The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices
The replica Ornstein-Zernike equations ion-molecular fluid adsorbed in a disordered ion-molecular matrix were applied. The explicit expressions for the potentials screening by the point particles are obtained. The analysis of the screening potentials for ion-dipolar case are presented. It is shown...
Gespeichert in:
| Datum: | 1999 |
|---|---|
| Hauptverfasser: | , |
| Sprache: | English |
| Veröffentlicht: |
Інститут фізики конденсованих систем НАН України
1999
|
| Schriftenreihe: | Condensed Matter Physics |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/120390 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices / M.F. Holovko, Z.V. Polishchuk // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 267-272. — Бібліогр.: 12 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-120390 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1203902025-02-09T20:55:13Z The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices Екрановані потенціали для іонно-дипольної системи, адсорбованої на іонно-дипольних невпорядкованих матрицях Holovko, M.F. Polishchuk, Z.V. The replica Ornstein-Zernike equations ion-molecular fluid adsorbed in a disordered ion-molecular matrix were applied. The explicit expressions for the potentials screening by the point particles are obtained. The analysis of the screening potentials for ion-dipolar case are presented. It is shown that fluid-fluid interionic potentials include connected and blocked parts. Розглядається метод репліки для рівнянь Орнштейна-Церніке для іонно-молекулярної рідини, адсорбованої в невпорядкованій іонно-молекулярній матриці. Отримано явні вирази для екранованих потенціалів для іонно-дипольного випадку. Показано, що флюїд-флюїд міжіонний потенціал розкладається на зв’язуючу і блоковану частини. 1999 The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices / M.F. Holovko, Z.V. Polishchuk // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 267-272. — Бібліогр.: 12 назв. — англ. 1607-324X DOI:10.5488/CMP.2.2.267 PACS: 61.20, 05.20 https://nasplib.isofts.kiev.ua/handle/123456789/120390 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The replica Ornstein-Zernike equations ion-molecular fluid adsorbed in a
disordered ion-molecular matrix were applied. The explicit expressions for
the potentials screening by the point particles are obtained. The analysis
of the screening potentials for ion-dipolar case are presented. It is shown
that fluid-fluid interionic potentials include connected and blocked parts. |
| author |
Holovko, M.F. Polishchuk, Z.V. |
| spellingShingle |
Holovko, M.F. Polishchuk, Z.V. The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices Condensed Matter Physics |
| author_facet |
Holovko, M.F. Polishchuk, Z.V. |
| author_sort |
Holovko, M.F. |
| title |
The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices |
| title_short |
The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices |
| title_full |
The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices |
| title_fullStr |
The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices |
| title_full_unstemmed |
The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices |
| title_sort |
screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120390 |
| citation_txt |
The screening potentials of ion-dipole system adsorbed in ion-dipolar disordered matrices / M.F. Holovko, Z.V. Polishchuk // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 267-272. — Бібліогр.: 12 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovkomf thescreeningpotentialsofiondipolesystemadsorbediniondipolardisorderedmatrices AT polishchukzv thescreeningpotentialsofiondipolesystemadsorbediniondipolardisorderedmatrices AT holovkomf ekranovanípotencíalidlâíonnodipolʹnoísistemiadsorbovanoínaíonnodipolʹnihnevporâdkovanihmatricâh AT polishchukzv ekranovanípotencíalidlâíonnodipolʹnoísistemiadsorbovanoínaíonnodipolʹnihnevporâdkovanihmatricâh AT holovkomf screeningpotentialsofiondipolesystemadsorbediniondipolardisorderedmatrices AT polishchukzv screeningpotentialsofiondipolesystemadsorbediniondipolardisorderedmatrices |
| first_indexed |
2025-11-30T16:34:11Z |
| last_indexed |
2025-11-30T16:34:11Z |
| _version_ |
1850233798454673408 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No 2(18), pp. 267–272
The screening potentials of ion-dipole
system adsorbed in ion-dipolar
disordered matrices
M.F.Holovko, Z.V.Polishchuk
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 290011 Lviv, Ukraine
Received September 4, 1998
The replica Ornstein-Zernike equations ion-molecular fluid adsorbed in a
disordered ion-molecular matrix were applied. The explicit expressions for
the potentials screening by the point particles are obtained. The analysis
of the screening potentials for ion-dipolar case are presented. It is shown
that fluid-fluid interionic potentials include connected and blocked parts.
