J. Phys. Chem. 1083, 87, 2821-2824
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Application of the Hypernetted Chain Approximation to the Electrical Double Layer. Comparison with Monte Carlo Results for 2:l and 1:2 Salts Marcel0 Lorada-Cassou Departamento de Fisica. UniversMad Aut6noma Metropolitana/Iztapalapa,M6xico 13, D.F., Mexico
and Douglas Henderson ISM Research Laboratory, Sen Jose, California 95193 (Received: November IO, 1982; I n Final Form: January 31, 1983)
The hypernetted chain approximation (HNC/MSA version) is applied to the electrical double layer problem. We consider an asymmetric binary salt consisting of a divalent and a monovalent ion. The ions, solvent, and electrode are modeled as charged hard spheres, a dielectric continuum, and a uniformly charged hard wall, respectively. The HNC/MSA results are in good agreement with recent computer simulations. Specifically we find that the double layer is generally narrower than that indicated by the Poisson-Boltzmann theory of Gouy and Chapman and that the density and potential profiles can be nonmonotonic which contrasts with the monotonic nature of these profiles in the Gouy-Chapman theory. This nonmonotonic behavior is more pronounced when the counterions are divalent and when the concentration is increased and results from the existence of layers of charge alternating sign. This also causes a nonmonotonic dependence of the diffuse layer potential and ionic adsorption isotherms on the electrode charge density.
Introduction Recently,' we have applied the hypernetted chain (HNC) approximation to the problem of a double layer consisting of an electrolyte solution in the presence of an electrode. In these studies we used the so-called primitive model of an electrolyte where the ions are modeled as charged hard spheres of equal diameter u and the solvent is modeled as a uniform dielectric medium. In our earliest studies we used the HNC approximation to describe both the interfacial region and the bulk electrolyte. Subsequently, Carnie e t al.2 have proposed using the HNC approximation for the interface and the mean spherical approximation (MSA) for the bulk electrolyte. Following Carnie et al., we call this hybrid approximation for HNC/MSA approximation. In an earlier paper3 we have compared the HNC/MSA results for 1:l and 2:2 salts with the Monte Carlo (MC) computer simulations of Torrie and V a l l e a ~ . ~ The agreement is very good. On the other hand, the use of the HNC approximation for both the interface and the bulk electrolyte yields less satisfactory results. Both the simulations and HNC/MSA results show that although the classic Poisson-Boltzmann theory of Gouy5 and Chapman6 (GC) is fairly satisfactory for 1:l salts, it is unsatisfactory for 2:2 salts (or equvalently, for 1:l salts in a nonaqueous solvent). We have generalized our HNC/MSA computer program to allow for ions of unequal charge. Thus, it is of interest to compare the HNC/MSA results for asymmetric salts with the Torrie-Valleau computer simulation results for 1:2 and 2:l salts.' These HNC/MSA and MC results are of interest not only because 2:l and 1:2 salts are an interesting practical application but also because we expect (1) D. Henderson, L. Blum, and W. R. Smith, Chem. Phys. Lett., 63, 381 (1979);D.Henderson and L. Blum, J.ElectroanaL Chem., 111,217 (1980). (2) S. L. Carnie, D. Y. C. Chan, D. J. Mitchell, and B. W. Ninham, J . Chem. Phys., 74, 1472 (1981). (3)M. Lozada-Cassou, R. Saavedra-Barrera, and D. Henderson, J . Chem. Phys., 77,5150 (1982). (4)G. M. Torrie and J. P. Valleau, J. Chem. Phys., 73, 5887 (1980). (5)G. Gouy, J.Phys., 9, 457 (1910). (6)D.L. Chapman, Phil. Mag., 25, 475 (1913). (7)G. M. Torrie and J. P. Valleau, J. Phys. Chem., 86,325 (1982). 0022-3654/83/2087-2821$01.50/0
that for this system the GC theory will exhibit both fairly good and rather poor results depending upon whether the counterion is monovalent or divalent. We expect this to be the case since it is reasonable to assume that, as soon as the electrode is charged, the double layer will be dominated by the counterions.
