1843
MIGRATION IN RAPIDDOUBLE-LAYER CHARGING
Migration in Rapid Double-Layer Charging by John Newman Inorganic Materials Research Division, Lawrence Radiation Laboratory, and Department of Chemical Engineering, University of California, Berkeley, California 947dO (Received September 30, 1968)
Potentials and charge movements are analyzed for the rapid injection of a large surface charge into the interface between an ideally polarizable electrode and a dilute electrolytic solution.
Introduction When an ideally polarizable electrode, which by definition does not pass a faradaic current, is charged rapidly in a dilute solution, the concentration and charge distributions are disturbed at distances much greater than the equilibrium thickness of the diffuse double layer. For example, if such an electrode in a 0.001 M KC1 solution is charged from zero to -10 pC/cm2, the disturbance reaches to about 5300a. The anions have been driven from this region, leaving the cations exposed and forming an extended region of charge. The cations are consequently driven, largely by their own charge, toward the electrode surface where they form a thin, diffuse layer. As Gibbs has pointed out, such nonequilibrium, transient effects are important only when the solution is dilute and the change of surface concentration is large. The change should also be effected in a time short compared to the relaxation time for reequilibw tion. Thus it is appropriate to consider charging the interface at 10 A/cm2 for 1 psec in a 1 mM solution. This paper treats the movement of charge and the decay of the observed potential after the charging period. In the first approximation, these processes can be described with neglect of diffusion. The mathematical problem is then hyperbolic and can be solved exactly by the method of characteristics. Figure 1 illustrates the characteristic curves in terms of a dimensionless time T and a dimensionless distance X from the surface. The surface was initially uncharged, and during the charging period the region with no anions increases linearly with time. Outside this region the initial, uniform concentration of both anions and cations prevails, and there is no net charge density. After the charging period, here taken to be T = T = 20, this region no longer increases in thickness. One set of characteristic curves is the lines of constant 7. The other set shown on the figure is the paths of the cations as they move toward the electrode. With neglect of diffusion, the cations accumulate at the electrode surface. One can, as a second approximation, treat the diffuse nature of this charge distribution. During the charging period it is also possible to treat
the diffuse nature of the boundary between the charged and charge-free regions. At long times it is necessary to treat the back-diffusion of the electrolyte, not as a second approximation, but as a primary factor in determining the decay of the potential. An analysis along these lines thus is not exact, but it does give considerable insight into the factors involved and a semiquantitative prediction of the decay of the observed potential.
Mathematical Formulation For a symmetric electrolyte with x+ = -2- = z, the fluxes of cations and anions in the direction x normal to the surface are
N+
=
-D+(ac+/ax)
+ (.zD+F/RT)~+G (1)
N-
=
-o-(ac-/ax)
- (ZD-F/RT)C-E
(2)
the conservation equations take the form
(3)
and Poisson’s equation is
aG/ax
( x F / € )(c+
=
- C-)
(5)
For charging a large planar electrode at a constant current density, the electric field G approaches a constant -B far from the electrode, where B is positive for the charging process considered here, which drives cations toward the electrode. It is convenient to use this constant in forming the dimensionless variables and parameters
c+
c=---
+ c- ,
2cm
c+
&=-, E
E=B’ Kat
7 = - = €
x’F’c,(D+ RTe
- c2Cm
Zzc,F x=-EB x ,
+ D-) t,
A = -&,RT €B2
Volume 79,Number 6 June lQ60
1844
JOHN NEWMAN
Neglect of Diffusion As a first approximation, let us neglect diffusion and treat the system behavior when the anions and cations move by migration alone. Far from the electrode the concentration remains uniform, and the anions and cations move with constant velocities. This depletes the solution of anions near the electrode, and in the absence of diffusion there is a growing region in which anions are virtually absent. In this region, which extends to X = 2t-7, we thus have C = Q. The remaining cations form a region of extended charge where the electric field increases by Poisson’s eq 9. Hence the cation speed increases as the cations approach the electrode surface. Then at X = 0, the cations pile up and in the absence of diffusion form a surface charge density u. In the region of extended charge and without diffusion, the electric field and charge density obey the equations aQ/ar = -2t+(aEQ/aX) X Figure 1. Characteristic curves for the charging and decay periods.
