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Electron density at the metal-liquid interface
307
J. Electroanal. Chem, 245 (1988) 307-312 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Short communication ELECTRON
DENSITY AT THE METAL-LIQUID
INTERFACE
R.R. SALEM D. I. Mendeleev Moscow Chemical and Technological Institute, 9 Miusskaya Moscow 125047 (U.S.S.R.) (Received
5th August 1987; in revised form 13th November
Square,
1987)
In recent years, a large number of papers have appeared which report on the study of the surface properties of metals at the interface with a vacuum [l-6]. In most of these papers, the approach is based on the Hohenberg-Kohn-Scham density functional method [7,8] in which the kinetic, exchange and correlation energies of the electron gas are described in terms of a functional of the free electron density, n (x). However, calculation of the surface energy of metals has shown that the model of non-interacting electrons in a rectangular potential well (the so-called “jellium” model) gives quite a good fit to experiment, while an improved description of an electron system with many-particle and quantum effects involved leads to a considerable discrepancy with experimental data, and even to negative surface energies beginning with a certain density n(x) [9] (a so-called “ultradense catastrophe”). The way out can be found by introducing pseudo-potential corrections [lo] (see also ref. 11) or by using a simpler approach in which only the Coulomb part, i.e. the electrostatic energy of the electrical double layer (EDL), is retained in the surface energy Es, and numerical agreement with experiment is attained by virtue of fitting parameters. Apparently, this approach is also valid to describe a metal-liquid system. The density functional method uses exponential probe functions [2] corrected sometimes for density oscillations inside the metal [12] (though the solution of the Thomas-Fermi problem is of power law form). It is demonstrated in refs. 13-15 that exponential decay of the electron density follows from a modified Thomas-Fermi model with a consistent allowance for quantum effects near the interface and with the use of exact one-electron functions. Using this consideration, a relatively simple model was proposed in refs. 16 and 17 to describe the electron density distribution at the metal-liquid interface, the subject of recent theoretical interest (for reviews see refs. 18-22). 0022-0728/88/$03.50
8 1988 Elsevier Sequoia S.A.
308
The model [16,17] takes into account the decay of the electron density outside the metal (x > 0) and its oscillations inside the metal (x < 0): x>o
n(x)
= n, exp( -x/R,)
n(x)=nv-(n,-n,)
xc0
(1) cos(2krx cos
+ 8) 8
exp( x/R
M)
nM is the density of free electrons Here, R, = A (2mW)-“2; RM = (n~~/4k,)“~; in the metal bulk (x = - co); n 0 = n M (x = 0); 6 is the phase between the incident and reflected electron density waves; W is the electron work function for the metal-medium interface; and a, is the Bohr radius. The distribution parameters PZ,, are determined from the continuity condition and from the condition that the number of electrons at the interface is conserved. In the physics of surface phenomena, and especially in electrochemistry, one often deals with the situation where the double layer caused by an inhomogeneous electron density distribution near the metal surface is subject to an external electric field. In this case, the condition that the number of electrons is conserved takes the form 0 u=e nM-n,(x)] dx - ejomn,(x) dx (3) -CO
J [
so that the electron (charge): PZ,=nM
distribution
parameters
2/?--(1+cu2)q
(1+ (Y2-
p’) + (1+ a2)(1+
(1+ a2 + p’) + (1+ c2)q
field
(4)
(Y2+ (1 -I- p)”
atan6=
will depend on the external
p)q
(5)
where (Y= 2 k,R,, /3 = R,/R, and the dimensionless charge q = a/en, R,. We now apply this distribution to describe the capacitance of the EDL formed at the metal-electrolyte solution interface (a surface-inactive electrolyte is assumed to play the role of the electrolyte solution). For not too high electrolyte concentrations [l&23], the sum of the capacitances of two series-connected capacitors, usually referred to as the compact layer and the Gouy-Chapman layer, can be represented as c-1
= C,i
+ CEC’
(6)
The second term on the right-hand side of this relation is described quite well by the Gouy-Chapman theory so that it can always be calculated, and we should concentrate on the term corresponding to the compact layer. In turn, the compact layer capacitance determined by the gap between the metal surface and the plane of maximum approach of solution ions depends significantly on the external field because the “electronic tail” extends beyond the metal, and the centre of mass of the negative electronic charge shifts (non-linearly) under the external field. One has, therefore, to distinguish between the “effective” and geometrical interface [4,24,25].
