J. Phys. Chem. 1983, 87, 4544-4547
4544
Anomalous Adsorption of Ions at an Electrode Douglas Henderson, Jorge Barojas,+ IBM Research Laboratory, Ssn Jose, California 95 193
and Lemer Blum Departmt of Physlcs, Universlfy of Puerto Rico, Rlo PMras, Puerto Rlco 0093 1 (Received:March 3, 1983; In Final Form: May 3 1, 1983)
Some general arguments based upon the behavior of correlation and thermodynamic functions near critical points leads us to expect that for some real electrolytes the interfacial excess ionic adsorption isotherms (even for co-ions) will have large maxima at temperatures and concentrations in the vicinity of a critical point of the electrolyte. Such predictions are beyond the scope of the Gouy-Chapman (GC) theory and most other theories of the double layer. The reason for the failure of these theories is pointed out. Some real systems which may exhibit this anomalous adsorption are given.
Introduction Specovius and Findeneggl have observed large maxima in the adsorption of ethylene on graphite near the critical point of ethylene. Henderson and Snook2have developed a simple theory for this anomalous adsorption using the surface Ornstein-Zernike (OZ) e q ~ a t i o n . ~ The Henderson-Snook theory of adsorption can be applied to ionic adsorption at an electrode. In this note we show that, if the electrolyte has a critical point, there will be large maxima in the excess ionic adsorption. Conventional theories of the double layer such as the GouyChapman (GC) theory: the mean spherical approximation (MSA)? and even the hypernetted chain approximation6 (HNC/MSA version) are all incapable of predicting this anomalous adsorption because of oversimplified descriptions of the bulk electrolyte. For example, in the GC theory, the excess adsorption isotherm is
ri =
Lm
[pi(x)
- pi] dx =
where pi(x) is the density profile of the ions of species i, pia= pi(-), x is the distance from the electrode, x = 0 is the distance of closest approach of the ions K = ((47rpe2/4 C ~ i p i ) l / ~ (2) I
0 = l / k T (T is the temperature and k is the Boltzmann constant), zie is the charge of an ion of species i, 6 is the dielectric constant of the solution, and b = PZeE/cK (3) In eq 3, E / 4 r is the chargelunit area of the electrode. In the MSA (4) ri = p { 3 ~ / [ 2 ( 1+ f b/K) where q = r p u 3 / 6 , p = x i p i , and u is the ionic diameter. For simplicity, we assume that all the ions have the same diameter. The HNC approximation does not yield an analytic result for ri. However, for monovalent ions, the numerical
'
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results obtained from the HNCIMSA are close to the results obtained from
Eguation 5 is virtually indistinguishablefrom eq 1 because, in most cases, 7 is small. None of the equations eq 1 , 4 , or 5 yields any anomalous behavior. For higher valence ions, the HNCIMSA excess adsorption isotherm for the co-ions has a small maximum but this is unrelated to the anomalous adsorption which we consider here. As far as we are aware, other theories of the double layer, such as the HNCIHNC, BGY, and MPB theories, have not been investigated in sufficient detail to know whether they would predict this anomalous adsorption. As we shall see, the question is not whether the theory predicts a critical point for the bulk electrode but whether the theory predicts a critical point when the "compressibility" route to thermodynamics is used. All of the theories mentioned above are based upon the primitive model of an electrolyte. That is, the ions are considered to be a system of charged hard spheres and the solvent is considered to be a uniform dielectric continuum, the dielectric constants of the interior of the spheres and of the continuum both being equal to t so that, outside the hard core, the ion-ion pair potential is equal to zizje2/u. We shall use this model throughout this note. Derivation of the Henderson-Snook Theory We consider the adsorption of a simple gas by a simple surface. By a simple gas and a simple surface, we mean a gas of spherical molecules which interact with each other (1)J. Specovius and G. H. Findenegg, Ber. Bunsenges. Phys. Chem., 84,690 (1980). (2)D.Henderson and I. K. Snook, J.Phys. Chem., 87,2956 (1983). (3)D.Henderson, F.F. Abraham, and J. A. Barker, Mol. Phys., 31, 1291 (1976). (4)G.Gouy, J.Phys., 9,457(1910);D.L. Chapman, Philos. Mag., 25, 475 (1913). (5) L. Blum, J. Phys. Chem., 81,136 (1977). (6)D.Henderson and L. Blum, J. Electroanal. Chem., 93,151(1978); 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);s. L. Carnie, D.Y. c. C h a , D. J. Mitchell, and B. w. Ninham, J. Chem. Phys., 74,1472 (1981);M. Lozada C., R. Saavedra B., and D. Henderson, ibid., 77,5150 (1982).
