37 1
J. Phys. Chem. 1982, 86,371-376
A Multistate Model for Unassociated Solvent Structure at a Polarizable Interface W. R. Fawcett’ and R. M. de Nobrlga OUelph-WaterkDoCentre for Graduate Work In Chemistry (ouelph Campus), Depat?ment of Chemistry, Unhrersny of G&ph, &farb, Cane& NlG 2W1 (Received: June 29, 1981; In Final Form: August 20, 1981)
ouelph.
A model for solvent structure at a polarizable metal/solution interface in which the solvent is represented as a monolayer of hard polarizable spheres with point dipoles is presented. It is assumed that the dipoles can take up any orientation with respect to the field at the interface, the dipolar population in a given orientation depending on the local field, chemical interactions between the dipole and the conducting substrate, and temperature. Equations for the dielectric properties of the monolayer are derived, and the resulting differential capacity against surface charge density curves are compared with those derived on the basis of the simpler three-state model. The significance of the results is discussed with respect to experimental data for the mercury/electrolyte solution interface in unassociated solvents; special emphasis is given to the possible effect of counterion nature on dipolar orientation and differential capacity at larger charge densities. Introduction Considerable success has been achieved recently in describing the dielectric properties of the inner layer at a polarizable metal/electrolyte solution interface on the basis of models in which the solvent in the layer is represented as a monolayer of molecules whose orientation depends on the local electrical fie1d.l The more successful treatments have assumed three possible orientations of solvent monomers in the case of relatively unassociated solvents,2such as dimethylf~rmamide~ and methanol? and four orientations of solvent monomers and clusters in the case of strongly associated solvents like water”’ and N-methylformamide.* According to the three-state model, the dipole vector in a solvent molecule may be oriented with the electrode’s field, against it, or perpendicular to it. This model is able to account for both maxima and minima on the inner-layer capacity against electrode charge density curves, one extremum, the principal extremum, corresponding to the condition that the net polarization of the monolayer in the directions of the field is zero. However, the theory predicts that the capacity curves are symmetrical with respect to the principal extremum,’ a result which is not found when capacity data are obtained for reasonable rangea of electrode charge density on either side of the principal extremum. A good example is the case of the Hg/methanol interface4 where the shape of the capacity curve on either side of the capacity minimum, assumed to be the principal extremum, depends on the nature of the predominant counterion at the outer Helmholtz plane. It is reasonable to expect the counterions to influence the orientation of adjacent solvent molecules to an extent which depends on their charge-to-radius ratio, that is, to the extent that they alter the local field. One way of accounting for these specific effects and the resulting asymmetry of the capacity curves is to introduce more allowed orientational states for the solvent molecules in the monolayer. Alternatively, one may assume that there is weak ionic specific adsorption, that is, displacement of some solvent molecules in the monolayer by ions; however, weak specific adsorption cannot be detected by (1) W. R. Fawcett, Isr. J. Chem., 18, 3 (1979). (2)W.R. Fawcett, J. Phys. Chem., 82, 1385 (1978). (3)W.R.Fawcett, B. M. Ikeda, and J. B. Sellan, Can. J . Chem., 57, 2268 (1979). (4)Z.Borkowska and W. R. Fawcett, Can. J. Chem., 69,710 (1981). (5) R. Parsone, J. Electroanal. Chem., 69,229 (1975). (6)B. B. Damaekin, J. Electroanal. Chem., 75, 359 (1977). (7) W.R.Fawcett, S. Levine, R. M. de Nobriga, and A. C. McDonald, J. Electroanal. Chem.,111, 163 (1980). (8) Z. Borkowska, W. R. Fawcett, and R. M. de Nobriga, J. Electroanul. Chem., 124, 263 (1981). 0022-3654/82/2086-0371$01.25/0
