Adsorption and Capillary Condensation on Rough Surfaces
The Journal of Physical Chemistry, Vol. 82, No. 12, 1978
1379
Adsorption and Capillary Condensation on Rough Surfaces J. R. Philip CSIRO Division of Environmental Mechanics, P.O. Box 821, Canberra City, A.C. T. 260 1, Australia (Received January 27, 1978) Publication costs assisted by Commonwealth Scientific and Industrial Research Organization
This work furnishes a quantitative basis for analyzing combined adsorption and capillary condensation on rough surfaces when neither process can be neglected. Interface configurations and isotherms are calculated for (a) adsorption only, (b) capillary condensation only, and (c) combined adsorption and capillary condensation, on a model rough surface. The adsorptive potential \k is calculated by integration over the solid of the intermolecular potential, and the capillary potential C through Kelvin’s equation. The interfaces for (c) are surfaces of constant +(= J! + C) and are found as solutions of the appropriate differential equation. For an illustrative example, capillary condensation influences the interface configurations, and the isotherm, over a wide range of relative vapor pressure; it more than doubles mean film thickness at the point of maximum augmentation, and its effect becomes negligible at film thicknesses greater than about two roughness heights.
I. Introduction The effect of the roughness of solid surfaces on adsorption may be both complicated and important. As well as directly modifying the adsorptive force field, surface roughness may also induce capillary Condensation. In this paper, we make use of some recent developments’--3in a first quantitative analysis of the influence of roughness on adsorption, of the associated capillary condensation, and of the interactions between the two. Philip’ developed a unitary approach to capillary condensation and adsorption. The basic concept of the unitary approach is that the equilibrium liquid-vapor interface is a surface of constant partial specific Gibbs free energy, CP = is made up of a capillary component, C, and an adsorptive component, 9,so that = C(K ) + 9 (1) where K is the mean curvature of the liquid-vapor interface. T h e original work’ used the “boundary layer approximation” \k = 9 ( v ) (2) where u is the normal distance from the solid surface to the interface (the “film thickness”); it was argued that the approximation is valid provided u is small compared with the radius of curvature, or some other characteristic length, of the surface. This proposition was confirmed through illustrative calculations for a number of configurations of the solid.2 Equation 1 constitutes the differential equation of the interface Q, = Q,*. When the boundary layer approximation, eq 2, applies, eq 1 is relatively simple, and solutions have been evaluated for various solid configurations.’ An inherent difficulty of the problem of adsorption and capillary condensation on rough surfaces is, however, that, in the circumstances of greatest interest, the film thickness is comparable to the characteristic length of the rugosities; that is, the criterion for the use of the boundary layer approximation, eq 2, does not hold. It is then necessary, first, to determine the adsorptive potential due to the rough surface, which will exhibit a more elaborate dependence on position than variation with u only; and, secondly, to solve the resulting, more complicated, form of eq 2. We begin, therefore, by investigating the adsorptive potential field due to a rough surface.
+,
+*
11. The Adsorptive Potential The adsorptive potential may be calculated by integrating the intermolecular potential over the solid volume. 0022-3654/78/2082-1379$01 .OO/O
An appropriate form of the intermolecular potential law is
d 9 = h(p) d V (3) where d 9 is the potential of the force field exerted by volume element d V of the solid a t a point a t distance p from it, It has been shown2that h(p) may be inferred from experimental data on F(v), the variation of adsorptive potential with the film thickness, u, on a single, effectively infinite, plane surface of the solid. The relation is h ( p ) = T - ‘ d/dv*[dF/dv]Iy=p
(4)
We shall employ the inverse power law h(p)=-yp+
E >
3
(5)
with y and t positive constants. Steele and Halsey4 used eq 5 with t = 6, Le., the inverse power law of London, to evaluate 9 between parallel plates and to form an integral expression for 9 in circular tubes. van der Waals or dispersion forces are approximated by t = 6, but electrostatic and other forces are involved also in a d ~ o r p t i o n .Philip’,2 ~ found that experimental data on adsorbed water films in the relative humidity range 0 . 7 0 . 9 9 could be empirically fitted by taking t = 4. He2l3l6 obtained exact solutions for inverse power law potentials for a wide range of solid configurations, giving special attention to the cases t = 6 and 4.
