Towards a simplified approach to the modelling of the star-like molecule fluids
A theoretical approach to considering a wide spectrum of equilibrium properties of fluids formed from the four-branched molecules (e.g. four-arm star polysterene samples, four-arm block copolymers, etc.) is presented and discussed. The proposed approach is within the framework of an associative...
Збережено в:
| Дата: | 2002 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики конденсованих систем НАН України
2002
|
| Назва видання: | Condensed Matter Physics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/120597 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Towards a simplified approach to the modelling of the star-like molecule fluids / Yu. Duda, A. Trokhymchuk // Condensed Matter Physics. — 2002. — Т. 5, № 2(30). — С. 227-247. — Бібліогр.: 43 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-120597 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1205972017-06-13T03:05:16Z Towards a simplified approach to the modelling of the star-like molecule fluids Duda, Yu. Trokhymchuk, A. A theoretical approach to considering a wide spectrum of equilibrium properties of fluids formed from the four-branched molecules (e.g. four-arm star polysterene samples, four-arm block copolymers, etc.) is presented and discussed. The proposed approach is within the framework of an associative version of integral equation theory and is based on an analytical solution of the four-site associative hard-sphere model. Results and discussion are explained by the comparison against Monte Carlo computer simulation data generated for a freely-joined tangent hard-sphere model of a star-like molecule fluid. It is shown that the proposed theory works well for the star-like molecule fluids in homogeneous phase where it predicts the structure for molecules with relatively long arms and at high densities. The obtained results qualitatively reproduce the most important experimental features of the solvation force induced between two macrosurfaces due to the presence of star-like aggregates. Запропоновано теоретичний підхід до вивчення широкого спектру рівноважних властивостей рідин, молекули яких мають форму зірок (як наприклад полістирен, та ряд кополімерів). Підхід у своїй основі ґрунтується на аналітичному розв’язку, отриманому в теорії інтегральних рівнянь для так званої асоціативної моделі силових центрів. Результати та висновки з теоретичного підходу підкріплені порівняннями з комп’ютерним експериментом, проведеним для моделі гнучких полімерів, що мають форму зірок. Показано, що запропонований теоретичний підхід працює добре для однорідної фази, де добре відтворює структурний фактор для відносно довгих кінцівок зірки і високих густин рідини. Отримані результати відтворюють найважливіші характеристики сольватаційної сили між двома плоскими поверхнями, зануреними у розчин з наявними зірковими молекулами, що спостерігається у лабораторному експерименті. 2002 Article Towards a simplified approach to the modelling of the star-like molecule fluids / Yu. Duda, A. Trokhymchuk // Condensed Matter Physics. — 2002. — Т. 5, № 2(30). — С. 227-247. — Бібліогр.: 43 назв. — англ. 1607-324X PACS: 05.20.Jj, 61.25.Em DOI:10.5488/CMP.5.2.227 http://dspace.nbuv.gov.ua/handle/123456789/120597 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
A theoretical approach to considering a wide spectrum of equilibrium properties
of fluids formed from the four-branched molecules (e.g. four-arm star
polysterene samples, four-arm block copolymers, etc.) is presented and
discussed. The proposed approach is within the framework of an associative
version of integral equation theory and is based on an analytical
solution of the four-site associative hard-sphere model. Results and discussion
are explained by the comparison against Monte Carlo computer
simulation data generated for a freely-joined tangent hard-sphere model of
a star-like molecule fluid. It is shown that the proposed theory works well for
the star-like molecule fluids in homogeneous phase where it predicts the
structure for molecules with relatively long arms and at high densities. The
obtained results qualitatively reproduce the most important experimental
features of the solvation force induced between two macrosurfaces due to
the presence of star-like aggregates. |
| format |
Article |
| author |
Duda, Yu. Trokhymchuk, A. |
| spellingShingle |
Duda, Yu. Trokhymchuk, A. Towards a simplified approach to the modelling of the star-like molecule fluids Condensed Matter Physics |
| author_facet |
Duda, Yu. Trokhymchuk, A. |
| author_sort |
Duda, Yu. |
| title |
Towards a simplified approach to the modelling of the star-like molecule fluids |
| title_short |
Towards a simplified approach to the modelling of the star-like molecule fluids |
| title_full |
Towards a simplified approach to the modelling of the star-like molecule fluids |
| title_fullStr |
Towards a simplified approach to the modelling of the star-like molecule fluids |
| title_full_unstemmed |
Towards a simplified approach to the modelling of the star-like molecule fluids |
| title_sort |
towards a simplified approach to the modelling of the star-like molecule fluids |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2002 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120597 |
| citation_txt |
Towards a simplified approach to the
modelling of the star-like molecule
fluids / Yu. Duda, A. Trokhymchuk // Condensed Matter Physics. — 2002. — Т. 5, № 2(30). — С. 227-247. — Бібліогр.: 43 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT dudayu towardsasimplifiedapproachtothemodellingofthestarlikemoleculefluids AT trokhymchuka towardsasimplifiedapproachtothemodellingofthestarlikemoleculefluids |
| first_indexed |
2025-07-08T18:11:54Z |
| last_indexed |
2025-07-08T18:11:54Z |
| _version_ |
1837103385242763264 |
| fulltext |
Condensed Matter Physics, 2002, Vol. 5, No. 2(30), pp. 227–247
Towards a simplified approach to the
modelling of the star-like molecule
fluids
Yu.Duda 1,2 , A.Trokhymchuk 2,3
1 Programa de Ingenieria Molecular, Instituto Mexicano del Petróleo,
Eje Central L.Cardenas 152, 07730, México D.F.