0 1983 American Chemical Society
Anomalous Adsorption of Ions at an Electrode
The Journal of Physical Chemistry, Vol. 87, No. 22, 1983 4545
and with a flat uniform surface by means of van der Waals forces. Henderson and Snook2used the surface OZ equation of Henderson et This equation is obtained by considering the surface to be one of the components of a mixture consisting of the gas molecules and a single giant molecule (the surface). This procedure permits the surface to be treated by the statistical mechanical techniques which have been developed for mixtures. In particular, the surface OZ equation is
h,(k) = E W ( k ) / U
- Pc'(k))
(6)
where Acc,(x) = c,(x) - c(,-m), since b(dp/dp) = 1 - pE8(0). The integral of hwd(X) is easily evaluated since charge neutrality requires that
Jmhwdx) dx = b/K
(17)
Therefore
where h,(x) and c,(x) are the surface total and direct correlation functions and c(r) is the direct correlation function of the bulk fluid. The tilde indicates that the Fourier transform has been taken. Now p(x)
= &,(XI
x
>0
(7)
and hence I' is that part of h,(O) which arises from the integration over x > 0. Using the fact that h,(x) = -1, x < 0, and /3(dp/dp) = 1 - pE(0) we have
+
where Ac,(x) = c,(x) p ( d p / d p ) = c,(x) - c,(--). To obtain a feeling for the properties of eq 8, we evaluate the numerator in a van der Waals type of approximation and assume that Ac,(x), x < 0, is the same as for hard spheres near a hard wall and c,(x) = -j3uw(z),x > 0, where &(x) is the interaction between the molecules and the wall. Thus
Again it is immediately obvious that near a critical point ri will become large for both the counterions and the co-ions. The question which naturally arises is why this anomaly is not predicted by the GC, MSA, or HNC/MSA theories. In the GC theory the ions are treated as points and, as a result, p = pkT (the perfect gas law). Hence, p ( d p / d p ) = 1. In the MSA f l ( d p / d p ) = (1 + 2 ~ ) ~ / ( 1T )- ~ .Since the HNC/MSA uses the MSA to describe the bulk fluid, the denominator in the HNC/MSA theory will again be (1 + 2d2/(1 - d4.Thus, the denominator is nonvanishing in each of these theories and there is no anomalous adsorption. However, it is well-known that the MSA for bulk electrolytes is not thermodynamically consistent. That is, different thermodynamic properties are obtained when different routes connecting correlation functions and thermodynamic properties are employed. If the "compressibility" route is chosen P ( d p / d p ) will be small and
B ( d p / d p ) = 1 - pE(0) = (1
The second term in the numerator of eq 9 is the Henry's law constant (to order 8). We see immediately that, in the vicinity of a critical point, I' will be large since p ( d p / d p ) is small. In fact, I' is singular a t the critical point. Henderson and Snook2 have found that eq 9 gives adsorption isotherms which are very similar to the experimental results of Specovius and Findenegg.'
Ionic Adsorption for a Double Layer This same treatment can be applied to the double layer. For simplicity, we condider a symmetric electrolyte, where the surface OZ equations are'
- PC',(kN
(10)
= Ewd(k)/{1 - pEd(k)1
(11)
h,(k) = E,(k)/{l hwd(k)
where
+ 277)2/(1 -
?7)4
(19)
and there is no critical point. However, if the "energy" route is chosen
where
r is related to
by K = 2 r ( 1 + ra) (21) We see that a critical point is possible if eq 20 is used. Equation 19 is the appropriate form to be used with eq 18 if a consistent solution of the MSA is sought. However, once the adsorption is written in the form of eq 18, we need not be consistent. We can use eq 20 or any other expression for @ ( d p / d p ) . In the MSA7 JOAc,.(x)
K
dx = 3 7 0
+ 21)/{2(1 - d41
(22)
h,,(x) = [h,,(x) + hwz(x)l/2
(12)
hwdb) = [hwl(x) - hw2(x)1/2
(13)
and c,(x) = 0, x > 0, because of the linearization in the ion-wall coupling parameter, b. We are of the opinion that the GC result
+4 r ) J P
(14)
&mc,.(z) dx = ( 2 / ~ ) [ ( 1+ b2/4)'/' - 11
cd(r) = [c11(r) - c12(r)1/2
(15)
etc. c,(r) =
[CII(~)
and the indexes 1 and 2 indicate which species of ion is considered. Proceeding as before (7) D. Henderson and L. Blum, J . Chem. Phys., 69, 5441 (1978).
