The multidensity integral equation approach in the theory of complex liquids
Recent development of the multi-density integral equation approach and its application to the statistical mechanical modelling of a different type of association and clusterization in liquids and solutions are reviewed. The effects of dimerization, polymerization and network formation are discussed....
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2001
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| Cite this: | The multidensity integral equation approach in the theory of complex liquids / M.F. Holovko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 291-295. — Бібліогр.: 31 назв. — англ. |
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| author | Holovko, M.F. |
| author_facet | Holovko, M.F. |
| citation_txt | The multidensity integral equation approach in the theory of complex liquids / M.F. Holovko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 291-295. — Бібліогр.: 31 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Recent development of the multi-density integral equation approach and its application to the statistical mechanical modelling of a different type of association and clusterization in liquids and solutions are reviewed. The effects of dimerization, polymerization and network formation are discussed. The numerical and analytical solutions of the integral equations in the multi-density formalism for pair correlation functions are used for the description of structural and thermodynamical properties of ionic solutions, polymers and network forming fluids.
|
| first_indexed | 2025-12-01T15:44:47Z |
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THE MULTIDENSITY INTEGRAL EQUATION APPROACH IN THE
THEORY OF COMPLEX LIQUIDS
M.F. Holovko
Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine,
Lviv, Ukraine
Recent development of the multi-density integral equation approach and its application to the statistical mechani-
cal modelling of a different type of association and clusterization in liquids and solutions are reviewed. The effects
of dimerization, polymerization and network formation are discussed. The numerical and analytical solutions of the
integral equations in the multi-density formalism for pair correlation functions are used for the description of struc-
tural and thermodynamical properties of ionic solutions, polymers and network forming fluids.
PACS: 05.20.-y
1. INTRODUCTION
Ten years ago I had a possibility to have in the Beke-
tov building in Kharkiv the seminar about the modern
state of liquid theory. On this seminar was A.I. Akhiez-
er. The subject was very interested for him. He put a lot
of questions that was important for understanding of
problems and development of theory. For the last ten
years we try to expand the techniques developed in the
theory of simple liquids to more complex liquids. This
short review of our activity in this field I would like to
devote to the unforgettable memory of A.I. Akhiezer.
The pair distribution function g(r) plays central role
in the modern liquid state theory. It establishes a bridge
between microscopic properties modeling by interparti-
cle interactions and macroscopical ones such as struc-
ture, thermodynamic, dielectric, kinetic and other prop-
erties. The essenceal progress in the liquid state theory
for the last decades is connected with the development
of the integral equation technique which is based on the
analytical or numerical calculation of the pair distribu-
tion function g(r) by the solution of the Ornstein-
Zernike (OZ) equation within different closures: the Per-
cus-Yevick (PY) approximation, the hypernetted chain
(HNC) one, the mean spherical approximation (MSA)
and its different modifications [1,2]. The background of
such closure relations is connected with diagram analy-
ses of the Mayer density expansions of the pair distribu-
tion function and their applicabilities are tested usually
by comparison with computer simulation results.
However such inegral equation approach is efficient
enough only for fluids having not so strong interparticle
attraction and needs an essential improvements for a
more complex fluids with strong interparticle attraction
which can lead to the clustering of particles into pairs or
larger groups such as chains, networks, self-assemlbing
agregates etc. Due to the clustering the δ-like intraparti-
cle distribution function appears and the distribution
function can be divided into the intra- and interparticle
parts
)()()( rgrgrg erintraint += . (1)
In addition, resulting from clusterization the corre-
sponding running integration numbers
∫=
R
drrrgRn
0
2)(4)( π ρ , (2)
which describes the average number of particles in the
sphere of radius R surrounding one of them which is
found in the center of this sphere, divides into bounded
(intra) and nonbounded (inter) parts
)()()( RnRnRn erintraint += , (3)
where ρ is the number density of particles.
Due to the saturation of bounding
bondraint nRn ≤)( (4)
where the number of bonds nbond is fixed. Specifically,
for pairs nbond = 1, for chains nbond = 2, for network
nbond = 4 etc.
