Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes
As an attempt to assess the effect of the long-range electrostatic interactions in solutions of electrolytes, a simple short-range model (SSM) of electrolytes made up of primitive water and primitive ions (i. e., ions whose Coulombic interaction with water has been replaced by a triangular-well...
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irk-123456789-1195462017-06-08T03:03:26Z Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes Vlcek, L. Nezbeda, I. As an attempt to assess the effect of the long-range electrostatic interactions in solutions of electrolytes, a simple short-range model (SSM) of electrolytes made up of primitive water and primitive ions (i. e., ions whose Coulombic interaction with water has been replaced by a triangular-well interaction) has been considered to compute the potential of mean force. The sizes of the primitive ions have been set so as to approximate realistic NaCl, LiI, and CsCl electrolytes. It is shown that despite the missing longrange Coulombic interaction the model captures the basic features of real electrolytes while the indirect, i.e. water mediated, potential of mean force in the SSM is in qualitative agreement with that of realistic models. Обчислення потенціалу середньої сили для спрощеної моделі з короткодіючими взаємодіями (кулонівська взаємодія між іонами і водою замінена взаємодією трикутної ями) розглядається як спроба визначити ефект далекодіючих електростатичних взаємодій у розчинах електролітів. Розміри таких розчинених іонів вибирались таким чином, щоб апроксимувати реальні NaCl, LiI, та CsCl розчини електролітів. Показано, що, незважаючи на нехтування далекодіючими кулонівськими взаємодіями, модель відображає основні характеристики реальних розчинів електролітів, тоді як непряма взаємодія між іонами опосередкована водою, тобто потенціал середньої сили якісно добре узгоджений з результатами реалістичних моделей. 2005 Article Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes / L. Vlcek, I. Nezbeda // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 261–270. — Бібліогр.: 17 назв. — англ. 1607-324X PACS: 05.20.Jj DOI:10.5488/CMP.8.2.261 http://dspace.nbuv.gov.ua/handle/123456789/119546 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
As an attempt to assess the effect of the long-range electrostatic interactions
in solutions of electrolytes, a simple short-range model (SSM) of
electrolytes made up of primitive water and primitive ions (i. e., ions whose
Coulombic interaction with water has been replaced by a triangular-well
interaction) has been considered to compute the potential of mean force.
The sizes of the primitive ions have been set so as to approximate realistic
NaCl, LiI, and CsCl electrolytes. It is shown that despite the missing longrange
Coulombic interaction the model captures the basic features of real
electrolytes while the indirect, i.e. water mediated, potential of mean force
in the SSM is in qualitative agreement with that of realistic models. |
| format |
Article |
| author |
Vlcek, L. Nezbeda, I. |
| spellingShingle |
Vlcek, L. Nezbeda, I. Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes Condensed Matter Physics |
| author_facet |
Vlcek, L. Nezbeda, I. |
| author_sort |
Vlcek, L. |
| title |
Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes |
| title_short |
Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes |
| title_full |
Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes |
| title_fullStr |
Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes |
| title_full_unstemmed |
Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes |
| title_sort |
potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2005 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119546 |
| citation_txt |
Potential of mean force between ions in infinitely diluted simple short-range models of aqueous electrolytes / L. Vlcek, I. Nezbeda // Condensed Matter Physics. — 2005. — Т. 8, № 2(42). — С. 261–270. — Бібліогр.: 17 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT vlcekl potentialofmeanforcebetweenionsininfinitelydilutedsimpleshortrangemodelsofaqueouselectrolytes AT nezbedai potentialofmeanforcebetweenionsininfinitelydilutedsimpleshortrangemodelsofaqueouselectrolytes |
| first_indexed |
2025-07-08T16:09:04Z |
| last_indexed |
2025-07-08T16:09:04Z |
| _version_ |
1837095661909049344 |
| fulltext |
Condensed Matter Physics, 2005, Vol. 8, No. 2(42), pp. 261–270
Potential of mean force between ions in
infinitely diluted simple short-range
models of aqueous electrolytes
L.Vlček 1 , I.Nezbeda 1,2
1 E. Hala Laboratory of Thermodynamics, ICPF, Acad. Sci.,
Rozvojova 135, 165 02 Prague 6 – Suchdol, Czech Republic
2 Physics Department, J. E. Purkyně University,
400 96 Ústı́ n. Lab., Czech Republic
Received January 6, 2005
As an attempt to assess the effect of the long-range electrostatic inter-
actions in solutions of electrolytes, a simple short-range model (SSM) of
electrolytes made up of primitive water and primitive ions (i. e., ions whose
Coulombic interaction with water has been replaced by a triangular-well
interaction) has been considered to compute the potential of mean force.
