J. Phys. Chem. 1988, 92, 6408-6413
6408
Temperature Dependence of the Primitive-Model Double-Layer Differential Capacitance: A Hypernetted Chain/Mean Spherical Approximation Calculation Luis Mier y Terin, Enrique Diaz-Herrera, Marcelo Lozada-Cassou,* Departamento de Fhica, Universidad Aut6noma Metropolitana-Iztapalapa,Apartado Postal 55-534, 09340 D.F., MPxico
and Douglas Henderson I B M Research Division, Almaden Research Center, S a n Jose, California 951 20-6099 (Received: December 29, 1987)
Calculations based on the hypernetted chain approximation (HNC/MSA version) for the differential capacitance are reported for the model double layer formed by charged hard spheres in a continuum dielectric medium and near a charged hard wall, The emphasis is on its temperature dependence. Rather than numerically differentiate a table of potential versus charge density, we formally differentiate the HNC/MSA equations and solve these equations simultaneously with the usual HNC/MSA equations. A finite element method that requires only a small number of iterations is used and is found to be very efficient. The resulting HNC/MSA values for Care considerably larger than the Gouy-Chapman (GC) values at high electrode charge density and, in contrast to the GC results, are found to decrease with increasing concentration.
Introduction Recent studies of some of the properties of the electrical double layer, formed in an electrolyte near an electrode, using the hypernetted chain ( H N C / M S A version) the modified Poisson-Boltzmann (MPBS version) a p p r o ~ i m a t i o n , ~ and the Born-Green-Yvon (BGY+EN version) approximation’*’* have all shown the predictions of the Poisson-Boltzmann theory of Gouy13 and Chapman are appreciably in error. All of the above applications of these approximations use the primitive model of the electrolyte, in which the solvent is represented by a uniform dielectric medium, and a primitive model of the electrode, in which the electrode is a uniform hard charged surface with zero charge depth. Image forces are sometimes considered but are generally ignored. The Gouy-Chapman (GC) approximation makes the further assumption that the ions are ( I ) Henderson, D.; Blum, L. J . Electroanal. Chem. 1978, 93, 151. Henderson, D.; Blum, L.; Smith, W. R. Chem. Phys. Lett. 1979,63, 381. Henderson, D.; Blum, L. J . Electroanal. Chem. 1980, 111, 217. (2) Carnie, S. L.; Chan, D. Y . C.; Mitchell, D. J.; Ninham, B. W. J . Chem. Phys. 1981, 74, 1472. (3) Lozada-Cassou, M . J . Chem. Phys. 1981, 75, 1412; Ibid. 1982, 77. 5258. (4) Lozada-Cassou, M.; Saavedra-Barrera, R.; Henderson, D. J . Chem. Phys. 1982, 77, 5 150. Lozada-Cassou, M.; Henderson, D. J . Phys. Chem. 1983. 87. 2821. (5) Barojas, J.; Henderson, D.; Lozada-Cassou, M. J . Phys. Chem. 1983, 87, 4547; Ibid. 1984, 88, 2926. (6) Carnie, S. L. Mol Phys. 1985, 54, 509. ( 7 ) Lozada-Cassou, M. J . Phvs. Chem. 1983,87, 3729. Gonzblez-Tovar, E.; Lozada-Cassou, M.; Henderson, D. J . Chem. Phys. 1985, 83, 361. (8) Alts, T.; Nielaba, P.; B’Aguanno, D.; Fortsmann, F. Chem. Phys. 1987, 1 1 1 , 223. (9) Outhwaite, C. W . J . Chem. Soc., Faraday Trans. 2 1978, 74, 1214. Levine, S.; Outhwaite, C . W. J . Chem. Soc., Faraday Trans. 2 1978, 74, 1670. Outhwaite, C. W.; Bhuiyan, L. B.; Levine, S. Chem. Phys. Lett. 1979, 64, 150. yBhuiyan, L. B.; Outhwaite, C . W.; Levine, S. Chem. Phys. Lett. 1979, 66, 321. Outhwaite, C. W.; Bhuiyan, L. B.; Levine, S . J . Chem. Soc., Faraday Trans. 2 1980, 76, 1388. Levine, S.; Outhwaite, C . W.; Bhuiyan, L. B. J . Electroanal. Chem. 1981, 123, 105. Outhwaite, C. W.; Bhuiyan, L. B.; Levine, S. Chem. Phys. Lett. 1981, 78, 413. Outhwaite, C. W.; Bhuiyan, L. B. J . Chem. Soc., Faraday Trans. 2 1982, 78, 775. Bhuiyan, L. B.; Outhwaite, C . W.; Levine, S. Mol. Phys. 1981, 42, 1271. Outhwaite, C. W.; Bhuiyan, L. B. J . Chem. Soc., Faraday Trans. 