111. Adsorption on Rough Surfaces A. A Model Rough Surface. An elementary model of an infinite plane rough solid surface is provided by the configuration depicted in Figure 1. The solid occupies the semiinfinite region with the rough surface as its upper boundary. We use rectangular Cartesian coordinates (x, y , z ) . a is the slope angle of the crests and valleys of the surface. Evidently a is a roughness parameter, increasing from the value 0 for a smooth plane surface through to the limiting value ‘/27r for an extreme rough surface with indefinitely steep and sharp crests and valleys. Bikerman,’ in his treatment of surface roughness, also denotes the slope angle by a. Note, however, that for our anisotropic surface the roughness factor r, introduced by W e n ~ e l , ~ ? ~ is equal to sec a; the relation given by Bikerman,’ r = sec2 a , is for isotropic surfaces. The crests are a t (x, y ) = (4nH cot a , H ) and the valley bottoms a t (x, y ) = ([4n + 2 ] H cot a, -H), with n taking all integral values from --co to +a. Coordinate z is parallel to the crest and valley lines. Because of the symmetry and 0 1978 American Chemical Society
1380
The Journal of Physical Chemistry, Vol. 82, No. 12, 1978
J. R. Philip
Figure 1. Section of the model rough sutface in the phne z = mnstant. Because of the periodicity and symmetry of the surface, Figures 2.4, and 6 show only the basic cell (the rectangle bounded by the heavier stippling).
periodicity of the configuration, it suffices to establish q in 0 5 x 5 2H cot a. In this section we calculate the distribution of the adsorptive potential q, and hence also the family of liquid-vapor interfaces, when only adsorption is operative (Le., when the surface tension at the interface, and hence the process of capillary condensation, is negligible). Before we proceed to the general case with 0 < a < 'f2r,we consider first the two limiting cases, a 0 (the smooth surface) and a 'I2*(the "extreme rough" surface). The solutions to these two cases represent hounds within which adsorptive behavior for 0 < a < will lie. B. Smooth Surface. The Limit a 0. In this case the liquid-vapor interface is planar, so that the mean film thickness, ir, is exactly equal t o y . For the inverse power law, eq 5 , the relation between @ and 8 is9
-
-
-
E>3
*=-
2ny y3-E 2)(€ - 3)
(E -
with the special values
-
C. Extreme Rough Surface. The Limit a 'I2r.In this limit the valleys are fully occupied by adsorbate, for all @ > -m. Here also the interface is planar and ir =y. For our model surface the volume fraction occupied by solid varies linearly withy in the range -H < y < H from 0 a t y = H to 1 a t y = -H. It follows that, in this case
F>H
q(F)=-
1
GfH
w
J F(v)du
>H
with F(v) as defined in section 11. For the inverse power law, eq 5, this hecomes, for all c > 3 except t = 4
V>H
ny
q(F)=(E
- Z)(E - 3 ) ( ~- 4 ) H
[ (F - H ) 4 f - (V and, for t = 4
F>H
*(V)=--ln:
W
We observe also that, for
Ti>H
q(F)=-
T+H V-H
ny t
=6
ny; 6(F2- P)'
X
+ H)'-E]
(10)
'U
,
0
2
X/H
Figure 2. The adsorptive potential due to the model rough surface for e = 4, a = ' 1 , ~ . The veRical numerals on the curves are values of the adsorptive potential '# in the dimensionless form -@H/(ay). The Ralic numerals (see section V) give values for water at 300 K of -IO5. '€"J kg-' m. In the absence of capillary condensation the equipotentiak represent equilibrium interface configurations and '# = a, the total
potential.
