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On the application of optimized cluster series for the description of the electrolyte model near a charged hard wall
ON THE APPLICATION OF OPTIMIZED CLUSTER SERIES FOR THE DESCRIPTION OF THE ELECTROLYTE MoDEL NEAR A CHARGED HARD WALL M. F. GOLOVKO,
S. N. BLOTSKY and 0. A. PKZIO
Lvov Department “Statistical Physics”, Institute for Theoretical Physics, Lvov-5. U.S.S.R. (Received
27 April 1988)
Abstract-An ion-dipole model of electrolyte solutions near a charged hard wall is considered. A method to derive the terms of optimized cluster expansion for the one particle distribution functions of ions and dipoles which urovide a set of aouroximations beyond the MSA is given. The third cluster coefficient approximation for io&all distributi& functions is in&tigated.
dipole moment (solvent). Let us denote the densities of ions p+, p- and that of dipoles--p,, the ions possess charges +q and the molecules the dipole moment p.. The system is electroneutral in total, ie p + + p _ = p,lZ. We assume that all the particles are of equal size O+ = CT- = u, = cr. Now, the system can be characterized by the set of reduced dimensionless parameters: the packing fraction q = s(pi + p,)a3/6, reverse ion temperature /?i+=q’/kTu, and the parameters LY =pJqa, E* = ((r3/kT)““4nz. The structure of the interface region will be given by one-particle distribution functions Q+(Z) and pJz, 9) describing the probability to find the Ion of type a at the distance z from the wall and correspondingly the dipole molecule s at the distance .z with orientation 9, where 9 is the angle between the dipole moment direction and the normal to the wall. Similarly to the case of ion-molecular systems in the bnlk[S, 127 the integral equation for the one-particle /distribution function of the system near the wall can be presented in the form:
A set of important results for the electrode-electrolyte interface description can be obtained on the basis of the models and methods, which have been developed and tested previously for bulk systems with electrostatic interactions and then general&d for inhomogeneous systems. The simplest model of an electrode as an ideal wall can be constructed starting from the multicomponent ion-molecular system where one of the ionic components is at infinite dilution and simultaneously the size of these ions goes to infinity. As a result of electroneutrality of the initial multicomponent system, the ideal impenetrable wall which is formed in this limit possesses a charge with definite surface density. This model reveals two main features of the electrode-its impenetrability and the presence of charge on the wall. On the other hand, the discreteness of the charge on the wall and other effects which appear at the consistent account of the surface medium are then neglected. The model considered here was first proposed in [l, 23, and successful investigations of the structural properties of the ion and ion-molecular electrolyte models near the wall have been given inC3-61 in the mean spherical (MSA) and generalized mean spherical (GMSA) approximations. However, the influence of the electrostatic effects on the properties of the interface in this framework is mainly due to the charge of the wall. In particular, MSA leads to the incorrect conclusion that the structure of the interface region near an uncharged wall does not depend on the charge of the ions and the dipole moment of the molecules. As with the case of the bulk ion-molecular systems, the consistent development beyond the MSA, ie the allowance for the many-particle correlations, can be done for the electrolyte near the walI within optimized cluster series[7, 8-J. First attempts to do this for the dipole solvent near the uncharged hard wall have been considered in[9, lo], and the appearance of the electrostatic image effects due to the account of manyparticle cluster coefficients was shown. The model which we consider in this paper for the electrolyte-wall system consists of the hard plane wall with homogeneous charge density z and a three-sort system of positively and negatively charged hard spheres (ions of the solute) and hard spheres with
where the index w denotes wall and x = a, s denote the ionic and solvent species. The function gk2(z, 9) is the one-particle distribution function of the reference system and the black circle is C,=,,,p,jdr,dR,. If one chooses the hard sphere model near the uncharged wall as the reference system, then gg(z, 9) =gi:(z)_ All other functions introduced in Equation (1) are: SCXW(z 79)-g
(z, 9)-g%(z)
-ks,,k
W&!(zN + 6JLdz, 8),
6c!y(r, a, 0,) =cl&. % QJ-&W --Ink zy(I 7 i-2XI RyYd”‘W) xy + 6E,,(r, QC, a,), 63
(2)
(3)
M. F. GOLOVKO
64
hx.Azr 9) = g,&,
9) - 1;
hg’(r, %, q = s!“:{r. a, a,) - 1, H,,k
~)=Kz!W+G,,(z,
(4)
9);
ff$!'(r, %, a,)= @J:)(r)+Gxy(r, R,, Cl,), w&, m;+,
(5)
~)=E,,(z, 9)-J%$xz);
%
$I=
E,,(r,a, a,)-Ec,O:(r),
05)
where E,,., and E,, denote the set of elementary diagrams of the corresponding distribution function. The optimized cluster expansion (Equation (I)) for an ion-dipole system near the wall is constructed on the basis of the screened potentials G,,(z, 9) and functionally depend on the screened potentials G,,(r, R,, 12,) of the corresponding bulk model. Both these functions are defined from the analytical solution of the MSA in each case[3, 111: W,,(z,
9) = HI;M,SA’(z,9),
H$t’(r, Q,, f$,) = H$?sA’(r, R,, n Y).