2 1983, 79, 707. Outhwaite, C. W. J . Chem. Soc., Faraday Trans. 2 1983, 79, 1315. ( I O ) Croxton, T . L.; McQuarrie, D. A . Chem. Phys. Lett. 1979, 68, 489; Mol. Phys. 1981, 42, 141. Croxton, T. L.; McQuarrie, D. A,; Patey, G . N.; Torrie, G . M.; Valleau, J. P. Can. J . Chem. 1981, 59, 1998. Henderson, D.; Blum, L.; Bhuijan, L. B. Mol. Phys. 1981, 43, 1185. ( 1 1 ) Blum, L.; Hernando, J.; Lebowitz, J. L. J . Phys. Chem. 1983, 87, 2825. Caccamo, C.; Pizzimeuti, G.; Blum, L. J . Chem. Phys. 1986,84, 3327. Bruno, E.; Ceccamo, C.; Pizzimeuti, G. J . Chem. Phys. 1987, 86. 5101. (12) Lozada-Cassou, M . Chem. Phys. Lett. 1981, 81, 472. (13) Gouy, G . J . Phys. (Les Ulis, Fr.) 1910, 9, 457. (14) Chapman, D. L. Philos. Mag 1913, 25, 475.
0022-3654/88/2092-6408$01 .SO/O
point charges. The effects of this latter approximation can be minimized by giving the ions a nonzero distance of closest approach to the electrode. This is usually called the stern correction. In the stern correction the ions still interact with each other as point charges. The G C approximation with the stern correction is the version used in this paper. Comparison with experiment does not provide an unambiguous test of these theories because semiempirical modification of the model parameters can force any of these theories into agreement with experiment mimicking in an ill-defined way the solvent and metal contributions to the double layer. Comparison with the simulation studies of Torrie and Valleau15 is the only satisfactory test of these theories. The HNC/MSA, MPB5, and BGY+EN approximations are all in satisfactory agreement with these simulation studies. Each of these theories can, in principle, be generalized to include solvent and metal contributions. However, this has not been done for any of the theories mentioned above but has been accomplished for the simpler mean spherical approximation (MSA), which is a linearized version of the hypernetted chain approximation, valid only for small electrode charge, where both solventi6and metal effect^"^^^ have been considered and where pleasing agreement with experiment is obtained. In this paper we use the HNC/MSA approximation to obtain the temperature dependence of the differential capacitance and related properties of the double layer. Previously, the differential capacitance has been obtained by numerical differentiation of the charge-potential relationship. In this paper we give a method of calculating the differential capacitance directly. The iterative method used in our previous studies sometimes required an excessive number of iteration^.^ In this paper we outline a new scheme for solving the HNC/MSA equations which requires only four or five iterations, confirming our previous belief that the excessive number of iterations in our earlier studies was a feature of the numerical method used and not a inherent characteristic of the H N C / M S A approximation. (15) Torrie, G. M.; Valleau, J. P. Chem. Phys. Lett. 1979, 65, 343; J . Chem. Phys. 1980, 73, 5807. Torrie, G . M.; Valleau, J. P.; Patey, G. N. J . Chem. Phys. 1982, 76, 4615. Torrie, G. M.; Valleau, J. P. J . Phys. Chem. 1982, 86, 3251. Valleau, J. P.; Torrie, G. M. J . Chem. Phys. 1984, 81, 6291. Torrie, G . M.; Outhwaite, C. W . J . Chem. Phys. 1984, 81, 6296. (16) Carnie, S . L.; Chan, D.Y. C. J . Chem. Phys. 1980, 73, 2349. Blum, L.; Henderson, D. J . Chem. Phys. 1981, 74, 1902. Blum, L.; Henderson, D.; Parsons, R. J . Electroanal. Chem. 1984, 161, 389. ( 1 7) Badiali, J. P.; Rosenberg, M. L.; Varicat, F.; Blum, L. J . Electroanal. Chem. 1983, 158, 253. (18) Schmickler, W.; Henderson, D. J . Chem. Phys. 1984, 80, 3381. Henderson, D.; Schmickler, W. J . Chem. Phys. 1985, 82, 2825. Schmickler, W . : Henderson, D. J . Chern. Phys. 1986, 85, 1650.