D. The General Case. 0 < a < 'f2r.It follows from the general solution for inverse power-law potentiah ahout infinitely long polygonal prisms3 that, for t > 3
Here pi = I r - 91, where ri is the position vector in the plane z = constant of the ith vertex of the surface, and r is the position vector of ( x , y); and 8, are the angles r ri makes with the two slopes of the surface which meet a t r;. The Bij are taken directed outward from the solid surface. g is an exact function of Bij which is, in general, an integral of an associated Legendre function, hut is elementary for t an even integer. We have, in particular, that
g(6; B j j )
=
ny
['I6
cot3 Ou
g ( 4 ; O j j ) = ny cot 0,
+
'14
cot S i j ] (14)
Equation 13 represents the formal exact solution, hut there are more rapidly convergent procedures for evaluating Wxn, y) than summing the series on the right of eq 13. The method adopted here is t o retain the exact "rough" configuration for xn - X C x C xn f X, hut to replace it outside that region by the mean surface y = 0. It is demonstrable that @ computed for this configuration converges rapidly as X increases to the value for the exact configuration. In the present work we take X = 4H cot a , i.e., one wavelength; this enables computation of q to adequate accuracy with relatively little labor. E. Illustratiue Calculations. Equipotentials, Interfaces, and Isotherms. Figure 2 shows the distribution of adsorptive potential, calculated by the method of foregoing subsection D for the case e = 4, a = ' f a r .The vertical numerals on the curves denote values of the potential in the dimensionless form - q H f ( r y ) . The italic numerals (discussed in section V below) give values for water at 300 K of -lO%H J kg-' m. Note that, in the ahsence of capillary condensation, the equipotentials represent equi-
The Journal of Physical Chemistry, Vol. 82, No. 72, 7978
Adsorption and Capillary Condensation on Rough Surfaces
1381
3
2 V -
I
'/ /
H
1
0
a105
2
1
0.7
0.5
0.4
mf100 10
0.1
0.3
1
0.05
1000
--6 0 H 3
--Q H
ny
ny
Figure 3. Adsorption on the model rough surface in the absence of capillary condensation (Le., \k E 0). Isotherms (a) for t = 4, (b) for c = 6. The numerals on the curves are values of the slope angle of the surface, a. The curve 0 is for the smooth surface: the curve ?r/2 is for the "extreme rough" surface. D is the mean film thickness. The abscissa is the potential in dimensionless form: (a) - @ H / ( s y ) ; (b) -6@H3/(7ry).
librium interface configurations and J! @, the total potential, Figure 3 gives the adsorption isotherms in the absence of capillary condensation, in dimensionless form (a) for t = 4 and (b) for t = 6. The curves for a = 0 we calculated from eq 7 and 8, those for a = 1/27r from eq 11 and 12, and those for a = 1/4sthrough computations of areas under equipotentials calculated by the method of foregoing the limiting subsection D. Note that for 0 C a C behavior of the isotherm at large -@ is connected with that for a = 0 through the relation lim T ( a ) P ( O )= sec a (15) @ -+-m
and behavior at small -@ is asymptotic to that for a = '/2?r; i.e. lim 7 ( a ) / F ('/zn) = 1 (16) @-+ 0
IV. Capillary Condensation on Rough Surfaces We next consider the operation of the process of capillary condensation alone on our model surface, with adsorptive forces negligible. In this case we have for an interface at equilibrium
C I CP = - u / ( p r )
(17)
where 0 is the surface tension at the interface, r is its radius of curvature, and p is the density of the liquid.'O Calculation of the interface configurations and isotherms is a matter of geometry. For simplicity, and consistent with the following section V, where we consider combined adsorption and capillary condensation, we take the contact angle as zero. Results for nonzero contact angle are qualitatively similar to those developed here, so long as it is less than the slope angle a. A. T h e Smooth Surface, a 0. T h e Extreme Rough Surface, CY 1 / 2 ~ . We begin with these limiting cases. For a 0 there is no capillary condensation for all negative @; i.e.
-
- -
@ -m. The interfaces are planar and there is no capillary condensation, so that the solution reduces to that given in subsection IIC above. In the limit as 0, for 0 < a < 1/2r, the interface becomes planar, there is no capillary condensation, and the limiting behavior of the isotherm is given by eq 16. D. Adsorption and Capillary Condensation of Water. In previous sections we have been ahle to express potentials in a dimensionless form which involved the properties of the adsorptive force field (section 111) or of the liquid (section IV). We were therefore able to develop illustrative examples without the need to limit ourselves to a particular liquid-solid combination (section 111) or a particular liquid (section IV). When adsorption and capillary condensation are both involved, a comparably general formulation is not possible, and in subsection E helow we take for our illustrative example of combined adsorption and capillary condensation on a rough surface the case of water a t 300 K. In a previous study' it was concluded from the somewhat diverse and incomplete data that the relation
The Journal of
-
Physical Chemistry, Vol. 82, No. 1 2
-
1978 1383
+
+
-
F = -1.38 X lO-'u-' J kg-'
J kg-'
(35)
The vertical numbers on the interface profiles of Figure 4 are values of -p*H/u. The italic numerals are these values multiplied by 7.15, which are therefore potentials for water expressed as -1OS+H J kg-' m. E. Illustrative Calculations. Interfaces and Isotherm. Figure 6 shows the family of liquid-vapor interfaces for combined adsorption and capillary condensation of water a t 300 K for the case a = 1/4r.These interface profiles were calculated by the methods of foregoing Subsection B. The (italic) numerals on the curves denote values of the potential in the form -lO"H* J kg-' m. The isotherm for this example of combined adsorption and capillary condensation was calculated by computing the areas under the interfaces shown in Figure 6. As indicated in subsection C above, the isotherm asymptotically approaches that for adsorption alone in the limit as 0. It is shown in Figure 7 as the curve marked "adsorption plus capillary condensation". In Figure 7 the isotherm is plotted in the reduced form D/H as a function of +H. The same information is presented more conventionally in Figure 8 as a family of isotherms for surfaces of different roughness heights, H,
+
-
1
2
r/H
Flgure 6. Combined adsorption and capillary condensation of water at 300 K on the model rough surface with a = 'I4=, The numerals on the curves are values of -iOS@HJ kgf m. The CUNeS represent both equipotentials and equilibrium interface configurations.