309
In this case, the total potential represented in the form [26,27]
drop in the compact
layer can generally
A+(c)=${(r(x;-x:)+P;}
be
(7)
where xp and x,” are the centre-of-mass coordinates of the ionic and electronic charge, respectively (i.e. the “effective” coordinates of the ionic and electronic “plates” of the dense layer) and P,"is the surface density of the bound charge dipole moment at the solvent molecules. The differential capacitance is expressed as
(8) However, it is more convenient to use the integral capacitance of the dense layer K HI K-1
H
=
‘+(‘) = e
(9)
E*
CT
where L" = [x7- (&T/a)] effectively takes into account the phenomena caused by the polarization of solvent molecules and by the shift of the centre of mass of the ionic plate. A theoretical analysis of the K,(u) dependences was performed in refs. 28 and 29, where it was shown that the most adequate state of the system is the one in which both XT and x,” shift. The rate of these shifts may be different, which makes the experimental K,(a) curve non-monotonic. It is therefore interesting to determine how L" varies in different solvents as a function of the metal surface charge. In order to evaluate the dependence of L" on u, we will use a semi-empirical approach in which the coordinate x,” is calculated and L" is determined from experimental data for different solvents for which z* = n2 is assumed constant and equal to - 2. The coordinate x,” describing the electronic response of the surface to a homogeneous external field can be calculated from the following relation derived in refs. 24, 29 and 30: xe”= -;j_W
x[n”(x)-no(x)] dx (IO) 00 where n”(x) and no(x) are the electron densities in the metal with and without a field, respectively, and u is an external charge supplied to the metal. By virtue of distributions (1) and (2), x: can be expressed in the form
_e (1”
xa= e
U
=
x[n:(x)-n&(x)]
--CC e
dx+~mx[n:(x)-rz:(x)]
dx
u[(2k,)2-R;z]2 x tan V(n& -n&) I
-tan
S”(n& - n&)4k,R;1
+ (n; - n,)[ R$(2kF)2]]
(11)
310
For u = 0, we have - R,,(l + (~~)a
2en,R,,R,
limx,” = lim - i
0+0
i R,,(l
+
,izqa2+(1+&+-Tq2]
a')[ a2 +
-eRtono
(1 + j3o)2] + en,R,,RMj$[l
i
+ a2 -
4P0 +
3/30”]
= [a2+(1+P0)2]2 (14
where ecp/ W is replaced by the equivalent term ea/KW, and K is the total (experimental) EDL capacitance. Figure 1 presents calculated xE=’ values for a system formed by mercury and a surface-inactive electrolyte dissolved in different solvents: water, methanol, ethanol, ethylene glycol and dimethyl formamide vs. the electron work function WHg,s for the interface between mercury and these solvents. It can be seen that the dependence is a straight line, x,” decreasing with increasing work function WHg,s. This is no surprise because the probability of electron escape from the metal decreases with increasing barrier height. Figure 2 shows x: values (divided by e* = 2) calculated from relation (11) for a mercury-aqueous solution system, and also experimental (47rK,)-’ values for the same system. It can be seen that as the negative charge u increases, the centre of mass of the electron density x,” shifts at different rates. In the positive region, dx,“/da is less than that in the negative region, but as the negative charge increases (for u = - 15 to - 20 &/cm2), dx,“/du decreases and may have an extremum. The coordinate x,” behaves approximately in the same way as predicted by self-consistent calculation for a metal-vacuum interface [4,24,29].
Fig. 1. The shift of the centre different solvents [31].
of mass of the electronic
charge
x,” vs. the electron
work function
for
311
Fig. 2. Dependence of L’/c, and x,“/c+ on the electrode charge 0. (xz was calculated from the approximation x,” = j{x [ n:( x) - n p (x)] dx.) DMF: dimethyl formamide; EG: ethylene glycol.