Since the clusterization is caused by the attraction
part of interparticle interaction for the description of
their contribution it is more convinient to use the activity
expansions instead of density ones [3]. In particular, in
order to reproduce the correct low-density limit for the
fluids with strong clusterization, an infinite number of
terms in the density expansion must be included, while
only a few terms of the activity expansions are enough
for this purpose [4,5]. Consider, for example, the series
in the activity Z for the pressure P and density ρ termi-
nated at n-th order terms
( ) ,...32
,...
3
3
2
2
3
3
2
2
n
n
n
n
ZnbZbZbZP
Z
Z
ZbZbZbZP
++++=
∂
∂=
++++=
βρ
β
(5)
where bn is the attraction part of the n-th virial coeffi-
cient, )(1 kT=β is the inverse temperature. In result of
the strong interparticle attraction 1≤Z and in the limit
0→Z , ∞→nb
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2001, № 6, p. 291-295. 35
.11...3
21
1
1
3
3
2
2
ρ
ρ
n
Zb
n
nZb
n
Zb
nn
Z
n
Zb
n
n
n
n
→−−−−
−−=
−
−
(6)
After elimination of Z from equations (5) we obtain
the equation of state which will be changed from the ide-
al gas equation 1=ρβ P to the equation of ideal gas of
pairs 21=ρβ P , ideal gas of trymers 31=ρβ P ,...,
and in general the ideal gas of n-mers nP 1=ρβ .
Moreover the summation of the infinite series in (5)
leads to the possibility of the self-assembling in the sys-
tem. In approximation 1
2
−= n
n bb the expansions (5) re-
duce to the following form
( ) ,
b-1
Z ,
1 2
22 ZZb
ZP =
−
= ρβ (7)
which is well known in the thermodynamical theory of
micellization [6]. For this reason 0ρ=Z can be identi-
fied with the monomer density of amphiphilic molecules
and the divergence point 21 bZ cc == ρ has the sense
of the critical micelle concentration (CMC).
From these considerations it might be expected that a
theory which combines the activity and density expan-
sions would be advantageous. The second and higher
terms in the expansions (5) can be interpreted as the
dimers density 2
21 Zb=ρ , the trymers density
3
32 Zb=ρ ,..., n-mers density n
nn Zb=− 1ρ correspond-
ly. Such interpretation suggests the possibility of the de-
scription of clusterization by the introducing the multi-
density formalism for this purpose. A consistent integral
equation theory for this description of the clusterization
in liquids has been proporsed by Wertheim [7,8]. This is
based on the multi-density formalism in which the de-
scription in terms of the activity and density expansions
are combined.
The multidensity formalism was reformulated in or-
der to treat the effects of clusterization in fluids with
spherically symmetric attraction and it was applied for
ionic liquids [5,9-19], chain and network forming fluids
[20,21] for the treatment of the percolation phenomena
in the network forming fluids [22], for the adsorption of
associative fluids in porous media [23], for the descrip-
tion of electronic structure of associative fluids [24] etc.
Short review of the development of the multi-density in-
tegral equation approach in the theory of complex fluids
was done in [25].
The recent progress in the application of the multi-
density integral equation approach to the modelling of a
different type of association and clusterization in liquids
and solution will be reviewed in this report. The general
scheme of this approach in the framework of the two-
density formalism with the applications to electrolyte
and polyelectrolyte solutions will be considered in the
second section. In the third section the possibility of the
multi-density formalism for the description of polymer-
ization and network formation are presented.
2. TWO-DENSITY APPROACHES: THE AP-
PLICATION TO IONIC SOLUTIONS
The general idea of the multi-density formalism is
conected with the separation of the potential of interpar-
ticle interaction U(r) into the bonding and non-bonding
parts
)()()( rUrUrU nonbbond += , (8)
where Ubond(r) is some short-range attractive interaction
which includes at least the potential energy minimum of
U(r). The nonbonding part Unonb(r) includes a repulsive
part and long-range tail of U(r).