During the charging period r
E The equations then become
ac -=A ar
azC ~
a x 2
-
ax + (t+ - t-)
aEQ ~
a2Q aEc + (t+ - t-) aQ -=A ar ax2 - ax ~
dE/dX = Q
(9)
+
where t+ = 1 - t- = D+/(D+ D-). The initial conditions are
C = l
and Q = O
at r = O
(10)
implying a zero charge on the electrode and no specific adsorption. Far from the electrode
E+-1,
C+1,
and Q + O
as X - + to
(11)
during the charging period. At the end of the charging period when the current is interrupted, the electric field E increases everywhere by 1 while the concentration and charge distributions remain unchanged. At the surface, which may be taken to be the inner limit of the diffuse layer, the fluxes of anions and cations are zero (see eq 12) implying the absence of faradaic reactions
A (aC/aX) = QE and A ( a Q / a X ) = CE at X
=
0
(12)
and specific adsorption of cations. The last conditions can be replaced by re,
rm
(C - 1) dX = (t+ - t-)r
The Journal of Physical Chemietrg
and
and aE/dX = Q
(14)
< T , the solution is
+ 2 t + t - ~ ) ~ ’ ~ ] / t +(15) Q = 1/2(1 - t+X + 2t+t-r)1’2 (16)
=
[t- - (1 - t+X
which satisfies the boundary conditions E = -1 and Q = 1/2 at X = 2t-7. When the current is interrupted at r = T , the electric field E increases by 1 and takes the value E = 0 for X > 2t-T. Subsequently, the region of extended charge no longer increases but has the constant thickness of 2t-T. During the decay period r > T , the charge density and electric field in this region are
t+E = I
+ t+d - [(1 + t+r’)
- t+(X - 2t-T) 1’” (17)
Q
=
1/2[(1
+ t+r’)’ - t+(X - 2t-T)]1’2 (18)
where r’ = r - T . This satisfies the condition a t r = T prescribed by eq 16 and the condition that E = 0 at X = 2t-T. The details of the derivation by the method of characteristics are interesting in themselves but are not essential to an understanding of the processes of charging and decay. Two fundamental works on hyperbolic systems and the method of characteristics are ref 1 and 2. Weber3 treated hyperbolic systems in the context of ion migration. For t+ = 0.5, the charge distribution is sketched in Figure 2 for the charging period and in Figure 3 for the (1) R . Courant and D. Hilbert, “Methods of Mathematical Physics,” Val. 11, Interscience Publishers, Inc., New York, N. Y., 1962. (2) R. Courant and K. 0. Friedrichs, “Supersonic Flow and Shock Waves,“ Interscience Publishers, Inc., New York, N. Y., 1948. (3) H. Weber, “Die partiellen Differential-gleichungen der mathematischen Physik,” Friedrich Vieweg und Sohn, Braunschweig, 1900,pp 481-506.
1845
MIGRATION IN RAPIDDOUBLE-LAYER CHARGING during the charging period r
I, E rm
< T and is
+ t+~’)3t+t-T + (1 + t+#) a - [( 1 + t+#) + 2t+t-T]8’2]
91 =
Figure 2. Distribution of extended charge during the charging period.
