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aqueous solution interface
J. Electroanal. Chem., 111 (1980) 163--180
163
© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
A MOLECULAR MODEL FOR THE DIELECTRIC PROPERTIES OF THE INNER LAYER AT THE MERCURY/AQUEOUS SOLUTION INTERFACE
W.R. FAWCETT, S. LEVINE, R.M. deNOBRIGA and A.C. McDONALD
Guelph--Waterloo Centre for Graduate Work in Chemlstry (Guelph Campus), Department of Chemistry, University of Guelph, Guelph, Ontario N I G 2W1 (Canada) (Received 9th August 1979; in revised form 29th January 1980)
ABSTRACT A four-state model for a monolayer of water molecules at a charged polarizable interface is presented. The states considered are water monomers and clusters, each of which can assume two orientations in the field due to the charge, i.e. in the direction of the field or against it. The clusters are composed of water molecules whose orientation with respect to one another is determined by hydrogen bonding and are assumed to have a net dipole moment much less than that of a m o n o m e r in the direction of the field. The model includes a detailed description of dipole--dipole interactions with consideration of both orientational and distortional components of molecular polarization. Expressions are developed for the equilibrium concentrations of the four components of the monolyaer, for the potential drop across the monolayer and its differential capacity. It is shown that the model provides a reasonable description of the inner layer capacity curves for the mercury/aqueous solution interface obtained in the absence of ionic specific adsorption in the temperature range 0--85°C. In addition, it is able to account for the fact that the maximum entropy of formation of the inner layer occurs at negative electrode charge densities, and that the temperature coefficient of the potential drop across the inner layer at the point of zero charge is positive.
INTRODUCTION
Molecular theories based on statistical mechanics have been proposed by many workers [1--14] to describe the solvent properties at the electrode/electrolyte solution interface, particularly the mercury/aqueous solution system. From the inception of such molecular models, a considerable degree of controversy among the different authors has been a characteristic feature. The models of the mercury/aqueous electrolyte interface discussed in this paper are based on a monolayer of water molecules, situated in v a c u o between the metal surface and a dielectric containing ions, i.e. the diffuse layer. The monolayer is identified with the inner S t e m region and specific ion adsorption is assumed absent. Prior to a paper by Damaskin and Frumkin [7], the water molecules in the monolayer were treated as separate kinetic entities, which usually assumed one of t w o orientations in the electrode field, with the oxygen pointing towards or away from the metal surface. Experimentally, a maximum in the differential capacity of the inner region (which, it should be noted, is n o t measured directly) is calculated from positive of the point of zero charge (pzc) for the mercury/
164
aqueous electrolyte interface in the absence of specific ionic adsorption. The two-state orientational model could account for this maximum with the postulate that the natural orientation of the water molecules in the monolayer at the pzc has the oxygen atom pointing away from the mercury. However, the experimental surface excess entropy, and the corresponding entropy of formation of the interface pass through a m a x i m u m on the negative side of the pzc [15], indicating that the preferred orientation of water on mercury corresponds to oxygen towards the metal. The behaviour of the temperature coefficient of the potential drop across the inner region is consistent with the latter orientation of the water molecules at the pzc. In most of the two-state models, the solvent monolayer is assumed to have an effective dielectric permittivity which accounts for all contributions to this permittivity other than that due to the permanent solvent dipoles. In the case of water, a typical value for this inner region permittivity is 6. Without such a permittivity, the potential drop across the monolayer region is found to be t o o large (by the factor 6). In contrast, Levine et al. [6] avoided the use of this reduction factor by considering polarizable water dipoles situated in a vacuum. In this treatment, the effective permittivity of the monolayer, which is derived b y statistical mechanics from a molecular model of the monolayer, varies significantly with electrode charge density. The lateral dipole--dipole interaction is n o w so strong that in the presence of an electrode field, the fractions of dipoles pointing parallel and anti-parallel to this field are nearly equal. Nevertheless, the potential drop across the inner region can still be t o o high. Also, although the two-state model of Levine et al. [6] reproduces the general shape of the experimental capacity--charge curve in the region of the pzc, it fails to account for this curve at the higher potential, away from the pzc. A molecular model of the inner region which treats the water molecules as polarizable dipoles must be inadequate since it neglects the effect of hydrogen bonding. Furthermore, the role of hydrogen bonding is probably masked by introducing an effective dielectric permittivity, which should therefore be avoided. Damaskin and Frumkin [7] have shown that the overall qualitative features of the differential capacity electrode charge curve for inner region can be reproduced if one assumes that associates or clusters of water molecules with a low net dipole m o m e n t perpendicular to the interface are present in the solvent layer immediately next to the electrode, in addition to individual water molecules. These clusters can be regarded as a manifestation of hydrogen bonding between the water molecules in the inner region. Parsons [8] and Damaskin [13] have developed further this model of a monolayer of clusters and single water molecules. The important achievement of this theory over previous ones is that the calculated capacity--charge curve contains the essential experimental features of a maximum and t w o minima. These authors assume an effective dielectric permittivity to the inner region and do not introduce the polarizability of their molecular units. In a series of papers, Levine et al. [16--18] have extended the theoretical treatment of the inner region described in ref. 6 to a theory of adsorption from a mixture of t w o polar solvents. In the present paper, this theory is applied to develop a model for the dielectric properties of the inner region in associated solvents, and to analyse the inner layer differential capacity data for the mercury/aqueous solution interface.
165 MODEL OF THE SOLVENT IN THE INNER LAYER
The model presented is a two-dimensional version of models developed earlier for the t h e r m o d y n a m i c properties of polymer solutions containing open chain polymers * of varying chain length [20]. The following assumptions are made. (1) The solvent in the inner region is present as a two-dimensional hexagonal lattice of fully occupied adsorption sites with a number density Ns per unit area. The solvent molecule is represented as a hard sphere with diameter d, so that Ns = 2/(\/3d2); the number of nearest neighbours (nn) for a given element of the monolayer (coordination number, c) is 6 and the nn distance is d. (2) One site is occupied by an unassociated solvent molecule and n sites by a cluster consisting of n elements (solvent molecules). Since the cluster is assumed to have an open configuration, each element at the end of chain has 5 nn and each internal element, 4 nn. It follows that the total number of nn interactions experienced by a cluster is qc = cn -- 2n + 2. In the following an unassociated solvent molecule or an element of a cluster is referred to as a particle. (3) Unassociated solvent molecules and clusters are oriented in one of two directions referred to as " u p " and " d o w n " . The " u p " position signifies that the positive end of the dipole is towards the metal, and the " d o w n " position, that the negative end is towards the metal (Fig. 1). The components of the permanent dipole moments in a direction perpendicular to the interface for unassociated monomers are Pm and --Pro in the " u p " and " d o w n " orientation respectively; the corresponding quantities for clusters are Pc and --Pc. (4) The induced dipole moments Pro1 and Pc1 for monomers and clusters respectively, refer to the " u p " direction. Here, Pc, is equally distributed among the n particles so t h a t each particle has an induced dipole m o m e n t of Pci/n. (5) The external field E = 4~ro in the " d o w n " direction is due to the charge density a on the metal. Here, Xm and Xc are the electric fields in the " d o w n " direction at a lattice site occupied by an unassociated solvent molecule and a particle of a cluster, respectively. (6) The induced dipole m o m e n t s are Pm~ = --am(E + Xm) and Pc~ = --~c(E + Xc) where ~m and ~c are the polarizabilities of a m o n o m e r and cluster, respectively. Let N*~, N~ be the numbers per unit area of monomers and clusters in " u p " positions and N~, N~ those in " d o w n " positions. The total numbers o f each species per unit area are Arm = N ~ + N ~ and Nc = N~ + N~. In order to estimate dipole--dipole interactions, one must distinguish the different types of neighbouring pairs among the monolayer elements. For monomers in the " u p " orientation,
cN*m : 2N~nm t t + Ntm*m + N ~ c + N~¢
(1 )
where Jvt..m*m,JVt..~m,N ~ t and N~c represent the numbers per unit area of neigh-
* It should be noted that Guggenheim and McGlashan have c o n s i d e r e d the thermodynamic properties of polymer solutions containing closed chain polymers [ 19 ]. Since the e s t i m a t e d entropy of mixing is virtually independent of polymer configuration, and depends almost entirely on polymer size, the assumption made here that the clusters have an open configuration is not expected to have any significant effect on the results.
166 ////////////i////////////
(b)
(a)
[c) Fig. 1. Schematic representation of (a) a cluster of solvent molecules in the "up" position, (b) a cluster in the "down" orientation and (c) a monolayer composed of clusters and monomers in both orientation at a charged interface. If it is assumed that a cluster is composed of three molecules with dipole vectors at an angle of 70.5 ° to the direction of the electrode field, the net dipole moment of the cluster would be one-third that of a monomer.