The diagrams appearing in the activity expansions
for the one-point density are classified with respect to
the number of associative bonds incident with the la-
belled white circle. Thus, the total number density of the
system is separated into two densities, the density of
nonbonding particles (monomers) ρ0 and the density of
bonding particles ρ1:
)1()1()1( 10 ρρρ += . (9)
Similarly the pair distribution function will be split-
ing into four terms
)2()12(ˆ)1(
)2()12(ˆ)1()2()12(ˆ)1(
)2()12()1()2()12()1()12(
0110
100001
00
ρρ
ρρρρ
ρρρρρ
g
gg
gg
+
++
==
(10)
or
)(ˆ)(ˆ)(ˆ)()( 11
2
100100 rgrgrgrgrg ααα +++= , (11)
where ρρα 0= is the fraction of a nonbonding part of
particles. In order to treat correctly the limit 0→α it is
advantageous to represent g(r) in the form [26]
)()()()()( 11100100 rgrgrgrgrg +++= . (12)
Due to saturation of bonding the restriction only by
pair formation leads to the self-consistent relation for α
∫
∞
+=
0
2
00
2 )()(41 drrrfrg asρπ αα , (13)
where 1))(exp()( −−= rUrf bondas β is the Mayer func-
tion for the associative interaction. At the sticky limit
follows
)()( RrBrf as −= δ (14)
and the equations (13) can be rewritten in the form
)(41
00
2
2
RgBRπ ρ
α
α =−
, (15)
where R is diameter of particles and g(R) is the contact
value at r = R of the pair distribution function.
The classification and topological reduction of the
diagrams for pair correlation function leads to the
Wertheim's modification of the OZ equation [7,8]
∫+= )()()()( 321331212 rrrdrr XhCCh ρ , (16)
where the coresponding matrices have the following
form:
. , , ,1
,
01
11
, ,
1111101001010000
1110
0100
1110
0100
ghghghgh
CC
CC
hh
hh
===−=
=
=
= XCh
(17)
36
As usually, the equation (16) should be supplement-
ed by closure relations. Among them we distinguish the
associative HNC (AHNC) closure
[ ],
, ,
,
2
1101100011
100010010001
00
00
as
tU
ftttgg
tggtgg
eg nonb
α
β
++=
==
= +−
(18)
where α βα βα β Cht −= ;
the associative PY (PYA) closure
),( ,
, ,
00
2
11111010
01010000
yfyegyeg
yegyeg
as
UU
UU
nonbnonb
nonbnonb
αββ
ββ
+==
==
−−
−−
(19)
where α βα βα β Cgy −= ;
the associative MSA (AMSA) closure
.for
,0 ,
for 0 ,1)(
00
2
11
100100
11100100
RrfgC
CCUC
Rrhhhrh
as
nonb
≥=
==−=
<===−=
α
β (20)
The analytical solution of AMSA for symmetrical
ionic systems was obtained [9] and also generalized for
nonsymmetrical case [12]. The essential feature of this
result is connected with appearance of new the screening
parameter BΓ instead of the usual inverse Debye-Huckel
screening length
21
2
2
4
= ∑
a
aaZe ρ
ε
π βκ . The param-
eter BΓ depends from the fraction of free ions α and the
sizes of ions R and is defined by
( ) ( ) ( ) .
1
14 222
R
R
kRRR
B
B
BB Γ+
Γ+
=Γ+Γ
α
(21)
The simple analysis of equation (21) suggests the
consideration of two regimes [18], namely the weak (
1→α ) and the strong ( 0→α ) association regimes.
The regime of the weak association is realized for
( )R
R
B
B
Γ+
Γ
> >≥
21
1 α
and corresponds to the traditional MSA-MAL (mass ac-
tion law) description of ion association [27] where equa-
tion (21) reduces to
( ) ( ) ( ) α222 14 kRRR BB =Γ+Γ . (22)
In this regime only the electrostatic contribution
from free ions is important and the electrostatic contri-
bution from ionic pair can be neglected.
The regime of the strong association is realized for
,
21
0
R
R
B
B
Γ+
Γ
< <≤ α
where equation (21) reduces to
( ) ( ) ( ).114 23 α−=Γ+Γ kRRR BB (23)
In this regime only electrostatic contribution from
the ion pairs is important and the contribution from the
free ions can be neglected.