Key words: replica Ornstein-Zernike equations, screening potentials,
porous media, ion-molecular system
PACS: 61.20, 05.20
The effects of quenched disorder on the fluid properties have received increased
attention in recent years. In theoretical studies the matrix-fluid system is considered
as a special binary mixture in which the matrix is treated as a rigid set of obsta-
cles,obtained by quenching an equilibrium configuration of particles (usually noted
as species “0”) with the fluid molecules (noted as species “1”) at equilibrium in the
presence of quenched particles. Resulting from the applying of the replica trick the
description of the original matrix-fluid system was replaced by a multicomponent
equilibrium (s+1) component mixture for which it was possible to use the standard
methods of liquid-state theory (the integral equations, cluster expansions ets.). The
specific feature of such (s+1)-component mixture is that a pair of fluid particles from
different replicas has no interaction and at the end of calculation it takes the limit
s → 0, assuming that there is no problem in performing the analytical continuation
for noninteger values of s.
In this way Given and Stell [1-2] recently proposed a set of coupled integral
equations, the so-called replica Ornstein-Zernike (ROZ) equations, relating the total
pair correlation functions hαβ(r) of fluid-matrix mixture to the corresponding direct
c© M.F.Holovko, Z.V.Polishchuk 267
M.F.Holovko, Z.V.Polishchuk
correlation functions cαβ(r). Using the Fourier-transform
f̂(k) = 4π
∞
∫
0
r2 dr
sin kr
kr
f(r) (1)
the formal solution of these ROZ equations can be represented in the following form
ĥ00(k) =
ĉ00(k)
1− ρ0ĉ00(k)
, (2)
ĥ01(k) =
ĉ01(k)
[1− ρ0ĉ00(k)] [1− ρ1ĉc11(k)]
, (3)
ĥc
11
(k) =
ĉc11(k)
1− ρ1ĉc11(k)
, (4)
ĥ12(k) =
ĉ12(k)
[1− ρ1ĉc11(k)]
2
+
ρ0ĉ01(k)ĉ10(k)
[1− ρ0ĉ00(k)] [1− ρ1ĉc11(k)]
2
, (5)
where due to the symmetry h01(r) = h10(r), c01(r) = c10(r), ρ0 and ρ1 are average
density of matrix and fluid particles correspondingly. The fluid-fluid pair correlation
functions are decomposed into two parts
h11(r) = hc
11(r) + h12(r), c11(r) = cc11(r) + c12(r), (6)
where hc
11
(r) and cc
11
(r) are the connected parts; h12(r) = hb
11
(r) and c12(r) = cb
11
(r)
are the correlation functions of two fluid molecules from different replica copies,
so-called blocked part of the fluid-fluid pair correlation functions.
For this time the application of ROZ equations was restricted the description
of simple fluids in random and quenched matrices [3-6].Only recently a study of
associating fluids [6-7] and ionic fluids [8-9] in porous media using ROZ equations was
initiated. In particular, in [8] the ROZ equations were applied for the investigation of
the screening potentials for the proposed in [10] model of ionic fluids adsorbed in an
electroneutral totally disordered matrix of ion. In this model both ionic subsystems
were presented as point charges interacting via Coulomb interactions. The important
influence of the blocking effects of the quenched matrix on the screening between
the fluid ions was shown. The obtained screening potentials can be used to build
the cluster expansions of pair correlation functions and free energy of the system
[11-12] or to develop the renormalization scheme for the long-range terms in ion-
ion correlations for the numerical solution ROZ equations [9]. However the model
considered and the results obtained should be, as usual, [11-12] revised in terms of
more civilized ion-molecular models which take into account explicitly the solvent
molecules and dielectric peculiarities of the matrix.
Investigating the screening potentials of such ion-molecular fluid-matrix model
is the aim of this article. We consider a general case of two electroneutral three-
component ion-molecular subsystems. The first one represents the matrix and the
other one is fluid. As regards the usual ion-molecular systems [11-12] the ions have
268
The screening potentials in disordered matrices
the charges ezαe and the molecules characterized by the generalized charges Qα
s
(▽),
where index α = 0 or 1 and denotes the quenched and fluid component respectively.
The index a denotes the ions and the index s denotes the molecules. For compactness
we will also use the indices x, y, which include ions and molecules
Qα
x(▽) =
{
ezαa , x = a
Qα
s (▽), x = s.
(7)
Thus, the electrostatic interactions between the particles in the model considered
can be represented in the following form
Φαβ
xy (r) = Qα
x(▽)Qβ
y(−▽)
1
R
. (8)
For simplicity, in this case we consider: the molecules with point dipole moment ~p s
[11-12] for which
Qα
s (▽) = (pαs ▽). (9)
In order to obtain the expressions for the screening potentials by point particles we
should put [11-12]
Cαβ
xy (r) = − 1
kBT
Φαβ
xy (r), hαβ
xy (r) = Gαβ
xy (r), (10)
where Gαβ
xy (r) are the screening potentials, Φαβ
xy (r) are given by equation (8) and
Φ12
xy(r) = 0 since the particles from different replicas do not interact.