(34)
holds reasonably well in the relative vapor pressure (p) range 0.7 5 p 5 0.99 for the adsorption of water on appropriate plane wetting solid surfaces. Equation 34 corresponds to adsorption of one monolayer at p = 0.72 and to a film thickness of m (100 A) at p = 0.99. It follows from eq 4 that eq 34 implies the following values for the parameters in eq 5 6 = 4, ry = 1.38 X J kg" m. Using these values, we can apply Figure 2 to the adsorption of water vapor on our model rough surface. The vertical numerals on the equipotentials are values of -*H/(ry). The italic numerals are these values multiplied by 1.38, and therefore represent adsorptive potentials for water expressed as -105aH J kg-' m. We have also, for water at 300 K, that u = 0.0715 N m-l, p = lo3 kg m-3, so that eq 17 becomes C = - 7.15 X lO"r-'
u 0
mi05
2
1
-lO%H
07
05
(Jki'rn)
Isotherms for water at 300 K on the model rough surface with a = 'I,= in the reduced form DIHagainst + t i . Comparison of the three isotherms: capillary condensation only; adsorption only; Flgure 7.
adsorption plus capillary condensation. in the form ~(p).Note that relation
p = exp(@/RT)
* and p are connected hy the (36)
where R (= 461.5 J kg-' K-I) is the gas constant for water vapor, and T is the absolute temperature. The H = 0 isotherm of Figure 8 is that for the smooth surface and was calculated from eq 7.
VI. Discussion A. Comparison of Equilibrium Interface Configurations. The italic numerals on the curve of Figures 2,4, and 6 give values (for water) of -105+H J kg-' m." Each set of curves is for the same set of values of +H, so that the three figures show comparable families of equilibrium interface configurations. Figure 2 gives the family calculated for adsorption only, Figure 4 that calculated for capillary condensation only, and Figure 5 that calculated with both mechanisms taken into account.
1384
J. R. Philip
The Journal of Physical Chemistry, Vol. 82, No. 12, 1978
all 0 < CY < l/pr, these isotherms are sigmoidal (with either @ or p as abscissa) and are bounded, with B < H for CP < 0 (cf. eq 27). Adsorption Plus Capillary Condensation. The maximum augmentation of adsorption by capillary condensation occurs at about @H = -4.4 X J kg-’ m, with D = 0.85H. The effect of capillary condensation on the isotherm decreases as @ increases toward zero: it becomes negligible when @H reaches about -0.7 X J kg-’ m. For different values of H, these points on the isotherm occur at different values of the relative vapor pressure, p . The augmentation due to capillary condensation is maximum at about p = exp(-0.318 X 10-9/H); and it becomes negligible at about p = exp(-0.051 X 10-9/H). The isotherms of Figure 8, plotted for various values of H , with abscissa p , exhibit kinks corresponding to the points of maximum augmentation due to capillary condensation. It will be noted (i) that capillary condensation influences the isotherm over a wide range of p values; (ii) that the p value for the maximum augmentation increases as H increases; (iii) and that, in consequence, the gross character of the isotherm varies greatly as H assumes different values. C . Simplifications in the Present Study. Some simplifications are appropriate in the present exploratory study, but are not inherent in the approach; they can be removed hy further, similar, investigations. Two simplifications are more basic: the approach applies directly only to rough surfaces which are regular and geometrically well-defined; and we use the continuum approximation (which may not be a serious limitation). (i) The Model Surface. The schematic character of our model rough surface must be recognized. We have studied only one form of roughness element. Similar calculations are feasible for other roughness elements; and more elaborate calculations of the same type can be made for surfaces rough in the z coordinate as well as in y , and we can expect qualitatively similar results. A more intrinsic limitation is that many real rough surfaces do not exhibit regularity. We should need, at the least, to specify such surfaces through