u2
decay period. During the charging period, Q is maintained at a value of 0.5 at X = 2t-r by the arrival of cations from the bulk solution. The presence of the free charge density increases the electric field and hence increases the speed of the cations as they move through this region. This results in a decrease in the concentration of cations, since they arrive at the electrode surface sooner. The cation trajectories are indicated in Figure 1. During the decay period an electric field continues to exist owing to the presence of the cations themselves. Hence the cations continue to migrate from the region of extended charge toward the electrode surface, as indicated in Figure 3. The (dimensionless) potential at X = 0, relative to the bulk solution but corrected for the ohmic drop which would exist if there were no charge layer, will be denoted by 0. The contribution to this potential due to the extended charge is =
/om
(E
=
+ 1) dX 2 [3t+t-r 3t+
+ 1 - (1 + 2t+t-r)3/2]
2-
and for
r
+ 2t+t-r)1/2 - l-J/t+
(21)
+ t+r’) + 2t+t-T]”2 - 1 - t+#) /t+
(22)
=
7
-u
= [(l
>T
~ 2 z T - u = { [(1
The Diffuse Part of the Double Layer In the absence of diffusion, cations pile up at X = 0 forming the surface charge density u. In reality these cations will form a diffuse layer, but one which is thin compared to the region of extended charge. Since anions are excluded from this region, Q = C , and eq 7 and 8 become
-1 _aQ- - A - a2Q -2t+ ar ax2
aEQ
ax
I n this region it is appropriate to use the variables e = E/u(r),
q = 2AQ/u2,
y = Xu/2A
0.5
I
I
1
I
I
I
(19) (26)
I
I
I
I 0.6
I
(27) For small values of A/u2, the right side of eq 25 can be dropped. The solution of the problem posed here then constitutes the basis of the inner expansion of a singularperturbation series, where the results of the preceding section are the basis of the outer expansion. With this approximation, integration of eq 25 subject to condition 27 gives dq/dy = 2qe
I
I
T= 20
Q
I
I 0.2
0.4
(24)
since these variables are now of order unity in the diffuse layer. Equations 23 and 9 and conditions 12 then become
delay = q 4
(20)
during the decay period r > T . This will represent an important contribution to the observed potential but will be modified by other considerations in the subsequent section. The dimensionless surface charge density is composed of a part u2 comprising the extended charge and a part u comprising the diffuse layer near the electrode surface 8s follows, for r < T
X/r
@I
r)
dX = L2 { (1 3t+
2eq
r‘=tO I
0.8
X/ T
Figure 3. Distribution of extended charge after the charging period (T = 20).
1.0
=
at y = 0
aq/dy = (aq/ae) ( a e l a y )
=
q(aq/ae)
(28)
A second integration gives q = ae/ay = e2 - em2
(29)
The constant of integration is evaluated by matching Volume 78, Number 6 June IOBO
JOHN NEWMAN
1846 with the outer solution; q is of order Ala2 and e is of order unity in the outer solution. Thus q - + O and e + e, as y + 0 0 , where e, is the value of e in the outer solution as X + 0. A third integration gives y
=
-in(--) 1 2e,
le-em e,
+
Fe
+
where F = (eo - e,) /(eo e,). After ment, the solutions for e and q are
rearrange-
The time required for the back-diffusion will be large compared to that for decay of extended charge if the solution is dilute and the charge is large. It will then at some time become appropriate to consider the diffuse layer of charge u to interact with a solution having the concentration Co. The value of Co will be changing so slowly with time that the diffuse layer is essentially in equilibrium, giving rise to a potential difference of @s =
The charge u has a greater spatial distribution than was assumed in the last section. Hence it makes an additional contribution to the potential @2
= 2A
/Om
(e - e,) dy = 2A In (1 - 3’)
-2A sinh-l [~/2(ACo)l/~]
(37)
At the same time there will be a liquid-junction or diffusion potential in the bulk solution between the concentrations of Coand 1. This value calculated from eq 34 is @4
=
- (t+ - t-)A in Co
(38)
(32)
Potential Decay Curves
where 1-F=
1 + T - U
for r
lfr-34
T
Back-Diffusion of Excluded Salt The potential should decay to the value corresponding to the equilibrium diffuse double layer a t the given charge density. However, u - t T as T-+ 00 and consequently @2 becomes infinitely large. As the extended charge decays to zero, the contribution of the diffuse layer is no longer small compared to that of the extended charge, and the above analysis is no longer valid. Instead, the diffuse layer begins to interact with the salt solution of anions which had been driven away from the electrode. When the extended charge has decayed, the salt solution from X > 2t-T begins to diffuse back into the essentially pure water left behind. If there is no charge density in the solution (& = 0) , eq 7 and 8 become
-
aBc/ax = (t+ - t-)A(a2c/ax2) ac/aT= 4 t + t - ~(a2c/ax2)
-
If eq 35 is solved subject to the conditions C = 0 for X < 2t-T for T = T C = 1 for X > 2t-T aclax = 0 at X = 0
t
C-1
0.1
as X + . o
The Journal of Physical Chemistry
0.5
1
2
5
10
20
50
100
Time, ps
then the concentration CO a t X = 0 approaches the equilibrium value as CO(T’)= erfc[3T(t-/t+AT’)’/2]
0.2
(36)
Figure 4. Potential a t the inner limit of the diffuse double layer. Charge produced in 1 m M solutions at 25“ by passing 10 A/cm2 for 1 psec. During the charging period, the potential is corrected for ohmic drop.
