Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles
The Replica Ornstein-Zernike (ROZ) equations were used to study the adsorption of ions from electrolyte mixtures. The adsorbent was represented as a quenched primitive model +1:-1 size symmetric electrolyte, while the mobile particles were ions differing in charge and/or size. The ROZ equations in...
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| Cite this: | Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles / M. Lukšič, G. Trefalt, B. Hribar-Lee // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 717-724. — Бібліогр.: 29 назв. — англ. |
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| citation_txt | Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles / M. Lukšič, G. Trefalt, B. Hribar-Lee // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 717-724. — Бібліогр.: 29 назв. — англ. |
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| description | The Replica Ornstein-Zernike (ROZ) equations were used to study the adsorption of ions from electrolyte mixtures.
The adsorbent was represented as a quenched primitive model +1:-1 size symmetric electrolyte, while
the mobile particles were ions differing in charge and/or size. The ROZ equations in hypernetted-chain (HNC)
approximation were tested against new Monte Carlo results in the grand canonical ensemble; good agreement
between the two methods was obtained. The ROZ/HNC theory was then used to study the exclusion
coefficients as a function of size and/or charge asymmetry of the annealed ions.
Реплiчне рiвняння Орнштейна-Цернiке (РОЦ) застосоване для вивчення адсорбцiї iонiв iз сумiшей
електролiтiв. Адсорбент розглядається у виглядi замороженої примiтивної +1:-1 моделi розмiрно
симетричного електролiту, тодi як рухомi частинки – це iони, що вiдрiзняються зарядом та/або роз-
мiрами. Данi, що отриманi з рiвняння РОЦ у гiперланцюжковому наближеннi (ГЛН), порiвняно з но-
вими результатами Монте-Карло моделювання у великому канонiчному ансамблi i виявлено хороше
узгодження мiж обома методами. Теорiю РОЦ/ГЛН далi було використано для вивчення коефiцiєнтiв
виключення, в залежностi вiд параметрiв розмiрної та/або зарядової асиметрiй нерухомих iонiв.
|
| first_indexed | 2025-12-07T16:12:59Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2009, Vol. 12, No 4, pp. 717–724
Application of Replica Ornstein-Zernike equations in
studies of the adsorption of electrolyte mixtures in
disordered matrices of charged particles∗
M. Lukšič1, G. Trefalt2, B. Hribar-Lee1
1 Faculty of Chemistry and Chemical Technology, University of Ljubljana,
5 Aškerčeva Str., SI–1000 Ljubljana, Slovenia
2 Jožef Stefan Institute, Jamova 39, SI–1000 Ljubljana, Slovenia
Received June 9, 2009, in final form June 19, 2009
The Replica Ornstein-Zernike (ROZ) equations were used to study the adsorption of ions from electrolyte mix-
tures. The adsorbent was represented as a quenched primitive model +1:−1 size symmetric electrolyte, while
the mobile particles were ions differing in charge and/or size. The ROZ equations in hypernetted-chain (HNC)
approximation were tested against new Monte Carlo results in the grand canonical ensemble; good agree-
ment between the two methods was obtained. The ROZ/HNC theory was then used to study the exclusion
coefficients as a function of size and/or charge asymmetry of the annealed ions.
Key words: Electrolyte mixtures, primitive model electrolyte, random porous material, Donnan exclusion
coefficient, Replica Ornstein-Zernike integral equation, Monte Carlo simulation
PACS: 82.60.-s, 02.30.Rz, 61.20.-p
1. Introduction
Partitioning of electrolytes between porous materials, adsorbents, and the bulk solution is
not only a matter of academic interest, but finds its practical application in many technological,
industrial and biological processes (desalination of water, ion exchange, membrane equilibria etc.)
[1]. In all these processes the selectivity of the adsorption of different kinds of ions is crucial:
the concentration ratio of competing ions in the adsorbent is different from that in the bulk
solution [2]. Apart from numerous experimental studies the phenomena have been extensively
studied theoretically, for review see, for example, [3,4]. While most of the earlier theories are based
on the empirical or semiempirical equations, the later theories have been developed that provide the
correct description of the phenomena based on the physical chemistry of these systems [1]. One of
the first theories of this kind is based on the classical Donnan equilibrium [5] where the rejection of
an electrolyte from the adsorbent is measured in terms of the so-called Donnan exclusion coefficient,
Γ, which can be for a certain ionic species A, for this purpose, defined as [6]:
ΓA =
cout
A − cin
A
cout
A
, (1)
where cA is the concentration of the ionic species A, and upper indexes “out” and “in” apply to
the bulk solution and adsorbent, respectively.