By expanding the Coulomb potential 1/r in a Fourier series the electrostatic
potentials Φαβ
xy (r) can be represented in the following form
Φαβ
xy (r) =
∑
k
1
V
4π
k2
Q̂α
x(k)Q
β
y (−k)eik̄r̄, (11)
where Q̂α
x(k) = ezαa for x = a and Q̂α
x(k) = Q̂s(k) = i(pα
s
k) for x = s.
Now according to expressions (2)–(5) and in assumption (10) the Fourier trans-
form of the screening potentials can be explicitly represented as follows:
Ĝ00
xy(k) = − 1
kBT
Q̂0
x(k)Q̂
0
y(−k)
(
ε0(k) +
κ2
0
k2
)
k2
, (12)
Ĝ01
xy(k) = − 1
kBT
Q0
x(k)Q
0
y(−k)
k2
[
ε0(k) +
κ2
0
k2
] [
ε1(k) +
κ2
1
k2
] , (13)
Ĝ11
xy(k) = Ĝc,11
xy (k) + Ĝ12
xy(k), (14)
Ĝc,11
xy (k) = − 1
kBT
Q̂x(k)Q
1
y(−k)
k2
(
ε1(k) +
κ2
1
k2
) , (15)
269
M.F.Holovko, Z.V.Polishchuk
Ĝ12
xy(k) =
1
kBT
Q1
x(k)Q
1
y(−k)
1
k2
×
ε0(k)−1
ε0(k)
1
(
ε1(k)+
κ2
1
k2
)2
+
κ2
0
ε0(k)k2
1
(
κ2
0
k2
+ ε0(k)
)(
ε1(k)+
κ2
1
k2
)2
,(16)
where κ0 =
[
4π 1
kBT
∑
aρ0ae
2(z0a)
2
]
1
2
, κ1 =
[
4π 1
kBT
∑
aρ1ae
2(z1a)
2
]
1
2
is the recipro-
cal Debye-Huckel radius of screening for the matrix and fluid ionic subsystem, re-
spectively. As usual [11-12], the molecular screening introduces the static dielectric
functions for the matrix and fluid subsystems, respectively
ε0(k) = 1 + 4π
1
kBT
ρ0
s
∫
dΩ0
s
1
k2
Q0
s
(k)Q0
s
(−k) = ε0 = 1 + 3y0 , (17)
ε1(k) = 1 + 4π
1
kBT
ρ1s
∫
dΩ1
s
1
k2
Q1
s (k)Q
1
s (−k) = ε1 = 1 + 3y1 , (18)
which have the meaning of the dielectic constant of matrix and fluid subsystem
respectively, y0 =
4
3
π 1
kBT
ρ0
s
(p0
s
)2, y1 =
4
3
π 1
kBT
ρ1
s
(p1
s
)2.
The expressions (12)–(16) are the main result of this article. For the ion-dipolar
case ε0(k) = ε0 and ε1(k) = ε1 the screening potentials in coordinate space have the
following form
G00
xy(R) = − 1
kBT
Q0
x(▽)Q0
y(−▽)
1
ε0r
exp
(
− κ0√
ε0
r
)
, (19)
G01
xy(r) = − 1
kBT
Q0
x(▽)Q1
y(−▽)
1
ε0ε1r
κ2
0
ε0
(
κ2
1
ε1
− κ2
0
ε0
)−1
×
[
exp
(
− κ0√
ε0
r
)
− κ2
1
ε0
κ2
0ε1
exp
(
− κ1√
ε1
r
)]
, (20)
Gc,11
xy (r) = − 1
kT
Q1
x(▽)Q1
y(−▽)
1
ε1r
exp
(
− κ1√
ε1
r
)
, (21)
G12
xy(r) =
1
kT
Q1
x(▽)Q1
y(−▽)
1
ε0ε21r
(ε0−1)
(
1−1
2
κ1√
ε1
r
)
exp
(
− κ1√
ε1
r
)
+
1
kT
Q1
x(▽)Q1
y(−▽)
1
ε0ε21r
κ4
0
ε2
0
(
κ2
0
ε0
−κ2
1
ε1
)−2
×
[
exp
(
− κ0√
ε0
r
)
−
(
1−
(
1−κ2
1ε0
κ2
0
ǫ1
)
κ1√
ε1
r
2
)
exp
(
− κ1√
ε1
r
)]
. (22)
The results obtained for ion-ion screening potential coincide with the result [8]
only for the case ε0 = 1. In order to understand the reason of such difference we
consider ionic mean force potentials at the infinite dilution of ions which play the
role of interionic potentials for considering the system in the McMillan-Mayer level
of a description [11]. At the infinite dilution of ions from expressions (19)–(23) we
270
The screening potentials in disordered matrices
have
W 00
ab
(r) = e2z0az
0
b
1
ε0r
, W 01
ab
(r) = e2z0ez
1
b
1
ε0ε1r
,
W c,11
ab
(r) = e2z1az
1
b
1
ε1r
, W 12
ab
(r) = −e2z1az
1
b
ε0 − 1
ε0ε21
1√
r
. (23)
At the same time for the model used in [8]
W 00
ab
(r) =
e2z0az
0
b
ε0r
, W 01
ab
= e2z0az
1
b
1
ε1r
, W c,11
ab
(r) =
e2z1az
1
b
ε1r
, W 12
ab
(r) = 0. (24)
We can see that both models coincide only for the case ε0 = 1. The principal
difference of the model (23) compared to (24) is the presence of the blocked part
W 12
ab
(r) in interionic potentials which describes the interactions through the dielec-
tric matrix. The expression (23) also corrects fluid-matrix interionic interactions.