1847
MIGRATION IN RAPIDDOUBLE-LAYER CHARGING Table I: Parameters for Charging at 10 A/cm2 for 1 psec in Various 1 mM Solutions at 25'
t+
A
L,disturbance distance, A Debye length, A T for 1psec e/xrn, rsec L2/D,psec
2001 0
1
I
I 0.5
I
t
I
1
KOH
KC1
HC1
0.2695 0.5317 7570 96.1 39.33 0.0254 201
0.4905 0.1606 5280 96.1 21.61 0.0463 140
0.8208 1.299 1857 96.1 61.47 0.0163 10.3
I
LO
Time, ms
Figure 5. Decay of potential to 1 msec.
rapidly decaying potential. At long times the backdiffusion of salt into the relatively pure solvent in the disturbance region determines the potential of the diffuse layer. At intermediate times, a completely satisfactory mathematical description has not been possible, but these two concepts can be superimposed qualitatively to provide a reasonably complete and accurate picture of the decay process. One may expect that the presence of the extended charge tends to continue to exclude anions, while the back-diffusion of salt increases the conductivity and speeds the decay of the extended charge. Although the potential at the end of the charging period can be quite high (see Figure 4), this does not lead to a reaction of cations. The field at the surface is determined by the charge and is no larger a t 7' = 0 than a t the final equilibrium. The magnitude of the potential is due to the large spatial extent of extended charge, rather than to an abnormally high field.
Discussion This analysis of rapid charging of double layers in dilute solutions treats the limited rates of ion movement and the departures from equilibrium in the charge region. In Anson's4 treatment of such problems, the diffuse charge layer was assumed to be in equilibrium with an electrically neutral solution whose concentration at the surface varied with time as a consequence of the charge injection. A similar model is used here for large decay times, but the establishment of conditions in the bulk solution a t early times is different when one considers the formation and decay of the extended charge. Anson, Martin, and Yarnitzky6 presented potential decay curves for hanging mercury drops following charge injection and discussed the pertinent literature. They pointed out that the usual assumption of an equilibrium double layer in a transient or alternatingcurrent experiment remains valid except under extreme conditions of dilute solutions, large charge changes, and
short response times. Their potential-decay curves with tapered capillary tips agree with the calculations reported here. For blunt tips, the back side of the drop will be shielded by the capillary, and the injected charge will follow the primary current distribution. The relaxation of this nonuniform charge distribution will involve a time constant on the order of 4?"oC/K NN 80msec, and this will obscure the transient due to the extended charge. Here we have used a double-layer capacity of C = 30 pF/cm2, a drop radius of ro = 0.1 em, and a conductivity of K = 1.5 X lod4mho/cm. We should mention here the expected consequences of making experiments under conditions different from those treated in this paper. If the initial charge is +5 @/em2 and the final charge is -10 pC/cm2, then the extended charge region will not form until the stored anions have been driven away from the surface, and by that time 5t+/t- pC/cm2 of cations will already have been driven to the surface. Thus if
- cfinal
< t+rinitial/t-
no region of extended charge will be formed, and departures from the final, equilibrium potential should be small. The region of extended charge, if it exists, can be treated by methods not essentially different from those used in the present paper. In the experiments of Anson and Martin, the current varies with time, decreasing exponentially with time throughout the experiment. This difference probably has little practical consequence because the experimental observation times are much larger than the time constant for the decay of current and are so large that the extended charge has decayed and the ohmic potential drop is negligible. The primary differences would occur in the early charging period and are not subject t o observation. In this paper it has been assumed that specific adsorption is absent, a good assumption for cations. Specific adsorption would modify the behavior of the diffuse layer and the observed potential, particularly at large times, but it would have little effect on the J. Phys. Chem., 71, 3605 (1967). F. C. Anson, R. F. Martin, and C. Yarnitzky, ibid., 73, 1835
(4) F. C. Anson,
(5)
(1969). Volume Y3,Number 6 June 1969
1848
L. BROUSSARD
extended charge since the ions must reach the surface before specific adsorption is possible. We should also like to mention in passing that it is possible to give a more accurate account of the diffuse nature of the boundary between the extended charge and the bulk solution during the charging period. This has not been included here since the potential a t the surface would not be changed significantly.