1988 American Chemical Society
Primitive-Model Double-Layer Differential Capacitance One might ask why we use the H N C / M S A equation at all. It has been stated19 quite categorically that the MPBS equation is appreciably more accurate than the HNC/MCA equation. The existing BYG+EN results are as yet too scattered for a final statement to be made about this approximation. We respectfully differ with these statements about the supposed superior accuracy of the MPBS approximation. We do agree that the MPBS approximation is a fine theory. However, the differences between the HNC/MSA, MPB5, and simulation results are small compared the errors in the G C approximation. Rather than worry about which of the successful theories is marginally better, theorists would do better to emphasize that all of the modern theories give essentially the same results and predict the same correction to the G C theory. Furthermore, careful examination of the predictions for the potential as a function of charge shows that, although the MPBS results are slightly closer to the simulation results than are the H N C / M S A results, the slope of the H N C / M S A charge vs potential curve is slightly better than that of the MPBS curve. This means that the HNC/MSA approximation will give slightly better differential capacitances. We do not attach great significance to this fact. However, we do feel that we must draw attention to it to redress the balance and to demonstrate the danger of basing conclusions about the merits of a theory on an examination of only one property. It should also be kept in mind that the MPB family of approximations has difficulty in producing physically reasonable solutions at high concentrations and high electrode charges. The H N C / M S A approximation is not without its deficiencies. The coupling of the H N C and MSA used in the H N C / M S A approach is ad hoc. However, the MPBS approach is not without its ad hoc features. A consistent H N C / H N C approach is more attractive and is preferable for ions of small diameter.z0 Nevertheless, for ions of the size considered here, the H N C / M S A approximation is very reliable. The H N C / M S A equation does not involve any pair-correlation functions. In some ways this is an advantage. Crude approximations for these functions need not be introduced. However, it does mean that image forces cannot be introduced in any easy way. Both the MPBS and BGY+EN equations do involve pair-correlation functions and so can deal with images, but they do not yield any fundamental prescription for obtaining these functions. The approximations for these pair functions used in the MPBS and BGY+EN equations are rather ad hoc. The H N C approximation does have the advantage that it can be generalized to give a consistent method of calculating pair, as well as singlet, functions. Such generalizations have been examined for charged hard spheres near two charged hard walls,z1 and uncharged hard spheres near an uncharged hard and charged hard spheres near a charged hard wallz3 and are promising. Finally, as was predictedz4 some years ago, even before it was observed in simulation studies, at high electrode charge the H N C / M S A and G C theories (and possibly some of the other theories) do not adequately account for pair correlations in the vicinity of the electrode and so must underestimate the potential at high electrode charge. Forstmann et al.25have generalized the H N C equations by using a local density approximation in place of the bulk direct correlation function and obtained pleasing results. The method of Lozada-CassouZ1or that of Plischke and Hen(19) Carnie, S. L.; Torrie, G. M. Adu. Chem. Phys. 1984, 56, 141. Torrie, G. M.; Valleau, J. P.; Outhwaite, C. W. J . Chem. Phys. 1984, 81, 6296. (20) Carnie, S. L.; Torrie, G. M.; Valleau, J. P. Mol. Phys. 1984, 53, 253. (21) Lozada-Cassou, M. J . Chem. Phys. 1984,80,3344. Lozada-Cassou, M.; Henderson, D. Chem. Phys. Lett. 1986, 127, 392. (22) Plischke, M.; Henderson, D. Proc. R . SOC.London, A 1986,404, 323. (23) Kjellander, R.; Marcelja, S. J. Chem. Phys. 1985,82, 2122; Chem. Phys. Lett. 1984, 112, 49; [bid. 1986, 127, 462. Plischke, M.; Henderson, D. J . Chem. Phys., in press. (24) Henderson, D.; Blum, L.; Smith, W. T. Chem. Phys. Lett. 1979, 63, 381. (25) Nielaba, P.; Forstmann, F. Chem. Phys. Lett. 1985, 117, 46. Alts, T.; Nielaba, P.; D’Aguanno, B.; Forstmann, F. Chem. Phys. 1987, 11 1, 223. Nielaba, P.; Alts, T.; D’Aguanno, B.; Forstmann, F. Phys. ReG. A 1986, 34, 1505. D’Aguanno, B.; Nielaba, P.; Alts, T.; Forstmann, F. J . Chem. Phys. 1986, 85, 3416.