the usual thermodynamic methods. In the present paper, ionic adsorption is assumed to be absent under all circumstances. The most general description of solvent structure in a monolayer a t a polarizable interface allows for an unspecified large number of dipole orientations. A multistate model describing the dielectric properties of interfacial solvent was presented earlier by Macdonald and Barlow.BJo This model considers the important role of imaging of discrete dipoles at the interface; however, it does not rely on entirely microscopic molecular properties and introduces a dielectric constant for the monolayer at very high fields, e,, to account for distortional polarization. A similar comment applies to the more recent work of Oldham et al.11J2 in which the point dipole approximation is dropped in a multistate treatment but an effective dielectric constant of the monolayer introduced. This procedure may be avoided by calculating the distortion polarization on the basis of the molecular polarizability and estimating its detailed role in determining dipole-dipole interactions and the local field acting a t elements of the monolayer.’ The effects of imaging are considered by introducing an effective coordination number which increases the extent of dipoledipole interactions beyond nearest neighbors. As a result the quantity equivalent to e, is treated as a function the average dipole environment and electrode field and not as a constant as it was previously.”12 The molecular approach was first used by Levine, Bell, and Smith13 in a two-state model for interfacial solvent structure and subsequently extended to three state^.^ In the present paper, the same method is used to develop a multistate model for interfacial solvent structure and dielectric properties. Thermodynamic Properties of t h e Monolayer A monolayer of solvent molecules represented as spheres of diameter d and arranged in hexagonal close packing are present at a polarizable electrode-electrolyte solution interface which is assumed to be planar. Each molecule is surrounded by six nearest neighbors and the molecular number density is NT where NT = 2/(3)’12 d2. All orien(9) J. R. Macdonald and C. A. Barlow, Jr., J . Chem. Phys., 36,3062 (1962). (10)J. R.Macdonald and C. A. Barlow, Jr., “Proceedingsof the First Australian Conference on Electrochemistry”, J. A. Friend and F. Gutmann, Ed., Pergamon, Oxford, 1965, pp 199-247. (11)K.B. Oldham and R. Parsons, Elektrokhimiya, 13, 866 (1977). (12)P.Dalrymple-Alford and K. B. Oldham, Can. J . Chem., 56,861 11978). . (13)S.Levine, G.M. Bell, and A. L. Smith, J. Phys. Chem., 73,3534 (1969).
0 1982 American Chemical Society
372
The Journal of Physical Chemistry, Voi. 86, No. 3, 1982
Fawcett and de Nobriga
METAL
f X
SOL UTlON
Flgwo 1. The model of a d e n t monolayer at a polarizable conducting wail with each molecule represented as a hard sphere with a permanent dipole moment p and polarizablllty a. The dipoles can assume any orlentation wlth respect to the Held due to the charge on the wail, the net dipole moment in the direction of the field being p, = p cos 6 where 0 Is the angle between the dipole vector and the vector
and (iii) the energy of the dipoles in the field of the electrode N d m P + mi)E Each dipole orientation has a nonelectrostatic contribution to its energy which depends on the specific chemical interactions between the atoms of the solvent molecule adjacent to the interface and the atoms of the electrode, which is called the residual energy, U,, In the random mixing approximation, the configurational entropy of the monolayer is given by
S = -CkNi In
(6)
(Ni/NT)
i
X.
Neglecting the van der Waals contribution to the energy of the monolayer, the Helmholtz free energy is given by
tations of the molecular dipole with respect to the interface are possible, the number per unit area in a given orientation i being Ni. It follows that
F = -(mp 2d3
NTce
+ mi)2+ CaiNid2/2+ NT(mp+ mi)E + 1
CNiUri+ CkTNi In (Ni/NT) (7)
n
i
5 Ni = NT i=l where n is the total number of orientations. Different orientations are distinguished by the fact that their dipole vectors are at different angles to the direction perpendicular to the interface (x direction); the angle made by the projection of the dipole vector on the plane of the interface (y,z plane) is not considered. The permanent dipole moment of a solvent molecule is p . I t follows that the net dipole moment in the x direction for orientation i is pi = p cos e (2) where 0 is the angle between the dipole vector and a vector in the x direction with ita positive end at the interface and negative end in the solution (Figure 1). The average net polarization due to permanent dipoles in the x direction is then mp =
