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Generalization of the UDCA theory and its application to the analysis of the ion-free layer thickness
49
J. Electroanal. Chem., 358 (1993) 49-62
Elsevier Sequoia !%A., Lausanne
JEC 02743
Generalization of the UDCA theory and its application to the analysis of the ion-free layer thickness R. Andreu, M. Molero, J.J. Calvente and J. Carbajo Departamento de Q&mica Fisica, Universidad de Sevilla, 41071-Sevilla (Spain)
(Received 20 November 1992)
Abstract
The modified Gouy-Chapman theory with two planes of closest approach is generalized for any electrolyte stoichiometry, considering also the influence of the interfacial dielectric constant. Application is made to some experimental systems to derive the cation and anion distances of closest approach from the ion-free layer thickness.
INTRODUCTION
Experimental evidence for the distance of closest approach to the electrode of an ion can be derived from the ion-free layer thickness [l-31. To avoid specific adsorption of the anions, the evidence is usually obtained at far negative charge densities on the electrode. As a consequence, the ion-free layer thickness is associated with the properties of the cation, and has recently been correlated with the hydrated radius of the cation for a series of chlorides [4]. A more puzzling situation is found when the ion-free layer thicknesses of electrolytes with a common cation are compared, since the nature of the strongly repelled anion also seems to play a significant role. The simplest theoretical tool that allows us to explore the simultaneous influence of cation and anion sizes on the double-layer properties was introduced by Valleau and Torrie in 1982 [5]. They developed a modified version of the Gouy-Chapman theory for a symmetrical electrolyte with unequal distances of closest approach to the electrode for the cation and the anion. Later, other workers extended its application to other simple stoichiometries 161,linearizing it [7,8] and including specific adsorption of ions [9]. 0022-0728/93/$06.00
0 1993 - Elsevier Sequoia S.A. All rights reserved
50
Determination of the ion-free layer thickness implies an analysis of the differences between Gouy-Chapman and experimental surface excesses. When the existence of two planes of closest approach is recognized, the possibility arises that the anion size influences the cation surface excess, even in the presence of a strong negative field, by fixing the thickness of the space region into which only one ionic species can penetrate. Consequently, the classical interpretation of the ion-free layer thickness must be revised. In this paper we shall generalize the Valleau-Torrie theory for any electrolyte stoichiometry and show how, in some situations, the distances of closest approach of cation and anion can be derived from the ion-free layer thickness. THEORY
The modified Gouy-Chapman
theory with unequal distances of closest approach
07~~2~) We shall consider an ideally polarizable electrode in contact with an electrolyte solution. The electrode is modelled as a flat hard wall, and the primitive model of an electrolyte is adopted for the solution side of the interphase. The solvent is characterized by its dielectric constant and its distance of closest approach to the electrode a,. Ions p (p stands for any ionic species) are defined through charge zp and distance of closest approach to the electrode a,. It is interesting to point out here that aP is not exactly equal to the radius rp, of the ion although it may have a similar value. While ap accounts for the ion-wall direct interactions, rP accounts for the ion-ion direct interactions, and therefore a value of a,, smaller than rP may explain partial desolvation of the ion in contact with the electrode. When the model is solved at the mean field level, the ion-ion direct correlation is lost and rP does not appear in the formulation of the theory. The local concentration c,(x) of an ion p at a distance x from the electrode is assumed to obey Boltzmann’s law in the form c&>
= cj exp[ -~z,eO#@)]
(1)
where cz stands for the ion concentration in the bulk, .zp is its charge number, e, is the electronic charge, p = (kT)-’ and 4(x) is the mean electrostatic potential at a distance x from the electrode. The mean potential is obtained by solving the Poisson-Boltzmann equation
d*+ -= dx*
49re,
- E
Czpcp” exp[ -z,&&,( P
-x)]
(2)
with additional restrictions imposed by the space regions inaccessible to each ion. Integration of eqn. (2) is now carried out for each of the three regions shown in Fig. 1, which are characterized by the ionic species that can enter them and their dielectric constant E.
Bulk solution
x=0,
a8 ,
a, ,
a.
J
,
00
Fig. 1. Schematic diagram of the interphase structure in the UDCA theory.