distribution functions’ for values of parameters such as H and CY, rather than by single values. Evidently a full analysis of the effect of roughness on interface configurations and isotherms for such surfaces will be more difficult. (ii) The Adsorptive Field. In the illustrative calculations for water we use eq 34 as a convenient empirical equation, with no implication of underlying physical significance. It suffices for our purpose that eq 34 correctly represents the order of magnitude of the adsorptive field. The distribution of \k may, of course, be calculated for any given h(P).l2 (iii) T h e Continuum Approximation. We work necessarily in the continuum approximation. This must be recognized as a limitation for film thicknesses less than a few molecular dimensions. Nevertheless, some validity in a statistical sense can be assigned to our results for small values of B. D. Concluding Remark. Despite the foregoing simplifications, this study provides a striking demonstration of the magnitude of the influence of surface roughness on adsorption and on capillary condensation, and of the character of the interactions between the two processes. Most real surfaces are rough. It is hardly surprising that attempts to interpret adsorption isotherms in terms of adsorption only, or in terms of capillary condensation only, are unsatisfactory when (i) the film thickness is comparable in magnitude (e.g., bears a ratio in the range 0.1-2) to the
I -
I
0 79
0 84
0 89
0 94
0 99
D
Flgure 8. Isotherms for adsorption plus capillary condensation of water at 300 K on the model rough surface with a = 1/47r, presented as 0, mean film thickness, against p , relative vapor pressure. Numerals on the curves are values of 109Hwhere H (m) is the roughness height. The curve 0 is for the smooth surface.
Comparison of the three figures is instructive. In Figure 2 (adsorption alone) the film thickness is reduced over the crests and increased in the valleys. In consequence the interface is smoother than the solid surface and the amplitude of deviations from the mean interface level decreases fairly rapidly as B increases. The interface is essentially planar when D reaches a value of about 2.5H (Le., @H = -0.58 X J kg-’ m). For capillary condensation only, Figure 4, the family of interfaces has totally different properties. First, the surface is only partly wetted for CPH < -2.53 X J kg-’ m. Secondly, for 0 > CPH 2 -2.53 X J kg- m, the interface is anchored at the crest lines and P < H . For combined adsorption and capillary condensation, Figure 6, we find that, as we should expect, surface tension reduces film thickness over the crests and increases it in the valleys more than does adsorption alone. In consequence the interface becomes planar much more rapidly as D increases than for adsorption only: the interface is essentially planar when B reaches 1.4H (Le., CPZ-Z = -1.39 X J kg-’ m). Because of adsorption, the interfaces of Figure 6 differ from those of Figure 4 in that they involve neither partial wetting nor limits to D due t o anchorage of interfaces on the crest lines. B. Comparison of Isotherms. The comparison is taken further in Figure 7. Adsorption Only. The “adsorption only” isotherm is a member of the family of isotherms shown in Figure 3a. For large -4the form of the isotherm is dominated by the limiting behavior described by eq 15; Le., the isotherm is like that for the smooth surface, but adsorption is greater in proportion to the greater surface area. For small -9, on the other hand, the isotherm asymptotically approaches that for the “extreme rough” surface (cf. eq 16). Capillary Condensation Only. This isotherm is a member of the family of isotherms shown in Figure 5. For
Unassociated Solvent Structure
at a Polarizable Interface
characteristic roughness height of the surface; and (ii) neither the surface tension of the adsorbate nor the adsorptive force field can be neglected. The present work furnishes a first quantitative basis for analyzing this class of problem.
References and Notes
The Journal of Physical Chemistry, Vol. 82, No.
