87
DOUBLE-LAYER RELAXATION FOLLOWING CHARGE INJ~CTION that T would be equal to the spin-spin relaxation time Tz. For B = 0, the three transitions relax differently and a unique Tz cannot be defined. Except for very small values of T,the widths of the satellites would be about 1.5 times that of the central peak, as can be seen from eq 3-5. This result can readily explain the W l spectra for PBLG solutions in C H r Clz. Since the splitting between the chlorine peaks is equal to B / T (eq 1 and 7), the value of T can be found by comparing the experimental and computed spectra. For CH2ClZ-PBLG solutions, T for chlorine is about 6 X 10+-8 X 10" sec (whereas TZfor pure see, as measured from the W l CHzCl2 is 5 X
line width), The correlation time T~ can be calculated from (8); its value is 6 X 10-12-8 X lo-'* sec. The W l line widths for the CH2ClCH2Cl solutions are similar to, but slightly broader than, those of the CH2Clz solutions for comparable concentrations of the polypeptide.
Acknowledgment. This work was supported by the National Institutes of Health. The broad-line nmr spectrometer was obtained through an Institutional Instrument Grant by the National Science Foundation. Valuable suggestions from Dr. G. Englert are gratefuliy acknowledged.
Theory of Relaxation of the Diffuse Double Layer followhg Coulostatic Charge Injection' by Stephen W. Feldberg Brookhaven Natioturl L a b o ~ a h y ,Upton, New York, 11979 (Rccekd May 18, 1069)
Quantitative relationships describing relaxation of the diffuse double layer (in the absence of specific adsorption) following coulostatic charge injection have been calculated using the technique of digital simulation. The variation of electric field and ion concentrations as a function of time and distance from the plane are presented for a system of interest. An empirical equation is derived which relates the basic parameters. The theoretical predictions are in accord with Anson's recent experimental results.
Recent studies by Anson, Martin, and Yarnitzkyl have demonstrated double-layer relaxation following coulostatic charge injection. Newmana has treated some of the theoretical aspects of the problem and Buck' has considered the case for small perturbations where the condition c+
+ c-
%bulk
(1)
prevails a t all points in solution. Buck has also presented a thorough review of the pertinent literature in this field. I n this paper I shall present the results of Digital Simulationb of double-layer relaxation for a 1:l electrolyte in the absence of specific adsorption. The variation of electric field and ion concentrations as a function of time and distance from the z2 plane6 are calculated and an empirical equation is derived which relates the basic parameters. The simulation was carried out for semiinfinite linear diffusion. Modification for spherical diffusion is presented in the Appendix. It is assumed that the diffusion coefficients
of the two ions are the same, that both ions have a common x2 plane, and that the electrode is initially at the point of zero charge. It is further assumed that the solvent is a continuous dielectric and that there is no dielectric saturation. Definition of symbols appears at the end of the paper.
A Physical picture When an electrode at the point of zero charge is instantaneously charged, coulostatically, an electric field is produced. If there are no mobile charge carriers (1) This work was performed under the auspices of the United States Atomic Energy Commission. (2) F. C. Anson, R. Martin, and C. Yarnitnky, J . Phye. C h . ,73, 1836 (1969). (3) J. Newman,ibid., 73,1843 (1969). (4) R. P. Buck, E&ctroad. chem., 23,219 (1969). (6) 8.W.Feldberg, ''ElectroanalyticalChembtry, "Vol. 3, A. J. Bard, Ed., Marcel Dekker, Ino., 1969, p 199. (6) The ZI plane is the plane of aloaeat approach for nonspecifically
adsorbed ions (see P. Delahay, "Double Layer and Electrode Kinetics," Interscience Publishers,New York, N. Y., 1966).
Vdumb P4, Numbw 1 January 8, 1970
88
-~ Figure 1. Variation of d c + / c b u l k , .\/c-/cmr, and 10 x field(z)/field (z = 0) with distance from the za plane. Abscissa units = (z - 2,). (8rzIF’cb.,r/RT.)”’. The concentration eUNe8 will always approach Unity at large z and may thus be distinguished from the field curve. Details of time, charge, and concentration are printed in each figure. This legend applies to all figures.
Figure 3.
Figure 4
flux of positive ions to be negative (away from the electrode). The change in charge distribution will of course &ect a change in the electric field. The field will no longer be constant (eq 2) hut must be calculated according to Gauss’ law
Figure 2.
in the dielectric medium or if the extant charge carriers have not had time to move, the electric field is constant, extending to the counter electrode. The magnitude of this field (assuming that the lines of force are all normal to the plane of the electrode) is
E = --4rrq
The variation of the electric field and the ion concentration as a function of time after charge injection are shown in Figures 1-7. The & potential is calculated as follows.