(23)
is fairly accurate for monovalent salts. It is less accurate for higher valence salts but that does not affect the character of our results. A more serious error in the GC theory is the approximation Ac,(x) = 0, x < 0, since, if this were true, there would be no anomalous adsorption at the point of zero charge (i.e., when b = 0). All of this suggests that a fairly useful adsorption isotherm could be
4546
Henderson et ai.
The Journal of Physical Chemistry, Vol. 87, No. 22, 1983
-
N
-{
0--
a
0.
-0.2
-
-
20 7 I
1 .o
I
2.0
I
I
I
l
5.0
l
1
10.0
TIT,
Flgure 1. Excess charge adsorbed by an electrode, calculated from eq 24, as a function of the tem erature at the crltlcal concentratlon (conc = 0.0157 for u = 4.25 The curves are labeled with the appropriate values of the electrode charge density (in C/m2) and T , Is the critical temperature. The upper and lower curves glve the counterlon and co-ion excess adsorption, respectively. The graph in the lower right comer gives excess charge adsorbed by an uncharged electrode where the counterions and co-Ions are adsorbed in equal numbers.
1).
obtained by substituting eq 20, 22, and 23 into eq 18 to obtain
I
I
0
1
I
0.2
0.1
conc (M/L?) Flgure 2. Excess charge adsorbed by an electrode, calculated from eq 24, as a function of the concentratlon for a constant electrode charge of 0.1 C/m2. The curves are labeled with the appropriate values of TIT,. The positive and negative curves give the counterion and
(24)
Equation 24 should at least be qualitatively correct and predicts anomalous ionic adsorption.
Results We have made calculations of the excess charge adsorption using eq 24. We adopt the values u = 4.25 A and c = 78.5 which have been used in computer simulations.s The critical point occurs (in the primitive model) when ap
p-
ap
=
(1 + 2.11)~ r3(1 (1 - 0)4 2 4 1
+ ru) = 0 + 2ru)
(25)
a2P
P B Z =
aP
41(i
+ 271~2+ 7) - r3(1 + ru)(i+ 4ra2 + 21722) = o (1- d 4 + 2ru)3
(26) where we are using the MSA energy equation thermodynamics to describe the electrolyte. We find that the critical point occurs when pa3 = 0.0145 or concentration = 0.157 M and pz2e2/tu= 12.73. For a 1:l salt and E = 78.5 this corresponds to a temperature of 39.4 K. For higher valence salts or solvents with a lower dielectric constant, the critical temperature will be much higher. In Figures 1-3, charge adsorption isotherms, resulting from eq 24, are plotted for states in the neighborhood of the critical point. The counterion isotherms have positive values whereas near the critical point the co-ion isotherms have negative values since the co-ion is of the opposite sign. (8) D. N. Card and J. P. Valleau, J. Chem. Phys., 52,6232 (1972); J. P. Valleau and L. K. Cohen, ibid., 72, 5935 (1980); J. P. Valleau, L. K. Cohen, and D. N. Card, ibid., 72, 5942 (1980).
I -
O
0.1
0.2
0.3
0.0
0.1
0.2
0.3
Chg den (c/m2)
Flgure 3. Excess charge adsorbed by an electrode as a function of the surface charge density for two concentrations, 0.05 and 0.1 M. The solid curves are calculated from eq 24 and are labeled with the approprlate values of TIT,. The broken curves glve the GouyChapman results for T = T,. The positive and negative curves give
the counterion and co-ion excess adsorption, respectively.