2 Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
3 Department of Chemistry and Biochemistry, Brigham Young University,
Provo, UT 84602, USA
Received March 29, 2002
A theoretical approach to considering a wide spectrum of equilibrium prop-
erties of fluids formed from the four-branched molecules (e.g. four-arm star
polysterene samples, four-arm block copolymers, etc.) is presented and
discussed. The proposed approach is within the framework of an asso-
ciative version of integral equation theory and is based on an analytical
solution of the four-site associative hard-sphere model. Results and dis-
cussion are explained by the comparison against Monte Carlo computer
simulation data generated for a freely-joined tangent hard-sphere model of
a star-like molecule fluid. It is shown that the proposed theory works well for
the star-like molecule fluids in homogeneous phase where it predicts the
structure for molecules with relatively long arms and at high densities. The
obtained results qualitatively reproduce the most important experimental
features of the solvation force induced between two macrosurfaces due to
the presence of star-like aggregates.
Key words: star-like molecules, associative integral equation theory,
computer simulation
PACS: 05.20.Jj, 61.25.Em
1. Introduction
In recent years much attention has been paid to the modeling and studying
various properties of branched polymers. One special class of branched polymers
that has recently received considerable attention from both theoretical scientists
and engineers is the star-like molecule (SLM) fluid or star polymers [1–19]. The
c© Yu.Duda, A.Trokhymchuk 227
Yu.Duda, A.Trokhymchuk
SLM fluid1 serves as a prototype example for tethered chains, polymer brushes, and
multibranched systems. Depending on the extension, the SLM aggregates can be
viewed as molecules or macromolecules, where more than two linear homopolymers,
which are called arms, are all chemically attached to a common microscopic seed.
The size of the seed is typically of the order of a bond length, i.e. significantly smaller
compared to the extension of the chain. The existence of the seed seems to be the
most important factor for the special identity of SLM fluids within the entire family
of polymers. The star molecules are also related to other objects such as colloidal
particles (in the limit of infinitely large number of arms) or linear polymer chains
(in the case when the number of arms is equal to one or two).
Besides experimental investigations of the branched polymers [1–7], computer
modeling, analytical and semi-analytical statistical-mechanics treatments provide a
valuable way of understanding these systems at a fundamental level [8–19]. How-
ever, an exact statistical-mechanics treatment of branched molecules, in general, is
complicated because of the large number of internal degrees of freedom, the cou-
pled intra- and intermolecular interactions between beads, etc. Thus, it is important
to introduce the simplified approaches to such complicated objects and to see if
the phenomena observed in the experiments can be reproduced and explained by
relatively simple models. Different models of SLM fluid have been developed and
various statistical-mechanics techniques have been adopted to study them. Basical-
ly, these are very sophisticated approaches such as molecular dynamics (MD) and
Monte Carlo (MC) computer simulations [8–12] as well as the mean-field theory
methods [13,14], renormalization group analysis [15–17], thermodynamic perturba-
tion theory [18], and scaling theory [19]. Most of these approaches have been devoted
to the structure and conformational properties of a single star molecule or to the
thermodynamics of the SLM fluid.