€
(5)
The ions will move according to
(3) In the beginning, def/& = 0; thus it is clear from eq 3 that if a positive charge is injected, the resulting negative field (eq 2) will cause the flux of negative ions to be positive (toward the electrode) and the The Journal of Phgd.4 Chmklry
Inspection of Figures 1-7 shows that there are two modes of relaxation (as has been pointed ~ u t * J . ~ ) . The first comprises neutralization of the charge on the electrode. In the particular case simulated, it is virtually complete after 15 rsec when the field a t infinity has become zero (Figures 1-5). The second relaxation mode reflects the redistribution of the ions
DOUBLE-LAYER RELAXATION FOLLOWINQ CHARGE INJECTION
89
Gouy-Chapman b potential, 110.6 mV (T = 25O), is approached, but not attained within the time domain of the simulation. Perhaps the critical phenomenon to note is that the concentration profile of the attracted ion (the anion when the electrode is positively charged) is not monotonic and exhibits a distinct minimum (Figure 4). The equilibrium Gouy-Chapman ion distribution, of course, exhibits no such minimum. During the second relaxation process the only observable change in the nature of the concentration profiles is the disappearance of the minimum. Two additional systems were simulated-me in which the attracted ion ww immobile, ie., its diffusion coefficient was zero, and the other in which the repulsed ion was immobile. In both cases there appears to be no second relaxation. Once the neutraliiation is complete, the ions are in their equilibrium configuration. The concentration profiles are at all times monotonic. Thus it seems that mobility of both ions is prerequisite to the appearance of the second relaxation.
Quantitating the Relaxation Phenomena An equation describing the first relaxation process can he deriveda,'
Figure 6
Equation 6 neglects the variation in field close to the electrode surface. In the early stages of the relaxation, however, the contribution of the field, E,, to the potential will be overwhelminglylarge, i.e.
There is no analytical solution for the second relaxation process. The calculations show, however, that a plot of (b)lV8. t-;"
is linear once the first relaxation process is complete.B The intercept of these plots is always within a per cent of the Gouy-Chapman 6 potential. Furthermore, the slopes for these plots are found to be proportional to when
'4
9'52
RTrc Figure 7.
in the double layer as they relax to the minimum energy as defined by the Gouy-Chapman equation' (Figures 5-7). It is the second, slower relaxation which is of primary interest. For the system shown here the
(7)
From these data it is possible to write the following empirical equation (7) See reference cited in footnote 6. (8) The resder may wish to verify this for the data presented in
Figures 1-7. The theoretical hpotentialis 110.6mV. Theresulting plot is typical subject to the limitations discussed in the paper. Volume 74,Number I
January 8, 1970
STEPHEN W. FELDBERG
90
Although a thorough correlation of theory and experiment remains to be done, the indications are that double-layer relaxation in the absence of specific adsorption may be explained by rather simple concepts of electrostatics, diffusion, and migration. Equation 8 is valid when the condition stated in eq 7 obtains and when E , = 0.
C. Anson. He, John Newman, and Richard Buck
Discussion
are thanked for their many helpful discussions.
Equation 8 should be considered as an algorithm valid from the lower limit stated in eq 7 a t least up to
Appendix The effects of electrode and cell geometry are significant during the first relaxation period when E , is finite. During the second relaxation the field is zero except in a region very close to the electrode surface, and the potential (+J will be independent of any practically attainable geometry. For spherical geometry consider a test electrode of radius rl and with a concentric spherical counter electrode of radius r2. If E, is redefined as being the field just outside the diffuse double layer, a t a distance S from the electrode surface where 6 > rl
v = (Ew)*r1 Substituting from eq 6 will give the complete expression
for V . Once the first relaxation is complete, then Vt
= (4dt
Nomenclature E 9 €
fj
F
electric field, V om-' charge density on the electrode, C cm-l dielectric constant, 1.111 X 10-18 X 77.9 C V-1 cm-1 for water at 25" flux of jth species, mol cm-l sec-l Faraday, 96,497 C/equiv gas constant X temperature, 2478 V C mol-' at 25' diffusion coefficient of jth species, em*sec-1 diffusion coefficient (= D1 = DZ , etc.), om8sec-l bulk concentration of electrolyte, mol em-* concentrationof jth species, mol om-* charge on jth species see eq 5 equilibrium value of qh calculated according to the Gouy-Chapman equation time measured after instantaneous charge injection, sec
..
+z
(+z)G.c,
t
(9) Dielectric saturation in water becomes significant at field strengths greater than 1 X 10' V/cm (see P. Delahay, "Double Layer and Electrode Kinetics," Interscience Publishers, New York, N. Y., 1966, p 49).