C(Nipi)/NT
(3)
1
The second component of the polarization is due to distortion and can be estimated from the component of the polarizability appropriate for orientation i, ai, and the value of the local field acting a t the given dipole in the x direction, d. Accordingly, the induced polarization is mi = C (aiNi)/NT i
(4)
The local field d is given by the sum of the field due to the charge on the electrode E; and the mean field in the same direction at a given dipole site due to surrounding dipoles, X.2J3If we assume random distribution of dipoles in the monolayer, X in the mean field approximation is given by (5)
where c, is the effective coordination number of one dipole with its neighbor^.^,'^ Three contributions to the electrostatic energy of the dipoles may be distinguished: (i) The dipole-dipole interaction energy given by
1
The omission of the van der Waals contribution is not important to the following derivation provided that it is independent of dipole orientation. Introducing eq 3-5, we can write the expression for the free energy as follows:
F=
(8) where A = 1 + ceCiaifi/d3,f i being the fraction of the surface covered by species i (fi = Ni/NT). The relationship between the number of dipoles in any two orientations i and j may be found by requiring that the free energy of the monolayer be independent of reorientation, the total number of dipoles in the i and j orientations being kept constant. Then
(E) aNi
=
N,.+N,
(9)
Thus, the condition for equilibrium in the monolayer between any two orientations may be written ajQ2
k T l n N i + Uri+piQ--
2
= ajd2
kT In Nj + Urj+ pj6 - - (10)
2 An expression for the surface concentration of any one orientation may be derived by defining the partition function for the monolayer: q =
Eai exp(-pid/kT) exp(aid2/2kT)
(11)
1
NTce
where ai = exp(-Uri/kT). Thus, it is easily shown from eq 10 and 11 that N. ai exp(-pid/kT) exp(aiG2/2kT) f,=--L (12) NT 9
i
Evaluation of the partition function requires a knowledge of the polarizability tensor for the solvent molecule, and of the dependence of U,i on the dipole's orientation with respect to the electrode field. The latter can only be determined on the basis of a detailed quantum mechanical
-(mp + mJ2 2d3 (ii) the energy of the induced dipoles Z:aiN?/2
Mullstate Model for Unassociated Solvent Structure
description of the solvent molecule’s interaction with the substrate in all orientations. The dielectric properties of the monolayer may be estimated by determining first the potential drop across the monolayer, A$:
The reaction field is then
Differentiating with respect to a, one obtains
where a (E = 4aa) is the charge density on the electrode. The reaction field X may be related to the partition function q through its derivative with respect to 6: aq _ dQ
1 ---piai exp(-piG/kT) exp(aiG2/2kT) + kTi 1 -EaiaiG exp(-piE/kT) exp(aiG2/2kT) (14) kT i
Combining eq 5 and 14 one obtains -c,kT a In q d3 a6
c,kT d2 In qo + (l+-- d3
aa
= 4ad
-a2X -a u2
[
-4acekT d3 no( 4a d3 a63
aa
E)]/
+-
c,kT d2 In qo Cea) dE2 + d3 = 0 (22)
~ ? F Nax T~~
+C, aa
(16)
from which it follows that a3 In qo/dG3 = 0 at an extremum. The latter derivative can be written d3 In 40 -a63
----(-)(a) z) 1 a3q0 3 q o a ~ 3 qo2
dqo a6
+
2 aqo 2(
(23)
Noting that
From eq 15
ax - = - - - --c,kT
(21)
(l+-- d3
The differential capacity of the inner layer is obtained by differentiating A 4 with respect to a:
ci
y) d3
dG2
I t is easily shown that the condition for an extremum is
x=--
1 =-
No. 3, 1982 373
The Journal of Physical Chemistry, Vol. 86,
d2 In q
a6
-4ac,kT a2 In q --
-
d3
d3
a62
= ai exp(-aiG)/qo (24) where qo = Ci ai exp(-qG) and ai = pi/kT, one may write 23 as follows: fi
(17)
1+--
Thus, in order to calculate Ci, one must estimate a2 In q / d e 2 , that is, d2q/aG2 as well as dq/aG and 6. It is obvious from the above that the local field Q is an important variable in describing solvent properties in the inner layer. On the basis of eq 12, the fraction of the monolayer containing solvent molecules in a given orientation is a function of 6 and temperature, the residual energy, dipole moment, and polarizability being characteristics of the molecule in the particular orientation. The potential drop A4 and electrode charge density a are not simply related to G (eq 13 and 15) so that they cannot be used conveniently as independent variables to describe solvent orientation and adsorption.