A more detailed model of electrolyte exclusion, in which the ion concentration profiles in the
adsorbent are evaluated from the properties of ions and the adsorbent, is based on the Poisson-
Boltzmann (PB) equation [7,8]. In contrast to the previous theories, different activity coefficients
inside and outside the adsorbent are taken into account [9]. Although the PB theory has successfully
described various experimental results it contains several statistical-thermodynamical approxima-
tions that can lead to unphysical results in some cases. In the last decades, new class of theories
∗Dedicated to the 100-th anniversary of Prof. M.M. Bogolyubov.
c© M. Lukšič, G. Trefalt, B. Hribar-Lee 717
M. Lukšič, G. Trefalt, B. Hribar-Lee
appeared in which the disordered porous materials filled with fluid are treated as partly quenched
systems in which some of the degrees of freedom are quenched and others are annealed. The sys-
tems differ from regular mixtures; the statistical-mechanical average which is needed to obtain
the free energy describing the confined fluid, becomes a double ensemble average [10–17]. The
thermodynamic and structural properties of these systems can be calculated using the computer
simulations and/or the replica integral equation theories. This work presents the continuation of
our previous studies of partly quenched systems containing charges [18–25]. The quenched “phase”
(we shall call it the matrix) is some frozen (quenched) equilibrium distribution of a symmetric
model +1:−1 electrolyte. Within such a matrix, a model mixture of two +1:−1 or +1:−1 and
+z:−1 electrolytes can anneal (come into thermodynamic equilibrium with the surrounding bulk
solution of the same chemical composition) [24]. We present the results for the thermodynamic
properties (the excess internal energy, and the mean activity coefficient) of a model electrolyte
mixture confined in matrices of different densities. We were mostly interested in what way the
density of the matrix of random structure effects the confined electrolyte activities and therefore
the Donnan exclusion coefficients. Furthermore, we investigated how the size and/or charge asym-
metry effects the adsorption of ions from electrolyte mixtures. The results were compared with the
Monte Carlo simulation results obtained in the grand canonical ensemble.
2. The model and methods
The model used here was similar to the one described in several previous papers [19,21,22,24].
The system studied consisted of two subsystems: the first, here called the matrix, was composed
of the quenched, and the second of the annealed ions. The assumption is that the matrix does not
respond to the presence of the annealed fluid. The modeled system was considered at a McMillan-
Mayer level of description; the solvent was treated as a dielectric continuum with the dielectric
constant of pure water under conditions of study. We use the notation as it has already become
traditional in describing such systems: the superscripts 0 and 1 correspond to the matrix and the
annealed fluid species, respectively.
The matrix represented as an electroneutral system of positively and negatively charged hard
spheres of equal sizes (σ0
+ = σ0
− = 4.25 Å ) is assumed to be formed by a rapid quench of a model
+1:−1 electrolyte at a certain temperature T0. It is therefore assumed that the structure of the
matrix corresponds to an equilibrium state of an ionic fluid of concentration c0 at T0, governed by
the interaction potential between pair of particles:
Umn
ij (r)
kBT
=
{
∞, r < (σm
i + σn
j )/2 ,
zm
i zn
j λB ·
1
r , r > (σm
i + σn
j )/2 ,
(2)
where λB is the Bjerrum length of the matrix (m = 0, n = 0): λB = λB,0 = e2
0/(4πε0εkBT0). ε0
is the permittivity of the vacuum, ε is the relative permittivity of the solvent in which the matrix
was equilibrated at temperature T = T0, and zi is the ion charge. Finally, r is the distance between
centers of charges i and j.
The annealed subsystem was modeled as a mixture of two +1:−1 or +1:−1 and +z:−1 elec-
trolytes with a common anion. The potential function between two annealed ions (m = n = 1) as
well as between a quenched and annealed ion (m = 1, n = 0 or m = 0, n = 1) is of the same form as
given by equation 2, but with Bjerrum length being λB = λB,1 = e2
0/(4πε0ε1kBT1). In general, the
conditions of matrix preparation, T0, ε, can be different from the conditions of observation, given
by T1 and ε1. For simplicity, the temperature of observation was in this study assumed to be the
same as the temperature of the quench, T1 = T0 = T = 298.15 K, and λB,0 = λB,1 = λB = 7.14 Å.