9)- 1= d!,JWev CG,,k W - 1. (10)
Then the corresponding perturbations for the direct correlation functions are given as follows: n,, a,))-
1)
- Gxy(r, a,, a,), ac,, = &J(z){exp
(11) (G,,(z,
8)) - I}-
G,,(z,
9). (12)
Now, by substitution of Equations (9H12) into Equation (1) and picking up the diagrams with only one field vertex, one obtains the analogue of the EXP3 approximation[l5,16] for the inhomogeneous model: g~~“)(z,
9) = g:%(z) exp (G,,(z, = &!(z)
exp
G,,(z,
P * g,,(z) = P *SKY’“(z) Po&v(z.
9)= P,&=“(z)
9) + GL?(z, 9))
+ 1 P. &?‘(W, IB 1
g::“““(z)ExpJ
The allowance for the diagrams with two field vertices produces the approximation which correspond to the first iterative step (It 1):
(14) By linearization
of EXP3 for the functions gxW and gxv
(cos 9),
= g::b (z){exp(G+,(z)+G%.,(z)) +exp(G_,(z)+G(3~(~)))/2,
&“e’Yz)EXPJ
=&W/2.
(17)
L exp (G,,(z, s 0
9)
+ GLz(z, 9)) sin 9 d 9, and correspondingly sity profiles are:
(18)
the charge and polarization
&&z)=Bi*Pfsl%z)
(exp (G+,(z)+
PE*XP&) =flY
P:s%)
s 0
den-
G%(z))
-exp(G_,(z)+G’~~(z))}/2, L exp (G,,(z,
(19) 9)
+ Gi$(z, 9)) cos 9 sin 9 d 9,
(20)
where: p: =p$,
/Y: =p,2/03kT,
G,,(z,
9) = G,,(z)
cos 9.
It is clear that within exponential approximations, contrary to MSA, the density profiles g$Y’)(z) and g~:“e”“‘L)(z)contain the contribution of electrostatic interactions besides the non-electrostatic contribution. Similarly, the charge and polarization density profiles (c.d.p. and p.d.p.) contain the non-electrostatic contribution besides the electrostatic one. It is important that unlike MSA the c.d.p. and p.d.p. within exponential approximations contain the non-linear dependence on the electric field on the wall E*. In order to manifest it one can compare the quantity p,(z) * gsw (none’)(~) within the MSA and EXP2 ap= p*wqs proximations: B: GAz)Pg%Yz)>
~:~~~~‘(z) = 8: 2 (13)
(15)
(16)
I?“*‘=
9)
* P * .&J’(z),
where B,(cos 9) are Legendre polynomials. It then follows that the density profiles ofions and dipoles are:
(8)
hZB:(r,% a,) = d$P:(r)exp CC,, (r, % WI - 1, (9)
~c~~‘=g~~‘(r)(exp(G,y(r,
one can obtain the EXP4 approximations[12] etc. As with the first iterative step, by applying Equation (14) and the analogous expression for the bulk function gry the following iterative approximations (It2) and higher can also be obtained. Now we shall separate the non electrostatic contribution in the one-particle distribution functions of ions and dipoles:
(7)
If one neglects the contribution of elementary diagrams 6E,, and JE,, in Equation (I) and the functions fit,,, 6c,,, then the one-particle distribution functions, which correspond to RHNC or its versions RLHNC and RQHNC[18], can be obtained from Equation (1) by iterative procedures. In particular, we shall choose as the zero-order approximation for the iterative procedure the exponential approximations for the bulk functions[13] and for the one-particle distribution function h,,(z, wc141:
kdz, 9) = gdz,
et al.