b o u r i n g pairs b e t w e e n a m o n o m e r in t h e " u p " p o s i t i o n a n d each o f t h e f o u r c o m p o n e n t s o f t h e m o n o l a y e r . In t h e case o f " u p " clusters, o n e m a y write:
qcN*c = 2Ntctc + Ntc*c + Ntctm + Ntc*m
(2)
T h e c o r r e s p o n d i n g e q u a t i o n s f o r t h e o t h e r t w o o r i e n t a t i o n s are
cN~ = N~m + 2N~m + N~c + g~c
(3)
and
qcN*c-- g c c~t + 2 N ~ + N c~t ~ m +Nc~m
(4)
The t o t a l n u m b e r o f n n i n t e r a c t i n g e l e m e n t s per units area is
c N J 2 = c(Nm + qN¢)/2
(5)
F o r r a n d o m d i s t r i b u t i o n o f t h e m o n o l a y e r c o m p o n e n t s a m o n g the available sites, it has been s h o w n t h a t [ 2 0 ]
N ~ m - eNtre2 ,2Na '
N ~ m - cNtmN*m Na '
g ~ c - qcNtmN*c ga '
N ~ m - cN~m2 2N a '
N ~ c - qcN*mNtc gq '
g~c-
2 t $ Ntc*c- q cN~N~c Nq
and
N~*c ~=
q2cN~c 2
qcN~mN~c Na '
N ~ c - qcgtmN*c Nq
Ntc*~-
q2cNtc 2 2Nq (6)
2Nq
S u b s t i t u t i o n o f these relations i n t o eqns. (1)--(4) gives eqn. (5). In t h e case o f m o n o m e r s , t h e t o t a l dipole m o m e n t s are p tm = P m + Pmi a n d P ~ = --Pm + Pmi ; similar relations h o l d f o r clusters. T h e d i p o l e - - d i p o l e interac-
167
tion energies for the 10 different pairs of elements are given by U~m-P~,
U~m-
d3 '
U~m~m- d3,
U~c-
nd 3 , t2
U~c-
nd 3 ,
U~c
and
n2d3
U~c
nd 3 ,
nd 3 ,
vii
p~c 2 n2d3
U~
(7)
The total dipole--dipole interaction energy among the nn per unit area is Vdd =
E ~ g k ] v kl i,]=m,c k,l= ~,$
(8)
Defining the mean net dipole moments in the " u p " direction due to monomers and clusters, M m = N~mP~m+ N~a~m
and
M c = g~p~ + N~Pc
it is easily shown that c U d d - 2Na d3 ( M m + qMc/n) 2
(9)
(10)
The dipole--dipole interaction energy may also be expressed in terms of the reaction fields Xm and X~: V d d = lI M m X m + IM~Xc 1
(11)
1
where the factor : occurs to avoid counting nn interaction twice. In the random mixing approximation used here, each neighbouring site is responsible for the same fraction of the mean field at a given site, and since X~ should n o t include the field due to other particles of the cluster, it is assumed that (12)
X c = qXm/n
Combining eqns. (10) and (12), it is easily shown that c Xm - Nqd3 (Mm + q M d n )
(13)
Since Mm and M~ depend on Am, eqn. (13) is more conveniently written in another form. Defining R ~ = N ~ -- N~m
and
Rc = N t -- N~
(14)
one may write Xm
-
C (R qRcpc d3B mPm + n
AE
)
(15)
where A = a m N m + a~qNc/n
(16)
and B = Nq +
C (o~mgm + q2acNc] n2 ]
(17)
168 As shown previously [18], the complete expression for the electrical contribution to the free energy of adsorption of the monolayer per unit area is eel =
Vdd "{-(Mm + Me) E + °~mgm (E + Xm) 2 + acNc (E + Xc) 2 2 2
(18)
where the first and second terms represent the total energy of the dipoles in the external field E, and the third and fourth terms account for the work done in producing the induced dipoles. In the above derivation, dipole--dipole interactions between pairs of particles which are not nn, and resulting from dipole images in the two media between which the monolayer is situated have been ignored [6,18]. It was explained that an approximate m e t h o d of accounting for these t w o factors in the case that the monolayer consists of monomers only [6] is to increase the coordination number c to an effective value c~, where ce is expected to have a value close to 15. In the present case, where both clusters and monomers are present in the monolayer, a corresponding increase in qc to qce is expected. Therefore, in the following, eqns. (15)--(17) are used with c replaced by Ce. In addition to Uel, t w o other contributions to the Helmholtz free energy of the monolayer need to be considered. It is postulated that "residual" energies in the " u p " and " d o w n " positions U ~ , U ~ , Uc*~and U~ describe the interaction of individual water molecules and clusters respectively, with the phases at the t w o boundaries of the monolayer region. The total residual energy per unit area is therefore:
V, = N*mV*mr+ N*mU~m,+ N*cV*c,+ g*cv*cr
(19)
Also, the number of configurationsg availableto a given composition of the monolayer is assumed to be given by Flory--Hugginsstatistics [20] so that --ln g = N ~ 1n
(N~_j)__
/nN~\
m + N*m
cNq
m~-~ [N*m~
/ Ns\
+Nc* In ~-N- s ) + -~- In[~qq)
/nN~ \
+ N~* ln[-~-s )
qcN¢ l n ( q )
(20)
2
Introducing the van der Waals interaction energy U w which is assumed to be independent of the dipole interactions, the total Helholtz free energy per unit area attributed to the monolayer is F = Ue, + U~ + U w - k T l n g
(21)
where k is Boltzmann's constant and T the absolute temperature. EQUILIBRIUM RELATIONS In this section the xiistribution of monomers and clusters among the four " u p " and " d o w n " positions is derived. At equilibrium, the chemical potentials of monomers in the " u p " and " d o w n " positions are equal. This condition is conveniently expressed [16--18] by the relation:
169
Noting that ON*m/aN~m = --1, and introducing the fractions of monomers in the " u p " and " d o w n " positions, f m = N ~ / N ~ and f ~ = N~/N~, differentiation of equation (21) yields: k T In
(23)
= U ~ -- U~r -- 2pm(E + X m )
The corresponding condition concerning the number of clusters in each orientation is (24)
*m 0 which leads to k T ln([-~t = U~¢r -- U~¢~-- 2p¢(E + X~) \f'~/
(25)
where f~ = nN~c/Ns and f~ = nNCc/N~ are the fractions of solvent molecules in the " u p " and " d o w n " cluster positions. Finally, there is equilibrium with respect to the association of n monomers into a cluster. In this regard, it is sufficient to consider n monomers in the " u p " position combining to form a cluster in the " u p " position [17]. The free energy condition may be expressed as
N, ,m,N
(26)
0
where N t = NCm + nNC¢, so that (3N~c/aN~) = --1/n. Equation (26) yields: n
k T In [_ r(fm)" f~ 7_j = U~ - - n U ~ r + p~(E + X ¢ ) - - n p m ( E (E + X¢) 2 --ac
ckT
2
nC~m(E + X m ) 2 +
( n - - q ) ln
2
(~q)
+ Zm)
daX2m +
ckT +-~-(n--q)
2Ce
(n - - q) + ( 1 - - n) k T
qckT ~ In
(27)
The condition that all Ns sites per unit area are occupied by particles is fCm + fCm + fCc + f~= l
(28)
Noting that Xm, X¢ and Nq depend on the four fractions fro, f~, fct and f~ of monomers and clusters in the two orientation states, the three equilibrium relations {23), (25) and (27) and the condition (28) constitute four equations which yield these fractions as functions of the electrode field E and, therefore, the surface charge density o. Because the equilibrium equations are non-linear, an iteration process is used to determine the fractions f¢~, f~, f~ and f¢~ numerically. DIFFERENTIAL CAPACITY OF THE MONOLAYER REGION The potential drop across the monolayer (from the metal phase to the solution phase) is A¢ = 4~rod + 41r(Mm + Me)
(29)
170 The differential capacity Ci of the monolayer region is given by 1 = ()A¢ = 47rd + 4fr( ~)Mm +aMcl C~ ao \ ao ao l
(30)
It is convenient to introduce st-
aNt S*~- aV*m ao ' aa ' where, from (28)
Sc*- aNc* and ao
So*- aNt* ao
(31)
(32)
S t + S ~ + nS~ + nS~ = 0
Then,
aMm~)o- P r o ( s t --S~m) -- amSm(E + Xm) -- amNm 41r + aXm~ ao ]
(33)
and
+
oo o
+ ) .oo
where sm = s t + s t
and
so = S¢' + s¢*
By differentiating the equilibrium relations (23), (25) and (27) with respect to o, one obtains together with eqn. (32) a set of four linear equations in the unknowns S t , S*m, S¢¢ and S~. The necessary equations are kT
--k
k T ~ c --kT
m
(35)
=__87fp m __ 2 P m ( _ ~ . )
¢ = - - 8 ~ p ¢ - - - -2pcq n \~-a/
(36)
and n k T (S~t) m -- k T (S~¢¢) ~ = 4~r(p¢ -- npm) + 4~rE(nam -- a¢) + 4~Xm (n am + lqPe __ np m _ ~cq (E + Xe) + nO~m(E + Xm) + da(n ---q) X m l n
+
n
c k T ( n -- q)
2Nq
ce
/
qn ¢)
~Xm ao
(37)
(Sm + qS¢)
The derivative of the reaction field Xm is given by Ip qP c , __ a Xmoa = Bd 3c¢% re(Stm __ S*m) + ~ (So -- S*¢) -- 47rA olmES m ee
acqEn Sol
(38)
171
Substituting eqn. (38) into eqns. (35)--(37), the four linear equations can be solved for the derivatives S~, S~, Set and S~; hence, the differential capacity may be calculated on the basis of eqn. (30) with eqns. (33) and (34). The integral capacity of the monolayer is defined by the equation
(39)
K~ = o/A¢
Because the dipole potential drop cannot be measured experimentally, the integral capacity has been defined in an alternate manner with the dipole potential at the pzc being assumed to equal zero. The corresponding "experimental" integral capacity is thus El e
=
(//(A¢ -- A•o )
(40)
where A¢o is the dipole potential at the pzc APPLICATION TO EXPERIMENTAL DATA
The data considered in this paper are those obtained by Grahame [21] for the mercury/water interface with NaF as electrolyte in the temperature range 0-85 ° C. Assuming that the capacity of the diffuse layer is given by the Gouy-Chapman theory, Grahame showed that the estimated inner-layer capacity was approximately independent of electrolyte concentration at a given temperature, and thereby, concluded that ionic specific adsorption was negligible from this solution over the polarizable range of mercury. Inner layer capacity data at 0 ° C show three extrema as a function of electrode charge density [21]. A capacity m a x i m u m occurs just positive of the pzc at +0.02 C m-2; minima are observed at both positive (0.10 C m -2) and negative charge densities (--0.13 C m-Z). As the temperature is raised the height of the m a x i m u m decreases and that of the positive minimum increases, until at the highest temperature, extrema are difficult to distinguish at positive charge densities; on the other hand, the value of the capacity at the negative minimum is
o
O3 t~
¢Y O2 o O
I OI
I
I
0
-0 I
o
O
O
I -0 2
~ - / C m-~
Fig. 2. F i t o f t h e cluster m o d e l t o d a t a for t h e i n n e r - l a y e r c a p a c i t y o f t h e H g / a q u e o u s N a F i n t e r f a c e at 2 7 3 K [21 ]. T h e p a r a m e t e r s used in o b t a i n i n g t h e fit are s u m m a r i z e d in T a b l e 1.
