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A comment on orientational models for the solvent at the electrode-electrolyte interface
J. Electroanal. Chem., 86 (1978) 425--428 © Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands
425
Short communication A COMMENT ON ORIENTATIONAL MODELS F O R THE SOLVENT AT THE ELECTRODE-ELECTROLYTE INTERFACE
I.L. COOPER and J.A. HARRISON
School o f Chemistry, The University, Newcastle upon Tyne, NE1 7RU (England) (Received 10th September 1976; in revised form 28th February 1977)
Introduction
A recent paper by Parsons [1] contains an extension of the familiar twoorientational state model [2--4] for the compact layer at the metal-electrolyte interface. His model purports to include, following a suggestion by Damaskin and Frumkin [5], both clusters and free water molecules within this layer. Since his resultant differential capacity curves, as a function of surface charge density, would appear to bear a certain similarity to those derived by Grahame [6--8] from his experimental data, it is crucial that this model be placed in its proper perspective. The purpose of this note is to indicate that our present understanding of the double layer problem is in a much less satisfactory state than would superficially appear to be the case. Orientational m o d e l s
Three major criticisms of orientational models for the solvent at the metalelectrolyte interface are outlined, the first two of which apply specifically to the treatment in ref. 1, and the third (and most serious) of which applies to all such molecular models for the interface. (a) It is apparent from a detailed examination of the mathematical analysis of ref. 1 that each water dipole is allowed to orientate i n d e p e n d e n t l y under the orientating influence of the external electric field and the randomizing effect of thermal motion. In the notation of ref. 1, the allowed components of dipole m o m e n t normal to the metal surface are +p, +-Pc (where Pc < P), and the probability of finding a particular normal component of dipole moment is dictated solely by a Boltzmann distribution over the available orientational energy states. In consequence, the model is simply a four-orientational state one, and the claim by Parsons that he is directly incorporating clusters of water molecules in the inner layer is incorrect. (b) It is a straightforward matter to show that the differential capacity for this four-state model has a singularity (i.e., is infinite) for certain values at the ratio p = Pc~P, using reasonable [1 ] values of physical parameters. Such e x t r e m e sensitivity within the model to what should be a relatively unimportant param-
426
eter * is disturbing, and must raise doubts as to the reasonableness of the model itself. Singularities in the (calculated) differential capacity have in fact been pointed out previously [9] in relation to the two-orientational state model (in the absence of a relative permittivity -- see (c) below) when no unjustifiable assumptions are made concerning an antiferroelectric alignment of water dipoles at room temperature. (Such an assumption is not, of course, made explicitly in refs. 3 and 4, but is implicit in a choice of numerical values which leads to an interaction energy between dipoles which is greater than thermal energy.) The origin of singularities in the differential capacity, with associated sensitivity of the results to values of molecular parameters, is almost certainly a consequence of the direct incorporation of individual molecular energies into a dipole monolayer which is subsequently treated as a macroscopic dielectric, despite the impossibility of performing a microscopic (statistical) averaging process across the region in question. In particular, the relative energy separation of the set of (two singly and one doubly) degenerate orientational energy levels at zero surface charge density is estimated in ref. 1 by fitting the fourorientational state model differential capacity to Grahame's compact layer differential capacity. Such a procedure ignores both the fact that Grahame's compact layer differential capacity is itself the consequence of a model [6], and that no simple physical relationship exists between orientational energies and a differential capacity. The extreme sensitivity of the model to the values of molecular parameters under such conditions should come as no surprise. (c) Since, as mentioned above, the compact layer differential capacity is a derived and not an experimentally measurable quantity, all models which attempt to explain the behaviour of this differential capacity should be assessed in the first instance on the reasonableness of their assumptions, since the possibility that the derived compact layer differential capacity itself is erroneous cannot be ruled out. For this reason, the c o m m o n assumptions inherent in orientational state models for the solvent at the metal-electrolyte interface must be subjected to close scrutiny. All orientational models for the compact layer assume the existence of a monolayer of water molecules lying between the metal surface and the distance of closest approach of (non-adsorbed) ions in the electrolyte, and this physical situation is simulated by a layer of point dipoles (which may be permitted to interact [3,4] and which may be assigned a polarizability [4] ) lying between two structureless, continuous, planar distributions of charge, of opposite sign, at the metal surface and the outer Helmholtz plane. The change in potential across this region resulting from a change in surface charge density is then attributed to changes in the (non-linear) polarization of this microscopic region. Certain authors [1--3] incorporate a relative permittivity (i.e. a dielectric constant) as a further parameter into the model, and this has been the subject of some debate [10]. A dielectric constant, er, is a macroscopic quantity, relating (through the constant of proportionality Co(Or- 1) the polarization across, for example, a homogeneous dielectric medium and the constant electric field Since p = Pc/P and Pc corresponds to the projection of the dipole moment on the surface normal for a specified dipolar orientation, only slight changes in orientation angle will produce significant changes in p.