Conclusions Large, nonequilibrium potentials should be produced by large, fast changes in the charge of an ideally polarizable electrode in a dilute solution. This phenomenon, which can be analyzed by neglecting diffusion, is associated with a charged region much thicker than the equilibrium diffuse charge layer. Such potentials should decay faster than can be observed in most experiments. Nonequilibrium potentials associated with the back-diffusion of salt into the space vacated by the extended charge should be smaller in magnitude but persist for longer times.
Acknowledgment. The problem treated here was posed to the author by Fred C. Anson. This work was supported by the United States Atomic Energy Commission.
Nomenclature A B Ci
c
Di e
E &
F F L
Ni !l
Q R t ti
T T X
X 2/ zi E K, U
uz 7
0
Dimensionless parameter; see eq 6 Magnitude of electric field far from electrode, V/cm Concentration of species i, mol/cma Dimensionless average concentration of cations and anions Diffusion coefficient of species i, cmz/sec Dimensionless electric field Dimensionless electric field Electric field, V/cm Faraday's constant, C/equiv See eq 33. Disturbance distance, cm Flux of species i , mol/cm2 sec Dimensionless charge density Dimensionless charge density Universal gas constant, J/mol deg Time, sec Transference number of species i Absolute temperature, OK Dimensionless time of charging Distance from inner limit of diffuse layer, om Dimensionless distance Dimensionless distance Charge number of species i Dielectric constant, F/cm Conductivity of bulk solution, mho/cm Dimensionless surface charge density in diffuse layer Dimensionless surface charge density in extended oharge Dimensionless time Dimensionless potential a t X = 0
The Disproportionation of Wustite by L. Broussard Esso Research Laboratories, Humble 0.11 and Refining Company, Baton Rouge Refinery, Baton Rouge, Louisiana (Received May 1 6 , 1 9 6 8 )
Mossbauer spectrometry provided a new tool for studies of the mechanism of disproportionation of normally prepared nonstoichiometric wustite, FeO,. The conclusions of these studies are supported by chemical analyses and by X-ray diffraction data. Initially, FeO, disproportionates by forming FeaO4 using Fe3+ cations already present in the structure plus Fez+ and 02- ions necessary to form Fes0.i. Stoichiometric wustite, which we designate as FeOl.o, is also formed during this initial stage but no a-Fe is formed. Finally, when all Fe3+cations initially present have been used to make Fe304, FeOm begins to disproportionate forming a-Fe and additional Fea04according to the generally accepted formula: 4Fe0 + Fe -I-FeaOl.
Introduction The properties of nonstoichiometric wustite, FeO,, have been studied by many investigators.l-13 It is generally recognized that stoichiometric wustite is difficult to prepare. Katsuri, et aZ.,l3 report synthesizing FeOl.oooat high pressures above 36 kbars a t 770" by the reaction of FeOl.osz with metallic Fe. Fischer, et produced stoichiometric wustite by decomposing nonstoichiometric wustite containing 1.2% precipiThe Journal of Physical Chemistrv
tated iron. We will describe the formation of FeOl.0 by decomposing nonstoichiometric wustite containing (1) E. R . Jette and F. Foote, J . Chem. Phys., 1 , 29 (1933); Trans. Amer. Inst. M i n . Met. Engrs., 105, 276 (1933). (2) L. S. Darken and R. W. Gurry, J. Amer. Chem. SOC.,6 7 , 1398 (1945). (3) J. BBnard, Bull. SOC.Chim., 16, D109 (1949). (4) P. K. Foster and A. J. E. Welch, Trans. Faraday Soc., 5 2 , 1626 (1956). (5) W. A. Fischer. A. Hoffmann, and R. Shimada, Arch. Eisenhuttenw., 2 7 , 521 (1956).