CG,,&)l,
(21)
(22)
the latter containing the saturation effect on the field E* expressed by the Langeve in function 2(x)= coth x - l/x. Similarly, we obtain: q~~xp2)(z) = q*(z)/&
gp’)(z)
= j3: tanh{G+,(z)} (23)
The behaviour of the set of density profiles within EXP2 has been discussed in detail in[ 14,171. It is clear that their behaviour is formed by the interplay of short-range interaction effects between the particles and the particles with a wall and on the other hand of long-range electrostatic effects between the particles and the electrostatic field on the wall. As with the case
Electrolyte model near a charged hard wall of the bulk ion-dipole electrolyte dependency on the value of the parameters x2 = 24qipt (?i = &/6) and ~~=8Q?,*j3(q~ = xp$/6) and E*, ci= Xi/N one can formally separate three regions of the behaviour of structural properties: Debye-like region, the region of solvation and short-range ionic ordering region. At high densities which correspond to the iondipole liquid and high concentration of ions (large x) short-range ionic order near the wall is formed. Here the c.d.p. q*(z) oscillate around the abscissa axis z, and the dipole subsystem produces only a weak disordering influence on the ionic order. On decreasing the ion concentration and correspondingly increasing the dipole parameter y, the c.d.p. decays more slowly and does not oscillate around the z-axis. The dipole subsystem essentially modulates Debye-like behaviour of q*(r). In other words, a layer of dipole molecules with predominant orientation between the ions arises which coincides with Debye-like form of the ion distribution functions near the wall, but essentially supplemented by the solvation effects. The increase of the field values E* is manifested in the strengthening of the polarization profile oscillations and promotes structuring of the electrolyte near the wall and polarization of dipoles in the vicinity of the wall as well as at larger distances. So, in total, EXP2 provides a qualitative description of the model structure_ However, it is similar to MSA and does not allow for the electrostatic effects in the limit E* -*O. It can be shown that in the limiting case of an uncharged wall E*+O and infinite dilution of ions Ci+O the ion-wall third cluster coefficient has the asymptotic of the electrostatic image interaction: GCJ+)= IW
_!5++0(~2))
42 2
65
Equation (13), are presented in Figs l-4. In the case of an uncharged wall (g* =0) the ion-wall interaction screened potentials are equal to zero and the functions p+(z)=p_(z) because of the equality of the sizes c+ =(T_. Here at low ion concentration, as with the solution of ions in strongly polar solvents (small values of x and large y), the cluster coefficient G!?(z) has the asymptotic form (Equation (24)) which is influenced by the oscillations due to solvation effects of the dipoles (Fig. 1). At high ion concentrations
-
IO
1.0
I
2.0
1.5
I
2.5
I
3.0
z
Fig. 2. Third cluster coefficient of the ion-wall distribution function of an ion-dipole system near a noncharged hard wall (Ef =O). Dipole molecules in an ionic melt (Ci = 0.98, I)= 0.4, /3,‘=50, a=O.1414).
(24)
and the dipole-wall coefficient G!:(z)l/z’-a result which was first obtained in[9]. Although the similar asymptotic behaviour can be eliminated for the higher cluster coefficients, the one-particle distribution function investigation within EXP3 clarifies the role of electrostatic image effects for non-point particles and their dependence on the ionic concentration. and the field values. The results which we obtained at the calculation of the third cluster coefficient ion-wall G$z)(z), given in
,
I
I
1
a.0
2.0
1.5
I
2.5
I
3.0
Z Fig. 3. Third cluster coefficient of the ion-wall distribution function of an ion-dipole system near the charged hard wall (E* ~2.5). Ions in strongly polar solvent (Ci=0.05, q =0.4, g: = 50, a = 0.25).
1.0
1.5
2.0
2.5
3.0
Z
Fig. 1. Third cluster coefficient of the ion-wall distribution function of an ion-dipole system near a non-charged hard wall (E*=O). Ions in a strongly polar solvent (C,=O.Ol, r]= 0.4, /3: = 50, tL= 0.1414).
Fig. 4. Third cluster coefficient of the ion-wall distribution function of an ion-dipole system near the charged hard wall (I? - 1.5). Dipole molecules in an ionic melt (Ci=0.98, q=o.q fi:=50, a=0.1414).