172 I
I
I
I
O3 ~'E
02 o o
I
i
I
Ol
0
-01
0
0
0
I -02
c - / C m-~
Fig. 3. Fit of the cluster model to inner-layer capacity data for the Hg/aqueous NaF interface at 298 K [21].
virtually independent of temperature (Figs. 2--5). In a previous paper dealing with inner layer structure in aprotic solvents [ 14], it was shown that the position of extrema on plots of C, against o could be used to extract information regarding the parameters describing solvent adsorption at the electrode. However, because of the complex nature of the equilibrium relations in the above model which considers species of different size, namely, monomers and clusters the interpretation of the extrema on the capacity curves is not simple. Fitting the above model to Grahame's data requires choosing 11 parameters, namely Pro, ~m, d, Ce, n, Pc, ac, U~r, U~r, U~r and U~. Three of these quantities were fixed on the basis of known experimental properties of water in the gas phase. Accordingly, P m = 1.84 Debyes (6.14 X 10 -3° C m), ~m = 1.5 × 10 -3 nm 3
I
l
I
1
O3 ~E
\o
\
J O2 o
o o
I
I
I
01
0
-01
o- /
o
o
o
I -02
C.i ~
Fig. 4. Fit of the cluster model to inner-layer capacity data for the Hg/aqueous NaF interface at 318 K [21].
173 I
o3
I
o\
i
o~
u_
\o O2 o
0 0
I
I
J
01
0 ~ " / C r6z
-0 1
o
0
0
J -0 2
Fig. 5. Fit of the cluster model to inner-layer capacity data for the Hg/aqueous NaF interface
at 338 K [21].
and d = 0.32 nm. The value chosen for the diameter of the water molecule represented as a sphere is similar to those used by Parsons [8] and Damaskin [13]. From the calculations of Levine et al. [6], the effective coordination number Ce should be close to 15, but it has been found t h a t small variations in this parameter can have a very large effect on calculated capacity curves [22]. For this reason, Ce was considered an adjustable parameter. Previous calculations have assumed that n = 2 or 3 [8,13] ; in the present analysis, n was varied to determine the best value. The dipole m o m e n t of the cluster Pc is also an adjustable parameter but, on the basis of hydrogen bonding models, it is certainly expected to be considerably less than that of an isolated water molecule [8]. One might argue that the polarizability of a cluster be set equal to " n " times that of a single solvent molecule. However, since the electrical field will distort the rather weak hydrogen bonds as well as the intramolecular bonds, ac was allowed to assume larger values than nam. Other than the four parameters just discussed, one must also determine the four residual energies. Since one is always dealing with a difference in residual energies, this determination actually involves only three independent parameters. For the sake of comparison, the residual energy of clusters in the down orientation, U~r is thus set equal to zero. The fitting of the inner-layer capacity vs. charge--density curves was performed by a ravine search m e t h o d in which one looks for a m i n i m u m in the variance, v, as a function of the seven independent variables. The variance is given by v-
1 ~ (np --nv) ,
[C,(ex) -- C,(calc)] 2
(41)
where np is the number of experimental points and nv the number of independent variables, the error in C,(ex) being assumed independent of o. Two dimensional cross-sections of the eight-dimensional space defining v were examined to determine the m i m i m u m in v with respect to a given adjustable parameter. It is important to note that a course grid mapping of v was initially performed for
174 TABLE 1 P a r a m e t e r s o b t a i n e d in f i t t i n g t h e cluster m o d e l t o e x p e r i m e n t a l d a t a for t h e H g / a q u e o u s N a F i n t e r f a c e in t h e t e m p e r a t u r e range 0 - - 6 5 ° C N u m b e r o f m o l e c u l e s in cluster, n Effective c o o r d i n a t i o n n u m b e r , c e Dipole m o m e n t o f c l u s t e r in t h e d i r e c t i o n of t h e e l e c t r o d e field, Pc a Cluster p o l a r i z a b i l i t y , ac Residual energ~ of w a t e r m o n o m e r in " u p " o r i e n t a t i o n , U~nr Residual e n e r g y of w a t e r m o n o m e r in ' d o w n ' o r i e n t a t i o n , U~mr Residual e n e r g y of w a t e r cluster in ' u p ' o r i e n t a t i o n , Uctr a Residual e n e r g y o f w a t e r cluster in ' d o w n ' o r i e n t a t i o n , Ue~r
2.46 15.365 1 . 9 1 4 X 1 0 - 3 ° - - 4 . 1 9 X 10 -33 t C m 1 . 3 0 9 × 10 -2 n m 3 3.62 k J tool -I 0.215 kJ mol -I 0 . 6 8 8 + 0 . 0 3 7 1 t k J tool -I 0 k J tool -1
a These p a r a m e t e r s varied linearly w i t h t e m p e r a t u r e in t h e range s t u d i e d ; t h e y are r e p o r t e d as linear f u n c t i o n s o f t h e Celsius t e m p e r a t u r e , t.
reasonable ranges of each parameter in order to ensure t h a t a local minimum was not involved. The number of experimental points was considerably greater than the number of parameters, a typical value being 30. In all cases the m i n i m u m in v was sufficiently sharp to determine the given parameter to within better than 1%, the fit being especially sensitive to n, c~, Pc, and ac. The best fits for data at 0, 25, 45 and 65°C are shown in Figs. 2--5, the corresponding parameters being summarized in Table 1. The fits are excellent in the range --0.10 < o ~< 0.10 C m -2, but the theoretical curves lie above the experimental ones at more negative charge densities. The present model is more successful in describing the experimental data than those proposed by Parsons [8] and Damaskin [13]. According to the present analysis, the dipole m o m e n t of the average cluster, and thus the conformation of monomers within it, changes with temperature. The cluster dipole m o m e n t at 0°C (1.91 × 10 -3° C m m) is considerably less than values obtained by Parsons (5.16 × 10 -30 C m) and Damaskin (4.42 × 10 -30 C m) who assumed that a cluster was made up of three monomers. Calculations based on the properties of water clusters in the gas phase suggest that the dipole m o m e n t of a dimer is zero [10,11] and t h a t of a trimer 4.24 × 10 -30 C m [23] ; the present result for clusters containing an average of 2.5 monomers seems reasonable within this context. The high value of the cluster polarizability is a reflection of the fact that the electrical field affects intermolecular bonding in the cluster. If the monomers in the cluster are represented as conducting spheres, the estimated polarizability is 1.01 × 10 -2 nm3; thus, the value f o u n d for the best fit (1.31 × 10 -2 nm 3) is n o t unreasonable large. Finally, the effective coordination number, an essential feature of the present model, is close to the value estimated by Levine et al. (15.2) [6]; however, it must be emphasized t h a t the fit to the experimental data is extremely sensitive to this parameter such t h a t departures of c~ from the quoted value by less than 0.1% resulted in unacceptable fits. The calculated distribution of solvent species in the monolayer at 25°C is
175 I
I
I
!
o6
~ m
04
--
g I'-(9
02--
b_
f: f:
I 0 ]
I 0
I -0 I
-0 2
0-/C niz Fig. 6. Plot of the fractions of molecules in the various orientations, m o n o m e r s
" u p " (f~),
m o n o m e r s " d o w n " ( / i ) , clusters " u p " ( t ) a n d clusters " d o w n " ( ~ ) as a f u n c t i o n o f elect r o d e charge d e n s i t y at 2 9 8 K.
shown in Fig. 6. As expected, clusters predominate at low electrode charge densities, the concentration of clusters in " u p " and " d o w n " orientations being equal at a charge density corresponding to the m a x i m u m on the capacity curve (+0.04 C m -2) (Fig. 3). At this point, the fraction of monomers in the " d o w n " orientation is greater than that in the " u p " orientation, resulting in a net negative polarization of the dipole monolayer. At the pzc, the fraction of the surface covered by " u p " clusters is greater than that covered by " d o w n " clusters, but the " d o w n " m o n o m e r fraction is still greater than the " u p " fraction; the net result is that the polarization of the monolayer is still negative but small, the estimated value of A¢ 0 being --0.017 V. The minima on the capacity curve correspond to the points where the surface concentrations of " u p " or " d o w n " clusters reach a maximum. Estimates of the surface potential by Parsons [8] and Damaskin [13] on the basis of similar models were --0.008 and --0.003 V respectively, at 0 ° C, whereas Trasatti obtained a value 6 f - - 0 . 0 7 0 V at 25°C on the basis of the analysis of t h e r m o d y n a m i c data using extrathermodynamic assumptions [24,25]. A critical test of any model for interfacical solvent properties is its ability to predict the temperature variation of the dielectric properties of the inner layer. As pointed out by Parsons [12] and Damaskin [13], a model which correctly describes the dependence of the inner-layer capacity on temperature and electrode charge density also describes the variation in entropy of formation of the monolayer with charge density. In this regard, the present model is quite successful (Fig. 7), especially at positive charge densities; however, at negative charge densities, the calculated values of a (l/Ci)/a T fall somewhat below the experimental ones. In addition, the calculated m a x i m u m in ~ (1/C,)/a T occurs close to the pzc, whereas the experimental m a x i m u m is at more negative values (~0.03 C m -2) [12]. The estimated temperature coefficient of the dipole potential at the pcz is positive in agreement with results obtained by Randles and Whitely [26] and Trasatti [24] (Fig. 8). This result constitutes a significant im-
176 i
i
i
16C 0
0
0
120 80 4O '~ ~E
O0
o
oJ o'~
o'~
oycW
o'o
-o'~
-o'~
Fig. 7. Plot of a(1/C1)/OTagainst electrode charge density o for the Hg/aqueous NaF interface for capacity data in the temperature range 273--388 K: ( ) according to present model, (o) experimental data.
provement with respect to the previous cluster models [8,13] which predict a negative value of a A¢0/a T. Both the values of A¢ 0 and a A¢0/a T obtained here are smaller than those estimated by Trasatti [24], but follow the same trend in that the magnitude of a A¢0/a T decreases with increasing temperature. Randles and Whitely [23] estimated that aAO0/aT is equal to 0.6 mV °C-1 in the tem-
] THISWOR-K- ~ /
~ 0
I ~
0
~
/
"
x
~TRASATTI X ~
X
-01
0
I
I
40 T/°C
80
Fig. 8. Plot of the potential drop across the solvent in the inner layer at the point of zero charge on Hg as a function of temperature according to the present model (o) and from calculations by Trasatti (X) [24].