2.1. The Replica Ornstein-Zernike theory
The derivation of the replica Ornstein-Zernike (ROZ) equations for a single electrolyte adsorp-
tion was in detail described elsewhere (see [22]) and will not be repeated here. In the case of
adsorption of an electrolyte mixture the general form of the equations in a matrix form remains
718
Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures
the same. The distribution of matrix particles obtained in terms of the pair correlation functions
follows from the integral equation
H
00
−C
00 = C
00
⊗ ρ
0
H
00 , (3)
where ρ
0 is a 2 × 2 diagonal matrix with diagonal elements ρ0
+ = ρ0
−, and the symbol ⊗ denotes
convolution. The correlation functions H and C are 2 × 2 symmetric matrices with the elements
f00
−−(r) = f00
++(r); f00
+−(r) = f00
−+(r), where f stands for h or c, h = g − 1 being the total, and c
being the direct correlation function [22].
The ROZ equations for the fluid-matrix and the fluid-fluid correlations read [22]:
H
10
−C
10 = C
10
⊗ ρ
0
H
00 + C
11
⊗ ρ
1
H
10
−C
12
⊗ ρ
1
H
10,
H
11
−C
11 = C
10
⊗ ρ
0
H
01 + C
11
⊗ ρ
1
H
11
−C
12
⊗ ρ
1
H
21,
H
12
−C
12 = C
10
⊗ ρ
0
H
01 + C
11
⊗ ρ
1
H
12 + C
12
⊗ ρ
1
H
11
− 2C12
⊗ ρ
1
H
21. (4)
Here C
12 and H
12 are the matrices containing the blocking parts of the relevant distribution
functions. ρ
1 is a 3 × 3 diagonal matrix with diagonal elements ρ1
+,A = ρ1
A, ρ1
+,B = ρ1
B, and ρ1
−.
The correlation functions H and C are also 3× 3 matrices. For clarity, the elements of the first of
the above equation 4 are in k-space given below:
h11
A,A
h11
A,B
h11
A,−
h11
B,A
h11
B,B
h11
B,−
h11
−,A
h11
−,B
h11
−,−
=
c11
A,A
c11
A,B
c11
A,−
c11
B,A
c11
B,B
c11
B,−
c11
−,A
c11
−,B
c11
−,−
+
c10
A,+
c10
A,−
0
c10
B,+
c10
B,−
0
c10
−,+ c10
−,− 0
ρ0
+ 0 0
0 ρ0
−
0
0 0 0
h01
+,A
h01
+,B
h01
+,−
h01
−,A
h01
−,B
h01
−,−
0 0 0
+
c11
A,A
c11
A,B
c11
A,−
c11
B,A
c11
B,B
c11
B,−
c11
−,A
c11
−,B
c11
−,−
ρ1
A
0 0
0 ρ1
B
0
0 0 ρ1
−
h11
A,A
h11
A,B
h11
A,−
h11
B,A
h11
B,B
h11
B,−
h11
−,A
h11
−,B
h11
−,−
−
h12
A,A
h12
A,B
h12
A,−
h12
B,A
h12
B,B
h12
B,−
h12
−,A
h12
−,B
h12
−,−
ρ1
A
0 0
0 ρ1
B
0
0 0 ρ1
−
h21
A,A
h21
A,B
h21
A,−
h21
B,A
h21
B,B
h21
B,−
h21
−,A
h21
−,B
h21
−,−
. (5)
Indexes A and B denote different cationic species of the annealed electrolyte mixture.
The integral equations written above can only be solved with the help of additional approxima-
tions, i. e. the so-called closure conditions. In this, as well as in many previous studies, we apply
the HNC closure condition in the form:
C
mn(r) = e[−βUmn(r)+Γmn(r)]
− 1 − Γmn(r), C
12(r) = eΓ12(r)
− 1 − Γ12(r). (6)
In this equation Γmn = H
mn
− C
mn, and the superscripts m, n assume values 0 and 1. Further,
U
mn are the matrices of interparticle pair potentials given by equation 2 and β = 1/(kBT ), where
kB is the Boltzmann constant.
The set of equations 4 and 6 must first be re-normalized before it can be solved numerically.