The sizes of the primitive ions have been set so as to approximate realistic
NaCl, LiI, and CsCl electrolytes. It is shown that despite the missing long-
range Coulombic interaction the model captures the basic features of real
electrolytes while the indirect, i.e. water mediated, potential of mean force
in the SSM is in qualitative agreement with that of realistic models.
Key words: aqueous electrolytes, simple short-range model of
electrolytes, primitive water, primitive ions, potential of mean force
PACS: 05.20.Jj
1. Introduction
Recent extensive investigations of the effect of the range of interactions on prop-
erties of polar and associating fluids have lead to an unambiguous conclusion that
the structure of pure fluids, given in terms of a complete set of the site-site correlati-
on functions, is determined primarily by short-range interactions (regardless of their
origin) and that the long-range part of the Coulombic interactions, regardless of
their strength, may be treated as a perturbation [1]. Although this conclusion seems
to remain valid also for a number of other properties of pure fluids, their validity for
mixtures is questionable and remains an open problem [2]. Particularly questionable
seems their extension to systems with free charges. Thus, the next evident step along
the path of investigations of the effect of the range of interactions on the properties
c© L.Vlček, I.Nezbeda 261
L.Vlček, I.Nezbeda
of fluids is to consider aqueous solutions of electrolytes.
With only a few attempts to account for the effect of the molecular nature of the
solvent, typically water (see e.g. [3] and references therein), solutions of electrolytes
have been so far considered at the McMillan-Mayer level [4]. It means that the
solute is considered explicitly as a collection of molecules but the solvent is treated
as dielectric continuum. Thus, to go beyond the McMillan-Mayer level means to
consider explicitly the molecular nature of the solvent as well.
In the light of the results for pure polar and associating fluids, and solutions
of non-electrolytes, the basic question to be asked is whether it is the long-range
part of the Coulombic interactions which gives rise to all observed properties of
electrolytes and is thus indispensable, or whether it can be omitted at a zeroth
level approximation (and at certain conditions). The latter possibility would make
it possible to use, at least in theoretical calculations, a simple model of electrolytes
with an explicit molecular model of water but without any long-range electrostatic
forces and, eventually, to develop a theory for such a model.
Recently we introduced a simple short-range model (SSM) of electrolytes and
investigated an infinitely dilute solution of spherical molecules in primitive water
[5]. In the SSM the solute molecule, called ion, is a hard sphere with a short-range
interaction tail (which may be either repulsive or attractive) to mimic the short-
range effects of the Coulombic interactions between an ion and a water molecule at
short separations. We carried out an imaginary hydration experiment on an infinitely
diluted solution (one ion in dense primitive water) whose results were in agreement
with realistic models [6] and that also provided a simple and direct interpretation of
the structure breaking/enhancement phenomena in terms of the site-site correlation
functions [7].
As a next step along the line of a potential use of the SSM, we consider in this
paper a pair of ions and compute the indirect part of the potential of mean force
(PMF). Although we set some parameters so as to roughly mimic the realistic ions,
this attempt should be viewed as a feasibility study examining the effect of the
long-range forces in an inverse way. In other words, instead of starting with a full
realistic model and then passing to its gradual simplification (shortening) we begin
immediately with a simple short-range model. The main reason for this approach is
primarily technical because otherwise the number of parameters to play with would
make the computations extremely time consuming and tedious.
