - Tb+) will change sign near the pzc for salts having Tb+ values near 0.5. The physical reason for this is that The Journal of Physical Chemistrg
FREDC. ANSON
the sign of AT+ and the sign of the change in salt concentration a t the surface are not independent. With the aid of eqs 20 and 21, eq 22 may be rewritten as
Integration of eq 23 is not straightforward because AT+ is a complex function of potential and therefore of time. However, a t potentials far from the pzc AT+ becomes essentially constant and equal to -Tb+ for positive potentials or Tb- for negative potentials. In the latter case, for example, integration of eq 23 gives
As an example, for a 1 mM solution of salt having Tb+ = 0.5 and D, = cm2/sec, injection of 10 pcoulombs/cm2 of positive charge with currents of or lo-' amp/cm2 leads to values of A E t r a n a of 0.9, 7.7, or 38.2 mv, respectively. These are the values that would be observed a t the instant that the current was turned off. They would, of course, decay toward zero as diffusion eliminates the concentration gradient of salt produced a t the electrode by the charge inje~tion.'~The magnitude of AEtrans is quite small even for rather dilute solutions. The considerable experimental problems associated with rapid injection of significant amounts of charge through dilute solutions having high resistances make the observation of A E t r a n s very difficult. With the advent of high-power pulse generators,14however, it may be possible to do so. Coulostatic Charge Injection. If, instead of injecting the charge with a constant current, the discharge of a small capacitor into the cell is employed, the resulting concentration perturbations are slightly different. For dilute solutions the resistance of the solution remains essentially constant throughout the charge injection so that the current will be given by
where E,,, is the voltage to which the injecting capacitor is charged, R, is the solution resistance, and Ci is the capacitance of the injecting capacitor, assumed to (13) If the current is turned off a t time torr, &Etrans will decay according to
(14) For example, Model 350 high-power pulse generator, Velonex Division of Pulse Engineering, Inc., Santa Clara, Calif.
DIFFUSEDOUBLELAYERIN DILUTESOLUTIONS
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be negligibly small compared with the capacity of the electrode. The experimerit can be considered as chronopotentiometry with an exponentially decreasing current. The problem is then to solve eq 11 again for the boundary conditions given by eq 17 and 25. The result is
where Csaltz=Ois given by eq 26. Barker2 presented an equation for the transient potential in the case of coulostatic charging which does not correspond to the combination of eq 26 and 28. The reason for the difference is that Barker considered the hypothetical case in lyhich the electrode can be considered to be charged instantaneously. The resistance where DI(X) is Dawson’s integral defined by15 in dilute electrolyte solutions prevents the realization of this idealized experiment in practice so that eq 26 DI(X) = e-AzLAevzdV (27) and 28 should represent a better approximation. As an example, for a 1 n i U solution of salt with and the other terms have all been previously defined. Tb- = 0.5 and D, = 10-i cm2/sec and a 0.03-cm2hangFigure 1 shows how Dawson’s integral depends on its ing mercury drop electrode, the effective solution argument. The perturbation in the salt concentration resistance is about 25 X lo3 ohms. Suppose that 10 at the electrode surface will follow this curve reaching pcoulombs/cm2 of charge is injected so that the elecits maximum value at a time given by (t/R,Ci)’/* = trode potential moves toward the pzc and the salt con0.92. centration a t the electrode surface increases. The 10 pcoulombs/cm2 of charge can be supplied with various combinations of Ecapand Ci. Typical combinations might be C, = 0.03 pf, E,,, = 10 v ; C, = 0.003 uf, E,,, = 100 v ; C , = 0.0003 pf, E,,, = 1000 v. I n each case the maximum value of AE,,,,, will occur at a time given by (t/R,C)”2 = 0.92 because of the properties of Damon’s integral. However, at times this early in the experiment the current flowing is still quite significant and the ohmic potential drop masks AEtrans. It is more realistic to calculate