177
perature range 0--35 ° C, whereas the present calculation gives a value of 0.35 mV o C_1. In view of the extrathermodynamic assumptions involved in the analyses by Randles and Whitely [26] and Trasatti [24,25], further assessment is n o t possible. The fact that the best values of Pc and U¢*~vary with temperature merits comment. The properties of a cluster at the metal/solution interface are determined not only by intermolecular hydrogen bonding but also by hydrogen bonding with solvent molecules in the layer adjacent to the monolayer being considered, and by interaction of the lone pairs of electrons in the water molecule with the metal. Apparently the relative contributions of these factors are such that the net dipole m o m e n t of a cluster perpendicular to the interface decreases with increase in temperature. The corresponding increase in Uc*~-- U¢*~indicates a destabilization of the " u p " orientation of the cluster with respect to the " d o w n " configuration, because of changing lone-pair interactions and hydrogen bonding with molecules b e y o n d the monolayer. If this assessment is correct, then it follows that the model does not accurately describe the change in free energy of the clusters with electrode charge density. As pointed o u t above, the high value of the cluster polarizability ae can be attributed to changes in intermolecular configuration in the cluster with the electrode field. The configurational change would then result in a change in U~ and U~ with o. This defect in the model is u n d o u b t e d l y the main reason for the poor fit obtained at higher negative charge densities (Figs. 2--5). DISCUSSION
The cluster models presented above and previously [8,13] for associated solvents, and a three-state model proposed for unassociated solvents [ 14], suggest that a large fraction of solvent molecules in the monolayer adjacent to the metal are oriented in directions other than that of the electrode field. As a result, the potential drop across the monolayer, and its change with electrode charge density are much less than would be estimated on the basis of two-state models [22,27]. Furthermore, these models appear to describe the properties of the interfacial solvent quite well on the basis of reasonable molecular parameters. The weakness of the cluster models is that one cannot experimentally determine the dipolar properties of the solvent clusters at the interface. It follows that fitting the theory to experimental data requires estimation of more adjustable parameters. The present model is more detailed than those presented previously [8,13] in that it treats dipole--dipole interactions on a molecular level. Thus, the introduction of an effective dielectric constant for the monolayer which is a feature of most work on this subject is avoided. Estimation of dipole--dipole interactions is especially difficult due to the necessity of considering imaging in the electrode and solution; however, on the basis of the above results and those obtained for unassociated solvents [14,22], the parameter ce introduced by Levine et al. [16--18] appears to be an effective way of accounting for image effects, and thus, of estimating the local field in the monolayer. The model presented by Damaskin [13] treats the contribution to dipole--dipole interactions due to polarizability through the introduction of an effective dielectric constant e
178 which is independent of electrode charge. This assumption is equivalent to assuming that the average molecular environment of a given dipole does not change with electrode charge density in so far as estimation of the polarization contribution to the local field is concerned. A particularly n o t e w o r t h y result of Damaskin's study is that an acceptable fit to experimental data could be obtained only when the permanent dipole m o m e n t of a water m o n o m e r on the surface was assumed to be twice the value measured in the gas phase. The larger value of Pm was attributed to chemical interaction between the solvent dipole and the substrate metal. Although one must consider a field-independent shift in the chemical potential of a given dipolar c o m p o n e n t of the monolayer due to this interaction, namely the residual energy, it is difficult to understand w h y the field-dependent contribution to the free energy should be so different from that estimated on the basis of the dipole m o m e n t and polarizability of the isolated molecule. Cooper and Harrison [28] have severely criticized the general approach taken here and in previous papers [1--14] in which the properties of a solvent monolayer in the inner layer are described on the basis of a molecular model and the remainder of the double layer, on the basis of continuum concepts. The present treatment relies on values of inner-layer capacity obtained from the experimental differential capacity after correction for the diffuse-layer capacity on the basis of the Gouy--Chapman theory. Although the Gouy--Chapman model is und o u b t e d l y incorrect at higher electrode charge densities, the diffuse-layer contribution to the total capacity is unimportant in these regions. In fact, Grahame [21] obtained acceptable agreement between inner-layer capacities calculated from experimental capacities measured at the same charge density b u t for different electrolyte concentrations. However, the present approach does n o t account for specific interactions between the molecules and ions on the boundary of the diffuse layer (oHp) and the solvent monolayer which solvates the electrode. Moreover, it ignores the possibility that ions on the oHp may retain their primary solvation sheath so that the thickness of the inner region would depend on the nature of the counter ion at the o H p and correspond on the average to more than one solvent molecule in a direction perpendicular to the interface [29]. Specific ionic effects are observed on inner-layer capacity curves for the alkali metal cations at far negative charge densities, but it is not clear whether these effects are due to cation-specific adsorption [30] or ionic size effects [29]. Ignoring the possibility that ions are present in the solvent monolayer being considered here, it is clear that the first layer of solvent molecules will have the major influence on the dielectric properties of the inner layer since it is most strongly influenced by the electrode field. In general, one should not ignore the influence of ions and solvent molecules b e y o n d the first monolayer, b u t they are expected to play a secondary role in determination of innerlayer properties if contact ionic specific adsorption is absent. In order to deal with these effects, the molecular model should be extended further from the electrode into the diffuse layer. Since one must then deal with species of different sizes and with ion--ion and ion--dipole interactions as well as dipole--dipole interactions, such an extension represents a much more difficult problem. In the meanwhile, results such as those obtained here may be considered to give a useful approximate description of solvent properties in the inner layer.
179
'E ?~ -20C
- 40C 012
011
0
-011
-012
,r'/c ~ 2
Fig. 9. The entropy of the monolayer, S according to the present model estimated from the temperature dependence of the free energy (eqn. 21) and plotted as a function of electrode charge density o.
Finally, as pointed out above, the experimentally determined maximum in the entropy of formation of the compact layer occurs at negative charge densities at approximately --0.05 C m-: [15]. To the extent that the present model is able to describe the charge and temperature dependence of the inner-layer capacity, it is also able to describe the variation in entropy of the inner-layer solvent with electrode charge density [12]. An analytical expression for the entropy of the monolayer could be derived by differentiating eqn. (21) with respect to temperature. In the present instance, the entropy was estimated by calculating the free energy of the monolayer on the basis of eqn. (21) and by determining the temperature derivative a F / a T = - - S numerically. The maximum entropy of the monolayer (Fig. 9) occurs at approximately the same charge density as reported by Harrison et al. [15]. It should be noted that this entropy maximum for the solvent monolayer does not coincide with the maximum in the configurational entropy of the monolayer (eqn. 20). According to the present model the configurational entropy maximum occurs at positive charge densities approximately at the point where the fraction of clusters in the " u p " and "down" orientations are equal (Fig. 6). The non-coincidence in the maxima for the total monolayer entropy and its configurational component implies that the internal energy of the monolayer varies significantly with temperature. This variation is due to a corresponding change in dipole--dipole interaction energy with temperature. Bockris and Habib [ 11 ] presented a model for the entropy of a water monolayer composed of monomers and dimers. However, this model can be criticised on fundamental grounds because of an incorrect formulation of the dipolar contribution to the potential drop across the monolayer, and of the equilibrium constant describing formation of dimers from monomers [22]. Although the model is able to describe variation in entropy with electrode charge, it does not properly describe the inner-layer capacity.
180 In summary, the present results serve to emphasize the importance of developing a detailed model for dipole--dipole interactions in describing the properties of solvent molecules at charged interfaces. In particular, the fact that the capacity maximum occurs at positive charge densities and the entropy maximum at negative charge densities can be explained on the basis of the present model; on the other hand, if dipole--dipole interactions are ignored [8] or treated approximately [13], one cannot account for the experimentally observed temperature dependence of the dipole potential. The present model can also be applied to describe inner-layer capacity curves observed for other associated solvents such as formamide [31] and N-methylformamide [32]. These systems will be considered in a subsequent paper. ACKNOWLEDGEMENT
The financial assistance of the Natural Sciences and Engineering Research Council, Canada, is gratefully acknowledged.