2. Basic definitions and computational details
The realistic potentials of water and ions used in the present simulations are
common site-site potentials composed of the Lennard-Jones (LJ) and Coulombic
terms,
u(1, 2) =
∑
i∈{1}
∑
j∈{2}
4εij
(
σij
rij
)12
−
(
σij
rij
)6
+
q
(1)
i q
(2)
j
∣
∣
∣r
(1)
i − r
(2)
j
∣
∣
∣
, (1)
262
Potential of mean force
where i and j denote interaction sites, εij and σij are the parameters of the LJ
potential, and qi is the charge on site i. Specifically, for compatibility with available
simulation data, for the water-water interaction we have chosen the SPC/E model
[8], and the LJ parameters of ion-water potentials are taken from reference [9];
the parameters of the ion-ion interaction, εII and σII, are derived from the usual
combining rules,
εij = (εiiεjj)
1/2 , σij = (σii + σjj)/2 . (2)
The parameters of the considered potentials are summarized in table 1.
Table 1. Parameters of the realistic models of water and ions.
model site εii/kB[K] σii[Å] εiO/kB[K] σiO[Å] qi[e]
SPC/E O 78.205 3.169 78.205 3.169 –0.8476
H 0.0 0.0 0.0 0.0 0.4238
Ions Li+ 83.037 1.505 80.583 2.337 1
Na+ 50.327 2.583 62.374 2.876 1
Cs+ 50.327 3.883 62.374 3.526 1
Cl− 50.327 4.401 62.374 3.785 –1
I− 50.327 5.167 62.374 4.168 –1
At the level of simple models the short-range repulsive interaction between the
sites of the water molecules are approximated by a hard sphere interaction and the
attractive interaction between the unlike sites by a square-well interaction to mimic
hydrogen bonding. Further, the primitive models derived in this way also respect the
geometry of the realistic parent models and there are recipes to set the parameters of
the models [10–12]. Primitive water PM/SPC-E, descending from the SPC/E model
is of the following form [12]:
uPM/SPC−E(1, 2) =
∑
i,j∈{O,H}
i=j
uHS
(∣
∣
∣r
(1)
i − r
(2)
j
∣
∣
∣ ; dij
)
+
∑
i,j∈{O,H}
i6=j
uSW
(∣
∣
∣r
(1)
i − r
(2)
j
∣
∣
∣ ; λ
)
, (3)
where the summation in the first term runs over the pairs of like sites and in the
second term over the pairs of unlike sites,
uHS(r12; σ) =
{
+∞, for r12 < σ,
0, for r12 > σ,
(4)
and
uSW(r12; λ) =
{
−εHB, for r12 < λ,
0, for r12 > λ.
(5)
The parameters of the model are given in table 2.
The ion-water molecule interaction is also considered in the form of a site-site
potential which, in the case of the PM/SPC-E solvent, assumes the form
uI−W = uHS
(∣
∣
∣rI − r
(2)
O
∣
∣
∣ ; dIO
)
+
∑
j∈{O,H}
uTW
(∣
∣
∣rI − r
(2)
j
∣
∣
∣ ; RTW
)
, (6)
263
L.Vlček, I.Nezbeda
rI-O/σI-O
1.0 1.5
u
I-
O
-2
-1
0
1
2
H
H
O
e
ee
e
ee
rI-H/σI-H
1.0 1.5
u
I-
H
-2
-1
0
1
2
(a) (b)
I
I
+
I
+
r
I-H
rI-O
Figure 1. Schematic representation of the solute and (5-site) water molecules
in the simple short-range models of aqueous solutions of electrolytes (top), and
the attractive solute–water interaction [bottom (a)] and repulsive solute–water
interaction [bottom (b)].
where rI refers to ions, the summation runs over appropriately chosen sites on the
water molecule, and
uTW(r; RTW) =
{
εTWqiqj
RTW−r
RTW
for 0 < r < RTW,
0 for r > RTW.