(I) 0 IX I a,. No ions are present in this region, so that eqn. (2) is simply d24 -= dx2
o (3)
which upon integration gives 4(x)
=4C”i>
+
gCxmui)
(4)
The electric field is constant throughout density uM on the metal by d+ -=--U dx
4~
this region and it is related to the charge
M (5)
EI
are obtained by solving eqn. (2) in the outer regions, 4(ai) and (d4/dx),=,, assuming the continuity of potential and electric displacement at ui, i.e. 4(“i)
l
t( 3-.;
=4(‘+)
=+(“i)
(6)
= en( 3_.,
(7)
(II) a, IX I uj. Only i ions populate this region and eqn. (2) takes the form d24 -=-dx2
45~~ EII
,icP
exp[
-zi&dx)l
which can be integrated once to give
(8)
52
When eqn. (9) is integrated again a general solution can be obtained potential profile in this region: 4(x)
=
2
for the
(10)
ln
I
where
and
(12) Equation (11) still applies when A, < 0; in this case it can conveniently be rearranged for computation using the relationships tanh-’ (iu) = i.arctan(u) and tanh(iu) = i.tan(u), where i = (-- 1)‘12. At the border between regions II and III the equivalent of continuity equations (6) and (7) are
(14) (III) aj sx s 01. Here the full Poisson-Boltzmann integrated with the boundary conditions
qzqrn)=o=(g)
expression (eqn. (2)) is first
(15)
x=m
to give
When eqn. (16) is integrated again, we obtain
(17) The above integral is solved numerically and 4(x> is obtained by an iterative procedure. The ionic distribution at any distance from the electrode x is estimated using eqn. (1). The computational sequence described below was followed to obtain the
53
potential profile. An arbitrary value for +(aj) is first selected, and the potential in region III is obtained by solving eqn. (17); (d4/dx>,_,+ is then calculated using eqn. (16) at x = aj, and (d4/dx),=,i is obtained from eqn. (14). Next, the solution in region II, in particular at x = a+, is found using eqns. (lo)-(12). Then, eqns. (6), (7) and (4) allow us to complete the calculation of the potential profile in region I. Finally, the charge density on the electrode is obtained from eqn. (5). When the results are required at a given u”, an iterative procedure is needed. The ion-free layer thickness
The Gibbs surface excess of an ion p is given by
(18) where I, and I, are the absolute surface excesses of p and solvent respectively. Computation of I” is performed by integrating the concentration excess function c,(x) - ci from the plane of closest approach of the solvent to the bulk of the solution:
r; = -
jx=“c; dx .X=lls
+ /x=m[ cP( x) - $1 dx .X=lZp
which can be rearranged
(19)
in the form
[iW[cp(x) -cp”] dx-r;=(a,-a,)c,O a
Specific adsorption is assumed to be absent and the integral is usually evaluated by means of the classic Gouy-Chapman equations and a plot of (I,“” - I,“) vs. cj, at a given a”, gives aP - a,, which represents an ion-free layer thickness. Our purpose is tocheck the influence that the presence of two distinct planes of closest approach has on the evaluation of the ion-free layer thickness. To this end we have computed the surface excess relative to the solvent on the basis of the UDCA theory r,UDC* =
/“=,(
cp( x)
-
,-;)uDcAdx
(21)
.X=tl,
We compared our results with those obtained from the Gouy-Chapman theory (IPGc). This comparison is easier to interpret if we define an apparent ion-free layer thickness hap by rGC hap =
p
_
rUDCA
P 0 CP
(22)
54
It follows from eqns. (201422) that hap = up - a, when the cation and anion have a common distance of closest approach. RESULTS AND DISCUSSION
Influence of ion-size asymmetry
Figure 2 shows a typical example illustrating how the differences between the Gouy-Chapman and UDCA surface excesses increase with electrolyte concentration at a rate that depends on the electrode charge density when the cation and anion have different planes of closest approach. The results are displayed as plots of the more familiar cation charge contribution Au+= z+F(I’~~- I’yDCA) vs. z+c”,. In this section, the dielectric constant throughout the solution side of the interphase will be kept at its bulk value (ei = lu = enI = 78.5). When the cation is the smaller of the two ions, plots of Au+ vs. z+ct are strictly linear for uM 2 0 and develop a slight curvature for uM I 0. If the cation is larger than the anion, the sign of g M should be reversed in the preceding statement. Such a curvature would be barely detectable in an experimental plot, so that it appears justified to fit the Au+ vs. z+ct relationships by straight lines with
i--
-2 +30 +10
0 -10 -30
10
5
0 0
1
2
3
4
5
6
z+c+O/mol dm” Fig. 2. Charge density due to differences between Gouy-Chapman and UDCA theories as a function of cation concentration. Charge densities on the electrode in PC cm-* are indicated in the figure. 1: 1 electrolyte; T=298K, a+-a,=2A, a_-a,=4A.
55 I’
011 -40
’
’
’
-30
’
’
-20
’
’
-10
’
’
0
’
’
10
’
’
’
’
20304050
I
’
’
”
Fig. 3. Ion-free layer thickness as a function of the charge density on the electrode for a given a + - (I, = 2 A and several a _ - a, values indicated in the figure. 1: 1 electrolyte; T = 298 K.
zero intercept and having slope FAaP. Unequal distances of closest approach lead to a sigmoidal dependence of hap on o”, as indicated in Fig. 3. Under the influence of a strongly repulsive field ( I uM 1 > + 20 /.LCcm-* in Fig. 3) the smaller ion is swept out from region II in the interphase and the ion-free layer extends to the plane of closest approach of the larger ion (hap = a _- a, in Fig. 3). However, when the smaller ion is attracted towards the electrode, hap remains larger than expected for an ion-free layer extending to the plane of closest approach of the smaller ion. In this situation the hap value reflects the physically different pictures behind the UDCA and Gouy-Chapman theories, and is a function of the distance between the two planes of closest approach. It is interesting to note that under attractive conditions for the smaller ion and at sufficiently high concentrations, a slight superequivalent adsorption is predicted in region II which would lead to a dramatic reversal of the composition in region III. InjIuence of electrolyte stoichiometry Differences between the Gouy-Chapman and UDCA surface excesses become larger when the charge number of the counter-ion (the smaller ion) is increased or the charge number of the co-ion (the larger ion) is decreased, as illustrated in Fig. 4. Asymmetry in the charge number appears to be more critical than its actual value, as shown by the very similar results obtained for 1: 1 and 2 : 2 salts. Previous considerations regarding the linearity of Au+ vs. z+c”+ for 1: 1 electrolytes are equally applicable to the 1: 2, 2 : 1, 2 : 2, 1: 3 and 3 : 1 stoichiometries.