427 within this medium, assuming implicitly that microscopic fluctuations have been " s m o o t h e d out". To attribute such a dielectric sontant to a region of molecular dimensions is to ignore the inherent local inhomogeneity which exists at the molecular level, and the inclusion of such a phenomenological parameter into what is intended to be a molecular calculation presupposes that the water dipoles are embedded in a fictitious homogeneous dielectric medium, which only serves to obscure the physical interpretation of the model in molecular terms. The microscopic distances involved between the dipoles and the planar boundary surfaces lead us to question the reasonableness of such an environment for the dipoles. The simulation of the structured metal surface by a planar distribution of charge can only be a first approximation to the true physical situation, and such an approximation would have a greater degree of validity if the distance of the dipoles from the surface was much larger than that in involved in this problem. A much more serious matter, however, is the replacement of discrete charges in the electrolyte by a continuous planar distribution of charge centred on the o.H.p. Since we are attempting to describe the inner layer in molecular terms, the local environment of this layer must bear some resemblance to reality if the model is to have a certain degree of validity. This local environment must, of necessity, include the neighbouring distribution of discrete ions in the electrolyte, which, even at high electrolyte concentration, cannot be centred on the o.H.p. Even if such an (incorrect) assumption were made, the (concentration-dependent) ion-ion separation would be sufficiently large -- on a molecular scale -- to prevent its effect on the adjacent dipole layer from being simulated by a continuous distribution of charge. This error is compounded by the absence from the analysis of neighbouring water dipoles, which must interact with the dipole layer and also be affected themselves by the presence of the metal surface and the ion distribution. The replacement of this environment by a structureless, continuous plane of charge a microscopic distance away from a dipole layer which is being treated in molecular terms cannot be reasonably taken as a first approximation to the physical situation which it is attempting to describe. The imposed " b o u n d a r y condition" at the o.H.p. can only have the consequence of rendering invalid all existing orientational models for the solvent at the metal-electrolyte interface. The origin of this (unphysical) b o u n d a r y c o n d i t i o n can, in fact, be traced, as discussed in detail elsewhere [11], to the formal separation by Grahame [6] of the observed double layer differential capacity into diffuse and compact layer contributions, pertaining to two independent (fictitious) parallel plate capacitors in series. Such an "equivalent circuit" description of a differential capacity is purely formal and has no bearing on the (effective) location of real charges within the electrolyte. The continuous distribution of charge on the o.H.p. which is invoked in molecular treatments of the so-called compact layer is of mathematical, as distinct from physical, origin, and does not refer to the ion distribution at all. It arises as a direct consequence of the arbitrary (and unjustifiable) partitioning of the electrolyte into two -- i n d e p e n d e n t -- dielectric phases, one of which is the monolayer constituting the compact layer itself, in conjunction with the association of "real" charge to this imaginary boundary surface, following the formal mathematical separation of the differential capac-
428
ity into two series contributions of parallel-plate capacitor type. We are aware that the present paper reaches a conclusion -- namely that orientational models of the compact layer display a degree of sophistication unwarranted by the approximate macroscopic procedure by which the compact layer is itself constructed, leading to serious inconsistencies at the molecular level -- which has the effect of raising doubts as to the validity of a seemingly well-established procedure in electrochemistry. We have recently completed [ 12] a study of the available experimental data on the metal-electrolyte interface, which shows that an internally consistent interpretation of the data can be formulated, and which does not rely, in contrast to the compact layer, on the introduction of a macroscopic dielectric medium of microscopic dimensions. REFERENCES
1 2 3 4 5 6 7 8 9 10 11 12
R. Parsons, J. Electroanal. Chem., 59 (1975) 229. R.J. Watts-Tobin, Phil. Mag., 6 (1961) 133. J.O'M. Bockris, M.A.V. Devanathan and K. Miiller, Proc. Roy. Soc., A274 (1963) 55. S. Levine, G.M. Bell and A.L. Smith, J. Phys. Chem., 75 (1969) 3534. B.B. D a m a s k i n and A . N . F r u m k i n , E l e c t r o c h i m . Aeta, 19 (1974) 173. D.C. Grahame, Chem. Rev., 41 (1947) 441. D.C. Grahame, J. A m e r . Chem. Soc., 76 (1954) 4819. D.C. Graharne, J. A m e r . Chem. Soc., 79 (1957) 2093. I.L. C o o p e r and J.A. Harrison, J. Eleetroanal. Chem., 66 (1975) 85. J.R. Macdonald and C.A. Barlow, J. Chem. Phys.. 39 (1963) 412. I.L. C o o p e r and J.A. Harrison, Electrochim. Acta, 22 (1977) 519. I.L. C o o p e r and J.A. Harrison, Electrochirn. A c t a , in press.