(7)
Here qi is the charge on the interaction site i, RTW is the range of the TW interaction,
and εTW is a parameter which relates the strength of the TW interaction to the SW
interaction. Since there is no obvious clue to set this parameter, εTW was set to such
a value that for dIO = 1.0, RTW = 2.0, and for charges qH and qO, the energy at
the contact is the same as the energy of the hydrogen bond, εHB. The ion-water
potential is schematically presented in figure 1.
In the case considered here we place charges at the O-site and H-sites only, which
is also a common practice for realistic models. We further set dIO = (dII + dOO)/2,
where dII are derived by the hybrid Barker-Henderson method (for details see, e.g.,
[10]). The parameters of the SSM are summarized in table 2.
The potential of the mean force, uMF, is defined by a generalization of the low-
density limit of the pair correlation function,
gion−ion(r) = e−βuMF = e−β[uion−ion+uindir] , (8)
264
Potential of mean force
Table 2. Parameters of the primitive models of water and ions.
model site dii[Å] dii/dOO diO[Å] diO/dOO qi
PM/SPC-E O 2.642 1.0 2.642 1.0 0.84
H(X) 2.087 0.79 1.638 0.621 0.42
Ions Li+ 1.255 0.475 1.948 0.737 1.0
Na+ 2.153 0.815 2.398 0.908 1.0
Cs+ 3.236 1.225 2.940 1.113 1.0
Cl− 3.670 1.389 3.156 1.194 -1.0
I− 4.306 1.630 3.475 1.315 -1.0
1 range of the SW attraction, λ.
where g is the pair correlation function, uion−ion is the potential of the direct ion-ion
interaction, and uindir is an indirect, water mediated part of uMF.
We performed Monte Carlo simulations in an NVT ensemble in a similar man-
ner for both realistic and primitive potentials. In both cases there were 511 water
molecules in the simulation box (SPC/E or PM/SPCE), and a pair of ions. The
perturbation methodology was used to compute uMF between the ions or ion-like
particles. The perturbation method is based on a relation for the free energy differ-
ence, ∆F , between a reference system, denoted by subscript 0, and the system of
interest [13]:
β∆F = β(F − F0) = − ln
Q
Q0
= − ln 〈exp(−β∆u)〉0 , (9)
where β = 1/(kBT ), T is temperature, kB is the Boltzmann constant, and ∆u is
the potential energy difference between the reference and perturbed system. The
reference system consists of water molecules and the pair of ions held at a con-
stant separation. The perturbation is generated by a virtual displacement of the
two solute particles: the interparticle separation is changed by ±∆r and the cor-
responding terms exp[−∆u/(kBT )] are accumulated. The PMF between the ions is
then obtained by integration of the averaged differences. To make the sampling of
the configurational space in the vicinity of the ion pair more efficient, the preferential
sampling was used [14].
The simulations were arranged in cycles, each cycle consisting of 3 × N one-
particle steps. After each cycle, virtual displacements were generated with increa-
sing/decreasing ion-pair separation. The length of the displacements was ∆r =
0.01 Å for realistic potentials and 0.005dOO for primitive potentials. Free energy
difference, (9), for each separation was obtained by averaging over 5 × 104 cycles
at every 2∆r from contact to 8.0 Å. All curves were normalized to zero for the
maximum ion-ion separation measured.