z_ : z_
I
I
0
I
1
2
3
4 z+c+’ /
I
,L
5
t1
mol dmJ
Fig. 4. Charge density due to the differences between the Gouy-Chapman and UDCA theories as a function of z+ c”,. Electrolyte stoichiometries are indicated in the figure. T = 298 K, gM = -20 PC a_-a,=4L cm-*, n+-a,=2A;
As before, a good estimate of the distance of closest approach of the larger ion can be obtained when the smaller ion is repelled from the electrode (aM z+ 0 for a + < a_ in Fig. 5). When the smaller ion is attracted towards the electrode, hai’ decreases at a different rate for each stoichiometry. Optimum conditions for the estimate of the distance of closest approach of the smaller ion are achieved when it bears a single charge and the larger ion is multicharged. Influence of dielectric constant
From a modellistic point of view we have divided the solution side of the interphase into three regions (I, II and III in Fig. 1). We shall now consider how a decrease in the dielectric constant of regions I and II affects hap. Combination of eqns. (5) and (7) shows that or has no influence on the UDCA surface excess derived at a given u”. Therefore we shall put lI = eu in the calculation below. It can be seen in Fig. 6 that differences between the Gouy-Chapman and UDCA surface excesses are reduced when eII is lowered. The decrease in Au+ is accompanied by an enhancement of the curvature in the Aa+ vs. z+ct plots when the smaller ion is attracted towards the electrode (Fig. 7). For lII 2 30, the curvature is relatively small and would probably not be detected in an experimental plot. However, if the dielectric saturation becomes more extensive, not only the curvature but also the appearance of negative Au+ values at low concentrations would be apparent in the analysis of a real system.
Fig. 5. Ion-free layer thickness as a function of the charge density on tte electrode for several electrolyte stoichiometries as indicated in the figure. T = 298 K; a + - a, = 2 A, a _ - a, = 4 A.
-5 0
76.6 33 6
. . I
1
0
1
2
3
1
I
4
5
z,c+*
/
mol dmd
Fig. 6. Charge density due to the differences between Gouy-Chapman and UDCA theories as a function of z+c: for several values of the dielectric constant between the two planes of closest approach. 1: 1 electrolyte; T = 298 K, oM = - 20 PC cm-*; a + - u, = 2 A; a _ - a, = 4 A.
58
0
I
I
-2
76zo3 ‘3
+30
-
15-
10 -*
S-
O-
L
1 0
I 1
I 2
3
4 z+c,’
5 /
mol dmJ
Fig. 7. Charge density due to the differences between Gouy-Chapman and UDCA theories as a function of z+ c”, for several values of tte charge density o~/~C crnmz on the electrode. 1:l electrolyte; T = 298 K, lII = 6; a + - a, = 2 A, a _ - a, = 4 A.
To facilitate comparison with our previous results we have continued to fit the Aa+ vs. z+c”, plots to straight lines passing through the origin, regardless of their possible curvature. For curved plots, the slope gives an average value Pp of the apparent ion-free layer thickness. As shown in Fig. 8, lower values of en lead to a steeper change in pp as a function of u M. However, this feature does not appear to be of practical value because the maximum variation (from cu = 6 to ln = 78.5) of pp in the experimentally accessible polarization range is rather small ( I 0.5 A). As a brief summary of our previous considerations, we can state that, within the framework of the UDCA theory, hap is determined mainly by the distances of closest approach of both anion and cation, and to a lesser extent by the electrolyte stoichiometry. Evidence for a lower dielectric constant in the vicinity of the electrode can be obtained from the curvature and negative ordinate intercept of the Au+ vs. z+ct plots. Comparison with experiment
Real electrolytes consist of ions of unequal size and therefore unequal distances of closest approach. In this section we shall fit some sets of experimental cation surface excesses r:, obtained at uM < 0, to the UDCA theory. Instead of proceeding to a direct comparison between I’: and I’yDCA,we shall compare their differences with respect to the value I’g’” calculated using Gouy-Chapman theory,
59
l.S-
6
-50
0
/
-40
I
I
I
-30
-20
-10
I1 0
10
I
I
I
20
30
40
cr"/pc cm
-¶
Fig. 8. Mean ion-free layer thickness as a function of the charge density on the electrode for several values of theodielectric const!nt between the two planes of closest approach. 1 : 1 electrolyte; T = 298 K; a+ -a,=2A;a_-a,=4A.
as measured by the ion-free layer thickness hap. In this way the physical picture behind the fit is clarified. Only a few determinations of the ion-free layer thickness appear in the literature [l-4,10-12], and all were carried out at negative charge densities on the electrode to avoid anion-specific adsorption. Table 1 shows some examples that will illustrate the application of the UDCA theory to the estimate of the cation and anion distances of closest approach. In all cases the solvent was water. A quick glance at Table 1 reveals that AaPdepends not only on the cation but also on the anion; the most obvious case is that of MgSO, and MgCl,.
TABLE 1 Experimental ion-free layer thickness AaP and UDCA distances of closest approach for the cation - a,) and the anion (a_ -a,)
(a,
NaH,PO,
NaClO,
Na,Mal a
/VP/‘&
2.4-4.6 b
2.5 c
2.2 d
1.7 e
4.4 =
(a+-
1.7
1.7
1.7
1.7
2.5
2.5
4.5-5.5
4.5
4.5
6.0
I 2.5
(a-
a b ’ d e
a,)/A - aJ/A
Sodium malonate. From ref. 3. From ref. 11. From ref. 13. From ref. 4.