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
R.J. W a t t s - T o b m , Phil. Mag., 6 ( 1 9 6 1 ) 133. N.F. M o t t a n d R.J. Watts-Tobin, E l e c t r o c h i m . A c t a , 4 ( 1 9 6 1 ) 79. J. O'M. B o c k n s , M.A.V. D e v a n a t h a n , a n d K. Muller, Proc. R o y . Soc., A 2 7 4 ( 1 9 6 3 ) 55. B.B. D a m a s k m , E l e k t r o k l n m i y a , 1 ( 1 9 6 5 ) 1 2 5 8 ; 2 ( 1 9 6 6 ) 828. J. O'M. B o c k n s , E. Gfleach a n d K. Muller, E l e c t r o c h i m , Acta., 12 ( 1 9 6 7 ) 1 3 0 1 . S. Levine, G.M. Bell a n d A.L. S m i t h , J. Phys. Chem., 73 ( 1 9 6 9 ) 3 5 3 4 . B.B. D a m a s k i n a n d A.N. F r u m k i n , l~lectroehlm. A c t a . , 19 ( 1 9 7 4 ) 173. R. Parsons, J. E l e c t r o a n a l . Chem., 59 ( 1 9 7 5 ) 229. I.L. C o o p e r a n d J.A. H a r n s o n , J. E l e c t r o a n a l . Chem., 66 ( 1 9 7 5 ) 85. J. O'M. B o c k n s a n d M.A. Habib, J. E l e c t r o a n a l . Chem., 6 5 ( 1 9 7 5 ) 4 7 3 . J. O'M. Bockris a n d M.A. Habib, E l e c t r o c l n m . Acta., 22 ( 1 9 7 7 ) 41. R. Parsons, E l e c t r o c h l m . A c t a . , 21 ( 1 9 7 6 ) 681, B.B. D a m a s k m , J. E l e c t r o a n a l . Chem., 75 ( 1 9 7 7 ) 359. W.R. F a w c e t t , J. Phys. Chem., 82 ( 1 9 7 8 ) 1 3 8 5 . J.A. H a r n s o n , J.E.B. R a n d l e s a n d D,J. Schiffrin, J. E l e c t r o a n a l . Chem., 4 8 ( 1 9 7 3 ) 3 5 9 . S. Levine a n d A.L. Smith, I V t h Int. Congr. Surface Active Agents, Bare., BII ( 1 9 6 8 ) 191. S. Levine, A.L. S m i t h a n d E. Matijevic, J. Colloid I n t e r f a c e ScL, 31 ( 1 9 6 9 ) 409. S. Levine, K. R o b i n s o n , A.L. S m i t h a n d A.C. Brett, Discuss. F a r a d a y Soc., 59 ( 1 9 7 5 ) 133. E.A. G u g g e n h e i m a n d M.L. McGlashan, Proc. R o y . Soe., 2 0 3 A ( 1 9 5 0 ) 435. I. P n g o g i n e , A. Bellemans a n d V. M a t h o t , Molecular T h e o r y o f S o l u t i o n s , N o r t h - H o l l a n d , A m s t e r d a m , 1957. D.C. G r a h a m e , J. A m e r . Chem. Soc.0 79 ( 1 9 5 7 ) 2 0 9 3 . W.R. F a w c e t t , Israel J. Chem.. 1 8 ( 1 9 7 9 ) 3. J. Del Bene a n d J.A. Pople, J. Chem. Phys., 52 ( 1 9 7 0 ) 4 8 5 8 . S. T r a s a t h , J. E l e c t r o a n a l . Chem., 82 ( 1 9 7 7 ) 391. A. de Battisti a n d S. Trasattl, Croat. Chem. Acta., 48 ( 1 9 7 6 ) 6 0 7 . J.E.B. R a n d l e s a n d K.S. Whltely, Trans. F a r a d a y Soe., 52 ( 1 9 5 6 ) 1 5 0 9 . R. Parsons, Trans. S.A.E.S.T., 13 ( 1 9 7 8 ) 239. I.L. C o o p e r a n d J.A. H a r r i s o n , J. E l e c t r o a n a l . Chem., 86 ( 1 9 7 8 ) 4 2 5 . W.R. F a w c e t t , J. E l e c t r o a n a l . Chem., 39 ( 1 9 7 2 ) 4 7 4 . B.B. D a m a s l u n , J. E l e c t r o a n a l . Chem., 6 5 ( 1 9 7 5 ) 799. E. D u t k i e w l c z a n d R. Parsons, J. E l e c t r o a n a l . Chem., 11 ( 1 9 6 6 ) 196. W.R. F a w e e t t a n d R.O. L o u t f y , J. E l e c t r o a n a l . Chem., 39 ( 1 9 7 2 ) 185.
163
© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
A MOLECULAR MODEL FOR THE DIELECTRIC PROPERTIES OF THE INNER LAYER AT THE MERCURY/AQUEOUS SOLUTION INTERFACE
W.R. FAWCETT, S. LEVINE, R.M. deNOBRIGA and A.C. McDONALD
Guelph--Waterloo Centre for Graduate Work in Chemlstry (Guelph Campus), Department of Chemistry, University of Guelph, Guelph, Ontario N I G 2W1 (Canada) (Received 9th August 1979; in revised form 29th January 1980)
ABSTRACT A four-state model for a monolayer of water molecules at a charged polarizable interface is presented. The states considered are water monomers and clusters, each of which can assume two orientations in the field due to the charge, i.e. in the direction of the field or against it. The clusters are composed of water molecules whose orientation with respect to one another is determined by hydrogen bonding and are assumed to have a net dipole moment much less than that of a m o n o m e r in the direction of the field. The model includes a detailed description of dipole--dipole interactions with consideration of both orientational and distortional components of molecular polarization. Expressions are developed for the equilibrium concentrations of the four components of the monolyaer, for the potential drop across the monolayer and its differential capacity. It is shown that the model provides a reasonable description of the inner layer capacity curves for the mercury/aqueous solution interface obtained in the absence of ionic specific adsorption in the temperature range 0--85°C. In addition, it is able to account for the fact that the maximum entropy of formation of the inner layer occurs at negative electrode charge densities, and that the temperature coefficient of the potential drop across the inner layer at the point of zero charge is positive.
INTRODUCTION
Molecular theories based on statistical mechanics have been proposed by many workers [1--14] to describe the solvent properties at the electrode/electrolyte solution interface, particularly the mercury/aqueous solution system. From the inception of such molecular models, a considerable degree of controversy among the different authors has been a characteristic feature. The models of the mercury/aqueous electrolyte interface discussed in this paper are based on a monolayer of water molecules, situated in v a c u o between the metal surface and a dielectric containing ions, i.e. the diffuse layer. The monolayer is identified with the inner S t e m region and specific ion adsorption is assumed absent. Prior to a paper by Damaskin and Frumkin [7], the water molecules in the monolayer were treated as separate kinetic entities, which usually assumed one of t w o orientations in the electrode field, with the oxygen pointing towards or away from the metal surface. Experimentally, a maximum in the differential capacity of the inner region (which, it should be noted, is n o t measured directly) is calculated from positive of the point of zero charge (pzc) for the mercury/
164
aqueous electrolyte interface in the absence of specific ionic adsorption. The two-state orientational model could account for this maximum with the postulate that the natural orientation of the water molecules in the monolayer at the pzc has the oxygen atom pointing away from the mercury. However, the experimental surface excess entropy, and the corresponding entropy of formation of the interface pass through a m a x i m u m on the negative side of the pzc [15], indicating that the preferred orientation of water on mercury corresponds to oxygen towards the metal. The behaviour of the temperature coefficient of the potential drop across the inner region is consistent with the latter orientation of the water molecules at the pzc. In most of the two-state models, the solvent monolayer is assumed to have an effective dielectric permittivity which accounts for all contributions to this permittivity other than that due to the permanent solvent dipoles. In the case of water, a typical value for this inner region permittivity is 6. Without such a permittivity, the potential drop across the monolayer region is found to be t o o large (by the factor 6). In contrast, Levine et al. [6] avoided the use of this reduction factor by considering polarizable water dipoles situated in a vacuum. In this treatment, the effective permittivity of the monolayer, which is derived b y statistical mechanics from a molecular model of the monolayer, varies significantly with electrode charge density. The lateral dipole--dipole interaction is n o w so strong that in the presence of an electrode field, the fractions of dipoles pointing parallel and anti-parallel to this field are nearly equal. Nevertheless, the potential drop across the inner region can still be t o o high. Also, although the two-state model of Levine et al. [6] reproduces the general shape of the experimental capacity--charge curve in the region of the pzc, it fails to account for this curve at the higher potential, away from the pzc. A molecular model of the inner region which treats the water molecules as polarizable dipoles must be inadequate since it neglects the effect of hydrogen bonding. Furthermore, the role of hydrogen bonding is probably masked by introducing an effective dielectric permittivity, which should therefore be avoided. Damaskin and Frumkin [7] have shown that the overall qualitative features of the differential capacity electrode charge curve for inner region can be reproduced if one assumes that associates or clusters of water molecules with a low net dipole m o m e n t perpendicular to the interface are present in the solvent layer immediately next to the electrode, in addition to individual water molecules. These clusters can be regarded as a manifestation of hydrogen bonding between the water molecules in the inner region. Parsons [8] and Damaskin [13] have developed further this model of a monolayer of clusters and single water molecules. The important achievement of this theory over previous ones is that the calculated capacity--charge curve contains the essential experimental features of a maximum and t w o minima. These authors assume an effective dielectric permittivity to the inner region and do not introduce the polarizability of their molecular units. In a series of papers, Levine et al. [16--18] have extended the theoretical treatment of the inner region described in ref. 6 to a theory of adsorption from a mixture of t w o polar solvents. In the present paper, this theory is applied to develop a model for the dielectric properties of the inner region in associated solvents, and to analyse the inner layer differential capacity data for the mercury/aqueous solution interface.
165 MODEL OF THE SOLVENT IN THE INNER LAYER
The model presented is a two-dimensional version of models developed earlier for the t h e r m o d y n a m i c properties of polymer solutions containing open chain polymers * of varying chain length [20]. The following assumptions are made. (1) The solvent in the inner region is present as a two-dimensional hexagonal lattice of fully occupied adsorption sites with a number density Ns per unit area. The solvent molecule is represented as a hard sphere with diameter d, so that Ns = 2/(\/3d2); the number of nearest neighbours (nn) for a given element of the monolayer (coordination number, c) is 6 and the nn distance is d. (2) One site is occupied by an unassociated solvent molecule and n sites by a cluster consisting of n elements (solvent molecules). Since the cluster is assumed to have an open configuration, each element at the end of chain has 5 nn and each internal element, 4 nn. It follows that the total number of nn interactions experienced by a cluster is qc = cn -- 2n + 2. In the following an unassociated solvent molecule or an element of a cluster is referred to as a particle. (3) Unassociated solvent molecules and clusters are oriented in one of two directions referred to as " u p " and " d o w n " . The " u p " position signifies that the positive end of the dipole is towards the metal, and the " d o w n " position, that the negative end is towards the metal (Fig. 1). The components of the permanent dipole moments in a direction perpendicular to the interface for unassociated monomers are Pm and --Pro in the " u p " and " d o w n " orientation respectively; the corresponding quantities for clusters are Pc and --Pc. (4) The induced dipole moments Pro1 and Pc1 for monomers and clusters respectively, refer to the " u p " direction. Here, Pc, is equally distributed among the n particles so t h a t each particle has an induced dipole m o m e n t of Pci/n. (5) The external field E = 4~ro in the " d o w n " direction is due to the charge density a on the metal. Here, Xm and Xc are the electric fields in the " d o w n " direction at a lattice site occupied by an unassociated solvent molecule and a particle of a cluster, respectively. (6) The induced dipole m o m e n t s are Pm~ = --am(E + Xm) and Pc~ = --~c(E + Xc) where ~m and ~c are the polarizabilities of a m o n o m e r and cluster, respectively. Let N*~, N~ be the numbers per unit area of monomers and clusters in " u p " positions and N~, N~ those in " d o w n " positions. The total numbers o f each species per unit area are Arm = N ~ + N ~ and Nc = N~ + N~. In order to estimate dipole--dipole interactions, one must distinguish the different types of neighbouring pairs among the monolayer elements. For monomers in the " u p " orientation,
cN*m : 2N~nm t t + Ntm*m + N ~ c + N~¢
(1 )
where Jvt..m*m,JVt..~m,N ~ t and N~c represent the numbers per unit area of neigh-
* It should be noted that Guggenheim and McGlashan have c o n s i d e r e d the thermodynamic properties of polymer solutions containing closed chain polymers [ 19 ]. Since the e s t i m a t e d entropy of mixing is virtually independent of polymer configuration, and depends almost entirely on polymer size, the assumption made here that the clusters have an open configuration is not expected to have any significant effect on the results.