The renormalization was carried out following the procedure described in [19,20]. The form of the
renormalized functions remains the same as in the case of single electrolyte adsorption [20], except
that κ2
1 = 4π
∑
ρ1
i (z
1
i )2λB,1 contains the sum over all (three) fluid species. A direct iteration on
a grid of 16,384 points with ∆r = 0.05 Å was used to solve the set of integral equations with the
related closure conditions given above.
2.1.1. Thermodynamic properties
The excess internal energy of a charged fluid inside a charged matrix was calculated from
equation 7 [26]:
βEex/N1 =
1
2
∑
i=A,B,−
∑
j=A,B,−
x1
i ρ
1
j
∫
drg11
ij (r)U11
ij (r) +
∑
i=A,B,−
∑
j=+,−
x1
i ρ
0
j
∫
drg10
ij (r)U10
ij (r), (7)
719
M. Lukšič, G. Trefalt, B. Hribar-Lee
where x1
i = ρ1
i /(ρ1
A + ρ1
B + ρ1
−). Another thermodynamic property of special interest for this study
is the excess chemical potential, µex
i,1 = ln γ1
i , of the adsorbed fluid. The equation has been derived
for the single electrolyte in the adsorbent within ROZ/HNC approximation [22]. For the electrolyte
mixture the equation reads:
ln γ1
i = −
∑
j=+,−
ρ0
jc
10
(s)ij(0) −
∑
j=A,B,−
ρ1
j [c
11
(s)ij(0) − c
12
(s)ij(0)] + 0.5
∑
j=+,−
ρ0
j
∫
drh10
ij (h10
ij − c10
ij )
+ 0.5
∑
j=A,B,−
ρ1
j
∫
dr[h11
ij (h11
ij − c11
ij ) − h12
ij (h12
ij − c12
ij )], (8)
where c
10
(s)ij(0) denotes the Fourier transform of the direct correlation function at k = 0.
2.2. The Grand Canonical Monte Carlo simulation
The matrix configuration was obtained using the canonical Monte Carlo simulation. After the
equilibration, the matrix ions were frozen in their positions. The annealed electrolyte ions were
then distributed within the matrix and the system was studied by the grand canonical Monte Carlo
(GCMC) method. The methodology of the method is well established and extensively described
in several previous papers and therefore is not repeated here [6,23,24,27,28]. The details of the
simulations are: the number of matrix particles was 1000 and the average number of a fluid cation
species distributed within the matrix varied from 50 to 400. The ions within the matrix were first
equilibrated over at least 106 GCMC steps. After the equilibration, the production run of 2·108
attempted configurations was carried out to obtain the average concentration of the adsorbed
electrolyte species.
The mean activity coefficients of the annealed electrolyte, γ1
±, were calculated from the equi-
librium relation
a1
± = γ1
± · ((c1
+)z+(c1
−)z
−)1/(z++z
−
) = (aout
± ) = γout
± · ((cout
+ )z+(cout
− )z
−)1/(z++z
−
) , (9)
where index + refers to one of the cation species, and out refers to the properties of the bulk
electrolyte mixture. The activity coefficients of the bulk electrolyte mixture, γout
± , were obtained
using the hypernetted-chain (HNC) theory which has proved to be very successful in describing
the properties of ionic fluids [24].
3. Results and discussion
3.1. Test of the method
The first step of this study was to test out the newly written ROZ equations in HNC approx-
imation in order to describe the thermodynamic and structural properties of electrolyte mixtures
adsorbed in electroneutral matrices. We compared the internal energy (equation 7) and excess
chemical potential (equation 8) obtained within ROZ/HNC theory with the newly obtained results
from GCMC simulations for two different mixtures of fluids: (i) zA = zB = +1, σ1
A = 5.04 Å (the
model for H+ ion), σ1
B = 3.87 Å (the model for Na+ ion), and σ1
− = 3.62 Å (the model for Cl−
ion) [29]; and (ii) zA = +1, zB = +2, σ1
A = 5.04 Å (the model for H+ ion), σ1
B = 7.03 Å (the model
for Ca2+ ion), and σ1
− = 3.62 Å (the model for Cl− ion) [29]. In all cases z1
− = −1. The results
for thermodynamic properties are collected in table 1, and the comparison between different pair
distribution functions is for one case shown in figure 1.