NaCl
Il.7
MgSO,
MgC1, 2.5 e
60
A perusal of the original Au+ vs. .z+c~ plots shows that the evidence for an extensive dielectric saturation is either absent or lies within experimental error. For our calculations we have chosen or = ln = 33, which appears to be a realistic value on the basis of the dielectric constant profile proposed by Levine and Fawcett [14]. This value does not introduce a relevant curvature in the A.o+ vs. .z+ct plots and gives distances of closest approach that differ from those obtained with a higher dielectric constant by no more than 0.5 A. To fit the hap values of NaClO,, NaH,PO,, sodium malonate (Na,Mal) and MgSO, to the UDCA theory, we start by determining (a+- a,) independently from the hap values of NaCl and MgCl,. If it is assumed that chloride can approach the electrode sufficiently close to satisfy u,,~ I a,, then hF1-= a+- a, (see Fig. 2). Once a+- a, is fixed in this way, we look for the u _- a, value that delivers the experimentally observed hap at a given u”. The perchlorate distance of closest approach thus obtained is in good agreement with the sum of its Pauling radius plus that of the water molecule. A similar procedure based on molecular models can be carried out with respect to malonate, assuming that the carboxylic groups are oriented towards the solution. Therefore the distances of closest approach for these two anions are coherent from a microscopic point of view. However, the sulphate distance of closest approach appears to be somewhat overestimated, although this may reflect the fact that sulphate interacts more strongly than perchlorate with the solvent and retains its primary solvation shell on approaching the electrode.
f
I
’
1
Iq
‘.
!5-
4-
3-
2-
l-
l O-
-1
0 -
-20
-15
-10
-5
5
0 d
10
/&+c cm-s
Fig. 9. Experimental ion-free layer thickness obtained from NaH,PO, solutions [3] as a function of the charge density on the electrode (0). The solid curves show calculated values for 1: 1 electrolyte at T=298K, a+--a,=L7Aand En = 33; a_ - a, is indicated in the figure.
61
Sodium dihydrogen phosphate is a most interesting electrolyte for examination of the charge dependence of hap owing to the low tendency of the anion to be specifically adsorbed. Parsons and Nobel [3] reported hap values in the range -20 ~LC cm-’ < uM < 6 PC cm-’ (Fig. 9). These authors emphasized the similarities between the Aspdependence on aM and that of the surface excess volumes derived by Hills and Payne [15]. They also pointed out that the abrupt decrease of AaPat positive charge densities probably originated with the onset of anion-specific adsorption. According to UDCA theory, an increase in hap with gM is to be expected when the anion is larger than the cation, and it is also expected that hap will decrease as the amount of specific adsorption increases. For uM I - 8 PC cme2, the hap values of Parsons and Zobel [3] can be fitted with anion distances of closest approach in the range 4.5-5.5 A (Fig. 9). Although these values are reasonable, it should be noted that Asp increases with uM faster than predicted by the theory. A tentative explanation for this discrepancy would be an increase in the distances of closest approach when the electrical field in the interphase decreases. CONCLUSIONS
Application of the UDCA theory shows that the ion-free layer thickness may contain information on the distances of closest approach of both anion and cation. As surface excesses are rather insensitive to the shortcomings of the Gouy-Chapman theory, it is expected that a similar conclusion can be reached from any electrostatic theory that accounts for differences in ion size. ACKNOWLEDGEMENTS
The authors wish to express their gratitude to the Spanish Comision Asesora de Investigation Cientifica y Tecnologica (CAICYT) for supporting this work (grant PS89-01161. REFERENCES 1 2 3 4 5 6 7 8 9 10
D.C. Grahame and R. Parsons, J. Am. Chem. Sot., 83 (1961) 1291. A.N. Frumkin, R.V. Ivanova and B.B. Damaskin, Dokl. Akad. Nauk SSSR, 157 (1964) 1202. R. Parsons and F.G.R. Zobel, J. Eiectroanal. Chem., 9 (1965) 333. M. Molero and R. Andreu, J. Electroanal. Chem., 322 (1992) 133. J.P. Valleau and G.M. Torrie, J. Chem. Phys., 76 (1982) 4623. L.B. Bhuiyan, L. Blum and D. Henderson, J. Chem. Phys., 78 (1983) 442. J.J. Spitzer, J. Colloid Interface Sci., 92 (1983) 198. M.J. Grimson, Chem. Phys. Lett., 95 (1983) 426. P. Nielaba, T. Knowles and F. Forstmann, J. Electroanal. Chem., 183 (1985) 329. B.B. Damaskin, A.N. Frumkin, V.F. Ivanova, N.I. Melenkova and V.F. Khonina, Elektrokhimiya, 4 (1968) 1336. 11 J.A. Harrison, J.E.B. Randles and D.J. Schiffrin, J. Electroanal. Chem., 25 (1970) 197.
62 12 B.S. Segel’man, V.F. Ivanova and B.B. Damaskin, Elektrokhimiya, 12 (1976) 451. 13 M. Pirez, M. Molero, M. Barrera and R. Andreu, 43rd Meeting of the I.S.E. C&doba, Argentina, September 1992, Abstract 6-009. 14 S. Levine and W.R. Fawcett, J. Electroanal. Chem., 99 (1979) 265. 15 G.J. Hills and R. Payne, Trans. Faraday Sot., 61 (1965) 326.