Inner-Layer Capacity against Charge Density Curves
It has been shown on the basis of simpler models that some significance can be attached to maxima and minima on inner-layer capacity against charge density curves.l According to the present model, an extremum occurs on the capacity curve when a(l/ci)_--4 a N ~ a2X d~ --0 (18)
aa
Ce
In general, one expects dipoles in a parallel orientation (ai = pia= 0) to predominate in the mono1ayer.l This conclusion is supported by the fact that the orientation of minimum energy for two contiguous dipoles in the absence of an external field is in a head-to-tail onf figuration.'^ It follows that f i C(H2P04-),that is, with increase in the anion’s ability to interact with interfacial solvent.21 However, the significance of the capacity maximum in the aqueous system is probably different? from that in unassociated solvents to which the three-state model is app1ied.lv2 Finally, it is clear that one cannot assess anionic specific adsorption solely on the basis of comparison of capacity curves. The original three-state model2 and the multistate version presented here overlook the interaction energy between dipoles in the “parallel” orientation. Because of dipole-dipole interactions, one expects the dipoles in the -), “parallel” orientation to line up head-to-tail (that is, in the configuration of lowest energy.14 If the total population of parallel dipoles remained constant over the polarizable range of the electrode, the contribution to the dipole-dipole interaction due to interactions parallel to the electrode would be included implicitly in the residual energy term. However, the population of “parallel” dipoles is significantly lower at the extremes of polarization so that some inaccuracy in estimated number densities and dielectric properties of the monolayer is expected. In order to treat this system more precisely, one should define a reaction field parallel to the surface, Y, as well as the reaction field perpendicular to the surface, X,treated in this paper. Y depends on the number density of dipoles parallel to the surface and on their orientation with respect to one another. It contributes to the free energy of these dipoles through the term p Y and is important when E and X are small. However, at large charge densities where E and X are large, Y is small in comparison. Therefore, changes in Yare not expected to result in large changes in the dipole distribution a t the extremes of polarization where the population of “parallel” dipoles decrease. In conclusion, the multistate model presented here provides an explanation for the asymmetry observed experimentally on inner-layer capacity against charge density
~~
(16) D. C. Grahame, J. Electrochem. SOC.,98, 343 (1951). (17) W. R. Fawcatt and M. D. Mackey, J. Chem. SOC.,Faraday Tram. I , 69, 634 (1973). (18) W. R. Fawcett, J. Electroanol. Chem., 39, 474 (1972). 119) R. Parsons and A. Stockton. J. Electroanol. Chem.,. 26.- ADD. - - 10 (igioj. (20) B. B. Damaskin, J. Electroanol. Chem., 65, 799 (1975).
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J. Phys. Chem. 1082, 86,376-382
376
curves. In addition, it allows one to introduce specific effects due to the counterions on the solution side of the inner layer. It is probable that the inner-layer dielectric properties can be described in a realistic way if at least three orientations are included in the ensemble. The exact number of states can be determined by using approximate quantum mechanical methods to estimate the most probable orientation of the molecule on a conducting substrate given ita electronic charge distribution.16*" Work (21) R. Parsons, Pure Appl. Chem., 18, 91 (1968).
in this direction with specific solvents to analyze data obtained at the mercury/nonaqueous solution interface is currently in progress.
Acknowledgment. Helpful discussion with Professor H. D. Hurwitz and Professor R. Parsons are gratefully acknowledged. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. (22) N. H. Cuong, A. Jenard, and H. D. Hurwitz, J. Electroanal. Chem., 103, 399 (1979).
Proton Transfers in Hydrogen-Bonded Systems. 4. Cationic Dimers of NH, and O H p Steve Schelner Depaflmnt of Chemistry and Bbchemlstry, Southern Illlnols Universl?y,Cerbondele. Illlnois 6290 1 (Recelved: Ju& 2 1, 198 1; I n Flnal Form: October 2, 1981)
Proton transfers along the hydrogen bonds in the systems (H3NHNH3)+and (H20HOH2)+are studied by ab initio molecular-orbital methods. The 4-31G basis set is used within the Hartree-Fock formalism to calculate transfer potentials for conformationsof each system involvil?g linear and angular deformationsof the hydrogen bond. Potentials obtained by using the rigid-molecule approximation are in excellent agreement with those calculated including full geometry optimizations at each stage of transfer. Transfer energy barriers for (NzH7)+ are found to vary with bond distortion in the same qualitative fashion as for the isoelectronic (02H5)+,although barriers in the former system are systematically lower. Certain calculated electronic properties of both (N2H7)+ and (02Hs)+are shown to correlate very well with the transfer energy barriers. Electron density rearrangements which accompany the transfer of the proton are examined in some detail by population analyses and by electron density difference maps to determine basic similaritiescharacteristic of proton transfers as well as fundamental differences between the (NzH7)+and (OzH6)+systems.