Figure 1 shows an excellent agreement between fluid ion-ion and fluid ion-matrix pair distribu-
tion functions for the model (i), i. e. HCl + NaCl: c0 = 1.0 M, c1
H+ = 0.1797 M, and c1
Na+ = 0.1987
M. The same agreement was obtained for other fluid/matrix concentrations, as well as for the
model (ii), i. e. the mixture of HCl and CaCl2. As a consequence, the results for the excess internal
energy and the activity coefficients that are in the ROZ/HNC theory calculated from the pair
distribution functions (equation 7 and 8) and are collected in table 1 also show a reasonably good
720
Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures
agreement with the GCMC simulation, as was previously observed for a single electrolyte annealed
fluid [21,22].
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25 30
g11
(r
)
r /Å
a.
0
0.5
1
1.5
2
2.5
0 5 10 15 20 25 30
g10
(r
)
r /Å
b.
Figure 1. (a) Fluid-fluid pair distribution function g11(r) and (b) fluid-matrix pair distribution
function g10(r) for model (i), i. e. HCl + NaCl. The lines show the ROZ/HNC results, and the
symbol results from the GCMC simulations. c0 = 1.0 M, cA = 0.1797 M, cB = 0.1987 M.
Table 1. The excess internal energy per annealed fluid particle, −Eex/N1kBT , and the mean
activity coefficient, γ±, as obtained from ROZ/HNC theory, and GCMC simulation. The pa-
rameters of the two models are given in the text.
−E/N1kBT γHCl
± γNaCl
±
cHCl cNaCl GCMC ROZ GCMC ROZ GCMC ROZ
0.0712 0.3171 0.77(7) 0.791 1.07(0) 1.061 0.90(9) 0.907
0.1435 0.2384 0.77(0) 0.783 1.07(5) 1.070 0.91(5) 0.914
0.2161 0.1593 0.76(2) 0.774 1.08(2) 1.078 0.92(3) 0.920
0.2892 0.0796 0.75(3) 0.765 1.09(0) 1.087 0.93(2) 0.927
−E/N1kBT γHCl
± γCaCl2
±
cHCl cCaCl2 GCMC ROZ GCMC ROZ GCMC ROZ
0.0278 0.1087 1.29(5) 1.248 1.03(6) 1.027 0.75(8) 0.775
0.0663 0.0961 1.19(2) 1.150 1.03(3) 1.035 0.76(6) 0.782
0.1220 0.0781 1.07(5) 1.096 1.04(0) 1.040 0.77(7) 0.787
0.2095 0.0500 0.93(9) 0.908 1.05(6) 1.064 0.79(3) 0.809
3.2. Donnan exclusion coefficients
Since, in general, the ions differ in size and charge we proceed with a systematic study of each
effect separately. First we studied the Donnan exclusion coefficient, Γ, of a cation B of different sizes
where all the other ions in an annealed electrolyte mixture were of the same size: zA = zB = +1,
z1
− = −1, σ1
A = σ1
− = 4.25 Å and σ1
B = 4.25, 5.00, 6.00, or 7.00 Å. The matrix was as described
above. The results for different mixture compositions (c0 = 1.0 M, I in = 0.5
∑
j=A,B,− c1
j (z
1
j )2 = 0.5
M) are as a function of X in
A,− = cin
A,−/(cin
A,− + cin
B,−) shown in figure 2a. As the size of the ion
increases, Γ increases to more positive values, which means that the big ion gets excluded from the
matrix to a larger extent. As previously established, this is a results of the excluded volume effect.
In figure 2b we show Γ of cation B, where σ1
B = 4.25 Å but its charge, zB varies from +1 to
+4. Again, c0 = 1.0 M, and I in = 0.5 M. With the increasing charge of the ion (increasing ion
charge density), the Donnan exclusion coefficient Γ decreases and finally (for zB = +4) becomes
negative, showing the actual sorption of the component: the concentration of the ion B is higher
in the matrix than in the bulk of the same chemical potential. This can be explained with the
721
M. Lukšič, G. Trefalt, B. Hribar-Lee
stronger electrostatic attraction between annealed ions with high charge density and oppositely
charged matrix ions.
0.25
0.3
0.35
0.4
0.45
0 0.2 0.4 0.6 0.8 1
Γ B
Xin
BX
a.a.a.a.a.a.a.a.a.
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0 0.2 0.4 0.6 0.8 1
Γ B
Xin
BXz+
b.