J. Electroanal. Chem., 358 (1993) 49-62
Elsevier Sequoia !%A., Lausanne
JEC 02743
Generalization of the UDCA theory and its application to the analysis of the ion-free layer thickness R. Andreu, M. Molero, J.J. Calvente and J. Carbajo Departamento de Q&mica Fisica, Universidad de Sevilla, 41071-Sevilla (Spain)
(Received 20 November 1992)
Abstract
The modified Gouy-Chapman theory with two planes of closest approach is generalized for any electrolyte stoichiometry, considering also the influence of the interfacial dielectric constant. Application is made to some experimental systems to derive the cation and anion distances of closest approach from the ion-free layer thickness.
INTRODUCTION
Experimental evidence for the distance of closest approach to the electrode of an ion can be derived from the ion-free layer thickness [l-31. To avoid specific adsorption of the anions, the evidence is usually obtained at far negative charge densities on the electrode. As a consequence, the ion-free layer thickness is associated with the properties of the cation, and has recently been correlated with the hydrated radius of the cation for a series of chlorides [4]. A more puzzling situation is found when the ion-free layer thicknesses of electrolytes with a common cation are compared, since the nature of the strongly repelled anion also seems to play a significant role. The simplest theoretical tool that allows us to explore the simultaneous influence of cation and anion sizes on the double-layer properties was introduced by Valleau and Torrie in 1982 [5]. They developed a modified version of the Gouy-Chapman theory for a symmetrical electrolyte with unequal distances of closest approach to the electrode for the cation and the anion. Later, other workers extended its application to other simple stoichiometries 161,linearizing it [7,8] and including specific adsorption of ions [9]. 0022-0728/93/$06.00
0 1993 - Elsevier Sequoia S.A. All rights reserved
50
Determination of the ion-free layer thickness implies an analysis of the differences between Gouy-Chapman and experimental surface excesses. When the existence of two planes of closest approach is recognized, the possibility arises that the anion size influences the cation surface excess, even in the presence of a strong negative field, by fixing the thickness of the space region into which only one ionic species can penetrate. Consequently, the classical interpretation of the ion-free layer thickness must be revised. In this paper we shall generalize the Valleau-Torrie theory for any electrolyte stoichiometry and show how, in some situations, the distances of closest approach of cation and anion can be derived from the ion-free layer thickness. THEORY
The modified Gouy-Chapman
theory with unequal distances of closest approach
07~~2~) We shall consider an ideally polarizable electrode in contact with an electrolyte solution. The electrode is modelled as a flat hard wall, and the primitive model of an electrolyte is adopted for the solution side of the interphase. The solvent is characterized by its dielectric constant and its distance of closest approach to the electrode a,. Ions p (p stands for any ionic species) are defined through charge zp and distance of closest approach to the electrode a,. It is interesting to point out here that aP is not exactly equal to the radius rp, of the ion although it may have a similar value. While ap accounts for the ion-wall direct interactions, rP accounts for the ion-ion direct interactions, and therefore a value of a,, smaller than rP may explain partial desolvation of the ion in contact with the electrode. When the model is solved at the mean field level, the ion-ion direct correlation is lost and rP does not appear in the formulation of the theory. The local concentration c,(x) of an ion p at a distance x from the electrode is assumed to obey Boltzmann’s law in the form c&>
= cj exp[ -~z,eO#@)]
(1)
where cz stands for the ion concentration in the bulk, .zp is its charge number, e, is the electronic charge, p = (kT)-’ and 4(x) is the mean electrostatic potential at a distance x from the electrode. The mean potential is obtained by solving the Poisson-Boltzmann equation
d*+ -= dx*
49re,
- E
Czpcp” exp[ -z,&&,( P
-x)]
(2)
with additional restrictions imposed by the space regions inaccessible to each ion. Integration of eqn. (2) is now carried out for each of the three regions shown in Fig. 1, which are characterized by the ionic species that can enter them and their dielectric constant E.
Bulk solution
x=0,
a8 ,
a, ,
a.
J
,
00
Fig. 1. Schematic diagram of the interphase structure in the UDCA theory.