Introduction Proton transfers from one molecule to another play an important role in a host of biological and chemical processes including photosynthesis> ATP formation? acidbase equilibria,6 and ionic conduction in ice.7 It would therefore be useful to have a good understanding of all of the factors involved in such transfers. For example, how are the energetics of a proton transfer influenced by the nature of the two molecules involved? In addition, the manner in which the energetics are affected by the relative orientation of the two molecules is largely undetermined at present. Another feature of perhaps more fundamental interest is the spatial reorientation of the electronic distribution that accompanies the motion of the proton. (1) Paper 1 of this series: Scheiner, S. J . Am. Chem. SOC.1981,103, 315-20. See also: Scheiner, S. Ann. N. Y. Acad. Sci. 1981,367,493-509. (2) Paper 2 Scheiner, S.;Harding, L. B. J.Am. Chem. SOC.1981,103, 2169-73. (3) Paper 3: Scheiner,S.; Harding, L. B. Chem. Phys. Lett. 1981, 79, 39-42. (4) Miller, K. R. Sci. Am. 1979,241,102-13. Sauer, K. Acc. C h m . Res. 1978,11, 257-64. Clayton, R. K.; Vermeglio, A. In "MembraneTrans-
duction Mechanisms";Cone, R. A., Dowling, J. E., Eds.; Raven Press: New York, 1979; pp 49-69. (5) Negrin, R. S.; Fwter, D. L.; Fillingame, R. H. J.B i d . Chem. 1980, 255,5643-8.Haddock, B. A; Jonee, C. W .Bacterial. Reu. 1977,41,47-99. Pozzan, T.; DiVirgilio, F.; Bragadin, M.; Miconi, V.; Azzone, G. F. BOC. Natl. Acad. Sci. U.S.A. 1979, 76,2123-7. Criddle, R. S.; Johnston, R. F.; Stack, R. J. Curr. Top. Bioenerg. 1979, 9, 89-145. (6) Caldin, E., Gold, V., Eds. "Proton-TransferReactions";Chapman and Hall: London, 1975 and references contained therein. (7) Chen, M.-S.; Onsager, L.; Bonner, J.; Nagle, J. J.Chem. Phys. 1974, 60, 405-19. Yanagawa, Y.; Nagle, J. F. Chem. Phys. 1979, 43, 329-39. 0022-3654/82/2086-0376$01.25/0
Quantum-mechanical methods are well suited to providing the necessary information as intermolecular geometries may be very precisely specified and electronic features obtained from analyses of the wave functions. The usefulness of such theoretical approaches for obtaining important information about proton transfers was first demonstrated by Clementi for the (H3N...HC1) system.8 Since that time a number of additional theoretical treatments of proton transfers in other systems have appeared."lg Paper 1 in this series reported energetics of proton transfers between the oxygen atoms of hydroxyl groups along bent as well as linear hydrogen bonds as calculated by ab initio molecular-orbital procedures. In order to distinguish features common to all proton-transfer processes from those particular to oxygen-containing systems, we focus our attention here upon proton transfers (8) Clementi, E. J. Chem. Phys. 1967,46,3851-80. (9) Merlet, P.; Peyerimhoff, S. D.; Buenker, R. J. J. Am. Chem. SOC. 1972,94,8301-8. (10) Delpuech, J.-J.; Serratrice, G.;Strich, A.; Veillard, A. Mol. Phys. 1975,29, 849-71; Chem. Commun. 1972,817-8. (11) Ikuta, S.; Haehimoto, S.; Imamura, M. Int. J. Quantum Chem. 1980, 18, 515-9. (12) Newton, M. D.; Ehrenson, S. J. Am. Chem. SOC.1971,93,4971-90. (13) Janoschek, R.; Weidemann, E. G.; Pfeiffer, H.; Zundel, G. J.Am. Chem. SOC.1972,94, 2387-96. (14) Kollman, P. A.; Allen, L. C. J.Am. Chem. SOC.1970,92,6101-7. (15) Kraemer, W. P.; Diercksen, G. H. F. Chem. Phys. Lett. 1970,5, 463-5. (16) Meyer, W.;Jakubetz, W.; Schuster, P. Chem. Phys. Lett. 1973, 21, 97-102.
0 1982 American Chemical Society