166 ////////////i////////////
(b)
(a)
[c) Fig. 1. Schematic representation of (a) a cluster of solvent molecules in the "up" position, (b) a cluster in the "down" orientation and (c) a monolayer composed of clusters and monomers in both orientation at a charged interface. If it is assumed that a cluster is composed of three molecules with dipole vectors at an angle of 70.5 ° to the direction of the electrode field, the net dipole moment of the cluster would be one-third that of a monomer.
b o u r i n g pairs b e t w e e n a m o n o m e r in t h e " u p " p o s i t i o n a n d each o f t h e f o u r c o m p o n e n t s o f t h e m o n o l a y e r . In t h e case o f " u p " clusters, o n e m a y write:
qcN*c = 2Ntctc + Ntc*c + Ntctm + Ntc*m
(2)
T h e c o r r e s p o n d i n g e q u a t i o n s f o r t h e o t h e r t w o o r i e n t a t i o n s are
cN~ = N~m + 2N~m + N~c + g~c
(3)
and
qcN*c-- g c c~t + 2 N ~ + N c~t ~ m +Nc~m
(4)
The t o t a l n u m b e r o f n n i n t e r a c t i n g e l e m e n t s per units area is
c N J 2 = c(Nm + qN¢)/2
(5)
F o r r a n d o m d i s t r i b u t i o n o f t h e m o n o l a y e r c o m p o n e n t s a m o n g the available sites, it has been s h o w n t h a t [ 2 0 ]
N ~ m - eNtre2 ,2Na '
N ~ m - cNtmN*m Na '
g ~ c - qcNtmN*c ga '
N ~ m - cN~m2 2N a '
N ~ c - qcN*mNtc gq '
g~c-
2 t $ Ntc*c- q cN~N~c Nq
and
N~*c ~=
q2cN~c 2
qcN~mN~c Na '
N ~ c - qcgtmN*c Nq
Ntc*~-
q2cNtc 2 2Nq (6)
2Nq
S u b s t i t u t i o n o f these relations i n t o eqns. (1)--(4) gives eqn. (5). In t h e case o f m o n o m e r s , t h e t o t a l dipole m o m e n t s are p tm = P m + Pmi a n d P ~ = --Pm + Pmi ; similar relations h o l d f o r clusters. T h e d i p o l e - - d i p o l e interac-
167
tion energies for the 10 different pairs of elements are given by U~m-P~,
U~m-
d3 '
U~m~m- d3,
U~c-
nd 3 , t2
U~c-
nd 3 ,
U~c
and
n2d3
U~c
nd 3 ,
nd 3 ,
vii
p~c 2 n2d3
U~
(7)
The total dipole--dipole interaction energy among the nn per unit area is Vdd =
E ~ g k ] v kl i,]=m,c k,l= ~,$
(8)
Defining the mean net dipole moments in the " u p " direction due to monomers and clusters, M m = N~mP~m+ N~a~m
and
M c = g~p~ + N~Pc
it is easily shown that c U d d - 2Na d3 ( M m + qMc/n) 2
(9)
(10)
The dipole--dipole interaction energy may also be expressed in terms of the reaction fields Xm and X~: V d d = lI M m X m + IM~Xc 1
(11)
1
where the factor : occurs to avoid counting nn interaction twice. In the random mixing approximation used here, each neighbouring site is responsible for the same fraction of the mean field at a given site, and since X~ should n o t include the field due to other particles of the cluster, it is assumed that (12)
X c = qXm/n
Combining eqns. (10) and (12), it is easily shown that c Xm - Nqd3 (Mm + q M d n )
(13)
Since Mm and M~ depend on Am, eqn. (13) is more conveniently written in another form. Defining R ~ = N ~ -- N~m
and
Rc = N t -- N~
(14)
one may write Xm
-
C (R qRcpc d3B mPm + n
AE
)
(15)
where A = a m N m + a~qNc/n
(16)
and B = Nq +
C (o~mgm + q2acNc] n2 ]
(17)
168 As shown previously [18], the complete expression for the electrical contribution to the free energy of adsorption of the monolayer per unit area is eel =
Vdd "{-(Mm + Me) E + °~mgm (E + Xm) 2 + acNc (E + Xc) 2 2 2
(18)
where the first and second terms represent the total energy of the dipoles in the external field E, and the third and fourth terms account for the work done in producing the induced dipoles. In the above derivation, dipole--dipole interactions between pairs of particles which are not nn, and resulting from dipole images in the two media between which the monolayer is situated have been ignored [6,18]. It was explained that an approximate m e t h o d of accounting for these t w o factors in the case that the monolayer consists of monomers only [6] is to increase the coordination number c to an effective value c~, where ce is expected to have a value close to 15. In the present case, where both clusters and monomers are present in the monolayer, a corresponding increase in qc to qce is expected. Therefore, in the following, eqns. (15)--(17) are used with c replaced by Ce. In addition to Uel, t w o other contributions to the Helmholtz free energy of the monolayer need to be considered. It is postulated that "residual" energies in the " u p " and " d o w n " positions U ~ , U ~ , Uc*~and U~ describe the interaction of individual water molecules and clusters respectively, with the phases at the t w o boundaries of the monolayer region. The total residual energy per unit area is therefore:
V, = N*mV*mr+ N*mU~m,+ N*cV*c,+ g*cv*cr
(19)
Also, the number of configurationsg availableto a given composition of the monolayer is assumed to be given by Flory--Hugginsstatistics [20] so that --ln g = N ~ 1n
(N~_j)__
/nN~\
m + N*m
cNq
m~-~ [N*m~
/ Ns\
+Nc* In ~-N- s ) + -~- In[~qq)
/nN~ \
+ N~* ln[-~-s )
qcN¢ l n ( q )
(20)
2
Introducing the van der Waals interaction energy U w which is assumed to be independent of the dipole interactions, the total Helholtz free energy per unit area attributed to the monolayer is F = Ue, + U~ + U w - k T l n g
(21)
where k is Boltzmann's constant and T the absolute temperature. EQUILIBRIUM RELATIONS In this section the xiistribution of monomers and clusters among the four " u p " and " d o w n " positions is derived. At equilibrium, the chemical potentials of monomers in the " u p " and " d o w n " positions are equal. This condition is conveniently expressed [16--18] by the relation:
169
Noting that ON*m/aN~m = --1, and introducing the fractions of monomers in the " u p " and " d o w n " positions, f m = N ~ / N ~ and f ~ = N~/N~, differentiation of equation (21) yields: k T In
(23)
= U ~ -- U~r -- 2pm(E + X m )
The corresponding condition concerning the number of clusters in each orientation is (24)
*m 0 which leads to k T ln([-~t = U~¢r -- U~¢~-- 2p¢(E + X~) \f'~/
(25)
where f~ = nN~c/Ns and f~ = nNCc/N~ are the fractions of solvent molecules in the " u p " and " d o w n " cluster positions. Finally, there is equilibrium with respect to the association of n monomers into a cluster. In this regard, it is sufficient to consider n monomers in the " u p " position combining to form a cluster in the " u p " position [17]. The free energy condition may be expressed as
N, ,m,N
(26)
0
where N t = NCm + nNC¢, so that (3N~c/aN~) = --1/n. Equation (26) yields: n
k T In [_ r(fm)" f~ 7_j = U~ - - n U ~ r + p~(E + X ¢ ) - - n p m ( E (E + X¢) 2 --ac
ckT
2
nC~m(E + X m ) 2 +
( n - - q ) ln
2
(~q)
+ Zm)
daX2m +
ckT +-~-(n--q)
2Ce
(n - - q) + ( 1 - - n) k T
qckT ~ In
(27)
The condition that all Ns sites per unit area are occupied by particles is fCm + fCm + fCc + f~= l
(28)
Noting that Xm, X¢ and Nq depend on the four fractions fro, f~, fct and f~ of monomers and clusters in the two orientation states, the three equilibrium relations {23), (25) and (27) and the condition (28) constitute four equations which yield these fractions as functions of the electrode field E and, therefore, the surface charge density o. Because the equilibrium equations are non-linear, an iteration process is used to determine the fractions f¢~, f~, f~ and f¢~ numerically. DIFFERENTIAL CAPACITY OF THE MONOLAYER REGION The potential drop across the monolayer (from the metal phase to the solution phase) is A¢ = 4~rod + 41r(Mm + Me)
(29)
170 The differential capacity Ci of the monolayer region is given by 1 = ()A¢ = 47rd + 4fr( ~)Mm +aMcl C~ ao \ ao ao l
(30)
It is convenient to introduce st-
aNt S*~- aV*m ao ' aa ' where, from (28)
Sc*- aNc* and ao
So*- aNt* ao
(31)
(32)
S t + S ~ + nS~ + nS~ = 0
Then,
aMm~)o- P r o ( s t --S~m) -- amSm(E + Xm) -- amNm 41r + aXm~ ao ]
(33)
and
+
oo o
+ ) .oo
where sm = s t + s t
and
so = S¢' + s¢*
By differentiating the equilibrium relations (23), (25) and (27) with respect to o, one obtains together with eqn. (32) a set of four linear equations in the unknowns S t , S*m, S¢¢ and S~. The necessary equations are kT
--k
k T ~ c --kT
m
(35)
=__87fp m __ 2 P m ( _ ~ . )
¢ = - - 8 ~ p ¢ - - - -2pcq n \~-a/
(36)
and n k T (S~t) m -- k T (S~¢¢) ~ = 4~r(p¢ -- npm) + 4~rE(nam -- a¢) + 4~Xm (n am + lqPe __ np m _ ~cq (E + Xe) + nO~m(E + Xm) + da(n ---q) X m l n
+
n
c k T ( n -- q)
2Nq
ce
/
qn ¢)
~Xm ao
(37)
(Sm + qS¢)
The derivative of the reaction field Xm is given by Ip qP c , __ a Xmoa = Bd 3c¢% re(Stm __ S*m) + ~ (So -- S*¢) -- 47rA olmES m ee
acqEn Sol
(38)
171
Substituting eqn. (38) into eqns. (35)--(37), the four linear equations can be solved for the derivatives S~, S~, Set and S~; hence, the differential capacity may be calculated on the basis of eqn. (30) with eqns. (33) and (34). The integral capacity of the monolayer is defined by the equation
(39)
K~ = o/A¢
Because the dipole potential drop cannot be measured experimentally, the integral capacity has been defined in an alternate manner with the dipole potential at the pzc being assumed to equal zero. The corresponding "experimental" integral capacity is thus El e
=
(//(A¢ -- A•o )
(40)
where A¢o is the dipole potential at the pzc APPLICATION TO EXPERIMENTAL DATA
The data considered in this paper are those obtained by Grahame [21] for the mercury/water interface with NaF as electrolyte in the temperature range 0-85 ° C. Assuming that the capacity of the diffuse layer is given by the Gouy-Chapman theory, Grahame showed that the estimated inner-layer capacity was approximately independent of electrolyte concentration at a given temperature, and thereby, concluded that ionic specific adsorption was negligible from this solution over the polarizable range of mercury. Inner layer capacity data at 0 ° C show three extrema as a function of electrode charge density [21]. A capacity m a x i m u m occurs just positive of the pzc at +0.02 C m-2; minima are observed at both positive (0.10 C m -2) and negative charge densities (--0.13 C m-Z). As the temperature is raised the height of the m a x i m u m decreases and that of the positive minimum increases, until at the highest temperature, extrema are difficult to distinguish at positive charge densities; on the other hand, the value of the capacity at the negative minimum is
o
O3 t~
¢Y O2 o O
I OI
I
I
0
-0 I
o
O
O
I -0 2
~ - / C m-~
Fig. 2. F i t o f t h e cluster m o d e l t o d a t a for t h e i n n e r - l a y e r c a p a c i t y o f t h e H g / a q u e o u s N a F i n t e r f a c e at 2 7 3 K [21 ]. T h e p a r a m e t e r s used in o b t a i n i n g t h e fit are s u m m a r i z e d in T a b l e 1.