The values of L" calculated from relation (9) with the x,” values calculated from relation (11) are represented for the three systems (Hg/H,O [19], Hg/EG [32] and Hg/DMF [33]), using experimental capacitances of the dense part of the EDL. It can be seen from Fig. 2 that as the negative charge increases, L" is ahead of x& i.e. the ionic plate of the dense layer moves away from the geometrical metal surface (X = 0) faster than the electronic plate does (x,“). From this consideration one may probably conclude that for a metal-electrolyte solution system the dependence of the compact layer capacitance on the electrode charge is determined not only by the shift of the centre of mass of the electronic charge xl, but also, to a great extent, by the shift of the second capacitor plate xp, which in turn is determined by the physical size of solvent molecules and their polarizability P,“. This conclusion differs from the conclusions of ref. 16, where it was assumed that the behaviour of the capacitance curve is governed by the response of the electronic plate only, which was, in all probability, a consequence of the incorrect expression for capacitance via x,. In the region of sufficiently large negative charge, one can hardly expect dipole reorientation, so that there is a possibility of avoiding the procedure of “hump” subtraction proposed in ref. 28 to obtain “base” capacitance curves K,(a). It is noteworthy that the curves of Fig. 2 were plotted by fitting the data to eqn. (9) for different solvents under the assumption that Q does not vary from solvent to solvent. This may not be exactly the case, but a variation of c* would change the scale of the L" variation only. The interpretation of the phenomenon considered above may depend on a
312
number of factors, such as the type of electron density deformation during charging and the repulsion of closed electron shells of solvent molecules from the “electronic tail” [28,29,34]. We believe that, in the region of large negative charges, the polarizability (a) of solvent molecules in the external field of the electrode can also be an important factor. ACKNOWLEDGEMENT
I am greatly indebted discussions.
to A.A. Komyshev
for critical remarks and fruitful
REFERENCES 1 N.D. Lang and W. Kohn, Phys. Rev., 61 (1970) 45555. 2 N.D. Smith, Phys. Rev., 181 (1969) 522. 3 A.V. Bennet in T.S. Jajadevaia and R. Vanselow @Is.), Advances in the Study of the Solid Surface, No. 9, Mir, Moscow, 1977, pp. 211-234 (Russian translation). 4 M.B. Partensky, Usp. Fii. Nauk, 128 (1979) 69. 5 J.H. Rose and J.E. Dobson, Solid State Commun., 37 (1981) 91. 6 V.F. Ukhov, R.M. Kobeleva, G.V. Dedkov and AI. Temrokov, Electron-Statistical Theory of Metals and Ionic Crystals, Nauka, Moscow, 1982 (in Russian). 7 P. Hohenberg and W. Kohn, Phys. Rev., 136 (1964) 864. 8 W. Kohn and L.J. Scham, Phys. Status Sohdi, 101 (1980) 239. 9 R.M. Kobeleva and B.R. Gelchinsky, Fiz. Met. Metalloved., 45 (1978) 25. 10 N.D. Lang and W. Kohn, Phys. Rev. B, 3 (1971) 1215. 11 M.A. Vorotyntsev and A.A. Komyshev, Elektrokhimiya, 20 (1982) 3. 12 S.N. Burmistrov and L.B. Dubovsky, Fiz. Tverd. Tela, 23 (1981) 3104. 13 S.F. Timashev, Elektrokhimiya, 15 (1979) 730. 14 H. GrPf, Nucl. Phys., A349 (1980) 339. 15 B.G. EngIer and J.N. Schwinger, Phys. Rev. A, 29 (1984) 2339. 16 G.A. Martynov and R.R. Salem, Can. J. Chem., 62 (1984) 1145. 17 G.A. Martynov and R.R. Salem, Adv. CoIIoid Interface Sci., 22 (1985) 229. 18 A.A. Komyshev in R.R. Dogonadze (Ed.), Proc. Int. Conf. on Electrodynamics and Quantum Phenomena at Interfaces, Telavi, 1984, Memiereba Pbl., Tbilisi, 1986. 19 W. Schmickler and D. Henderson, Progr. Surf. Sci., 22 (1986) 323. 