The AMSA was applied for the description of ther-
modynamic properties of ionic solutions and it was
shown that it satisfactory reproduces the properties of
nonaqueous electrolyte solutions of solvent of relative
permittivities in range 3620 << ε [18]. For ionic solu-
tions of lower permittivity the AMSA was modified by
including the effect of ion trimer and tetramers [19]. The
possibility of such modification of AMSA is enough
promising for the description and interpretation of ther-
modynamical and transport properties of nonaqueous
electrolyte solutions with the enough low dielectric per-
mittivity [28].
3. THE MULTIDENSITY APPROACHES:
CHAIN AND NETWORK FORMATIONS
For the particles having more than one bonding state,
the formation of chains, rings, networks and more com-
plex agregates is possible. Such agregates can be consid-
ered as a collection of monomers (segments) bonded at
asymmetric attraction sites. In general for the particles
with M bonding sites the density is separated into 2M
densities of different bonding states. The diagram analy-
sis leads to the generalized version of the OZ equation
which has the form similar to (16) where in general case
h, C, X are the matrices MM 22 × . In general 2M-1 self-
consistent relations are needed instead of the relation
(13) for the pairing case. Some simplification can be
connected with the approximation that the bond creation
between two particles is independent of the existence of
other bonds. As a result, the fraction of the particles that
have Mn ≤ bonded neighbors can be given by the bi-
nomial distribution
,)1(
)( nMnMn
M pp
n
Mn
x −−
==
ρ
ρ
(24)
where α−= 1p , α is the fraction of particles nonbond-
ed by one fixed site.
For example, for the particles with two attractive
sites A and B (one is donor, the other is acceptor) the as-
sumption (24) leads to the ideal chain approximation
(ICA) [29] and m=1/α can be consider as the mean
chain length.
The analytical solution of the OZ-like equation in
polymer PY (PPY) approximation for the chain forming
fluids in ICA approximation for the case when bond
length L = R was obtained and discussed in [29]. The
generalization of this result for the case RL ≤ was con-
sidered in [20]. In the ICA approximation the formation
of the ring polymers is neglected. This approximation
can be used to describe a system of chain polymers,
polydisperse in length that is characterized by a pre-
scribed mean chain length m. As example of such case
the application of PPY theory for the description of the
structure of liquid sulfur was discussed [30].
For the molecules with four attractive sites A, B, C
and D (the two are donors and the two are acceptors) the
assumption (24) leads to the ideal network approxima-
tion (INA) [21,22]. The analytical solution of the OZ-
like equation for the network forming fluids in network
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 2000, №2.
Серия: Ядерно-физические исследования (36), с. 3-6.
37
Fig. 1. The predicted from NPY-INA approximation
structure factor of network forming fluids at 25.0=η
for different strength of bonding Bs: Bs = 0 (dotted
line); Bs = 0.1 (dashed line); Bs = 30 (solid line)
PY (NPY) and INA approximation was done in [21].
The structure factor calculated for the model of associat-
ing hard spheres with four symmetrical bonding sites is
presented in Fig. 1. It can be display the strong changes
in S(k) caused by increasing of association
constant Bs. For the network forming fluid similar as for
chain forming fluid [20] at small number region of k the
pre-peak appears connected with the forming of relative-
ly large clusters. Due to the correlation between them,
the ordering in mesoscopic scale appears, so-called in-
termediate-range order [31].
The number of the bonding states of molecules can
principally change the thermodynamic properties of flu-
ids [25]. For example, the formation of finite m-mers
leads to increasing of the liquid-gas critical temperature
and the decreasing of critical density. For network form-
ing fluids there are a new mechanism of critically con-
nected with the network formation.