Figure 2. (a) Donnan exclusion coefficient Γ for different sizes of cation B, σ1
B: 4.25 Å (full
circles), 5.0 Å (empty circles), 6.0 Å (full squares), 7.0 Å (empty squares). (b) Γ for different
charges of cation B, zB: +1 (full circles), +2 (empty circles), +3 (full squares), and +4 (empty
squares). Other parameters are described in the text. All results were obtained using ROZ/HNC
theory.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Γ H
Xin
HCl
a.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Γ H
Xin
HCl
b.
Figure 3. Donnan exclusion coefficient Γ for H+ cation in (a) model (i), i. e. HCl + NaCl, and
(b) model (ii), i. e. HCl + CaCl2. I in = 0.5 M, and c0 = 0.2 M (full circles), 1.0 M (empty
circles), 2.0 M (full squares), and 5.0 M (empty squares). All the results were obtained using
ROZ/HNC theory.
Both effects, excluding volume and electrostatics, result from the interaction with matrix parti-
cles. It is therefore to be expected that the matrix concentration importantly effects the exclusion
coefficients of ions. In figure 3 we show the Donnan exclusion coefficient for H+ ion in models (i),
i. e. HCl + NaCl (figure 3a), and (ii), i. e. HCl + CaCl2 (figure 3b); I in = 0.5 M, and c0 varies
from 0.2 M to 5.0 M.
As the matrix concentration increases the Donnan exclusion coefficient increases to more posi-
tive values suggesting that the volume exclusion effect prevails over the electrostatic attraction.
722
Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures
3.3. Preferential adsorption
For practical applications of electrolyte mixture adsorption, such as water softening and water
deionization, the mixture composition in the adsorbent versus the mixture composition in the bulk
is of particular interest. Figure 4 shows such ion-exchange isotherms obtained using ROZ/HNC for
models (i), i. e. HCl + NaCl, and (ii), i. e. HCl + CaCl2, for different matrix concentrations.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
X
in H
C
l
Xout
HCl
a.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
X
in H
C
l
Xout
HCl
b.
Figure 4. Ion-exchange isotherms (X in
HCl as a function of Xout
HCl) for (a) model (i), i. e. HCl +
NaCl, and (b) model (ii), i. e. HCl + CaCl2. I in = 0.5 M, and c0 = 0.2 M (full circles), 1.0 M
(empty circles), 2.0 M (full squares), and 5.0 M (empty squares). All results were obtained using
ROZ/HNC theory.
For the lowest matrix concentration studied here (c0 = 0.2 M) almost no preferential adsorption
is observed (X in
HCl ≈ Xout
HCl). At higher matrix concentrations, the larger cation (H+ in model (i)
and Ca2+ in model (ii)) gets partly excluded from the matrix. Note that the charge densities of the
model H+ and Ca2+ ions are approximately the same and the partition of the mixture components
is, therefore, determined by the excluded volume effect. As expected, the phenomenon is more
pronounced at higher matrix concentrations.
4. Conclusions
The ROZ/HNC equations for an annealed electrolyte mixture in an electroneutral matrix with
charges were tested against GCMC simulation results. Good agreement was obtained for structural
and thermodynamic properties, such as excess internal energy, and excess chemical potential. The
theory was then used to study the excluded volume and electrostatic effect in the process of the
adsorption of electrolyte mixtures with a common anion. The results show that in most cases
studied here the ions get excluded from the adsorbent, in other words, the excluded volume effect
prevails. Only in the case of highly charged annealed ions, the sorption occurs; the electrolyte gets
sucked into the adsorbent. A more systematic study of the matrix structure (quenching) and charge
density on the adsorption isotherms will be a subject of future work. We will pay special attention
to the charged matrices that will be used as a model to study the ion exchange phenomena and
will enable a direct comparison with experiments.
Acknowledgements
M. L. and B. H.-L. appreciate the financial support of the Slovenian Research Agency through
grant P1–0201, and G. T. through grant PR–02485.
723
M. Lukšič, G. Trefalt, B. Hribar-Lee
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Застосування граничних рiвнянь Орнштейна-Цернiке при
вивченнi адсорбцiї сумiшей електролiтiв у невпорядкованих
матрицях заряджених частинок
М. Лукшiч1, Г. Трефальт2, Б. Хрiбар-Лi1
1 Факультет хiмiї та хiмiчних технологiй, Унiверситет Любляни, Любляна, Словенiя
2 Iнститут iм. Йожефа Стефана, вул. Ямова 39, SI–1000 Любляна, Словенiя
Отримано 9 червня 2009 р., в остаточному виглядi – 19 червня 2009 р.