20 J.-P. Badiah, Ber. Bunsenges. Phys. Chem., 91 (1987) 276. 21 V.J. Feldman, M.B. Partenskii and M.M. Vorob’ev, Progr. Surf. Sci., in press. 22 A.A. Komyshev in R.R. Dogonadze, E. Kalman, A.A. Komyshev and J. Ulstrup (Eds.), The Chemical Physics of Solvatation, Part C, Elsevier, Amsterdam, 1986, Ch. 6. 23 P. Delahay, The Double Layer and Electrode Kinetics, Mir, Moscow, 1967 (Russian translation). 24 M.B. Partenskii, Poverkhnost’, 10 (1982) 15. 25 A.K. Theophilou and A. Modinos, Phys. Rev. B, 6 (1972) 801. 26 M.B. Partenskii and M.M. Vorob’ev, Dokl Akad. Nauk SSSR, 278 (1984) 343. 27 M.B. Partenskii, V.J. Feldman and M.M. Vorob’ev, Dokl Akad. Nauk SSSR, 5 (1986) 1107. 28 A.A. Komyshev and M.A. Vorotyntsev, J. Electroanal. Chem., 167 (1984) 1. 29 V.J. Feldman, A.A. Komyshev and M.B. Partenskii, Solid State Commun., 53 (1985) 157. 30 P.G. Dzhavakhidze, A.A. Komyshev and G.J. Tsitsushvih, Solid State Commun., 52 (1984) 401. 31 A.M. Brodsky and Yu.Ya. Gurevich, The Theory of Electron Emission from Metals, Nauka, Moscow, 1973 (in Russian). 32 V.A. Chagehshvih, J.I. Dzaparidze and B.B. Damaskin, Elektrokhimiya, 13 (1977) 1300. 33 R.R. Salem, Elektrokhimiya, 11 (1975) 1096. 34 D. Price and J.W. Halley, J. EIectroanal. Chem., 150 (1983) 347.
J. Electroanal. Chem, 245 (1988) 307-312 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
Short communication ELECTRON
DENSITY AT THE METAL-LIQUID
INTERFACE
R.R. SALEM D. I. Mendeleev Moscow Chemical and Technological Institute, 9 Miusskaya Moscow 125047 (U.S.S.R.) (Received
5th August 1987; in revised form 13th November
Square,
1987)
In recent years, a large number of papers have appeared which report on the study of the surface properties of metals at the interface with a vacuum [l-6]. In most of these papers, the approach is based on the Hohenberg-Kohn-Scham density functional method [7,8] in which the kinetic, exchange and correlation energies of the electron gas are described in terms of a functional of the free electron density, n (x). However, calculation of the surface energy of metals has shown that the model of non-interacting electrons in a rectangular potential well (the so-called “jellium” model) gives quite a good fit to experiment, while an improved description of an electron system with many-particle and quantum effects involved leads to a considerable discrepancy with experimental data, and even to negative surface energies beginning with a certain density n(x) [9] (a so-called “ultradense catastrophe”). The way out can be found by introducing pseudo-potential corrections [lo] (see also ref. 11) or by using a simpler approach in which only the Coulomb part, i.e. the electrostatic energy of the electrical double layer (EDL), is retained in the surface energy Es, and numerical agreement with experiment is attained by virtue of fitting parameters. Apparently, this approach is also valid to describe a metal-liquid system. The density functional method uses exponential probe functions [2] corrected sometimes for density oscillations inside the metal [12] (though the solution of the Thomas-Fermi problem is of power law form). It is demonstrated in refs. 13-15 that exponential decay of the electron density follows from a modified Thomas-Fermi model with a consistent allowance for quantum effects near the interface and with the use of exact one-electron functions. Using this consideration, a relatively simple model was proposed in refs. 16 and 17 to describe the electron density distribution at the metal-liquid interface, the subject of recent theoretical interest (for reviews see refs. 18-22). 0022-0728/88/$03.50
8 1988 Elsevier Sequoia S.A.