the value of AE,,,,, a t a time when the current has decayed to the point that the ohmic potential drop is, say, 10 inv. For the E,,, values of 10, 100, and 1000 v the times at which the ohmic drop falls to 10 niv can be calculated from eq 25 to be 3.2 msec, 0.69 msec, and SG psec, respectively. At these times the values of AE,,,,, calculated Dawson’s integral as a function of its argument. Figure 1. The maximum value of the integral is 0.541 which from eq 26 and 2s and the tabulated values for Dawoceurs for a n argument of 0.92. sons’s integral15 are 3.6, 5.3, and 15.4 niv, respectively. Thus even a t these late times in the experiment the remaining ohmic potential drop exceeds or is comparable As was true with constant current charge injection, eq 26 predicts that for certain combinations of Ecap to the transient potential perturbation for all reasonable experimental conditions. Calculations for both and AT+ the surface concentration of salt will be relower and higher salt concentrations lead to the same duced to zero. However, it suffices to make certain conclusion and it appears that, for a given high voltage that C, remains nonzero when (t/R,Ci)’’? = 0.92 and source, coulostatic charge injection is less suitable than Damon’s integral attains its maximum value of 0.541 injection by a current step for experimental measureto assure that the salt concentration at the electrode ments of AEtrans.By the same token, these calcularemains nonzero :at all times in the experiment. tions show that in coulostatic experiments with dilute The transient potential change produced by the solutions of reactants capable of undergoing charge charge injection is given as before by integration of the transfer a t the electrode, complications arising from second term in eq 10. If me again consider experiments far from the pzc SO that ATi can be considered approxi(15) “Handbook of Mathematical Functions,” Applied Mathemately constant and equal to -Tb+ or Tb-, this intematics Series 55, National Bureau of Standards, Kashington, D. C., gration leads to 1964, p 319. Volume 7i,Number 11 October 1967
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the concentration perturbations produced during the injection generally will be small and will die away quickly, so that the standard mass transfer analysis16 of the resulting data should be possible if appropriate time scales are chosen for the experiments. Finally, it is of interest to apply eq 26 and 28 to the recent coulostatic experiments of Delahay, de Levie, and Guiliani’’ in which double-layer capacitances were evaluated in extremely dilute solutions of salts. Because very small charges were injected in each experiment and relatively long times were allowed to elapse before the resulting potential changes were measured, the calculations show that the magnitudes of AEtrans mere completely negligible at the times the potentials were measured. However, at’ shorter times the values of AEtrans should have been significant compared to the size of the potential changes being measured and it is possible that this effect may have contributed t o some of the drifting potentials observed by Delahay, et uZ.17 As an example, a t a potential 0.2 v negative of the pzc for the M solution of XaF employed by Delahay, et u1.,17 one calculates that the measured equilibrium
The Journal of Physical Chemistry
change in potential for the experimental conditions employed was ca. 1.4 mv. The value of A E t r a n a calculated from eq 26 and 28 under the same conditions is 1.2 mv after 1 msec, 0.3 mv after 10 msec, and 0.09 mv after 100 msec. These values for AEtrsns were obtained by using the following parameters (taken from the ) eq 26: Co = lo-* paper of Delahay, et ~ 1 . ’ ~ in mole/cm3. E,,, = 1.0 v, R, = lo6 ohm (estimated), Ci = 500 pf, A = 0.041 cm2. D, was assumed to be cm2/sec. AT+ and TI,+ were both taken to be 0.5,
Acknowledgment. Very helpful discussions with Dr. Roger Parsons, Dr. Janet Jones, and Mr. Joseph Christie are gratefully acknowledged. This work was supported in part by the U. S. Army Research Office (Durham). The author is an Alfred P. Sloan Foundation Research Fellow. (16) P. Delahay, J . Phys. Chem., 66, 2204 (1962).
(17) P. Delahay, R. de Levie, and A,-M. Guiliani, Electrochim. Acta, 11, 1141 (1966).