425
Short communication A COMMENT ON ORIENTATIONAL MODELS F O R THE SOLVENT AT THE ELECTRODE-ELECTROLYTE INTERFACE
I.L. COOPER and J.A. HARRISON
School o f Chemistry, The University, Newcastle upon Tyne, NE1 7RU (England) (Received 10th September 1976; in revised form 28th February 1977)
Introduction
A recent paper by Parsons [1] contains an extension of the familiar twoorientational state model [2--4] for the compact layer at the metal-electrolyte interface. His model purports to include, following a suggestion by Damaskin and Frumkin [5], both clusters and free water molecules within this layer. Since his resultant differential capacity curves, as a function of surface charge density, would appear to bear a certain similarity to those derived by Grahame [6--8] from his experimental data, it is crucial that this model be placed in its proper perspective. The purpose of this note is to indicate that our present understanding of the double layer problem is in a much less satisfactory state than would superficially appear to be the case. Orientational m o d e l s
Three major criticisms of orientational models for the solvent at the metalelectrolyte interface are outlined, the first two of which apply specifically to the treatment in ref. 1, and the third (and most serious) of which applies to all such molecular models for the interface. (a) It is apparent from a detailed examination of the mathematical analysis of ref. 1 that each water dipole is allowed to orientate i n d e p e n d e n t l y under the orientating influence of the external electric field and the randomizing effect of thermal motion. In the notation of ref. 1, the allowed components of dipole m o m e n t normal to the metal surface are +p, +-Pc (where Pc < P), and the probability of finding a particular normal component of dipole moment is dictated solely by a Boltzmann distribution over the available orientational energy states. In consequence, the model is simply a four-orientational state one, and the claim by Parsons that he is directly incorporating clusters of water molecules in the inner layer is incorrect. (b) It is a straightforward matter to show that the differential capacity for this four-state model has a singularity (i.e., is infinite) for certain values at the ratio p = Pc~P, using reasonable [1 ] values of physical parameters. Such e x t r e m e sensitivity within the model to what should be a relatively unimportant param-
426
eter * is disturbing, and must raise doubts as to the reasonableness of the model itself. Singularities in the (calculated) differential capacity have in fact been pointed out previously [9] in relation to the two-orientational state model (in the absence of a relative permittivity -- see (c) below) when no unjustifiable assumptions are made concerning an antiferroelectric alignment of water dipoles at room temperature. (Such an assumption is not, of course, made explicitly in refs. 3 and 4, but is implicit in a choice of numerical values which leads to an interaction energy between dipoles which is greater than thermal energy.) The origin of singularities in the differential capacity, with associated sensitivity of the results to values of molecular parameters, is almost certainly a consequence of the direct incorporation of individual molecular energies into a dipole monolayer which is subsequently treated as a macroscopic dielectric, despite the impossibility of performing a microscopic (statistical) averaging process across the region in question. In particular, the relative energy separation of the set of (two singly and one doubly) degenerate orientational energy levels at zero surface charge density is estimated in ref. 1 by fitting the fourorientational state model differential capacity to Grahame's compact layer differential capacity. Such a procedure ignores both the fact that Grahame's compact layer differential capacity is itself the consequence of a model [6], and that no simple physical relationship exists between orientational energies and a differential capacity. The extreme sensitivity of the model to the values of molecular parameters under such conditions should come as no surprise. (c) Since, as mentioned above, the compact layer differential capacity is a derived and not an experimentally measurable quantity, all models which attempt to explain the behaviour of this differential capacity should be assessed in the first instance on the reasonableness of their assumptions, since the possibility that the derived compact layer differential capacity itself is erroneous cannot be ruled out. For this reason, the c o m m o n assumptions inherent in orientational state models for the solvent at the metal-electrolyte interface must be subjected to close scrutiny. All orientational models for the compact layer assume the existence of a monolayer of water molecules lying between the metal surface and the distance of closest approach of (non-adsorbed) ions in the electrolyte, and this physical situation is simulated by a layer of point dipoles (which may be permitted to interact [3,4] and which may be assigned a polarizability [4] ) lying between two structureless, continuous, planar distributions of charge, of opposite sign, at the metal surface and the outer Helmholtz plane. The change in potential across this region resulting from a change in surface charge density is then attributed to changes in the (non-linear) polarization of this microscopic region. Certain authors [1--3] incorporate a relative permittivity (i.e. a dielectric constant) as a further parameter into the model, and this has been the subject of some debate [10]. A dielectric constant, er, is a macroscopic quantity, relating (through the constant of proportionality Co(Or- 1) the polarization across, for example, a homogeneous dielectric medium and the constant electric field Since p = Pc/P and Pc corresponds to the projection of the dipole moment on the surface normal for a specified dipolar orientation, only slight changes in orientation angle will produce significant changes in p.