265
L.Vlček, I.Nezbeda
3. Results and discussion
In [5] we set, for simplicity, RTW = 1.4dOO, i.e. the same value as λ for primitive
water. It is, however, evident that the range of the ion-water interaction should
considerably affect the properties of the system. Since there is neither any rule
nor at least a hint where to truncate this interaction, several following values for
R∗
TW ≡ RTW/dOO are considered: 1.5, 2.0, 2.5, and 3.0. Density is set to that of
pure water at ambient conditions, and β = 6; at these conditions the structure of
primitive water is very close to that of real water at ambient conditions [12].
Figure 2. Dependence of the indirect part of the PMF on the range of the ion-
water interaction in the SSM electrolyte. (Full line: cation-cation; dashed line:
anion-anion; dotted line: anion-cation; dashed-double dotted line: hard sphere-
hard sphere).
First, we assess a qualitative dependence of the ion-ion PMF on the range of the
ion-water interaction. Simulations with two equal-sized ions with diameter dII = dOO
and the range R∗
TW equal to 1.5 and 2.0 have been carried out. The former case
corresponds to the boundary between the first and second hydration shells, in the
latter case the ion-water interaction reaches half of the second hydration shell. The
results are shown in figure 2 along with the PMF between two hard spheres of the
same diameter. From the comparison of these results it follows that for smaller, near
to contact separations, the PMF of all the ion pairs is qualitatively correct: like ions
are attracted while unlike are repelled. At larger separations, however, the systems
with R∗
TW = 1.5 exhibit a minimum for oppositely charged ions and for identically
charged ions a maximum at the ion separation rII ≈ 2. Comparing it with the shape
of the PMF for the system with R∗
TW = 2.0 it is clear that these extremes are
artifacts of the too short range of the ion-water interactions. Moreover, for larger
ions (of the size dII/dOO > 1), the range R∗
TW = 1.5 would be too short and most
of the Coulombic-like interactions would be shielded by the HS repulsions between
the cores of the ion and water oxygen. Therefore, in further studies only the values
R∗
TW = 2.0 and higher are considered.
266
Potential of mean force
It is also worth to note the difference in the indirect interaction between the
pair of cations and anions. The former are, in the used model, attracted one to
another more strongly than the latter. Since the ions themselves are symmetrical,
the difference should be attributed to the asymmetry of the water molecules.
Figure 3. Comparison of the PMF in a realistic model of NaCl (the leftmost
graph) with that in the SMM electrolyte of different ion-water interaction range
(graphs on the left). [Full line: cation-cation; dashed line: anion-anion; dotted
line: anion-cation].
Figure 4. The same as figure 3 for LiI.
Figure 5. The same as figure 3 for CsCl.
267
L.Vlček, I.Nezbeda
To study the effect of the ion size, i.e. the range of the HS repulsion, on the PMF,
we use a realistic representation of ions with the HS diameters related to real ions
(see table 2). Besides the common pairs of NaCl, the LiI and CsCl pairs are chosen
to represent ions with the maximal and minimal difference between their diameters,
respectively. In figures 3, 4, and 5 we compare the indirect part of the PMF of
realistic ions to that of primitive models with two ranges of the TW potential,
R∗
TW = 2.0 and 2.5. (We have also considered the range R∗
TW = 3.0 but the results
do not show any different and significant trend; they are therefore not shown here for
the sake of space.) It is seen that the shape of the primitive potential is in qualitative
agreement with the realistic one for the both RTW considered, and it improves with
the increasing range of the TW interaction. A certain (quantitative) difference is
best seen for larger anions, in which case the shorter range causes a substantial
part of the TW interaction to be shielded by the HS repulsion between the ion
and oxygen. Since the longer range also means stronger water-ions interactions,
the strength of the indirect interaction also increases. However, the actual strength
can also be changed by varying the parameter εTW and, therefore, the absolute
value of the PMF is not very important. The figures corresponding to primitive
potentials show that the range of the indirect interaction is larger than the range of
the direct effect of the TW potential. This means that beyond the separation given
by RTW, the PMF between ions is given purely by the indirect, water-mediated part
of interaction, which leads to unphysical behavior: like ions are attracted and unlike
are repelled. This contrasts with realistic electrolyte solutions in which both direct
and indirect interactions between ions are long-ranged and for oppositely charged
ions are in a fine balance and approximately compensate each other. This could
mean certain limitations for the use of finite range potentials mimicking infinite
Coulombic interactions. As for the PMF dependence on the size of the ionic hard
cores, the comparison of the indirect interactions for Li+ pairs and Cs+ pairs shows
that smaller cations are attracted strongly as well as the interaction of two Cl− ions
is weaker than for I−. This is a direct consequence of a stronger interaction between
water and ions as they approach shorter separations.