The methods of integral equation theory [20–28] in conjunction with computer
simulation methods [8–11] seem to be one of the most suitable tools of investigating
the dense SLM systems. The information about interstar correlations provided by
these techniques may lead to a deeper insight and understanding of the diffraction
experimental data on dense star polymer solutions. Computer simulations are the
best possibility since the generated data may be considered exact for a given mod-
el. Particularly, when the modeling opportunities are practically unlimited. With
the development of new generations of computers, inaccuracies introduced by the
system size could be overcome though some methodological problems, like space
sampling, still exist. The main problem faced by the integral equation theory con-
cerns the appropriate modeling: the model must be reasonably adequate but still
has to permit an analytical or at least a numerical (semi-analytical) treatment un-
der certain approximations. An advantage of the integral equation theory is that
this approach is significantly less time consuming. Therefore, a promising strategy
is to test the theoretical predictions versus computer experiment for some chosen
1For an approach outlined in this study as well as for the results and discussions presented,
the term “star-like molecule fluids” is more appropriated than “star polymers” and abbreviation
“SLM” will be used throughout
228
Simplified approach to the star-like molecule fluids
conditions and properties and then try to exploit this model under the conditions
and for the properties that are difficult to reach by computer modeling.
The main goal of the present contribution is to share with readers a simplified ap-
proach to the fluid systems composed of star aggregates. Our consideration includes
both the modelling and semi-analytical treatment aspects. The model considered
here belongs to the class of the so-called associative hard-sphere models. The the-
oretical tool for treating such models is a multi-density formalism realized through
the associative version of the integral equation theory [22,23]. We will apply this
approach to a wide spectrum of properties of the SLM fluids in an homogeneous
and inhomogeneous phase and will try to find out how the theoretical predictions
are related to both the computer simulations and to the observation. To further
proceed with the results and discussion, in the following section we briefly intro-
duce the fundamentals of the theoretical scheme and outline the details of computer
simulation procedure.
2. Fundamentals of theoretical approach
2.1. Associative integral equation theory
The theoretical approach employed to study the properties of the SLM fluid
is based on the associative version of integral equation theory combined with the
four-site associative (FSA) model.
Four-site associative model of the SLM fluid. This model consists of N
hard-sphere particles, which we will call beads, in a volume V and of diameter d
(without any loss of generality in some parts of our presentation we assume d = 1).
The particle total number density is defined as ρ = N/V . Each of the beads has four
interaction spots (or sites) placed on their surface. The sites are of two types, denoted
as a and b. Two neighboring beads can form an associative complex (aggregate) due
to the site-site (a−b) interaction only. Bonding between the same type of sites is not
allowed. The mutual location of the sites determines the aggregate conformation,
while the strength of the site-site attraction together with the number (density)
of beads are responsible for the aggregate space extension. In the present study,
we restrict ourselves to the freely-joined formations: attraction sites are randomly
located on the bead surface and they are independent of each other, i.e. bonded
state of a given site on the bead does not affect a bonding ability of any other site
on the same bead.
The pairwise interaction, U(1, 2), between two beads, 1 and 2, consists of the
centre-centre hard-sphere repulsion plus the site-site attraction and can be written
in a form:
U(1, 2) = UHS(r) +
(
∑
a,b
Ua
b (1, 2) +
∑
a,b
Ua
b (2, 1)
)
, (2.1)
where 1 and 2 denote the position and orientations of two particles; UHS(r) is the
hard-core interaction between any two particles with r being their centre-to-centre
separation; Ua
b (1, 2) denotes the potential for a short-range and highly directional
229
Yu.Duda, A.Trokhymchuk
Figure 1. Two-dimensional sketch of the different star-like aggregates that could
be formed within the four-site associative (FSA/SLM) model used in WOZ/APY
theory (right panel) and the star-like molecule within the tangent hard-sphere
(THS/SLM) model employed in MC simulations (left panel).
attraction between site a on particle 1 and site b on particle 2. For the sake of
convenience, in what follows we use the superscripts for sites a and the subscripts
for sites b.
To provide an analytical treatment of the FSA model, the site-site attraction,
Ua
b (1, 2), is chosen to be infinitesimally short-ranged so that the corresponding
orientation-averaged f -function [26,29] (an associative analog of the classical Mayer
function) is proportional to the Dirac δ-function [30]:
fa
b (r) = Ka
b δ(r − 1), (2.2)
whereKa
b is the stickiness parameter of the site-site (a−b) attraction. At a fixed bead
number density, ρ, by an appropriate choice of the stickiness parameter K a
b , the FSA
model could reproduce various types of aggregates, i.e. linear polymers, symmetrical
and nonsymmetrical networks etc. Recently, some of these possibilities have been
studied and analyzed [29,31]. In the present study, we restrict our attention to the
case when all site-site interactions are the same, i.e., K a
b ≡ K, that topologically
corresponds to a star-like formations. In what follows, we will refer to this model as
the FSA/SLM model. The schematic two-dimensional presentation of the FSA/SLM
model is shown in figure 1.