66
M. F. GOLOVKO
(large x and small y), the function G$z)(z) oscillates around the abscissa axis and reflect the ionic shortrange ordering for dense ionic systems (Fig. 2). At E*#O, the ion-wall screened potentials are non-zero and the functions p+(z) #p_(z). In the region of low values of x and large y the cluster coefficients G!;)(z) become negative near the wall and oscillate around the z-axis (Fig. 3). Up to the first minimum G’:!,.(z) < G?k(z) showing the preferential ordering of anions near the positively charged wall, and providing a strengthening of the EXP2 effect[14, 171. At high ion concentrations (large x) and at non-zero field E* #O the cluster coefficients behave similarly to the previous case but here G’:‘,(z)> G?L(z) up to the first minimum of G’$,,(z), weakening the effect of the EXP2 approximation[14, 171. In conclusion we would like to mention that although the EXP3 possesses some advantages compared with EXP2 the former needs further correction also. In particular at E* = 0, qi = 0, ie the case of an uncharged solute near uncharged wall, G!:)(z) = 0. So the EXP3 doesn’t contain the effect of the polar medium and the higher approximations are necessary to account this effect.
et al.
2. J. K. Percus, J. stat. Phys. 15, 423 (1976). 3. L. Blum and D. Henderson, J. them. Phys. 74, 1902 (1981). 4. F. Vericat, L. Blum and D. Henderson, J. them. Phys. 77,
5808 (1982). J. electroanal. 5. F. Vericat, L. Blum and D. Henderson Chem. 150, 315 (1983). 6. L. Blum, J. Hernando and J. L. Lebowitz, J. phys. Chem. 87, 2825 (1983). in The Chemical 7. M. F. Golovko and I. R. Yukhnovsky, Physics of Soloation (Edited by R. R. Dogonadze, E. KBlmBn, A. A. Kornychev and J. Ulstrup), Elsevier, Amsterdam., Part A, Ch. 5 (1985). I. I. Kuriljak, 0. A. Pizio and E. N. 8. M. F. Golovko, Sovjak, Problems of Contemporary Statistical Physics, p. 82. Naukova Dumka, Kiev (1985). Mol. Phys. 53, 1399 9. J.-P. Badiali and M. E. Bound-Hir, (1984). 10. J.-P. Badiali, M&c. Phys. 55, 939 (1985). 11. F. Vericat and L. Blum, J. stat. Phys. 32, 593 (1980). Phys12. M. F. Golovko, Preprint Institute for Theoretical ics N187P, Kiev (1984). 13. H. C. Andersen, D. Chander and J. D. Weeks J. them.
Phys. 56, 3812 (1972). 14. M. F. Golovko, 0. A. Pizio and S. N. Blotsky, Preprint Institute for Theoretical Physics N176P, Kiev (1984). 15. G. Stell and J. J. Weis, Phys. Rev. A16, 757 (1977). 16. M. F. Golovko and 0. A. Pizio, Ukrain. Fir. J. 21, 653
(1976). REFERENCES 1. D. Henderson, F. F. Abraham Phys. 31, 1291 (1976).
and J. A. Barker,
M&c.
and 0. A. Pizio, Proceedings of the 17. M. F. Golovko International Conference Telavi-84, Tbilisi, Metsniereba, 187. 18. L. Blum and F. Vericat, in The Chemical Physics of Salvation, Part A, Ch. 4, Elsevier, Amsterdam (1985).
S. N. BLOTSKY and 0. A. PKZIO
Lvov Department “Statistical Physics”, Institute for Theoretical Physics, Lvov-5. U.S.S.R. (Received
27 April 1988)
Abstract-An ion-dipole model of electrolyte solutions near a charged hard wall is considered. A method to derive the terms of optimized cluster expansion for the one particle distribution functions of ions and dipoles which urovide a set of aouroximations beyond the MSA is given. The third cluster coefficient approximation for io&all distributi& functions is in&tigated.