The Journal of Physical Chemistry. Vol. 92, No. 22, 1988 6409 dersonz3also accounts for pair correlations in the vicinity of the electrode. In this paper we do not worry about this problem because it does not appear until the electrode charge is far in excess of any reasonable experimental value.
Theory The H N C equations for the primitive electrolyte/primitive electrode are
where g i ( x ) = hi(x) + 1 is the reduced density profile for ions of species i and is zero for x < a / 2 (the distance of closest approach). The quantity 6 = l / k T (k is Boltzmann’s constant and T is the temperature).
-4ae ZZ. J
+(x) =
t
i
X - t ) g j ( t )dt p. J ~ ( @
is the mean electrostatic potential, t is the dielectric constant, pi is the number of ions of species i divided by the volume, zi is the valence (including sign) of the ions of species i, e is the magnitude of the electron charge, and tu(x,t)is an integral over the bulk direct correlation function of the electrolyte: (3)
If the H N C values for cij(s) are used, one obtains the H N C (or H N C / H N C ) equations. If the MSA values for cij(s)are used, one obtains the HNC/MSA equations. As mentioned earlier, we use the latter approach here. Finally, a is the ionic diameter. For simplicity, we assume all the ions are of equal diameter. The GC theory results from the approximation lu(x,t) = 0. For the GC theory, (1) can be solved analytically. However, in general a numerical solution is required. Either the charge density, u, or the electrostatic potential, q0,of the electrode can be specified and the equations solved. Either choice is acceptable. We use both choices here. If +o is given, (1) is written as In gi(x) = -eziP+o
+ 2apA(x) +
K ( x , t ) dt
+
J
where A ( x ) , K ( x , t ) , and L ( x , t ) are given in the Appendix. If u is specified, (4) can be solved for a series of q0 and the final value of +o is then determined iteratively by requiring that the electroneutrality condition (5) produces the specified value of u. This is a tedious procedure. It is more conventient to write (1) in terms of u, i.e. In g i ( x ) =
2aezi@ua
-+ 2apA(x) + t
where
L’(x,t) = a, t Ix - a = 2a - x
I’ + t - -[a2 1 + ra
=
- ( x - t)2]
a - 2(x - r ) , x
+
+a 5t
(7)
The solution of (6) gives the distribution of ions for a given u directly. Evaluation of
6410
The Journal of Physical Chemistry, Vol. 92, No. 22, 1988
Mier y Tersn et al. where the superscript a is used to emphasize the approximate character of the functions Y , ( x ) . The functions $J(x) have simple mathematical forms and are defined piecewise over the finiteelement mesh. The unknown coefficients, (qijJ,are the values of the solution functions at the nodes of the elements. In the collocation version of the weighted residuals method,26 the residual functions L [F]are reduced to zero at the nodes of the elements:
provides a check on the accuracy of the calculation. The differential capacitance is given by
~ L [ F ( x ) ] d ( x- x,) dr = 0
One could calculate C by computing a and for a sequence of potentials or charges and differentiating numerically. This means that several potentials (charges) must be considered even if the capacitance were desired at only one potential (charge). It is more efficient, and probably more accurate, to calculate C by calculating dgj(t)/&b0 from (4) and
and then integrating dg,(t)/d+,. The solution of (4) and (10) is no more time-consuming than the solution of (4) by themselves. If instead of (4), (6) is solved for a given u, then from (8) we obtain
for i = 1, ..., N,where 6(x - xi) is the Dirac delta function. Here, this method is used to solve (4) and (10) or (6) and (12). Solving these equations on an unbounded domain is clearly impractical. For this reason we assume that the reduced density profile, gi(x), is equal to unity for x greater than a cutoff value of R. To solve (4) and (10) or (6) and (12), the domain, a / 2 I x I R, was divided to form a finite-element mesh. Over each element three nodal points were identified, two of them being the end points of the element and the third being the middle point. Correspondingly, three local quadratic Lagrange shape functions can be defined over each finite element:
41(0 = (1/2)F(E - 1)
(15a)
42(0 = 1 - E2 43(8 = (1/2)E(E + 1)
(1 j b )
Solving (1 2) and integrating agj(t)/du in (1 1) give the differential capacitance, C, for a given a in a more direct manner.