Multi-density formalism. FSA/SLM model introduced above belongs to the
class of the so-called associative hard-sphere models. The theoretical approach to
treat such models is a multi-density formalism realized through the associative
version of integral equation theory pioneered by Wertheim [22,23]. According to
Wertheim, the division of the potential of interaction (2.1) into an associative and
non-associative parts allows one to classify the diagrammatic representation of the
grand partition function in terms of the cluster diagrams which account for the for-
mation of the aggregates of various size and conformation. Therefore, the graphs
defining the total number density are classified according to the bonded state of the
labeled point 1. Then, the total local number density can be presented as follows:
̺(1) = ˆ̺0(1) + ˆ̺G(1) + ˆ̺G′(1) + ˆ̺Γ(1), (2.3)
230
Simplified approach to the star-like molecule fluids
where the captured densities are defined as follows:
ˆ̺0(1) = ̺00(1) + ̺F0 (1) + ̺F
′
0 (1) + ̺Γ0 (1), (2.4)
ˆ̺G(1) = ̺0G(1) + ̺FG(1) + ̺F
′
G (1) + ̺ΓG(1), (2.5)
ˆ̺G′(1) = ̺0G′(1) + ̺FG′(1) + ̺F
′
G′(1) + ̺ΓG′(1), (2.6)
ˆ̺Γ(1) = ̺0
Γ
(1) + ̺F
Γ
(1) + ̺F
′
Γ
(1) + ̺Γ
Γ
(1). (2.7)
The function, ̺αi (1), represents the partial densities of the individual bead 1 with a
given bonded state of its sites. The diagrammatic analysis similar to that developed
in [22–25] leads to the relation of the partial densities, ̺α
i (1), with the partial one-
particle direct correlation functions, cαi (1), which in turn define the corresponding
two-particle direct correlation functions:
cαβij (1, 2) =
δcαi (1)
δσΓ−β
Γ−j (2)
, (2.8)
where σΓ−β
Γ−j (2) are the density parameters of labeled point 2. Two-particle direct
correlation functions, cαβij (1, 2), are related to the partial pair correlation functions,
hαβ
ij (1, 2), through an associative version of the Ornstein-Zernike (OZ) equation
formulated by Wertheim [22,23] and due to this is abbreviated in present study as
WOZ equation:
hαβ
ij (1, 2) = cαβij (1, 2) +
∑
l,m
∑
γε
∫
d3cαγil (1, 3)σ
Γ−γ−ε
Γ−l−m(3)h
εβ
mj(3, 2). (2.9)
Here and everywhere below, the indices {α, β, γ, ε} are used to denote the sites of
type a while {i, j, l,m} are reserved for the sites of type b and are assigned the values
0, F , Γ and 0, G, Γ, respectively.
Density parameters and self-consistency relations. The density parame-
ters, σα
i (1), are defined as the linear combinations of the partial densities:
σα
i (1) =
i
∑
j=0
α
∑
β=0
̺βj (1). (2.10)
These parameters have to be evaluated from the self-consistency relations. As a part,
these relations include
σΓ
Γ−G(1) = ̺(1)XG(1); σΓ−F
Γ
(1) = ̺(1)XF (1), (2.11)
where quantities, XG(1) and XF (1), are the fractions of the particles not bonded
at sites G and F , respectively. In the case of a star-like topology there is relation:
XG(1) = XF (1), and each of the fraction can be calculated as follows:
Xa = Xb = 2
/
[
1 +
√
1 + 32πρKy0000(r = d)
]
. (2.12)
231
Yu.Duda, A.Trokhymchuk
In this equation, y00
00
(r = d) is a contact value of the partial cavity correlation
function. The partial cavity correlation functions, yαβ
ij (r), are related to the partial
radial distribution functions, gαβ
ij (r), by:
gαβij (r) = hαβ
ij (r) + δi,0δj,0δ
α,0δβ,0
= e(r)yαβij (r) + e(r)
∑
a,b
yα−a,β
i,j−b (r)fa
b (r)[1− δα,0][1− δj,0]
+ e(r)
∑
a,b
yα,β−a
i−b,j (r)fa
b (r)[1− δi,0][1− δβ,0], (2.13)
where e(r) = exp[−UHS(r)/kT ], while δij and δαβ stand for the Kronecker symbol.