The multidensity integral equation theory was refor-
mulated for studying connectedness properties in order
to understand a peculiarity of the network forming flu-
ids. The division of the potential of interparticle interac-
tion into the blocked and connectedness parts
)()()( * rUrUrU += + (25)
leads to the similar separation also for the pair and direct
correlation functions
).()()( ),()()( ** rrrrrr CCChhh +=+= ++ (26)
The connectedness pair and direct correlation func-
tions satisfy the OZ equation similar to (16). The mean
cluster size is given by
∫ ++= )(1 rhrdS ρ (27)
As the percolation transition is approached S increases
and becomes infinite at the percolation threshold. The
connectedness version of the OZ equation supplemented
by the NPY-like closure and INA ap proximation was
solved analytically [22]. In this approximation
Fig. 2. The phase diagram of network forming fluid
in NPY-INA approximation: the spinodal (broken line)
and percolation (solid line) curves
2
2
)31(
)24(
3
4
α
αα
−
−+=S (28)
It is seen that ∞→S when 31→α . The spinodal
(broken line) and percolation (solid line) curves in the
coordinats the density 3
6
1 Rπ ρη = and the temperature
1* )( −∈= βT is presented in Fig. 2, є is the square-well
parameter of intersite bonding. The Fig. 2 shows that
liquid phase including the critical point is inside the per-
colation region.
4. CONCLUSIONS
The characteristic features of numerous complex liq-
uids are connected with associating the molecules into a
different clusters caused by strong interparticle attrac-
tion. The starting point of the theory of such liquids is
the combined cluster expansions for pair correlation
function in which the activity expansions are used to de-
scribe the contribution of the bonding part of the inter-
particle interactions while the usual density expansion is
used to describe of nonbonding part of interactions. The
diagram analysis of these cluster expansions leads to the
multi-density integral equation approach which is flexi-
ble enough to treat different associative features of liq-
uids such as dimerization, polymerization, network for-
mation, self-assembling etc.
The possibilities of the theory are tested by compar-
ing with computer simulations. It is shown that the mul-
tidensity approach essentially improves the integral
equation theory for ionic systems. The analytical solu-
tion of AMSA is useful for the description of thermody-
namic and kinetic properties of nonaqueous electrolyte
solutions in a wide range of ionic concentration special-
ly for the solvents with enough low dielectric permittivi-
ty.
The multidensity integral equations are solved for
polymerization and network formation cases. It is shown
that structure factor for chain and network forming flu-
38
ids exhibits a peculiarity at small wave number connect-
ed with the forming of relatively large molecular clus-
ters. The multidensity integral equation theory is refor-
mulated for studying the connectedness properties of
network forming fluids. The gas-liquid critical point is
predicted to exist for network case, and a region of liq-
uid state is inside the percolation region.
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40
M.F. Holovko
PACS: 05.20.-y
1. Introduction
3. THE MULTIDENSITY APPROACHES: CHAIN AND NETWORK FORMATIONS
REFERENCES
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| id | nasplib_isofts_kiev_ua-123456789-80035 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-01T15:44:47Z |
| publishDate | 2001 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Holovko, M.F. 2015-04-09T16:07:11Z 2015-04-09T16:07:11Z 2001 The multidensity integral equation approach in the theory of complex liquids / M.F. Holovko // Вопросы атомной науки и техники. — 2001. — № 6. — С. 291-295. — Бібліогр.: 31 назв. — англ. 1562-6016 PACS: 05.20.-y https://nasplib.isofts.kiev.ua/handle/123456789/80035 Recent development of the multi-density integral equation approach and its application to the statistical mechanical modelling of a different type of association and clusterization in liquids and solutions are reviewed. The effects of dimerization, polymerization and network formation are discussed. The numerical and analytical solutions of the integral equations in the multi-density formalism for pair correlation functions are used for the description of structural and thermodynamical properties of ionic solutions, polymers and network forming fluids. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Kinetic theory The multidensity integral equation approach in the theory of complex liquids Article published earlier |
| spellingShingle | The multidensity integral equation approach in the theory of complex liquids Holovko, M.F. Kinetic theory |
| title | The multidensity integral equation approach in the theory of complex liquids |
| title_full | The multidensity integral equation approach in the theory of complex liquids |
| title_fullStr | The multidensity integral equation approach in the theory of complex liquids |
| title_full_unstemmed | The multidensity integral equation approach in the theory of complex liquids |
| title_short | The multidensity integral equation approach in the theory of complex liquids |
| title_sort | multidensity integral equation approach in the theory of complex liquids |
| topic | Kinetic theory |
| topic_facet | Kinetic theory |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80035 |
| work_keys_str_mv | AT holovkomf themultidensityintegralequationapproachinthetheoryofcomplexliquids AT holovkomf multidensityintegralequationapproachinthetheoryofcomplexliquids |