(6) (7) (8)
(9) (10)
(1) J. R. Philip, J . Chem. Phys., 66, 5069 (1977). (2) J. R. Philip, J . Chem. Phys., 67, 1732 (1977). The present eq 4 correctly states the relation for h(p), which was misprintedin ref 2. (3) J. R. Philip, J . Aust. Math. Soc. B , in press. (4) W. A. Steele and G. D. Halsey, J . Phys. Chem., 59, 57 (1955). (5) See, for example, the following reviews: A. Sheludko, Adv. Colloid
(11) (12)
72, 1978 1385
Interface Sci., 1, 391 (1967); J. Clifford, Water, Compr. Treat., 5. 75 (1975). J. R. Philip, 2.Angew. Math. Phys., in press. J. J. Bikerman, "Physical Surfaces", Academic Press, New York, N.Y., 1970, Chapter V. R. N. Wenzel, Ind. Eng. Chem., 28, 988 (1936). Wenzel's usage of r i s not to be confused with our use of rfor the radius of curvature of the liquid-vapor interface. Cf. eq 5 of ref 2. Strictly, p is the excess density of the liquid over that of the vapor (or over that of the air, if air is present). With Figure 2 interpreted as showing interface configurations for adsorption in the absence of capillary condensation. The method of ref 2 applies for h@)an inverse power law or a linear combination of inverse power laws, such as the Lennard-Jones potential.
A Three-State Model for Unassociated Solvent Structure at a Polarizable Interface W. I?. Fawcett Guelph-Waterloo Centre for Graduate Work in Chemistry (Guelph Campus), Department of Chemistry, University of Guelph, Guelph, Ontario, Canada (Received December 19. 1977; Revised Manuscript Received March 9, 1978)
A three-state model for solvent structure at a polarizable electrode/solution interface is presented. The three
orientations assumed for solvent molecules in the monolayer adjacent to the interface are with the electrode field, against it, and perpendicular to it. The dielectric properties of the monolayer are derived assuming the molecules can be represented as hard spheres with both permanent and induced dipole moments, and accounting for dipole-dipole interactions within the monolayer. It is shown that the model can account for the inner layer capacity curves often observed for unassociated solvents, that is, for capacity curves which are characterized by a single maximum or minimum. The results of applying the model to data for the mercury/ethylene carbonate and mercury/methanol interface are presented.
Introduction An understanding of solvent structure a t the electrode/solution interface is an essential component in the development of a theory for electrosorption and electrocatalysis. The orientation of solvent molecules in the solvation sheath of an electrode and, thus, the dielectric properties of this monolayer are determined by the electrical fields due to the charge on the electrode and on adjacent ions, and by covalent interactions among the solvent molecules, between solvent molecules and the electrode material, and between solvent molecules and adjacent ions. The first model'v2 for the electrostatic properties of the solvation sheath of the electrode assumed that solvent molecules could assume two orientations, namely, in the direction of the electrode field or against it. This model accounts qualitatively for the presence of a hump on the differential capacity-potential curves obtained for the mercury-solution interface with water3 and some other solvent^.^ Macdonald and Barlows developed a model for the dielectric properties of water a t the mercury surface in which all orientations of solvent dipoles were considered but dipole-dipole interactions were ignored. Both the two-state6J and multistate models8 were developed further to account for dipole-dipole interactions and molecular polarizability but no treatment of the hydrogen bonding expected in protic solvents such as water was included in these models. More recently, Damaskin and FrumkingJoand Parsons" have shown that, if one assumes that clusters of solvent molecules with a net dipole moment per molecule in the direction of the electrode field less than that of a free solvent molecule are also present in the solvent monolayer 0022-3654/78/2082-1385$01 .OO/O
at the electrode surface, one can account qualitatively for all the features of the differential capacity-potential curve for the mercury/aqueous solution interface. The presence of clusters a t the interface may be regarded as an expression of the fact that intermolecular hydrogen bonding is important in determining water orientation at a charged interface. This treatment may be classified as a four state model in which single solvent molecules and clusters may each assume two orientations, in the direction of and against the electrode field. Parsons12 also applied the four-state model to interfacial capacity data for the mercury/nonaqueous solution interface and showed that the features of the inner layer capacity-potential curve could be described with reasonable parameters in the cases of formamide and methanol. In the case of ethylene carbonate, an aprotic solvent, a good fit to the experimental data was obtained on the basis of the two-state model of Watts-Tobin.1,2 Since the majority of aprotic solvents are unassociated in the bulk, there is little reason to assume that they would form associates a t an interface. However, it is quite probable that more than the two orientations considered by Watts-Tobin1i2are important constituents of the surface monolayer in unassociated solvents. Because of specific interactions between the metal atoms in the electrode and solvent molecules, the predominant species could easily be one with its dipole vector a t an angle to the surface other than 90'. From a statistical point of view, the easiest way of analyzing such a situation is to assume that three orientations of solvent dipoles at the surface are possible, namely, in the direction of the electrode field, against it, and perpendicular to it. In the present paper, the dielectric
0 1978 American Chemical Society