(1jc)
such that
E=
and from (6), we find
(14)
2.u - XI - x, Xr
- XI
where x, and xr are the positions of the left and right end points of the element, respectively. In order to improve the accuracy with which a solution is calculated, the positions of the nodes in the finite-element mesh, {x,), may be chosen to concentrate elements in those portions of the domain where the solution varies more r a ~ i d l y .In ~ this work, the domain was divided into several subdomains, and a uniform grid was constructed over each one of them. The density of nodes in each subdomain and the value of R depend on the concentrati~n.~ For our case, accordingly with (13), the unknown functions, h,(x), were represented by N
Numerical Procedure The numerical technique used in this work is based on the finite-element method.26 A simplified version of this technique has been applied before to the solution of the Percus-Yevick (PY) and H N C integral equations for the 6-12 Lennard-Jones fluidZ7 and to the MSA for the ~ q u a r e - w e land l ~ ~Y~~~k~a w a ~fluids. ~*~l The algorithm reduces a set of equations L[Y(x)] = 0 to a system of algebraic equations by subdividing the domain into a number of subdomains, or elements, of appropriate size and shape. The solution functions, Y,(x), are approximated by a set of N linearly independent basis functions { p ( x ) ] : .v = xqlJ@(X) /=I
(13)
(26) See, for example: Strang, G.; Fix, G. J. An Analysis of the Finite Element Method; Prentice Hall: Englewood Cliffs, NJ, 1973. (27) Mier y Terin, L.; Falls, A. H.; Scriven, L. E.; Davis, H. T. Proceedings of the Eighth Symposium on Thermophysical Properties; Sengers, J. V . , Ed.; American Society of Mechanical Engineers: New York, (1982); Vol. I, p 4 5 . (28) Quifiones, S. E. B S c . Thesis, Universidad de Guadalajara, 1983. (29) Mier y Terln, L.; Fernlndez-Fassnacht, E.; Quifiones, S. Phys. Left. A 1985, 107A, 329. (30) Mier y Terin, L.; Fernlndez-Fassnacht, E. Reu. Mex. Fis. 1986, 32, S241. (31) Mier y Terin. L.; Fernindez-Fassnacht. E. Phys. Lett. A 1986, 117A, 43.
h , ( x ) = CW,,$J(X), i = 1, ..., n J=
1
(17)
where n is the number of ionic species. Substitution of (17) into (4) or (6), and the use of the minimization criteria expressed in (14), results in a set of algebraic equations for the coefficients wlJ,F ( w ) = 0. This nonlinear set of equations was solved by Newton's method, which provides a , set of linear equations for the expansion coefficients, w ( ~ + ' )at the (K+l)th iteration in terms of the coefficients w(Q, at the kth iteration: J . ( w ( K + l ) - w(K)) = -F(w(K)) (18) where J is the Jacobian of the system. Gauss elimination was used to solve (18). The Newton process is continued until the Euclidean norm of the difference between successive iterations becomes less than a prescribed small number A:
IIdK+') - w(QI(
n N E
[
(w?I(~+I) l=lJ=l
- w , ( " ) ~ ]'I2 5 A
(1 9)
Once a solution for certain values of the parameters in (4) or ( 5 ) is found, initial estimates for a solution at other values for
the parameters can be found easily by a first-order continuation technique, similar to Euler's method. In our case, we chose the potential, +bo, (or charge, u ) of the electrode as the changing parameter. Initial estimate? for a larger potential, $0'. or charge
Primitive-Model Double-Layer Differential Capacitance
The Journal of Physical Chemistry, Vol. 92, No. 22, 1988 6411 02 > \
s=
-20
1
01
II L 0
2
L h
1
4
6 Iteration
Figure 1. Typical convergence of the solutions of HNC/MSA equation. The logarithm of the Euclidean norm as a function of the number of iterations, when $o = 0.05 V, is shown. The upper curve corresponds to the convergence process when the initial guess function is the solution of the HNC/MSA equation for the &, = 0.01 V case. The lower curve is for the case in which the initial guess function is obtained through the parametric continuation method described in the text. Both curves are Newton-Raphson iterations. u’, were generated from the converged Jacobian of a previous
solution w o at a lower value
Jio or u by
or
Figure 2. Helmholtz potential as a function of the electrode charge density for a model monovalent electrolyte ( a = 4.25 A, Z+ = 12-1= 1, 6 = 78.5) for two concentrationsand the following values of temperatures: (a) 273.16 K; (b) 298 K; (c) 325 K; (d) 350 K. The solid and dashed curves correspond to the HNC/MSA and PB theories, respectively.