308
The model [16,17] takes into account the decay of the electron density outside the metal (x > 0) and its oscillations inside the metal (x < 0): x>o
n(x)
= n, exp( -x/R,)
n(x)=nv-(n,-n,)
xc0
(1) cos(2krx cos
+ 8) 8
exp( x/R
M)
nM is the density of free electrons Here, R, = A (2mW)-“2; RM = (n~~/4k,)“~; in the metal bulk (x = - co); n 0 = n M (x = 0); 6 is the phase between the incident and reflected electron density waves; W is the electron work function for the metal-medium interface; and a, is the Bohr radius. The distribution parameters PZ,, are determined from the continuity condition and from the condition that the number of electrons at the interface is conserved. In the physics of surface phenomena, and especially in electrochemistry, one often deals with the situation where the double layer caused by an inhomogeneous electron density distribution near the metal surface is subject to an external electric field. In this case, the condition that the number of electrons is conserved takes the form 0 u=e nM-n,(x)] dx - ejomn,(x) dx (3) -CO
J [
so that the electron (charge): PZ,=nM
distribution
parameters
2/?--(1+cu2)q
(1+ (Y2-
p’) + (1+ a2)(1+
(1+ a2 + p’) + (1+ c2)q
field
(4)
(Y2+ (1 -I- p)”
atan6=
will depend on the external
p)q
(5)
where (Y= 2 k,R,, /3 = R,/R, and the dimensionless charge q = a/en, R,. We now apply this distribution to describe the capacitance of the EDL formed at the metal-electrolyte solution interface (a surface-inactive electrolyte is assumed to play the role of the electrolyte solution). For not too high electrolyte concentrations [l&23], the sum of the capacitances of two series-connected capacitors, usually referred to as the compact layer and the Gouy-Chapman layer, can be represented as c-1
= C,i
+ CEC’
(6)
The second term on the right-hand side of this relation is described quite well by the Gouy-Chapman theory so that it can always be calculated, and we should concentrate on the term corresponding to the compact layer. In turn, the compact layer capacitance determined by the gap between the metal surface and the plane of maximum approach of solution ions depends significantly on the external field because the “electronic tail” extends beyond the metal, and the centre of mass of the negative electronic charge shifts (non-linearly) under the external field. One has, therefore, to distinguish between the “effective” and geometrical interface [4,24,25].
309
In this case, the total potential represented in the form [26,27]
drop in the compact
layer can generally
A+(c)=${(r(x;-x:)+P;}
be
(7)
where xp and x,” are the centre-of-mass coordinates of the ionic and electronic charge, respectively (i.e. the “effective” coordinates of the ionic and electronic “plates” of the dense layer) and P,"is the surface density of the bound charge dipole moment at the solvent molecules. The differential capacitance is expressed as
(8) However, it is more convenient to use the integral capacitance of the dense layer K HI K-1
H
=
‘+(‘) = e
(9)
E*
CT
where L" = [x7- (&T/a)] effectively takes into account the phenomena caused by the polarization of solvent molecules and by the shift of the centre of mass of the ionic plate. A theoretical analysis of the K,(u) dependences was performed in refs. 28 and 29, where it was shown that the most adequate state of the system is the one in which both XT and x,” shift. The rate of these shifts may be different, which makes the experimental K,(a) curve non-monotonic. It is therefore interesting to determine how L" varies in different solvents as a function of the metal surface charge. In order to evaluate the dependence of L" on u, we will use a semi-empirical approach in which the coordinate x,” is calculated and L" is determined from experimental data for different solvents for which z* = n2 is assumed constant and equal to - 2. The coordinate x,” describing the electronic response of the surface to a homogeneous external field can be calculated from the following relation derived in refs. 24, 29 and 30: xe”= -;j_W
x[n”(x)-no(x)] dx (IO) 00 where n”(x) and no(x) are the electron densities in the metal with and without a field, respectively, and u is an external charge supplied to the metal. By virtue of distributions (1) and (2), x: can be expressed in the form