dipole moment (solvent). Let us denote the densities of ions p+, p- and that of dipoles--p,, the ions possess charges +q and the molecules the dipole moment p.. The system is electroneutral in total, ie p + + p _ = p,lZ. We assume that all the particles are of equal size O+ = CT- = u, = cr. Now, the system can be characterized by the set of reduced dimensionless parameters: the packing fraction q = s(pi + p,)a3/6, reverse ion temperature /?i+=q’/kTu, and the parameters LY =pJqa, E* = ((r3/kT)““4nz. The structure of the interface region will be given by one-particle distribution functions Q+(Z) and pJz, 9) describing the probability to find the Ion of type a at the distance z from the wall and correspondingly the dipole molecule s at the distance .z with orientation 9, where 9 is the angle between the dipole moment direction and the normal to the wall. Similarly to the case of ion-molecular systems in the bnlk[S, 127 the integral equation for the one-particle /distribution function of the system near the wall can be presented in the form:
A set of important results for the electrode-electrolyte interface description can be obtained on the basis of the models and methods, which have been developed and tested previously for bulk systems with electrostatic interactions and then general&d for inhomogeneous systems. The simplest model of an electrode as an ideal wall can be constructed starting from the multicomponent ion-molecular system where one of the ionic components is at infinite dilution and simultaneously the size of these ions goes to infinity. As a result of electroneutrality of the initial multicomponent system, the ideal impenetrable wall which is formed in this limit possesses a charge with definite surface density. This model reveals two main features of the electrode-its impenetrability and the presence of charge on the wall. On the other hand, the discreteness of the charge on the wall and other effects which appear at the consistent account of the surface medium are then neglected. The model considered here was first proposed in [l, 23, and successful investigations of the structural properties of the ion and ion-molecular electrolyte models near the wall have been given inC3-61 in the mean spherical (MSA) and generalized mean spherical (GMSA) approximations. However, the influence of the electrostatic effects on the properties of the interface in this framework is mainly due to the charge of the wall. In particular, MSA leads to the incorrect conclusion that the structure of the interface region near an uncharged wall does not depend on the charge of the ions and the dipole moment of the molecules. As with the case of the bulk ion-molecular systems, the consistent development beyond the MSA, ie the allowance for the many-particle correlations, can be done for the electrolyte near the walI within optimized cluster series[7, 8-J. First attempts to do this for the dipole solvent near the uncharged hard wall have been considered in[9, lo], and the appearance of the electrostatic image effects due to the account of manyparticle cluster coefficients was shown. The model which we consider in this paper for the electrolyte-wall system consists of the hard plane wall with homogeneous charge density z and a three-sort system of positively and negatively charged hard spheres (ions of the solute) and hard spheres with
where the index w denotes wall and x = a, s denote the ionic and solvent species. The function gk2(z, 9) is the one-particle distribution function of the reference system and the black circle is C,=,,,p,jdr,dR,. If one chooses the hard sphere model near the uncharged wall as the reference system, then gg(z, 9) =gi:(z)_ All other functions introduced in Equation (1) are: SCXW(z 79)-g
(z, 9)-g%(z)
-ks,,k
W&!(zN + 6JLdz, 8),
6c!y(r, a, 0,) =cl&. % QJ-&W --Ink zy(I 7 i-2XI RyYd”‘W) xy + 6E,,(r, QC, a,), 63
(2)
(3)
M. F. GOLOVKO
64
hx.Azr 9) = g,&,
9) - 1;
hg’(r, %, q = s!“:{r. a, a,) - 1, H,,k
~)=Kz!W+G,,(z,
(4)
9);
ff$!'(r, %, a,)= @J:)(r)+Gxy(r, R,, Cl,), w&, m;+,
(5)
~)=E,,(z, 9)-J%$xz);
%
$I=
E,,(r,a, a,)-Ec,O:(r),
05)
where E,,., and E,, denote the set of elementary diagrams of the corresponding distribution function. The optimized cluster expansion (Equation (I)) for an ion-dipole system near the wall is constructed on the basis of the screened potentials G,,(z, 9) and functionally depend on the screened potentials G,,(r, R,, 12,) of the corresponding bulk model. Both these functions are defined from the analytical solution of the MSA in each case[3, 111: W,,(z,
9) = HI;M,SA’(z,9),
H$t’(r, Q,, f$,) = H$?sA’(r, R,, n Y).
9)- 1= d!,JWev CG,,k W - 1. (10)
Then the corresponding perturbations for the direct correlation functions are given as follows: n,, a,))-
1)
- Gxy(r, a,, a,), ac,, = &J(z){exp
(11) (G,,(z,
8)) - I}-
G,,(z,
9). (12)
Now, by substitution of Equations (9H12) into Equation (1) and picking up the diagrams with only one field vertex, one obtains the analogue of the EXP3 approximation[l5,16] for the inhomogeneous model: g~~“)(z,
9) = g:%(z) exp (G,,(z, = &!(z)
exp
G,,(z,
P * g,,(z) = P *SKY’“(z) Po&v(z.