0 IO >
d/’
0 oe
The rapid convergence of the procedure used here contrasts with that of the traditional Picard method for which thousands of iterations can be necessary to obtain solutions of similar accuracy! With the convergence criterion A = lO-’O, solutions were obtained in two to four iterations for steps in q0 as large as 25 V. Figure 1 illustrates a typical example of how the Euclidean norm, defined by (19), depends on the value of I I w ( ~ + I ) - w(uII before each iteration when the extrapolation scheme of (20) is used. The upper curve shows the same quantity when a previous solution is used as the initial estimate. After convergence was attained, the value of the potential on the electrode, q0,[if (4) is used] or the value of the charge density on the electrode, u, [if (6) is used] computed with (8) and (5), respectively, agreed with the value of q0 or u originally used to solve the equations within lo-’%. Once the numerical solutions for the reduced density profiles are found for particular values of the parameters, (10) or (1 2) can be solved by use of a finite-element algorithm similar to that described above. The method is powerful. On the CDC CYBER 180-830 computer, 141 central processor seconds are required to generate a solution for a particular set of parameters values with 131 nodes per function. Solution of (10) or (12) for the calculation of the differential capacitance requires 35 additional central processor seconds. What is perhaps more important is that, if a reasonable initial guess is provided, the computing time is independent of the values selected for the parameters. This was not the case with our previous caIc~Iations.~ Results We have made calculations for 1:1, 2:2, 1.2, and 2:l electrolyte solutions near a charge electrode for various values of the temperature and concentration of the solution and for various electrode charge densities. When we refer to a z1:z2 electrolyte, we mean that the valence of the counterion is z2. In all our calculations the dielectric constant of the solvent, t, is 78.5. Although the dielectric constant is certainly a function of the temperature, we
0 06
0 04
0 02
0
01
02
-2
03
0 /cm Figure 3. Helmholtz potential as a function of the electrode charge density for a model divalent electrolyte ( a = 4.25 A, Z+ = 1Z-I = 2, t = 78.5) for two concentrationsand several temperatures. The labels refer to the same values as in Figure 2. The solid and dashed curves correspond to the HNC/MSA and PB theories, respectively. have left e a constant in this calculation since including a temperature dependence would introduce an element of empiricism we prefer to avoid. In Figures 2 and 3 the Helmholtz potential = + ( a / 2 ) is plotted as a function of the electrode charge density, o, for two concentrations and four temperatures for 1 : 1 and 2:2 electrolytes. The HNC/MSA results are always less than the G C results. For the 2:2 case the H N C / M S A values of qH have a negative slope at high charge density. This is confirmed by computer simulat i o n ~ .Both ~ ~ the G C and H N C / M S A values of qHincrease as T increases with the difference between the G C and HNC/MSA results decreasing as T increases. We do not plot any results for the 2.1 and 1:2 cases since the double layer is dominated by the
+”
6412
Mier y Terfin et al.
The Journal of Physical Chemistry, Vol. 92, No. 22, 1988 1
30 ‘
\
I994 M
0 1
/
\
/
--,
C
I
I
01
02
---L 03
0-/ c m 2 Figure 4. Total differential capacitance as a function of the electrode charge density for a model monovalent electrolyte ( a = 4.25 A,Z+ = 12-1 = 1 , 6 = 78.5, T = 298 K) for various concentrations. The solid and dashed curves correspond to the HNC/MSA and PB theories, respectively.