(I) 0 IX I a,. No ions are present in this region, so that eqn. (2) is simply d24 -= dx2
o (3)
which upon integration gives 4(x)
=4C”i>
+
gCxmui)
(4)
The electric field is constant throughout density uM on the metal by d+ -=--U dx
4~
this region and it is related to the charge
M (5)
EI
are obtained by solving eqn. (2) in the outer regions, 4(ai) and (d4/dx),=,, assuming the continuity of potential and electric displacement at ui, i.e. 4(“i)
l
t( 3-.;
=4(‘+)
=+(“i)
(6)
= en( 3_.,
(7)
(II) a, IX I uj. Only i ions populate this region and eqn. (2) takes the form d24 -=-dx2
45~~ EII
,icP
exp[
-zi&dx)l
which can be integrated once to give
(8)
52
When eqn. (9) is integrated again a general solution can be obtained potential profile in this region: 4(x)
=
2
for the
(10)
ln
I
where
and
(12) Equation (11) still applies when A, < 0; in this case it can conveniently be rearranged for computation using the relationships tanh-’ (iu) = i.arctan(u) and tanh(iu) = i.tan(u), where i = (-- 1)‘12. At the border between regions II and III the equivalent of continuity equations (6) and (7) are
(14) (III) aj sx s 01. Here the full Poisson-Boltzmann integrated with the boundary conditions
qzqrn)=o=(g)
expression (eqn. (2)) is first
(15)
x=m
to give
When eqn. (16) is integrated again, we obtain
(17) The above integral is solved numerically and 4(x> is obtained by an iterative procedure. The ionic distribution at any distance from the electrode x is estimated using eqn. (1). The computational sequence described below was followed to obtain the
53
potential profile. An arbitrary value for +(aj) is first selected, and the potential in region III is obtained by solving eqn. (17); (d4/dx>,_,+ is then calculated using eqn. (16) at x = aj, and (d4/dx),=,i is obtained from eqn. (14). Next, the solution in region II, in particular at x = a+, is found using eqns. (lo)-(12). Then, eqns. (6), (7) and (4) allow us to complete the calculation of the potential profile in region I. Finally, the charge density on the electrode is obtained from eqn. (5). When the results are required at a given u”, an iterative procedure is needed. The ion-free layer thickness
The Gibbs surface excess of an ion p is given by
(18) where I, and I, are the absolute surface excesses of p and solvent respectively. Computation of I” is performed by integrating the concentration excess function c,(x) - ci from the plane of closest approach of the solvent to the bulk of the solution:
r; = -
jx=“c; dx .X=lls
+ /x=m[ cP( x) - $1 dx .X=lZp
which can be rearranged
(19)
in the form
[iW[cp(x) -cp”] dx-r;=(a,-a,)c,O a
Specific adsorption is assumed to be absent and the integral is usually evaluated by means of the classic Gouy-Chapman equations and a plot of (I,“” - I,“) vs. cj, at a given a”, gives aP - a,, which represents an ion-free layer thickness. Our purpose is tocheck the influence that the presence of two distinct planes of closest approach has on the evaluation of the ion-free layer thickness. To this end we have computed the surface excess relative to the solvent on the basis of the UDCA theory r,UDC* =
/“=,(
cp( x)
-
,-;)uDcAdx
(21)
.X=tl,
We compared our results with those obtained from the Gouy-Chapman theory (IPGc). This comparison is easier to interpret if we define an apparent ion-free layer thickness hap by rGC hap =
p
_
rUDCA
P 0 CP
(22)
54
It follows from eqns. (201422) that hap = up - a, when the cation and anion have a common distance of closest approach. RESULTS AND DISCUSSION
Influence of ion-size asymmetry
Figure 2 shows a typical example illustrating how the differences between the Gouy-Chapman and UDCA surface excesses increase with electrolyte concentration at a rate that depends on the electrode charge density when the cation and anion have different planes of closest approach. The results are displayed as plots of the more familiar cation charge contribution Au+= z+F(I’~~- I’yDCA) vs. z+c”,. In this section, the dielectric constant throughout the solution side of the interphase will be kept at its bulk value (ei = lu = enI = 78.5). When the cation is the smaller of the two ions, plots of Au+ vs. z+ct are strictly linear for uM 2 0 and develop a slight curvature for uM I 0. If the cation is larger than the anion, the sign of g M should be reversed in the preceding statement. Such a curvature would be barely detectable in an experimental plot, so that it appears justified to fit the Au+ vs. z+ct relationships by straight lines with
i--
-2 +30 +10
0 -10 -30
10
5
0 0
1
2
3
4
5
6
z+c+O/mol dm” Fig. 2. Charge density due to differences between Gouy-Chapman and UDCA theories as a function of cation concentration. Charge densities on the electrode in PC cm-* are indicated in the figure. 1: 1 electrolyte; T=298K, a+-a,=2A, a_-a,=4A.
55 I’
011 -40
’
’
’
-30
’
’
-20
’
’
-10
’
’
0
’
’
10
’
’
’
’
20304050
I
’
’
”
Fig. 3. Ion-free layer thickness as a function of the charge density on the electrode for a given a + - (I, = 2 A and several a _ - a, values indicated in the figure. 1: 1 electrolyte; T = 298 K.
zero intercept and having slope FAaP. Unequal distances of closest approach lead to a sigmoidal dependence of hap on o”, as indicated in Fig. 3. Under the influence of a strongly repulsive field ( I uM 1 > + 20 /.LCcm-* in Fig. 3) the smaller ion is swept out from region II in the interphase and the ion-free layer extends to the plane of closest approach of the larger ion (hap = a _- a, in Fig. 3). However, when the smaller ion is attracted towards the electrode, hap remains larger than expected for an ion-free layer extending to the plane of closest approach of the smaller ion. In this situation the hap value reflects the physically different pictures behind the UDCA and Gouy-Chapman theories, and is a function of the distance between the two planes of closest approach. It is interesting to note that under attractive conditions for the smaller ion and at sufficiently high concentrations, a slight superequivalent adsorption is predicted in region II which would lead to a dramatic reversal of the composition in region III. InjIuence of electrolyte stoichiometry Differences between the Gouy-Chapman and UDCA surface excesses become larger when the charge number of the counter-ion (the smaller ion) is increased or the charge number of the co-ion (the larger ion) is decreased, as illustrated in Fig. 4. Asymmetry in the charge number appears to be more critical than its actual value, as shown by the very similar results obtained for 1: 1 and 2 : 2 salts. Previous considerations regarding the linearity of Au+ vs. z+c”+ for 1: 1 electrolytes are equally applicable to the 1: 2, 2 : 1, 2 : 2, 1: 3 and 3 : 1 stoichiometries.