9)= P,&=“(z)
9) + GL?(z, 9))
+ 1 P. &?‘(W, IB 1
g::“““(z)ExpJ
The allowance for the diagrams with two field vertices produces the approximation which correspond to the first iterative step (It 1):
(14) By linearization
of EXP3 for the functions gxW and gxv
(cos 9),
= g::b (z){exp(G+,(z)+G%.,(z)) +exp(G_,(z)+G(3~(~)))/2,
&“e’Yz)EXPJ
=&W/2.
(17)
L exp (G,,(z, s 0
9)
+ GLz(z, 9)) sin 9 d 9, and correspondingly sity profiles are:
(18)
the charge and polarization
&&z)=Bi*Pfsl%z)
(exp (G+,(z)+
PE*XP&) =flY
P:s%)
s 0
den-
G%(z))
-exp(G_,(z)+G’~~(z))}/2, L exp (G,,(z,
(19) 9)
+ Gi$(z, 9)) cos 9 sin 9 d 9,
(20)
where: p: =p$,
/Y: =p,2/03kT,
G,,(z,
9) = G,,(z)
cos 9.
It is clear that within exponential approximations, contrary to MSA, the density profiles g$Y’)(z) and g~:“e”“‘L)(z)contain the contribution of electrostatic interactions besides the non-electrostatic contribution. Similarly, the charge and polarization density profiles (c.d.p. and p.d.p.) contain the non-electrostatic contribution besides the electrostatic one. It is important that unlike MSA the c.d.p. and p.d.p. within exponential approximations contain the non-linear dependence on the electric field on the wall E*. In order to manifest it one can compare the quantity p,(z) * gsw (none’)(~) within the MSA and EXP2 ap= p*wqs proximations: B: GAz)Pg%Yz)>
~:~~~~‘(z) = 8: 2 (13)
(15)
(16)
I?“*‘=
9)
* P * .&J’(z),
where B,(cos 9) are Legendre polynomials. It then follows that the density profiles ofions and dipoles are:
(8)
hZB:(r,% a,) = d$P:(r)exp CC,, (r, % WI - 1, (9)
~c~~‘=g~~‘(r)(exp(G,y(r,
one can obtain the EXP4 approximations[12] etc. As with the first iterative step, by applying Equation (14) and the analogous expression for the bulk function gry the following iterative approximations (It2) and higher can also be obtained. Now we shall separate the non electrostatic contribution in the one-particle distribution functions of ions and dipoles:
(7)
If one neglects the contribution of elementary diagrams 6E,, and JE,, in Equation (I) and the functions fit,,, 6c,,, then the one-particle distribution functions, which correspond to RHNC or its versions RLHNC and RQHNC[18], can be obtained from Equation (1) by iterative procedures. In particular, we shall choose as the zero-order approximation for the iterative procedure the exponential approximations for the bulk functions[13] and for the one-particle distribution function h,,(z, wc141:
kdz, 9) = gdz,
et al.