Solving the set of self-consistency relations given through equations (2.9)–(2.13), one
will reproduce all the density parameters required for the solution of equation (2.9).
Approximations applied to solve the WOZ equation. Equation (2.9) is
solvable analytically subject to a set of approximations. First of all, an associative
analog of the Percus-Yevick (APY) closure is formulated in a form [26]:
cαβij (r) = yαβij (r)[e(r)− 1] + e(r)
∑
a,b
yα−a,β
i,j−b (r)fa
b (r)[1− δα,0][1− δj,0]
+ e(r)
∑
a,b
yα,β−a
i−b,j (r)fa
b (r)[1− δi,0][1− δβ,0]. (2.14)
To proceed further, the ideal network approximation [26,29] is applied. In accordance
with this approximation, we neglect the part of the intramolecular correlations re-
sponsible for the formation of the ring-like complexes with respect to all (a − b)
pairs of the sites. The most developed conformation in such a case is the network
consisting of crossing polymer chains built along each of the (a− b) directions and
with each polymer branch being described within the ideal chain approximation
[24]. Therefore, any pair of atoms in such a network is assumed to remain singly
connected with changing particle density or strength of the associative interaction.
The WOZ equation (2.9) together with the APY closure conditions (2.14) and
with the density relations (2.10)–(2.12) form a closed set of equations to be solved.
We refer to this as the WOZ/APY theory. The method used for the solution is based
upon Wertheim-Baxter factorization technique [30,32]. The general scheme of the
analytical solution is similar to that presented in [24].
Identification of the star-like aggregates. The size (total number of beads)
of the complexes formed in the FSA/SLM fluid is characterized by the mean cluster
size:
M =
4(4 +Xb − 2X2
b )
3(1− 6Xb + 9X2
b )
, (2.15)
where Xb is the fraction of the beads not bonded at site b and is defined according
to equation (2.12). The above expression (2.15) was derived within the percolation
theory in [27]. In the present study, we identify M as the number of beads per star-
like molecule or as the size of a star-like aggregate. Thus, we assume thatM = ξm+1
with ξ being the functionality or number of arms. For the FSA/SLM model ξ is equal
to 4. Parameter m is the length of each arm.
232
Simplified approach to the star-like molecule fluids
2.2. Monte Carlo simulations
Tangent hard-sphere model. The star-like aggregates in computer simula-
tions are modeled as a collection of freely-joined tangent hard spheres (THS) and
below we will refer to this model as the THS/SLM fluid (see figure 1). The THS/SLM
model of a functionality ξ = 4 arms and with m = 2, 4 6 and 8 beads per each arm
have been simulated at different volume fraction, η = πρ/6, occupied by star-like
aggregates. Number density, ρ, has the same meaning as for the FSA/SLM model
and corresponds to the total (arm beads+central beads) number density of beads.
The central and arm beads are of the same size. We used no less than 160 star-like
molecules that is equivalent to the total number of hard-sphere monomers in the
box, N = (ξm+1)×160. Some test runs with a larger number of star-like aggregates
have been performed but no finite-size effect was observed.
Canonical ensemble simulations. Bulk Monte Carlo (MC) simulations were
performed in the canonical ensemble with a cubic simulation cell supplemented by a
standard periodic boundary conditions applied in all three directions. In most cases,
the initial configurations were generated through the random insertion of a central
bead followed by the growth of the rest of a molecule. At high densities the growth
cycles have been combined with an attempt to move a randomly chosen molecule
and to add one bead to a randomly chosen molecule. Trials have been continued
until all star molecules are completely grown to the desired size (desired number
of participating beads) [33]. After an initial configuration has been generated, the
system is allowed to evolve by moving a randomly chosen single molecule. The
scheme is adjusted so that about 30–35% of the attempted moves are accepted. This
simulation algorithm corresponds to a slightly modified version of the Dickman-Hall
algorithm [34], appropriately adapted to the star molecules. At this stage, a random
number of beads, mx (mx 6 m), of each of the arms of a randomly chosen star-like
molecule is deleted. Following this, a chain is regrown by adding beads sequentially.
In a straightforward application of the Metropolis algorithm [35], trial configurations
are accepted if they are free of overlap.