_e (1”
xa= e
U
=
x[n:(x)-n&(x)]
--CC e
dx+~mx[n:(x)-rz:(x)]
dx
u[(2k,)2-R;z]2 x tan V(n& -n&) I
-tan
S”(n& - n&)4k,R;1
+ (n; - n,)[ R$(2kF)2]]
(11)
310
For u = 0, we have - R,,(l + (~~)a
2en,R,,R,
limx,” = lim - i
0+0
i R,,(l
+
,izqa2+(1+&+-Tq2]
a')[ a2 +
-eRtono
(1 + j3o)2] + en,R,,RMj$[l
i
+ a2 -
4P0 +
3/30”]
= [a2+(1+P0)2]2 (14
where ecp/ W is replaced by the equivalent term ea/KW, and K is the total (experimental) EDL capacitance. Figure 1 presents calculated xE=’ values for a system formed by mercury and a surface-inactive electrolyte dissolved in different solvents: water, methanol, ethanol, ethylene glycol and dimethyl formamide vs. the electron work function WHg,s for the interface between mercury and these solvents. It can be seen that the dependence is a straight line, x,” decreasing with increasing work function WHg,s. This is no surprise because the probability of electron escape from the metal decreases with increasing barrier height. Figure 2 shows x: values (divided by e* = 2) calculated from relation (11) for a mercury-aqueous solution system, and also experimental (47rK,)-’ values for the same system. It can be seen that as the negative charge u increases, the centre of mass of the electron density x,” shifts at different rates. In the positive region, dx,“/da is less than that in the negative region, but as the negative charge increases (for u = - 15 to - 20 &/cm2), dx,“/du decreases and may have an extremum. The coordinate x,” behaves approximately in the same way as predicted by self-consistent calculation for a metal-vacuum interface [4,24,29].
Fig. 1. The shift of the centre different solvents [31].
of mass of the electronic
charge
x,” vs. the electron
work function
for
311
Fig. 2. Dependence of L’/c, and x,“/c+ on the electrode charge 0. (xz was calculated from the approximation x,” = j{x [ n:( x) - n p (x)] dx.) DMF: dimethyl formamide; EG: ethylene glycol.
The values of L" calculated from relation (9) with the x,” values calculated from relation (11) are represented for the three systems (Hg/H,O [19], Hg/EG [32] and Hg/DMF [33]), using experimental capacitances of the dense part of the EDL. It can be seen from Fig. 2 that as the negative charge increases, L" is ahead of x& i.e. the ionic plate of the dense layer moves away from the geometrical metal surface (X = 0) faster than the electronic plate does (x,“). From this consideration one may probably conclude that for a metal-electrolyte solution system the dependence of the compact layer capacitance on the electrode charge is determined not only by the shift of the centre of mass of the electronic charge xl, but also, to a great extent, by the shift of the second capacitor plate xp, which in turn is determined by the physical size of solvent molecules and their polarizability P,“. This conclusion differs from the conclusions of ref. 16, where it was assumed that the behaviour of the capacitance curve is governed by the response of the electronic plate only, which was, in all probability, a consequence of the incorrect expression for capacitance via x,. In the region of sufficiently large negative charge, one can hardly expect dipole reorientation, so that there is a possibility of avoiding the procedure of “hump” subtraction proposed in ref. 28 to obtain “base” capacitance curves K,(a). It is noteworthy that the curves of Fig. 2 were plotted by fitting the data to eqn. (9) for different solvents under the assumption that Q does not vary from solvent to solvent. This may not be exactly the case, but a variation of c* would change the scale of the L" variation only. The interpretation of the phenomenon considered above may depend on a
312
number of factors, such as the type of electron density deformation during charging and the repulsion of closed electron shells of solvent molecules from the “electronic tail” [28,29,34]. We believe that, in the region of large negative charges, the polarizability (a) of solvent molecules in the external field of the electrode can also be an important factor. ACKNOWLEDGEMENT
I am greatly indebted discussions.