172 I
I
I
I
O3 ~'E
02 o o
I
i
I
Ol
0
-01
0
0
0
I -02
c - / C m-~
Fig. 3. Fit of the cluster model to inner-layer capacity data for the Hg/aqueous NaF interface at 298 K [21].
virtually independent of temperature (Figs. 2--5). In a previous paper dealing with inner layer structure in aprotic solvents [ 14], it was shown that the position of extrema on plots of C, against o could be used to extract information regarding the parameters describing solvent adsorption at the electrode. However, because of the complex nature of the equilibrium relations in the above model which considers species of different size, namely, monomers and clusters the interpretation of the extrema on the capacity curves is not simple. Fitting the above model to Grahame's data requires choosing 11 parameters, namely Pro, ~m, d, Ce, n, Pc, ac, U~r, U~r, U~r and U~. Three of these quantities were fixed on the basis of known experimental properties of water in the gas phase. Accordingly, P m = 1.84 Debyes (6.14 X 10 -3° C m), ~m = 1.5 × 10 -3 nm 3
I
l
I
1
O3 ~E
\o
\
J O2 o
o o
I
I
I
01
0
-01
o- /
o
o
o
I -02
C.i ~
Fig. 4. Fit of the cluster model to inner-layer capacity data for the Hg/aqueous NaF interface at 318 K [21].
173 I
o3
I
o\
i
o~
u_
\o O2 o
0 0
I
I
J
01
0 ~ " / C r6z
-0 1
o
0
0
J -0 2
Fig. 5. Fit of the cluster model to inner-layer capacity data for the Hg/aqueous NaF interface
at 338 K [21].
and d = 0.32 nm. The value chosen for the diameter of the water molecule represented as a sphere is similar to those used by Parsons [8] and Damaskin [13]. From the calculations of Levine et al. [6], the effective coordination number Ce should be close to 15, but it has been found t h a t small variations in this parameter can have a very large effect on calculated capacity curves [22]. For this reason, Ce was considered an adjustable parameter. Previous calculations have assumed that n = 2 or 3 [8,13] ; in the present analysis, n was varied to determine the best value. The dipole m o m e n t of the cluster Pc is also an adjustable parameter but, on the basis of hydrogen bonding models, it is certainly expected to be considerably less than that of an isolated water molecule [8]. One might argue that the polarizability of a cluster be set equal to " n " times that of a single solvent molecule. However, since the electrical field will distort the rather weak hydrogen bonds as well as the intramolecular bonds, ac was allowed to assume larger values than nam. Other than the four parameters just discussed, one must also determine the four residual energies. Since one is always dealing with a difference in residual energies, this determination actually involves only three independent parameters. For the sake of comparison, the residual energy of clusters in the down orientation, U~r is thus set equal to zero. The fitting of the inner-layer capacity vs. charge--density curves was performed by a ravine search m e t h o d in which one looks for a m i n i m u m in the variance, v, as a function of the seven independent variables. The variance is given by v-
1 ~ (np --nv) ,
[C,(ex) -- C,(calc)] 2
(41)
where np is the number of experimental points and nv the number of independent variables, the error in C,(ex) being assumed independent of o. Two dimensional cross-sections of the eight-dimensional space defining v were examined to determine the m i m i m u m in v with respect to a given adjustable parameter. It is important to note that a course grid mapping of v was initially performed for
174 TABLE 1 P a r a m e t e r s o b t a i n e d in f i t t i n g t h e cluster m o d e l t o e x p e r i m e n t a l d a t a for t h e H g / a q u e o u s N a F i n t e r f a c e in t h e t e m p e r a t u r e range 0 - - 6 5 ° C N u m b e r o f m o l e c u l e s in cluster, n Effective c o o r d i n a t i o n n u m b e r , c e Dipole m o m e n t o f c l u s t e r in t h e d i r e c t i o n of t h e e l e c t r o d e field, Pc a Cluster p o l a r i z a b i l i t y , ac Residual energ~ of w a t e r m o n o m e r in " u p " o r i e n t a t i o n , U~nr Residual e n e r g y of w a t e r m o n o m e r in ' d o w n ' o r i e n t a t i o n , U~mr Residual e n e r g y of w a t e r cluster in ' u p ' o r i e n t a t i o n , Uctr a Residual e n e r g y o f w a t e r cluster in ' d o w n ' o r i e n t a t i o n , Ue~r
2.46 15.365 1 . 9 1 4 X 1 0 - 3 ° - - 4 . 1 9 X 10 -33 t C m 1 . 3 0 9 × 10 -2 n m 3 3.62 k J tool -I 0.215 kJ mol -I 0 . 6 8 8 + 0 . 0 3 7 1 t k J tool -I 0 k J tool -1
a These p a r a m e t e r s varied linearly w i t h t e m p e r a t u r e in t h e range s t u d i e d ; t h e y are r e p o r t e d as linear f u n c t i o n s o f t h e Celsius t e m p e r a t u r e , t.
reasonable ranges of each parameter in order to ensure t h a t a local minimum was not involved. The number of experimental points was considerably greater than the number of parameters, a typical value being 30. In all cases the m i n i m u m in v was sufficiently sharp to determine the given parameter to within better than 1%, the fit being especially sensitive to n, c~, Pc, and ac. The best fits for data at 0, 25, 45 and 65°C are shown in Figs. 2--5, the corresponding parameters being summarized in Table 1. The fits are excellent in the range --0.10 < o ~< 0.10 C m -2, but the theoretical curves lie above the experimental ones at more negative charge densities. The present model is more successful in describing the experimental data than those proposed by Parsons [8] and Damaskin [13]. According to the present analysis, the dipole m o m e n t of the average cluster, and thus the conformation of monomers within it, changes with temperature. The cluster dipole m o m e n t at 0°C (1.91 × 10 -3° C m m) is considerably less than values obtained by Parsons (5.16 × 10 -30 C m) and Damaskin (4.42 × 10 -30 C m) who assumed that a cluster was made up of three monomers. Calculations based on the properties of water clusters in the gas phase suggest that the dipole m o m e n t of a dimer is zero [10,11] and t h a t of a trimer 4.24 × 10 -30 C m [23] ; the present result for clusters containing an average of 2.5 monomers seems reasonable within this context. The high value of the cluster polarizability is a reflection of the fact that the electrical field affects intermolecular bonding in the cluster. If the monomers in the cluster are represented as conducting spheres, the estimated polarizability is 1.01 × 10 -2 nm3; thus, the value f o u n d for the best fit (1.31 × 10 -2 nm 3) is n o t unreasonable large. Finally, the effective coordination number, an essential feature of the present model, is close to the value estimated by Levine et al. (15.2) [6]; however, it must be emphasized t h a t the fit to the experimental data is extremely sensitive to this parameter such t h a t departures of c~ from the quoted value by less than 0.1% resulted in unacceptable fits. The calculated distribution of solvent species in the monolayer at 25°C is
175 I
I
I
!
o6
~ m
04
--
g I'-(9
02--
b_
f: f:
I 0 ]
I 0
I -0 I
-0 2
0-/C niz Fig. 6. Plot of the fractions of molecules in the various orientations, m o n o m e r s
" u p " (f~),
m o n o m e r s " d o w n " ( / i ) , clusters " u p " ( t ) a n d clusters " d o w n " ( ~ ) as a f u n c t i o n o f elect r o d e charge d e n s i t y at 2 9 8 K.