4. Conclusions
As a feasibility study to explore a possibility to extend the recently introduced
simple short-range model of electrolytes beyond the infinitely dilute range, we have
evaluated the indirect (i.e., water mediated) part of the potential of mean force for
three types of electrolytes mimicking realistic NaCl, LiI, and CsCl electrolytes. The
main ingredient of the model is primitive water used as a molecular model of the
solvent. This model, despite its simplicity, has been found to describe the structure of
pure water at a semiquantitative level [12], as well as to be able to correctly describe
the hydration of, at least, small spherical particles, such as the hard cores of ions
[15]. Moreover, a good theoretical description of the thermodynamic properties of
pure primitive models and their mixtures with hard-body particles is available [16].
The purpose for focussing on the indirect part of the PMF has been an attempt
268
Potential of mean force
to isolate the effect of the molecular nature of the solvent on the potential of mean
force between ion pairs. No systematic investigation of the PMF dependence on the
strength of the ion-water potential has therefore been carried out. A question about
the convenience of the considered choice of this strength remains thus open and
further study determining optimal values will be needed.
To study the effect of the truncation of the Coulombic interactions at a simple
level, we used a triangular-well potential of varying range for the ion-water interac-
tion. It turns out that if the range of this interaction extends at least to the second
hydration shell, then the model captures, qualitatively, the main features of the in-
direct part of the PMF of real electrolytes. However, in our previous general study
[2] it was found that in real electrolyte solutions a subtle balance exists between the
direct and indirect parts of the ion-ion interaction and that a simple truncation of
the long-range Coulombic interactions may lead to an unphysical behavior. It will
be therefore necessary to study these effects in a systematic way to find the basis of
such a balance and to find a suitable way to minimize the artifacts of truncation;
the model considered in this paper seems an ideal model to be used for this purpose.
5. Acknowledgement
This work was supported by the Grand Agency of the Czech Republic (Grant
No. 203/02/0764).
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Потенціал середньої сили між іонами у безмежно
розведених спрощених моделях водних
електролітів
Л.Влцек 1 , І.Незбеда 1,2
1 Лабораторія термодинаміки Гала Е., Прага,
Чеська республіка
2 Фізичний факультет університету Я.Е. Пуркіне,
Усті над Лабом, Чеська республіка
Отримано 6 січня 2005 р.
Обчислення потенціалу середньої сили для спрощеної моделі з
короткодіючими взаємодіями (кулонівська взаємодія між іонами
і водою замінена взаємодією трикутної ями) розглядається як
спроба визначити ефект далекодіючих електростатичних взаємодій
у розчинах електролітів. Розміри таких розчинених іонів вибирались
таким чином, щоб апроксимувати реальні NaCl, LiI, та CsCl розчини
електролітів. Показано, що, незважаючи на нехтування далеко-
діючими кулонівськими взаємодіями, модель відображає основні
характеристики реальних розчинів електролітів, тоді як непряма
взаємодія між іонами опосередкована водою, тобто потенціал
середньої сили якісно добре узгоджений з результатами реаліс-
тичних моделей.
Ключові слова: водні електроліти, проста короткосяжна модель
електролітів, примітивна вода, примітивні іони, потенціал
середньої сили
PACS: 05.20.Jj
270
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