to A.A. Komyshev
for critical remarks and fruitful
REFERENCES 1 N.D. Lang and W. Kohn, Phys. Rev., 61 (1970) 45555. 2 N.D. Smith, Phys. Rev., 181 (1969) 522. 3 A.V. Bennet in T.S. Jajadevaia and R. Vanselow @Is.), Advances in the Study of the Solid Surface, No. 9, Mir, Moscow, 1977, pp. 211-234 (Russian translation). 4 M.B. Partensky, Usp. Fii. Nauk, 128 (1979) 69. 5 J.H. Rose and J.E. Dobson, Solid State Commun., 37 (1981) 91. 6 V.F. Ukhov, R.M. Kobeleva, G.V. Dedkov and AI. Temrokov, Electron-Statistical Theory of Metals and Ionic Crystals, Nauka, Moscow, 1982 (in Russian). 7 P. Hohenberg and W. Kohn, Phys. Rev., 136 (1964) 864. 8 W. Kohn and L.J. Scham, Phys. Status Sohdi, 101 (1980) 239. 9 R.M. Kobeleva and B.R. Gelchinsky, Fiz. Met. Metalloved., 45 (1978) 25. 10 N.D. Lang and W. Kohn, Phys. Rev. B, 3 (1971) 1215. 11 M.A. Vorotyntsev and A.A. Komyshev, Elektrokhimiya, 20 (1982) 3. 12 S.N. Burmistrov and L.B. Dubovsky, Fiz. Tverd. Tela, 23 (1981) 3104. 13 S.F. Timashev, Elektrokhimiya, 15 (1979) 730. 14 H. GrPf, Nucl. Phys., A349 (1980) 339. 15 B.G. EngIer and J.N. Schwinger, Phys. Rev. A, 29 (1984) 2339. 16 G.A. Martynov and R.R. Salem, Can. J. Chem., 62 (1984) 1145. 17 G.A. Martynov and R.R. Salem, Adv. CoIIoid Interface Sci., 22 (1985) 229. 18 A.A. Komyshev in R.R. Dogonadze (Ed.), Proc. Int. Conf. on Electrodynamics and Quantum Phenomena at Interfaces, Telavi, 1984, Memiereba Pbl., Tbilisi, 1986. 19 W. Schmickler and D. Henderson, Progr. Surf. Sci., 22 (1986) 323. 20 J.-P. Badiah, Ber. Bunsenges. Phys. Chem., 91 (1987) 276. 21 V.J. Feldman, M.B. Partenskii and M.M. Vorob’ev, Progr. Surf. Sci., in press. 22 A.A. Komyshev in R.R. Dogonadze, E. Kalman, A.A. Komyshev and J. Ulstrup (Eds.), The Chemical Physics of Solvatation, Part C, Elsevier, Amsterdam, 1986, Ch. 6. 23 P. Delahay, The Double Layer and Electrode Kinetics, Mir, Moscow, 1967 (Russian translation). 24 M.B. Partenskii, Poverkhnost’, 10 (1982) 15. 25 A.K. Theophilou and A. Modinos, Phys. Rev. B, 6 (1972) 801. 26 M.B. Partenskii and M.M. Vorob’ev, Dokl Akad. Nauk SSSR, 278 (1984) 343. 27 M.B. Partenskii, V.J. Feldman and M.M. Vorob’ev, Dokl Akad. Nauk SSSR, 5 (1986) 1107. 28 A.A. Komyshev and M.A. Vorotyntsev, J. Electroanal. Chem., 167 (1984) 1. 29 V.J. Feldman, A.A. Komyshev and M.B. Partenskii, Solid State Commun., 53 (1985) 157. 30 P.G. Dzhavakhidze, A.A. Komyshev and G.J. Tsitsushvih, Solid State Commun., 52 (1984) 401. 31 A.M. Brodsky and Yu.Ya. Gurevich, The Theory of Electron Emission from Metals, Nauka, Moscow, 1973 (in Russian). 32 V.A. Chagehshvih, J.I. Dzaparidze and B.B. Damaskin, Elektrokhimiya, 13 (1977) 1300. 33 R.R. Salem, Elektrokhimiya, 11 (1975) 1096. 34 D. Price and J.W. Halley, J. EIectroanal. Chem., 150 (1983) 347.