z_ : z_
I
I
0
I
1
2
3
4 z+c+’ /
I
,L
5
t1
mol dmJ
Fig. 4. Charge density due to the differences between the Gouy-Chapman and UDCA theories as a function of z+ c”,. Electrolyte stoichiometries are indicated in the figure. T = 298 K, gM = -20 PC a_-a,=4L cm-*, n+-a,=2A;
As before, a good estimate of the distance of closest approach of the larger ion can be obtained when the smaller ion is repelled from the electrode (aM z+ 0 for a + < a_ in Fig. 5). When the smaller ion is attracted towards the electrode, hai’ decreases at a different rate for each stoichiometry. Optimum conditions for the estimate of the distance of closest approach of the smaller ion are achieved when it bears a single charge and the larger ion is multicharged. Influence of dielectric constant
From a modellistic point of view we have divided the solution side of the interphase into three regions (I, II and III in Fig. 1). We shall now consider how a decrease in the dielectric constant of regions I and II affects hap. Combination of eqns. (5) and (7) shows that or has no influence on the UDCA surface excess derived at a given u”. Therefore we shall put lI = eu in the calculation below. It can be seen in Fig. 6 that differences between the Gouy-Chapman and UDCA surface excesses are reduced when eII is lowered. The decrease in Au+ is accompanied by an enhancement of the curvature in the Aa+ vs. z+ct plots when the smaller ion is attracted towards the electrode (Fig. 7). For lII 2 30, the curvature is relatively small and would probably not be detected in an experimental plot. However, if the dielectric saturation becomes more extensive, not only the curvature but also the appearance of negative Au+ values at low concentrations would be apparent in the analysis of a real system.
Fig. 5. Ion-free layer thickness as a function of the charge density on tte electrode for several electrolyte stoichiometries as indicated in the figure. T = 298 K; a + - a, = 2 A, a _ - a, = 4 A.
-5 0
76.6 33 6
. . I
1
0
1
2
3
1
I
4
5
z,c+*
/
mol dmd
Fig. 6. Charge density due to the differences between Gouy-Chapman and UDCA theories as a function of z+c: for several values of the dielectric constant between the two planes of closest approach. 1: 1 electrolyte; T = 298 K, oM = - 20 PC cm-*; a + - u, = 2 A; a _ - a, = 4 A.
58
0
I
I
-2
76zo3 ‘3
+30
-
15-
10 -*
S-
O-
L
1 0
I 1
I 2
3
4 z+c,’
5 /
mol dmJ
Fig. 7. Charge density due to the differences between Gouy-Chapman and UDCA theories as a function of z+ c”, for several values of tte charge density o~/~C crnmz on the electrode. 1:l electrolyte; T = 298 K, lII = 6; a + - a, = 2 A, a _ - a, = 4 A.
To facilitate comparison with our previous results we have continued to fit the Aa+ vs. z+c”, plots to straight lines passing through the origin, regardless of their possible curvature. For curved plots, the slope gives an average value Pp of the apparent ion-free layer thickness. As shown in Fig. 8, lower values of en lead to a steeper change in pp as a function of u M. However, this feature does not appear to be of practical value because the maximum variation (from cu = 6 to ln = 78.5) of pp in the experimentally accessible polarization range is rather small ( I 0.5 A). As a brief summary of our previous considerations, we can state that, within the framework of the UDCA theory, hap is determined mainly by the distances of closest approach of both anion and cation, and to a lesser extent by the electrolyte stoichiometry. Evidence for a lower dielectric constant in the vicinity of the electrode can be obtained from the curvature and negative ordinate intercept of the Au+ vs. z+ct plots. Comparison with experiment
Real electrolytes consist of ions of unequal size and therefore unequal distances of closest approach. In this section we shall fit some sets of experimental cation surface excesses r:, obtained at uM < 0, to the UDCA theory. Instead of proceeding to a direct comparison between I’: and I’yDCA,we shall compare their differences with respect to the value I’g’” calculated using Gouy-Chapman theory,
59
l.S-
6
-50
0
/
-40
I
I
I
-30
-20
-10
I1 0
10
I
I
I
20
30
40
cr"/pc cm
-¶
Fig. 8. Mean ion-free layer thickness as a function of the charge density on the electrode for several values of theodielectric const!nt between the two planes of closest approach. 1 : 1 electrolyte; T = 298 K; a+ -a,=2A;a_-a,=4A.
as measured by the ion-free layer thickness hap. In this way the physical picture behind the fit is clarified. Only a few determinations of the ion-free layer thickness appear in the literature [l-4,10-12], and all were carried out at negative charge densities on the electrode to avoid anion-specific adsorption. Table 1 shows some examples that will illustrate the application of the UDCA theory to the estimate of the cation and anion distances of closest approach. In all cases the solvent was water. A quick glance at Table 1 reveals that AaPdepends not only on the cation but also on the anion; the most obvious case is that of MgSO, and MgCl,.
TABLE 1 Experimental ion-free layer thickness AaP and UDCA distances of closest approach for the cation - a,) and the anion (a_ -a,)
(a,
NaH,PO,
NaClO,
Na,Mal a
/VP/‘&
2.4-4.6 b
2.5 c
2.2 d
1.7 e
4.4 =
(a+-
1.7
1.7
1.7
1.7
2.5
2.5
4.5-5.5
4.5
4.5
6.0
I 2.5
(a-
a b ’ d e
a,)/A - aJ/A
Sodium malonate. From ref. 3. From ref. 11. From ref. 13. From ref. 4.