To obtain a fairly smooth radial distribution function, runs of approximately 5 ·
104 simulation cycles are required. Additionally, an initial one quarter of this number
of cycles is necessary to equilibrate the system. To simulate the non-uniform (near
a single wall) SLM fluid, we used the same algorithm but applied to computational
cell which is elongated in Z direction, i.e. parallelepipedic in a shape. In this case,
the periodic boundary conditions are applied in X and Y directions only. The length
of computational cell along Z direction, Lz, was chosen large enough to have the
homogeneous (bulk) region in the center of the cell. For the slit-like (between two
walls) confinement, the length of computational cell, Lz, was equal to the gap width,
H . The density profiles, ρ(z), are calculated in MC runs by counting the number
of beads, Nz, in the slabs of thickness, ∆z = 0.05, parallel to the XY plane using
ρ(z) = 〈Nz〉/v0, where v0 is the slab volume, v0 = LxLy∆z, and z is the coordinate
of the center of the slab.
233
Yu.Duda, A.Trokhymchuk
3. Results and discussion
The results and discussion presented bellow consist of two main parts. First one
is aimed at making comparison between the FSA/SLM model used in theoretical
treatment and the THS/SLM model simulated in a computer experiment. The sec-
ond part concerns the physics beyond the theoretical model of the SLM fluid. To
compare the theoretical predictions for the FSA/SLM model with computer simula-
tion data for the THS/SLM model, the attraction parameter, K, that serves as an
input in the WOZ/APY theory, was adjusted to make the mean cluster size, M , of
the FSA/SLM model equal to the number of beads per star-like molecule employed
in MC simulations of the THS/SLM fluid.
3.1. Star-like molecule fluid in a bulk
The bulk properties and an information regarding correlations in the SLM fluid
can be analyzed by means of the total radial distribution function and the total
structure factor.
Total radial distribution function. One of the most important results provid-
ed by the WOZ/APY theory is the total radial distribution function (RDF) of the
FSA/SLM model. In particular, by factorizing equation (2.9) and using an iteration
method due to Perram [36], the partial radial distribution function, g αβ
ij (r), can be
obtained [29]. These partial RDFs are similar to those in Wertheim theory of chem-
ical association [22,23]; they do not have a physical meaning that might follow from
the indices (0 – unbonded, F or G – partially bonded and Γ – completely bonded)
[20]. However, they are used to calculate the total radial distribution function, g(r),
which for the FSA/SLM fluid is the following superposition of the partial functions
[26]:
g(r) =
1
ρ2
∑
i,j
∑
α,β
σα
i g
Γ−α,Γ−β
Γ−i,Γ−j (r)σβ
j . (3.1)
Figure 2, as an example, shows the total RDFs resulting from the WOZ/APY
theory for the FSA/SLM fluid and from the MC simulations for the THS/SLM flu-
id. We can see a remarkable qualitative and quantitative agreement between the
theoretical results and simulation data between both sets of RDFs for almost all pa-
rameters considered. The WOZ/APY theory for the FSA/SLM model agrees better
with simulation data for the THS/SLM model at higher densities. Some quantitative
disagreement was observed [40] at low densities, ρ, especially for the smaller star
entities. Since the ideal network approximation [26,29] employed for the FSA/SLM
fluid has the same physical origin as the ideal chain approximation in the case of lin-
ear chains [24], the prediction of the WOZ/APY theory is more satisfactory for the
stars with longer arms (m = 6 in the case of figure 2) similar to its greater success
for the longer linear chains [24]. Fortunately, the case of longer arms is of utmost in-
terest with a view to future applications of the WOZ/APY theory to inhomogeneous
problems.
234
Simplified approach to the star-like molecule fluids
1 2 3
0.8
1.0
1.2
1.4
r/d
m = 2
m = 4
1 2 3 4
0.8
1.0
1.2
1.4
r/d
m = 6
1 2 3
r/d
Figure 2. Total radial distribution function, g(r), of a four-armed star-like
molecule fluid with various arm length, m, indicated in the figure. Number den-
sity of fluid beads is fixed at ρ = 0.5. The solid lines and symbols correspond
to WOZ/APY theory results for the FSA/SLM model and MC data for the
THS/SLM model, respectively.
Structure factor. A knowledge of the total RDF enables us to calculate the
total structure factor, S(k), which is related to the Fourier transform of g(r) as
follows:
S(k) = 4πρ
∫
[g(r)− 1]
sin kr
kr
r2dr. (3.2)
However, evaluation of S(k) for the SLM fluid from the definition (3.2) by numerical
integration is not the most productive way. Due to the long-range nature of the
density fluctuations in the SLM fluid and due to the discontinuity-like behavior
of g(r) at r ≈ 2d (depicted by both FSA/SLM and THS/SLM model fluids) the
accuracy of numerical procedure is questionable. In such a case, an advantage of
analytical solution of the FSA/SLM model is evident. Particularly, in present case
the structure factor can be calculated implicitly from the knowledge of Wertheim-
Baxter factor q-functions as it was discussed in [26].