shown in Fig. 6. As expected, clusters predominate at low electrode charge densities, the concentration of clusters in " u p " and " d o w n " orientations being equal at a charge density corresponding to the m a x i m u m on the capacity curve (+0.04 C m -2) (Fig. 3). At this point, the fraction of monomers in the " d o w n " orientation is greater than that in the " u p " orientation, resulting in a net negative polarization of the dipole monolayer. At the pzc, the fraction of the surface covered by " u p " clusters is greater than that covered by " d o w n " clusters, but the " d o w n " m o n o m e r fraction is still greater than the " u p " fraction; the net result is that the polarization of the monolayer is still negative but small, the estimated value of A¢ 0 being --0.017 V. The minima on the capacity curve correspond to the points where the surface concentrations of " u p " or " d o w n " clusters reach a maximum. Estimates of the surface potential by Parsons [8] and Damaskin [13] on the basis of similar models were --0.008 and --0.003 V respectively, at 0 ° C, whereas Trasatti obtained a value 6 f - - 0 . 0 7 0 V at 25°C on the basis of the analysis of t h e r m o d y n a m i c data using extrathermodynamic assumptions [24,25]. A critical test of any model for interfacical solvent properties is its ability to predict the temperature variation of the dielectric properties of the inner layer. As pointed out by Parsons [12] and Damaskin [13], a model which correctly describes the dependence of the inner-layer capacity on temperature and electrode charge density also describes the variation in entropy of formation of the monolayer with charge density. In this regard, the present model is quite successful (Fig. 7), especially at positive charge densities; however, at negative charge densities, the calculated values of a (l/Ci)/a T fall somewhat below the experimental ones. In addition, the calculated m a x i m u m in ~ (1/C,)/a T occurs close to the pzc, whereas the experimental m a x i m u m is at more negative values (~0.03 C m -2) [12]. The estimated temperature coefficient of the dipole potential at the pcz is positive in agreement with results obtained by Randles and Whitely [26] and Trasatti [24] (Fig. 8). This result constitutes a significant im-
176 i
i
i
16C 0
0
0
120 80 4O '~ ~E
O0
o
oJ o'~
o'~
oycW
o'o
-o'~
-o'~
Fig. 7. Plot of a(1/C1)/OTagainst electrode charge density o for the Hg/aqueous NaF interface for capacity data in the temperature range 273--388 K: ( ) according to present model, (o) experimental data.
provement with respect to the previous cluster models [8,13] which predict a negative value of a A¢0/a T. Both the values of A¢ 0 and a A¢0/a T obtained here are smaller than those estimated by Trasatti [24], but follow the same trend in that the magnitude of a A¢0/a T decreases with increasing temperature. Randles and Whitely [23] estimated that aAO0/aT is equal to 0.6 mV °C-1 in the tem-
] THISWOR-K- ~ /
~ 0
I ~
0
~
/
"
x
~TRASATTI X ~
X
-01
0
I
I
40 T/°C
80
Fig. 8. Plot of the potential drop across the solvent in the inner layer at the point of zero charge on Hg as a function of temperature according to the present model (o) and from calculations by Trasatti (X) [24].
177
perature range 0--35 ° C, whereas the present calculation gives a value of 0.35 mV o C_1. In view of the extrathermodynamic assumptions involved in the analyses by Randles and Whitely [26] and Trasatti [24,25], further assessment is n o t possible. The fact that the best values of Pc and U¢*~vary with temperature merits comment. The properties of a cluster at the metal/solution interface are determined not only by intermolecular hydrogen bonding but also by hydrogen bonding with solvent molecules in the layer adjacent to the monolayer being considered, and by interaction of the lone pairs of electrons in the water molecule with the metal. Apparently the relative contributions of these factors are such that the net dipole m o m e n t of a cluster perpendicular to the interface decreases with increase in temperature. The corresponding increase in Uc*~-- U¢*~indicates a destabilization of the " u p " orientation of the cluster with respect to the " d o w n " configuration, because of changing lone-pair interactions and hydrogen bonding with molecules b e y o n d the monolayer. If this assessment is correct, then it follows that the model does not accurately describe the change in free energy of the clusters with electrode charge density. As pointed o u t above, the high value of the cluster polarizability ae can be attributed to changes in intermolecular configuration in the cluster with the electrode field. The configurational change would then result in a change in U~ and U~ with o. This defect in the model is u n d o u b t e d l y the main reason for the poor fit obtained at higher negative charge densities (Figs. 2--5). DISCUSSION
The cluster models presented above and previously [8,13] for associated solvents, and a three-state model proposed for unassociated solvents [ 14], suggest that a large fraction of solvent molecules in the monolayer adjacent to the metal are oriented in directions other than that of the electrode field. As a result, the potential drop across the monolayer, and its change with electrode charge density are much less than would be estimated on the basis of two-state models [22,27]. Furthermore, these models appear to describe the properties of the interfacial solvent quite well on the basis of reasonable molecular parameters. The weakness of the cluster models is that one cannot experimentally determine the dipolar properties of the solvent clusters at the interface. It follows that fitting the theory to experimental data requires estimation of more adjustable parameters. The present model is more detailed than those presented previously [8,13] in that it treats dipole--dipole interactions on a molecular level. Thus, the introduction of an effective dielectric constant for the monolayer which is a feature of most work on this subject is avoided. Estimation of dipole--dipole interactions is especially difficult due to the necessity of considering imaging in the electrode and solution; however, on the basis of the above results and those obtained for unassociated solvents [14,22], the parameter ce introduced by Levine et al. [16--18] appears to be an effective way of accounting for image effects, and thus, of estimating the local field in the monolayer. The model presented by Damaskin [13] treats the contribution to dipole--dipole interactions due to polarizability through the introduction of an effective dielectric constant e
178 which is independent of electrode charge. This assumption is equivalent to assuming that the average molecular environment of a given dipole does not change with electrode charge density in so far as estimation of the polarization contribution to the local field is concerned. A particularly n o t e w o r t h y result of Damaskin's study is that an acceptable fit to experimental data could be obtained only when the permanent dipole m o m e n t of a water m o n o m e r on the surface was assumed to be twice the value measured in the gas phase. The larger value of Pm was attributed to chemical interaction between the solvent dipole and the substrate metal. Although one must consider a field-independent shift in the chemical potential of a given dipolar c o m p o n e n t of the monolayer due to this interaction, namely the residual energy, it is difficult to understand w h y the field-dependent contribution to the free energy should be so different from that estimated on the basis of the dipole m o m e n t and polarizability of the isolated molecule. Cooper and Harrison [28] have severely criticized the general approach taken here and in previous papers [1--14] in which the properties of a solvent monolayer in the inner layer are described on the basis of a molecular model and the remainder of the double layer, on the basis of continuum concepts. The present treatment relies on values of inner-layer capacity obtained from the experimental differential capacity after correction for the diffuse-layer capacity on the basis of the Gouy--Chapman theory. Although the Gouy--Chapman model is und o u b t e d l y incorrect at higher electrode charge densities, the diffuse-layer contribution to the total capacity is unimportant in these regions. In fact, Grahame [21] obtained acceptable agreement between inner-layer capacities calculated from experimental capacities measured at the same charge density b u t for different electrolyte concentrations. However, the present approach does n o t account for specific interactions between the molecules and ions on the boundary of the diffuse layer (oHp) and the solvent monolayer which solvates the electrode. Moreover, it ignores the possibility that ions on the oHp may retain their primary solvation sheath so that the thickness of the inner region would depend on the nature of the counter ion at the o H p and correspond on the average to more than one solvent molecule in a direction perpendicular to the interface [29]. Specific ionic effects are observed on inner-layer capacity curves for the alkali metal cations at far negative charge densities, but it is not clear whether these effects are due to cation-specific adsorption [30] or ionic size effects [29]. Ignoring the possibility that ions are present in the solvent monolayer being considered here, it is clear that the first layer of solvent molecules will have the major influence on the dielectric properties of the inner layer since it is most strongly influenced by the electrode field. In general, one should not ignore the influence of ions and solvent molecules b e y o n d the first monolayer, b u t they are expected to play a secondary role in determination of innerlayer properties if contact ionic specific adsorption is absent. In order to deal with these effects, the molecular model should be extended further from the electrode into the diffuse layer. Since one must then deal with species of different sizes and with ion--ion and ion--dipole interactions as well as dipole--dipole interactions, such an extension represents a much more difficult problem. In the meanwhile, results such as those obtained here may be considered to give a useful approximate description of solvent properties in the inner layer.
179
'E ?~ -20C
- 40C 012
011
0
-011
-012
,r'/c ~ 2
Fig. 9. The entropy of the monolayer, S according to the present model estimated from the temperature dependence of the free energy (eqn. 21) and plotted as a function of electrode charge density o.
Finally, as pointed out above, the experimentally determined maximum in the entropy of formation of the compact layer occurs at negative charge densities at approximately --0.05 C m-: [15]. To the extent that the present model is able to describe the charge and temperature dependence of the inner-layer capacity, it is also able to describe the variation in entropy of the inner-layer solvent with electrode charge density [12]. An analytical expression for the entropy of the monolayer could be derived by differentiating eqn. (21) with respect to temperature. In the present instance, the entropy was estimated by calculating the free energy of the monolayer on the basis of eqn. (21) and by determining the temperature derivative a F / a T = - - S numerically. The maximum entropy of the monolayer (Fig. 9) occurs at approximately the same charge density as reported by Harrison et al. [15]. It should be noted that this entropy maximum for the solvent monolayer does not coincide with the maximum in the configurational entropy of the monolayer (eqn. 20). According to the present model the configurational entropy maximum occurs at positive charge densities approximately at the point where the fraction of clusters in the " u p " and "down" orientations are equal (Fig. 6). The non-coincidence in the maxima for the total monolayer entropy and its configurational component implies that the internal energy of the monolayer varies significantly with temperature. This variation is due to a corresponding change in dipole--dipole interaction energy with temperature. Bockris and Habib [ 11 ] presented a model for the entropy of a water monolayer composed of monomers and dimers. However, this model can be criticised on fundamental grounds because of an incorrect formulation of the dipolar contribution to the potential drop across the monolayer, and of the equilibrium constant describing formation of dimers from monomers [22]. Although the model is able to describe variation in entropy with electrode charge, it does not properly describe the inner-layer capacity.
180 In summary, the present results serve to emphasize the importance of developing a detailed model for dipole--dipole interactions in describing the properties of solvent molecules at charged interfaces. In particular, the fact that the capacity maximum occurs at positive charge densities and the entropy maximum at negative charge densities can be explained on the basis of the present model; on the other hand, if dipole--dipole interactions are ignored [8] or treated approximately [13], one cannot account for the experimentally observed temperature dependence of the dipole potential. The present model can also be applied to describe inner-layer capacity curves observed for other associated solvents such as formamide [31] and N-methylformamide [32]. These systems will be considered in a subsequent paper. ACKNOWLEDGEMENT
The financial assistance of the Natural Sciences and Engineering Research Council, Canada, is gratefully acknowledged.
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