Theory We have solved the HNC equations
for the case where the kernel, @i,(x,t),is obtained from the MSA (eq 3 of ref 3). We follow Carnie et a1.2and refer to these equations with this choice of e i j ( x , t ) as the HNC/ MSA equations. The density pi is the number of ions of species i per unit volume in the bulk. We assume that z1 = 2 and z2 = -1. When the electrode has a positive charge, the monovalent ion is the counterion and we refer to this as the 2:l case. When the electrode has a negative charge, the divalent ion is the counterion and we refer to this as the 1:2 case. The density is given in terms of the molarity (the number of moles/liter of the divalent ions). The bulk density of the monovalent ion is twice the divalent ion density so that the charge neutrality condition
Eqpi = 0 i
is satisfied. Our method of numerical solution of eq 1has been described fully in our earlier HNC/MSA papers3 The function gi(x) is the reduced density profile of the ions of species i (normalized so that g;(x) = 1when x is large) and x is the distance from the electrode with the origin so that the distance of closest approach is half the ionic diameter, u/2. Thus the density profile, pi(x), is pigi(x) and the charge profile, q i ( x ) , is ziepgi(x). From g i ( x ) we can calculate the potential profile
(3) @ 1983 American Chemical Society
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The Journal of Physical Chemistry, Vo/. 87, No. 15, 1983 1001
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Lozada-Cassou and Henderson
I
x(A1
Figure 3. Density profiles for a 2:l electrolyte. The points and curves have the same meaning as in Figure 1. X(A)
Flgure 1. Density profiles for a 2:l electrolyte. The counterion is monovalent and the coion is divalent. The points give the Monte Carlo results and the solid and broken curves give the HNC/MSA and GC results, respectively.
0 10
r', \
-
21
X
c
9
2 1 016cm2 0 05 M
0.00 -
Figure 4. Potential profiles for a 21 electrolyte. The points and curves have the same meaning as in Figure 1.
X ( X
Figure 2. Potential profile for a 2:1 electrolyte. The points and curves have the same meaning as in Figure 1.
where h,(t) = g j ( t ) - 1, the diffuse layer potential @, = 4(a/2), and the ionic adsorption isotherms
Results The density and potential profiles for four states are shown in Figures 1-8. All calculations were performed with u = 4.25 A, = 78.5, and T = 298 K. The HNC/MSA Tesults for the 2:l case are in excellent agreement with the MC results. Even the interesting shape of the MC coion density profile at 0.5 M is reproduced by the HNC/MSA results. The GC results for the density profiles in the 2:l case are fairly good. However, the GC potential profiles for the 2:l case are significantly high and are much less satisfactory. For the 1:2 case shown in Figures 5-8, both the GC density and potential profiles are unsatisfactory. The HNC/MSA are not perfect but are appreciably better than the GC results. This is especially true at the higher concentration.
XiA)
Flgure 5. Density profiles for a 1:2 electrolyte. The counterion is divalent and the coion is monovalent. The points and curves have the same meaning as in Figure 1.
Generally speaking the MC and HNC/MSA double layer layer is narrower than the GC prediction. The ap-
The Journal of Physical Chemistry, Vol. 87, No. 15, 1983
HNC Application to the Electrical Double Layer
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0
1:2 -0.18 clm’ 0.05 M
-0.2 I
0.496M
-0.1 I 1
1
1
0.1
0.2
Charge den (cimz)
\
-0.121 0
I
,
20
30
1
10
x(h)
Figure 9. Diffuse layer potential as a function of charge density. For positive charge the monovalent ions are the counterions whereas for negative charge the divalent ions are the counterions. The points give the Monte Carlo results and the solid and broken curves give the HNCIMSA and GC results, respectively.
Flgure 8. Potential profiles for a 1:2 electrolyte. The points and curves have the same meaning as in Figures 1 and 5.
1:2
-0.15 c/m2 0.496M
8
4
0
12
X(A)
Figue 7. Density profiles for a 1:2 electrolyte. The points and curves have the same meaning as in Figures 1 and 5. 0.025
r-----l
0
8
4
12
x(h)
Flgure 8. Potential profile for a 1:2 electrolyte. The points and curves have the same meaning as in Figures 1 and 5.