The resulting structure factor for the total number density of beads fixed at
ρ = 0.5 is plotted in figure 3. We intend to show the changes in S(k) caused by an
increasing of the arm length. It is seen that the curves for arm length, m = 2, 4 and
8, differ mainly in the magnitudes of the maxima and minima of S(k). Particularly,
the magnitude of oscillations versus wavelength number increases when arms become
longer. Another interesting aspect of the results presented in figure 3 is the difference
between SLM fluid structure factor and that for the spherical monomer fluid, i.e. for
hard-sphere (HS) fluid of the same number density. Besides the shift towards larger
k-values in the position of the main peak of S(k) for SLM fluid, the appearance of
a new peak (pre-peak) at 1 < kd < 3 is observed. The genesis of this pre-peak with
an increasing arm length reveals its relation to the size of SLM aggregates. More
generally, we can conclude that pre-peak is an indication of the density fluctuations
due to the formation of the stable structure units with the size larger than the size
of basic units, i.e. beads in the case of the SLM fluids.
From the full expression for S(k), its low-k limit can be expressed. The value
of S(k = 0) is an important quantity because it is connected with the isothermal
235
Yu.Duda, A.Trokhymchuk
0 5 10 15 20
0.0
0.5
1.0
1.5
0 5 10 15 20 0 5 10 15 20 25
0.0
0.5
1.0
1.5
m = 2
kd
m = 4
kd
m = 8
HS kd
Figure 3. Total structure factor, S(k), of a four armed star-like molecule fluid
with different arm length, m, indicated in the figure. Number density of fluid
particles (beads) is fixed at ρ = 0.5. The solid lines show the results for the
FSA/SLM fluid while the thin solid line corresponds to HS fluid.
compressibility by S(k = 0) = ρkTχT . Therefore, the behavior of S(k = 0) is closely
related to the compressibility pressure equation [26].
3.2. Star-like molecule fluid under confinement
The structure of molecular fluids next to the substrate is a subject of great fun-
damental and technological interest. This topic is connected with many important
processes including adsorption, adhesion, manufacturing of thin films, membrane
separation, and others. Very little theoretical work has been reported for inhomo-
geneous properties of SLM fluids. The aim of this section is to get insight both into
the ordering of the SLM fluid next to a single wall and into local density of the SLM
fluid confined into a space between two rigid surfaces.
HAB approach to modelling a single-wall confinement. Recently, the set
of papers was published [28,37] where the classical method developed by Henderson,
Abraham and Barker [38] for inhomogeneous hard-sphere fluid had been extended
to the inhomogeneous polymerizing fluids. The attractive feature of this approach,
which we will abbreviate as the HAB, is a possibility to preserve an analytical
treatment without the notable loss of accuracy even for more sophisticated models
of adsorbed fluids. A necessary requirement is to know the correlation functions of
the same model fluid in the bulk phase.
The planar wall (W) confinement is formed in the HAB approach as a result of
a limiting procedure applied to the (SLM fluid plus HS fluid) mixture in which the
concentration of hard-sphere particles is taken to tend to zero whereas their diameter
tends to infinity. Then, the fluid-wall correlations are described by the associative
version of the HAB integral equation [28,39]:
hαW
j0 (z) = cαWj0 (z) +
∑
lm
∑
β,γ
∫
dr′cαγjl (r
′)σγβ
lmh
βW
m0
(|r− r′|), (3.3)
where hαW
j0 (z) and cαWj0 (z) are the fluid-wall partial pair and direct correlation func-
tions, respectively. Both the density parameters, σγβ
lm, and the partial direct corre-
236
Simplified approach to the star-like molecule fluids
0 3 6 9 12
0.3
0.6
0.9
1.2
z/d
m = 2
0 3 6 9 12 15
0.3
0.6
0.9
1.2
z/d
m = 4
Figure 4. Total normalized density profile, gW(z), of a four-armed star-like
molecule fluid next to a hard wall. The arm length, m, is variable as it is in-
dicated in the figure. Bulk number density of fluid particles (beads) is fixed at
ρ = 0.42. The solid lines and symbols correspond to the WOZ/APY theory results
for the FSA/SLM model and MC data for the THS/SLM model, respectively.
lation functions of the bulk fluid, cαβij (r), are the same as those already defined in a
previous section.