Реплiчне рiвняння Орнштейна-Цернiке (РОЦ) застосоване для вивчення адсорбцiї iонiв iз сумiшей
електролiтiв. Адсорбент розглядається у виглядi замороженої примiтивної +1:−1 моделi розмiрно
симетричного електролiту, тодi як рухомi частинки – це iони, що вiдрiзняються зарядом та/або роз-
мiрами. Данi, що отриманi з рiвняння РОЦ у гiперланцюжковому наближеннi (ГЛН), порiвняно з но-
вими результатами Монте-Карло моделювання у великому канонiчному ансамблi i виявлено хороше
узгодження мiж обома методами. Теорiю РОЦ/ГЛН далi було використано для вивчення коефiцiєнтiв
виключення, в залежностi вiд параметрiв розмiрної та/або зарядової асиметрiй нерухомих iонiв.
Ключовi слова: сумiшi електролiтiв, примiтивна модель електролiту, випадковi пористi матерiали,
коефiцiєнт виключення Доннана, реплiчне iнтегральне рiвняння Орнштейна-Цернiке, моделювання
Монте-Карло
PACS: 82.60.-s, 02.30.Rz, 61.20.-p
724
|
| id | nasplib_isofts_kiev_ua-123456789-120552 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T16:12:59Z |
| publishDate | 2009 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Lukšič, M. Trefalt, G. Hribar-Lee, B. 2017-06-12T10:52:34Z 2017-06-12T10:52:34Z 2009 Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles / M. Lukšič, G. Trefalt, B. Hribar-Lee // Condensed Matter Physics. — 2009. — Т. 12, № 4. — С. 717-724. — Бібліогр.: 29 назв. — англ. 1607-324X PACS: 82.60.-s, 02.30.Rz, 61.20.-p DOI:10.5488/CMP.12.4.717 https://nasplib.isofts.kiev.ua/handle/123456789/120552 The Replica Ornstein-Zernike (ROZ) equations were used to study the adsorption of ions from electrolyte mixtures. The adsorbent was represented as a quenched primitive model +1:-1 size symmetric electrolyte, while the mobile particles were ions differing in charge and/or size. The ROZ equations in hypernetted-chain (HNC) approximation were tested against new Monte Carlo results in the grand canonical ensemble; good agreement between the two methods was obtained. The ROZ/HNC theory was then used to study the exclusion coefficients as a function of size and/or charge asymmetry of the annealed ions. Реплiчне рiвняння Орнштейна-Цернiке (РОЦ) застосоване для вивчення адсорбцiї iонiв iз сумiшей електролiтiв. Адсорбент розглядається у виглядi замороженої примiтивної +1:-1 моделi розмiрно симетричного електролiту, тодi як рухомi частинки – це iони, що вiдрiзняються зарядом та/або роз- мiрами. Данi, що отриманi з рiвняння РОЦ у гiперланцюжковому наближеннi (ГЛН), порiвняно з но- вими результатами Монте-Карло моделювання у великому канонiчному ансамблi i виявлено хороше узгодження мiж обома методами. Теорiю РОЦ/ГЛН далi було використано для вивчення коефiцiєнтiв виключення, в залежностi вiд параметрiв розмiрної та/або зарядової асиметрiй нерухомих iонiв. M. L. and B. H.-L. appreciate the nancial support of the Slovenian Research Agency through grant P1-0201, and G. T. through grant PR-02485. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles Застосування граничних рiвнянь Орнштейна-Цернiке при вивченнi адсорбцiї сумiшей електролiтiв у невпорядкованих матрицях заряджених частинок Article published earlier |
| spellingShingle | Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles Lukšič, M. Trefalt, G. Hribar-Lee, B. |
| title | Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles |
| title_alt | Застосування граничних рiвнянь Орнштейна-Цернiке при вивченнi адсорбцiї сумiшей електролiтiв у невпорядкованих матрицях заряджених частинок |
| title_full | Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles |
| title_fullStr | Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles |
| title_full_unstemmed | Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles |
| title_short | Application of Replica Ornstein-Zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles |
| title_sort | application of replica ornstein-zernike equations in studies of the adsorption of electrolyte mixtures in disordered matrices of charged particles |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/120552 |
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