Figure 10. Ionic adsorption isotherms. For positive charge the monovalent ions are the counterions whereas for negative charge the divalent ions are the counterions. The solid and broken curves give the HNCIMSA and GC results, respectively. There are no Monte Carlo results.
parent exception to this is the 0.5 M 1:2 results. Here the MC and HNC/MSA interfacial region is as wide as the GC prediction. However, there are some interesting features of the MC profiles which are also present in the HNC/ MSA results and which are not present in the GC results. At 0.5 M concentration in the 1:2 case, there is a layering of charge. The double layer (if it is restricted to the electrode and the adjacent region of opposite charge) is narrower than the GC estimate. However, the interfacial region is about the same thickness as in the GC theory. The charge in the initial layer exceeds the electrode charge. Thus, the initial layer is followed by a layer of opposite sign. We could speak of a “triple layer”. The “triple layer” manifests itself in the potential profile as a change in sign. This layering is not confined to the 1:2 case (or the 2:2 case where it is also seen). It is present in 1:l and 2 : l solutions but a t higher concentration^.^ A t high concen(8)L. Blum, J. L. Lebowitz, and D. Henderson, J . Chem. Phys., 72, 4249 (1980). (9)L. B. Bhuiyan, C. W. Outhwaite, and S. Levine, Chem. Phys. Lett., 66,321(1979);C.W.Outhwaite, L. B. Bhuiyan, and S. Levine, J. Chem. Soc., Faraday Trans. 2, 76,1388 (1980).
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trations (physically attainable only in fused salts) there would presumably be several layers of counterions and coions. In Figures 9 and 10 we have plotted the diffuse layer potential and ionic adsorption isotherms as functions of electrode charge density. The GC estimates of c$o are monotonic and too large in magnitude. The HNC/MSA values are in quite good agreement with the MC results and are nonmonotonic when the divalent ions are the counterions. The total potential drop across the double layer aE 2t
V=-+&
(5)
is monotonic in accord with the exact theorem of Blum et aL8 The coion adsorption isotherm also shows nonmonotonic behavior. This is especially apparent in Figure 10 when the counterions are divalent. There is some experimental evidencelo for this. The experimental results show even stronger deviations from the GC results and become negative although there is some doubt about this because of the experimental uncertainty. As is seen in Figure 5, the HNC/MSA coion profile underestimates the deviations from the GC theory. Torrie and Valleau do not report values for their ionic adsorption isotherms. However, it is clear from Figure 5 that the MC isotherm for the coions at negative electrode charge would show even larger deviations from the GC results and might even be negative. Differences in ionic radii can also give rise to deviations from the conventional GC adsorption isotherms.ll
Conclusions The HNC/MSA equations satisfactorily account for the interfacial properties of a simple model of 2:l and 1:2 electrolyte solutions near an electrolyte. The GC theory (10) R. Parsons and S. Trasatti, Tram. Faraday Soc., 65,3314 (1969). (11) J. P. Valleau and G. M. Torrie, J . Chem. Phys., 76, 4623 (1982).
Lozada-Cassou and Henderson
is not satisfactory, particularly when the counterions are divalent. At higher densities, especially when the counterions are divalent, there is a layering of charge present in the interfacial region. This gives rise to nonmonotonic behavior not only of the density and potential profiles but also of the dependence of the diffuse layer potential and the ionic adsorption isotherms upon the electrode charge. The diffuse layer potential is virtually inaccessible experimentally. However, there is experimental evidence for such behavior for the ionic adsorption isotherms. Of the alternative recent theories of the double layer only the modified Poisson-Boltzmann (MPB) theory has been applied to 2:l and 1:2 solutions.12 Most of the MPB results are similar to the HNC/MSA results reported here. Charge oscillation or layering is seen, as is nonmonotonic behavior in the diffuse layer potential. The MPB density profiles are somewhat in error in showing a peak near contact in the coion profile in the 1:2 case. This peak is not present in either the HNCIMSA or MC profiles. In earlier studies of the MPB approximation for the double layer, it has been convenient to assume that the radial distribution function gij vanishes for x L ,x, < 0. If the correct condition, gij = 0 for xi < a/2, xi < a/2, is used, this spurious peak in the MPB coion profile is eliminated.13 Outhwaite and Bhuiyan refer to this latest version as the MPB5 approximation. It is satisfying that the MC, MPB5, and HNC/MSA results are in close agreement.
Acknowledgment. The authors are grateful to Drs. G. Torrie and J. Valleau for sending their Monte Carlo results in advance of publication. They are also grateful to Drs. C. W. Outhwaite and L. B. Bhuiyan for sending a preprint of their MPB5 paper. This work was supported in part by NSF Grant CHE80-01969 to D.H. (12) L. B. Bhuiyan, C. W. Outhwaite, and S. Levine, Mol. Phys., 42, 1271 (1981). (13) C. W. Outhwaite, private communication; C. W. Outhwaite and L. B. Bhuiyan, preprint.