’/c
E L
Figure 6. Total differential capacitance as a function of temperature for a model monovalent electrolyte (a = 4.25 A,Z+ = IZ-I = 1, t 78.5) for three concentrations and an electrode charge density of 0.01 C/m2. The solid and dashed lines correspond to the HNC/MSA and PB theories, respectively.
l
OO5
0401
0 01 M
l
0 3
31
02 C3 0 /em2
Figure 5. Total differential capacitance as a function of the electrode charge density for a model divalent electrolyte ( a = 4.25 A,Z+ = lZ-1 = 2, t = 78.5, T = 298 K) for two concentrations. The solid and dashed curves correspond to the HNC/MSA and PB theories, respectively.
counterions; thus, the 2:l results are much the same as the 1:l results and the 1.2 results are much the same as the 2:2 results. In Figures 4 and 5 the total differential capacitance C is plotted
as a function of the electrode charge density for T = 298 K and three concentrations for 1 :1 and 2:2 electrolytes. Again the 2:1 and 1:2 results are not shown since they are much like the 1:l and 2:2results, respectively. For the GC theory C always increases with increasing concentration whereas this is true for the H N C / M S A theory only for low electrode charge density. At higher electrode charge density the reverse is true. The H N C / M S A values of the capacitance at the higher values of u
20
40
,
,
60
,
1
,
80 T P C
Figure 7. Total differential capacitance as a function of temperature for a model monovalent electrolyte (a = 4.25 A,Z+ = 12-1= 1 , t = 78.5) for three concentrations and an electrode charge density of 0.20 C/m2. The solid and dashed lines correspond to the HNC/MSA and PB theo-
ries, respectively. increase more rapidly with increasing u than do those of the GC theory. This is especially true for 2:2 electrolytes at low concentrations. In Figures 6 and 7 , the differential capacitance for 1 :1 electrolyte is plotted for low and high values of u. Similar plots are made for a 2:2 electrolyte in Figures 8 and 9 . Again 2:l and 1:2 results are not plotted since they are much like the 1:l and 2:2 results. The slopes of the G C and H N C / M S A curves are fairly similar. However, at high u, the GC capacitance increases with increasing concentration whereas the reverse is true for the H N C / M S A values. Summary
Most electrochemical studies have tended to be at room temperature. However, interest in the study of the effects of tem-
Primitive-Model Double-Layer Differential Capacitance
usually highly degenerate. The contribution of the discrete solvent molecules to the capacitance has been calculated from an integral equation only in the limit of small g,I6 where a good description of the temperature variation is obtained.'* Given the absence of a theory proven to be applicable for all values of u, we feel the use of the "primitive model" is reasonable. We find that our method of calculating C i s efficient and takes only slightly longer than the calculation of the potential. In addition, the use of the finite-element method dramatically improves the efficiency of the calculation over the earlier Picard iteration m e t h ~ d . ~ The values of C at high charge density are found to be rather different in the H N C / M S A scheme than in the G C approximation. The HNC/MSA values at large are considerably larger than the G C values and decrease with increasing concentration. The GC values of C increase with increasing concentration. Comparison of these values of C with Monte Carlo and other theories is not possible since C has not been determined in these studies.
7 70,
0
.
80C
I
1.40-
I .oo
005M -----_------------
0 05M
1
0.60b
--___
0.005 M
i
0
.
The Journal of Physical Chemistry, Vol. 92, No. 22, 1988 6413
Acknowledgment. We gratefully acknowledge the support of CONACYT, MBxico. 2
0 20 ~
"
" 40"
"
ao
' 60
TT I
Appendix
Figure 8. Total differential capacitance as a function of temperature for a model divalent electrolyte ( a = 4.25 A, Z+ = IZ-I = 2, c = 78.5) for three concentrations and an electrode charge density of 0.01 C/mZ. The
solid and dashed lines correspond to the HNC/MSA and PB theories, respectively . N
=o, c1
K ( x , t ) = - [ a 2 - IX - t12] 2
c2 + 3a
-
-[[a3
c3
-[a5
4 50-
sa3
3a 2
x>-
-
IX
IX
- ti3] -
- t15], x - a
= 0, otherwise
L ( x , t ) = -2t, 3 50L
r
= a - x - t - -[a2 1
L
= -2x, I l
5
I
0
w 20
I 40
1
60
1
1