We should note that interionic interactions have got Coulomb form with effective
dielectric constant
εef = ε1
ε1ε0
ε0(ε1 − 1) + 1
. (25)
Since ε1ε0 > ε0(ε1 − 1) + 1, εef > ε1. In particular εef = ε1 for ε0 = 1 and
εef = ε0 for ε1 = 1. Nevertheless, according to the scheme of describing the effects of
quenched disorder on the fluid properties we should separate the interionic potential
into connected and blocked parts
W 11
ab
(r) = W c,11
ab
(r) +W b,11
ab
(r). (26)
It is not difficult to show that the ROZ equations (2)–(5) with approximations
(10) for the model (23) lead to the interionic screening potentials in the form (19)–
(22).
The analysis of the result (19)–(22) shows that as for the usual ion-dipole case
[11–12] resulting from ionic screening, all electrostatic interactions (ion-ion, ion-
dipole and dipole-dipole) decay exponentially. As in [8] connected and blocked parts
of the fluid-fluid screening potentials have opposite signs and different asymptotics.
The connected part decay as 1
r
exp
(
− κα√
εα
r
)
, the blocked part has more long-range
asymptotic exp
(
− κ1√
ε1
r
)
. Hence, the screening potentials G11
xy(r) can change their
signs for intermediate distances. Such interparticle ordering was observed in [8–10]
for ionic systems and we see that resulting from ionic screening the similar ordering
is possible for dipolar molecules as well.
References
1. Given J.A., Stell G. // J. Chem. Phys., 1992, vol. 97, p. 4573.
2. Given J.A., Stell G. // Physica A, 1994, vol. 209, p. 495.
3. Lomba E., Given J.A., Stell G., Weis J.J., Levesque D. // Phys. Rev. E, 1993, vol. 48,
p. 233.
271
M.F.Holovko, Z.V.Polishchuk
4. Vega C., Kaminsky R.D., Monson P.A. // J. Chem. Phys., 1993, vol. 99, p. 3003.
5. Kierlik E., Rosinberg M.L., Tarjus G., Monson P.A. // J. Chem. Phys., 1997, vol. 106,
p. 264.
6. Trokhymchuk A.D., Pizio O., Holovko M.F., Sokolowski S. // J. Chem. Phys., 1996,
vol. 100, p. 17004.
7. Trokhymchuk A.D., Pizio O., Holovko M.F., Sokolowski S. // J. Chem. Phys., 1997,
vol. 106, p. 220.
8. Hribar B., Pizio O., Trokhymchuk A., Vlachy V. // J. Chem. Phys., 1997, vol. 107,
p. 6335.
9. Hribar B., Pizio O., Trokhymchuk A., Vlachy V. // J. Chem. Phys., 1998, vol. 109,
p. 2480.
10. Bratko D., Chakraborty A.K. // J. Chem. Phys., 1996, vol. 104, p. 7700.
11. Yukhnovsky I.R., Holovko M.F. The Statistical Theory of the Classical Equilibrium
Systems. Kiev, Naukova Dumka, 1980 (in Russian).
12. Golovko M.F., Yukhnovsky I.R. In: The Chemical Physics of Solvation. Amsterdam,
Elsevier, 1985, vol. A, p. 207–262.
Екрановані потенціали для іонно-дипольної
системи, адсорбованої на іонно-дипольних
невпорядкованих матрицях
М.Ф.Головко, З.В.Поліщук
Інститут фізики конденсованих систем НАН Укpаїни,
290011 Львів, вул. Свєнціцького, 1
Отримано 4 вересня 1998 р.
Розглядається метод репліки для рівнянь Орнштейна-Церніке для
іонно-молекулярної рідини, адсорбованої в невпорядкованій іонно-
молекулярній матриці. Отримано явні вирази для екранованих по-
тенціалів для іонно-дипольного випадку. Показано, що флюїд-флю-
їд міжіонний потенціал розкладається на зв’язуючу і блоковану
частини.
Ключові слова: метод репліки для рівнянь Орнштейна-Церніке,
екрановані потенціали, пористе середовище, іонно-молекулярна
система
PACS: 61.20, 05.20
272
|