NaCl
Il.7
MgSO,
MgC1, 2.5 e
60
A perusal of the original Au+ vs. .z+c~ plots shows that the evidence for an extensive dielectric saturation is either absent or lies within experimental error. For our calculations we have chosen or = ln = 33, which appears to be a realistic value on the basis of the dielectric constant profile proposed by Levine and Fawcett [14]. This value does not introduce a relevant curvature in the A.o+ vs. .z+ct plots and gives distances of closest approach that differ from those obtained with a higher dielectric constant by no more than 0.5 A. To fit the hap values of NaClO,, NaH,PO,, sodium malonate (Na,Mal) and MgSO, to the UDCA theory, we start by determining (a+- a,) independently from the hap values of NaCl and MgCl,. If it is assumed that chloride can approach the electrode sufficiently close to satisfy u,,~ I a,, then hF1-= a+- a, (see Fig. 2). Once a+- a, is fixed in this way, we look for the u _- a, value that delivers the experimentally observed hap at a given u”. The perchlorate distance of closest approach thus obtained is in good agreement with the sum of its Pauling radius plus that of the water molecule. A similar procedure based on molecular models can be carried out with respect to malonate, assuming that the carboxylic groups are oriented towards the solution. Therefore the distances of closest approach for these two anions are coherent from a microscopic point of view. However, the sulphate distance of closest approach appears to be somewhat overestimated, although this may reflect the fact that sulphate interacts more strongly than perchlorate with the solvent and retains its primary solvation shell on approaching the electrode.
f
I
’
1
Iq
‘.
!5-
4-
3-
2-
l-
l O-
-1
0 -
-20
-15
-10
-5
5
0 d
10
/&+c cm-s
Fig. 9. Experimental ion-free layer thickness obtained from NaH,PO, solutions [3] as a function of the charge density on the electrode (0). The solid curves show calculated values for 1: 1 electrolyte at T=298K, a+--a,=L7Aand En = 33; a_ - a, is indicated in the figure.
61
Sodium dihydrogen phosphate is a most interesting electrolyte for examination of the charge dependence of hap owing to the low tendency of the anion to be specifically adsorbed. Parsons and Nobel [3] reported hap values in the range -20 ~LC cm-’ < uM < 6 PC cm-’ (Fig. 9). These authors emphasized the similarities between the Aspdependence on aM and that of the surface excess volumes derived by Hills and Payne [15]. They also pointed out that the abrupt decrease of AaPat positive charge densities probably originated with the onset of anion-specific adsorption. According to UDCA theory, an increase in hap with gM is to be expected when the anion is larger than the cation, and it is also expected that hap will decrease as the amount of specific adsorption increases. For uM I - 8 PC cme2, the hap values of Parsons and Zobel [3] can be fitted with anion distances of closest approach in the range 4.5-5.5 A (Fig. 9). Although these values are reasonable, it should be noted that Asp increases with uM faster than predicted by the theory. A tentative explanation for this discrepancy would be an increase in the distances of closest approach when the electrical field in the interphase decreases. CONCLUSIONS
Application of the UDCA theory shows that the ion-free layer thickness may contain information on the distances of closest approach of both anion and cation. As surface excesses are rather insensitive to the shortcomings of the Gouy-Chapman theory, it is expected that a similar conclusion can be reached from any electrostatic theory that accounts for differences in ion size. ACKNOWLEDGEMENTS
The authors wish to express their gratitude to the Spanish Comision Asesora de Investigation Cientifica y Tecnologica (CAICYT) for supporting this work (grant PS89-01161. REFERENCES 1 2 3 4 5 6 7 8 9 10
D.C. Grahame and R. Parsons, J. Am. Chem. Sot., 83 (1961) 1291. A.N. Frumkin, R.V. Ivanova and B.B. Damaskin, Dokl. Akad. Nauk SSSR, 157 (1964) 1202. R. Parsons and F.G.R. Zobel, J. Eiectroanal. Chem., 9 (1965) 333. M. Molero and R. Andreu, J. Electroanal. Chem., 322 (1992) 133. J.P. Valleau and G.M. Torrie, J. Chem. Phys., 76 (1982) 4623. L.B. Bhuiyan, L. Blum and D. Henderson, J. Chem. Phys., 78 (1983) 442. J.J. Spitzer, J. Colloid Interface Sci., 92 (1983) 198. M.J. Grimson, Chem. Phys. Lett., 95 (1983) 426. P. Nielaba, T. Knowles and F. Forstmann, J. Electroanal. Chem., 183 (1985) 329. B.B. Damaskin, A.N. Frumkin, V.F. Ivanova, N.I. Melenkova and V.F. Khonina, Elektrokhimiya, 4 (1968) 1336. 11 J.A. Harrison, J.E.B. Randles and D.J. Schiffrin, J. Electroanal. Chem., 25 (1970) 197.
62 12 B.S. Segel’man, V.F. Ivanova and B.B. Damaskin, Elektrokhimiya, 12 (1976) 451. 13 M. Pirez, M. Molero, M. Barrera and R. Andreu, 43rd Meeting of the I.S.E. C&doba, Argentina, September 1992, Abstract 6-009. 14 S. Levine and W.R. Fawcett, J. Electroanal. Chem., 99 (1979) 265. 15 G.J. Hills and R. Payne, Trans. Faraday Sot., 61 (1965) 326.