CG,,&)l,
(21)
(22)
the latter containing the saturation effect on the field E* expressed by the Langeve in function 2(x)= coth x - l/x. Similarly, we obtain: q~~xp2)(z) = q*(z)/&
gp’)(z)
= j3: tanh{G+,(z)} (23)
The behaviour of the set of density profiles within EXP2 has been discussed in detail in[ 14,171. It is clear that their behaviour is formed by the interplay of short-range interaction effects between the particles and the particles with a wall and on the other hand of long-range electrostatic effects between the particles and the electrostatic field on the wall. As with the case
Electrolyte model near a charged hard wall of the bulk ion-dipole electrolyte dependency on the value of the parameters x2 = 24qipt (?i = &/6) and ~~=8Q?,*j3(q~ = xp$/6) and E*, ci= Xi/N one can formally separate three regions of the behaviour of structural properties: Debye-like region, the region of solvation and short-range ionic ordering region. At high densities which correspond to the iondipole liquid and high concentration of ions (large x) short-range ionic order near the wall is formed. Here the c.d.p. q*(z) oscillate around the abscissa axis z, and the dipole subsystem produces only a weak disordering influence on the ionic order. On decreasing the ion concentration and correspondingly increasing the dipole parameter y, the c.d.p. decays more slowly and does not oscillate around the z-axis. The dipole subsystem essentially modulates Debye-like behaviour of q*(r). In other words, a layer of dipole molecules with predominant orientation between the ions arises which coincides with Debye-like form of the ion distribution functions near the wall, but essentially supplemented by the solvation effects. The increase of the field values E* is manifested in the strengthening of the polarization profile oscillations and promotes structuring of the electrolyte near the wall and polarization of dipoles in the vicinity of the wall as well as at larger distances. So, in total, EXP2 provides a qualitative description of the model structure_ However, it is similar to MSA and does not allow for the electrostatic effects in the limit E* -*O. It can be shown that in the limiting case of an uncharged wall E*+O and infinite dilution of ions Ci+O the ion-wall third cluster coefficient has the asymptotic of the electrostatic image interaction: GCJ+)= IW
_!5++0(~2))
42 2
65
Equation (13), are presented in Figs l-4. In the case of an uncharged wall (g* =0) the ion-wall interaction screened potentials are equal to zero and the functions p+(z)=p_(z) because of the equality of the sizes c+ =(T_. Here at low ion concentration, as with the solution of ions in strongly polar solvents (small values of x and large y), the cluster coefficient G!?(z) has the asymptotic form (Equation (24)) which is influenced by the oscillations due to solvation effects of the dipoles (Fig. 1). At high ion concentrations
-
IO
1.0
I
2.0
1.5
I
2.5
I
3.0
z
Fig. 2. Third cluster coefficient of the ion-wall distribution function of an ion-dipole system near a noncharged hard wall (Ef =O). Dipole molecules in an ionic melt (Ci = 0.98, I)= 0.4, /3,‘=50, a=O.1414).
(24)
and the dipole-wall coefficient G!:(z)l/z’-a result which was first obtained in[9]. Although the similar asymptotic behaviour can be eliminated for the higher cluster coefficients, the one-particle distribution function investigation within EXP3 clarifies the role of electrostatic image effects for non-point particles and their dependence on the ionic concentration. and the field values. The results which we obtained at the calculation of the third cluster coefficient ion-wall G$z)(z), given in
,
I
I
1
a.0
2.0
1.5
I
2.5
I
3.0
Z Fig. 3. Third cluster coefficient of the ion-wall distribution function of an ion-dipole system near the charged hard wall (E* ~2.5). Ions in strongly polar solvent (Ci=0.05, q =0.4, g: = 50, a = 0.25).
1.0
1.5
2.0
2.5
3.0
Z
Fig. 1. Third cluster coefficient of the ion-wall distribution function of an ion-dipole system near a non-charged hard wall (E*=O). Ions in a strongly polar solvent (C,=O.Ol, r]= 0.4, /3: = 50, tL= 0.1414).
Fig. 4. Third cluster coefficient of the ion-wall distribution function of an ion-dipole system near the charged hard wall (I? - 1.5). Dipole molecules in an ionic melt (Ci=0.98, q=o.q fi:=50, a=0.1414).
66
M. F. GOLOVKO
(large x and small y), the function G$z)(z) oscillates around the abscissa axis and reflect the ionic shortrange ordering for dense ionic systems (Fig. 2). At E*#O, the ion-wall screened potentials are non-zero and the functions p+(z) #p_(z). In the region of low values of x and large y the cluster coefficients G!;)(z) become negative near the wall and oscillate around the z-axis (Fig. 3). Up to the first minimum G’:!,.(z) < G?k(z) showing the preferential ordering of anions near the positively charged wall, and providing a strengthening of the EXP2 effect[14, 171. At high ion concentrations (large x) and at non-zero field E* #O the cluster coefficients behave similarly to the previous case but here G’:‘,(z)> G?L(z) up to the first minimum of G’$,,(z), weakening the effect of the EXP2 approximation[14, 171. In conclusion we would like to mention that although the EXP3 possesses some advantages compared with EXP2 the former needs further correction also. In particular at E* = 0, qi = 0, ie the case of an uncharged solute near uncharged wall, G!:)(z) = 0. So the EXP3 doesn’t contain the effect of the polar medium and the higher approximations are necessary to account this effect.
et al.
2. J. K. Percus, J. stat. Phys. 15, 423 (1976). 3. L. Blum and D. Henderson, J. them. Phys. 74, 1902 (1981). 4. F. Vericat, L. Blum and D. Henderson, J. them. Phys. 77,
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