427 within this medium, assuming implicitly that microscopic fluctuations have been " s m o o t h e d out". To attribute such a dielectric sontant to a region of molecular dimensions is to ignore the inherent local inhomogeneity which exists at the molecular level, and the inclusion of such a phenomenological parameter into what is intended to be a molecular calculation presupposes that the water dipoles are embedded in a fictitious homogeneous dielectric medium, which only serves to obscure the physical interpretation of the model in molecular terms. The microscopic distances involved between the dipoles and the planar boundary surfaces lead us to question the reasonableness of such an environment for the dipoles. The simulation of the structured metal surface by a planar distribution of charge can only be a first approximation to the true physical situation, and such an approximation would have a greater degree of validity if the distance of the dipoles from the surface was much larger than that in involved in this problem. A much more serious matter, however, is the replacement of discrete charges in the electrolyte by a continuous planar distribution of charge centred on the o.H.p. Since we are attempting to describe the inner layer in molecular terms, the local environment of this layer must bear some resemblance to reality if the model is to have a certain degree of validity. This local environment must, of necessity, include the neighbouring distribution of discrete ions in the electrolyte, which, even at high electrolyte concentration, cannot be centred on the o.H.p. Even if such an (incorrect) assumption were made, the (concentration-dependent) ion-ion separation would be sufficiently large -- on a molecular scale -- to prevent its effect on the adjacent dipole layer from being simulated by a continuous distribution of charge. This error is compounded by the absence from the analysis of neighbouring water dipoles, which must interact with the dipole layer and also be affected themselves by the presence of the metal surface and the ion distribution. The replacement of this environment by a structureless, continuous plane of charge a microscopic distance away from a dipole layer which is being treated in molecular terms cannot be reasonably taken as a first approximation to the physical situation which it is attempting to describe. The imposed " b o u n d a r y condition" at the o.H.p. can only have the consequence of rendering invalid all existing orientational models for the solvent at the metal-electrolyte interface. The origin of this (unphysical) b o u n d a r y c o n d i t i o n can, in fact, be traced, as discussed in detail elsewhere [11], to the formal separation by Grahame [6] of the observed double layer differential capacity into diffuse and compact layer contributions, pertaining to two independent (fictitious) parallel plate capacitors in series. Such an "equivalent circuit" description of a differential capacity is purely formal and has no bearing on the (effective) location of real charges within the electrolyte. The continuous distribution of charge on the o.H.p. which is invoked in molecular treatments of the so-called compact layer is of mathematical, as distinct from physical, origin, and does not refer to the ion distribution at all. It arises as a direct consequence of the arbitrary (and unjustifiable) partitioning of the electrolyte into two -- i n d e p e n d e n t -- dielectric phases, one of which is the monolayer constituting the compact layer itself, in conjunction with the association of "real" charge to this imaginary boundary surface, following the formal mathematical separation of the differential capac-
428
ity into two series contributions of parallel-plate capacitor type. We are aware that the present paper reaches a conclusion -- namely that orientational models of the compact layer display a degree of sophistication unwarranted by the approximate macroscopic procedure by which the compact layer is itself constructed, leading to serious inconsistencies at the molecular level -- which has the effect of raising doubts as to the validity of a seemingly well-established procedure in electrochemistry. We have recently completed [ 12] a study of the available experimental data on the metal-electrolyte interface, which shows that an internally consistent interpretation of the data can be formulated, and which does not rely, in contrast to the compact layer, on the introduction of a macroscopic dielectric medium of microscopic dimensions. REFERENCES
1 2 3 4 5 6 7 8 9 10 11 12
R. Parsons, J. Electroanal. Chem., 59 (1975) 229. R.J. Watts-Tobin, Phil. Mag., 6 (1961) 133. J.O'M. Bockris, M.A.V. Devanathan and K. Miiller, Proc. Roy. Soc., A274 (1963) 55. S. Levine, G.M. Bell and A.L. Smith, J. Phys. Chem., 75 (1969) 3534. B.B. D a m a s k i n and A . N . F r u m k i n , E l e c t r o c h i m . Aeta, 19 (1974) 173. D.C. Grahame, Chem. Rev., 41 (1947) 441. D.C. Grahame, J. A m e r . Chem. Soc., 76 (1954) 4819. D.C. Graharne, J. A m e r . Chem. Soc., 79 (1957) 2093. I.L. C o o p e r and J.A. Harrison, J. Eleetroanal. Chem., 66 (1975) 85. J.R. Macdonald and C.A. Barlow, J. Chem. Phys.. 39 (1963) 412. I.L. C o o p e r and J.A. Harrison, Electrochim. Acta, 22 (1977) 519. I.L. C o o p e r and J.A. Harrison, Electrochirn. A c t a , in press.