Near the critical point both the counterions and the co-ions are adsorbed in large numbers. The difference between the magnitudes of the adsorbed counterion and co-ion charges is equal to the magnitude of the charge on the electrode so that charge neutrality is preserved. The fact that the anomalous adsorption does not affect this dif-
J. phys. Chem. 1083, 87, 4547-4548
ference means that such properties as the potential and capacitance which depend upon the difference in the counterion and co-ion profiles are not affected (or relatively little affected) by the anomalous adsorption. In contrast, the GC co-ion isotherm is always positive. Eventually, at high enough temperatures the co-ion charge adsorption isotherms predicted from eq 24 also become positive since at these high temperatures Odp/dp N 1 and eq 24 is only slightly different from eq 1. However, this change in sign of the co-ion adsorption isotherm does not occur until surprisingly high temperatures ( T 3Tc to 6Tc for electrode charge densities of 0.1-0.3 C/m2, respectively). N
Conclusions On quite general grounds, we have seen that anomalous highly ionic adsorption of both co-ions and counterions can be expected near a critical point of an electrolye. This anomalous adsorption will occur even at the point of zero charge (i.e., when b = 0). All of these predictions are based upon the primitive model of an electrolyte. It is uncertain whether these predictions are applicable to real electrolytes with molecular solvents. However, Friedmans has suggested that the primitive model critical point may correspond to a lower consolute point in some real solutions. (9)H.Friedman, Annu. Rev. Phys. Chem., 32, 179 (1981).
4547
Even accepting that the primitive model critical point corresponds to a critical point in some real solutions, it seems unlikely that the real critical point would be characterized by the vanishing of the first and second derivatives of the osmotic coefficient with respect to concentrations. Even so, we would argue that the theory outlined in this note is still applicable because the real (for our argument) characterization of the critical point is the vanishing of 1 - pE,(O) which results from the extremely long-range character of the correlation functions near the critical point. The substitution P ( d p / d p ) = 1 - pE,(O) is a feature of the primitive model. Thus, we expect that the anomalous ionic adsorption predicted by eq 20 can be observed in real solutions. An experimental investigation of the ionic adsorption in such solutionss as KI in SO2,NH4GaC14or NH4FeC14in diethyl ether, and Ga2C14in benzene may yield interesting and heretofore unexpected results. We expect the above systems to yield large and striking deviations from the GC charge adsorption isotherms if experiments are performed near the critical point of these solutions. However, since the approach of eq 24 to the GC result as the temperature is increased is gradual, small but measurable deviations from the GC predictions may be seen in states far from the critical point. Acknowledgment. This work was supported in part by NSF Grant No. CHE 80-01969.
Comparison of Monte Carlo and HNCIMSA Excess Charge Adsorption Isotherms for an Electrlcal Double Layer Contalnlng Divalent Ions Jorge Barojas,t Douglas Henderson, IBM Research Laboratory, Sen Jose, Callfornh 95193
and Marcel0 Lorada-Cassou Depattamento de Flsca, UnlversMed Aut6noma MetropolItanaNztapalapa,Apartado Postal 55-534, 09340 M6xlco DF (Recelvd: March 28, 1983; I n Final Form: June 10. 1983)
An excess charge adsorption isotherm for a 2:l electrolyte is calculated from the recent Monte Carlo data of Torrie and Vdeau. The errors in the Gouy-Chapman isotherms are even larger than indicated by the HNC/MSA calculations.
Lozada-Cassou and Henderson' have applied the HNC/MSA integral equations to the electrical double layer formed by a 2:l salt near an electrode and have found generally good agreement with the Monte Carlo (MC) simulations of Torrie and Valleau.2 In both studies, the electrolyte is modeled as a system of charged hard spheres in a uniform dielectric medium. Among the properties calculated by Lozada-Cassou and Henderson' were the excess charge adsorption isotherms:
where zie and pi(x) are, respectively, the charge and density profile of an ion of species i, x is the distance from the
electrode, pi = p i ( - ) , and u is the hard-sphere diameter. Henderson and Lozada-Cassou made no comparison with the MC simulations since values of the excess charge adsorption isotherms were not included in the paper of Tome and Valleau.2 However, Lozada-Cassou and Henderson did observe that when the divalent ions were the counterions, the MC co-ion density profile was even larger than the HNC/MSA profile. They further speculated that, because the co-ion profile exceeds the bulk co-ion density over an extended region, the co-ions might actually be adsorbed. In order to examine this speculation, we have calculated excess charge adsorption isotherms from the MC data.2 The MC, HNC/MSA, and Gouy-Chapman3 (GC) ad-
Permanent address: Departamento de Fisica, Universidad Autijnoma in Metropolitana/Iztapalapa, Apartado Postal 55-534, MBxico DF.
(1) M. Lozada-Cassou and D. Henderson, J. Phys. Chem., 87,2821 (1983). (2)G.M.Torrie and J. P. Valleau, J.Phys. Chem., 86,3251 (1982).
QQ22-3654I83/2Q87-4547$01.5010
0 1983 American Chemical Society