Similarly to the bulk case, the APY closure is the most popular approximation
to solve associative HAB equation (3.3). It reads:
gαWj0 (z) = yαWj0 (z) + fW(z)yαWj0 (z); cαWj0 (z) = fW(z)yαWj0 (z), (3.4)
where yαWj0 (z) are the partial cavity-wall correlation functions, gαW
j0 (z) are the nor-
malized partial local densities that are defined through the partial pair correlation
functions as follows: gαW
j0 (z) = hαW
j0 (z)+δj0δ
α0. Function, fW(z)=exp[−βUW(z)]−1,
is the Mayer function for the bead-wall interaction:
UW(z) =
{
∞, if z < 0
0, if z > 0,
(3.5)
where z is the distance between the center of any bead and the wall.
The set of equations (3.3)–(3.4) can be solved analytically in Laplace space by
a standard factorization method of reduction to the Baxter q-functions [30,32]. By
combining Baxter method with an iteration procedure suggested by Perram [36],
the solution for local density distribution could be easy inverted into a real space.
Local density near a single wall. The total normalized local density distri-
bution that incorporates the density of all (arm + central) beads of the star-like
aggregates, gW(z) ≡ ρW(z)/ρ, is the following combination of the partial particle
densities:
gW(z) = g0W00 (z) +
1
σΓ
Γ
∑
i,α
σα
i g
Γ−α,W
Γ−i,0 (z). (3.6)
In order to estimate the relevance of theoretical prediction, in figure 4 we show both
the results of equations (3.3)–(3.6) for the FSA/SLM model and MC simulation
data for the THS/SLM model. Two models being treated by different theoretical
tools result in a quite similar local density behavior. An exception has to be made
for the close vicinity of the wall. It is worth noting that we have an interplay of two
237
Yu.Duda, A.Trokhymchuk
0.1 0.2 0.3 0.4 0.5 0.6
0.0
0.4
0.8
1.2
m = 8
m = 1
HS
bulk density, r
wall density
rW(z = 0)
0.1 0.2 0.3 0.4 0.5 0.6
-0.2
-0.1
0.0
0.1
0.2
m = 8
m = 1
HS
adsorption coefficient
G
bulk density, r
Figure 5. Wall density (left side) and excess surface coverage (right side) of a
four-armed SLM fluid next to a hard wall vs. bulk density, ρ. The curves from
top to bottom on each part of the figure are the WOZ/APY theory results for
the FSA/SLM model of arm length, m = 1, 2, 4 and 8. Symbols on the left part
are MC simulation data for for THS/SLM model with m = 2.
factors which compensate one another: (i) the approximations involved in theoretical
treatment (PY approximation and the ideal network approximation) and (ii) the
differences in the modelling. Similarly to the homogeneous case, there is a tendency
to improve the agreement with an increase of the arm length.
The wall or contact density, ρW(z = 0), of the SLM fluid as a function of the
bulk number density, ρ, and at various arm lengths is plotted in figure 5. We can see
that for the same bulk density, the wall density decreases with the increase of the
arm length. In the case of the fluid of spherical molecules, the wall density is always
higher than the bulk density. In contrast, for the SLM fluid this is true starting from
some certain value of ρ. For example, it corresponds to ρ = 0.3 for m = 1. However,
this density increases with an increase of the arm length, e.g. ρ = 0.5 when m = 8.
Another property closely related to the wall density is the excess surface coverage
defined as follows:
Γ = ρ
∫
∞
0
dz[gW(z)− 1] = ρ lim
s→0
[
g̃W(s)−
1
s
]
. (3.7)
Very often Γ is called as the adsorption isotherm, adsorption coefficient or excess
surface coverage and provides an important overall surface-coveraging characteri-
zation of the adsorption. Baxter solution for the Laplace transform, g̃W(s), allows
to take the limit in equation (3.7) and an analytical expression for Γ is available
[28]. The results for Γ in the case of the FSA/SLM fluid vs the bulk density and
at different arm length are shown in figure 5 as well. One can see that an excess
surface coverage of the SLM fluid is always lower than for HS fluid of the same
bulk number density and decreases as the arms become longer. Moreover, the excess
238
Simplified approach to the star-like molecule fluids
-3 -2 -1 0 1 2 3
0.3
0.4
0.5
0.6
total fluid density
-3 -2 -1 0 1 2 3
0.0
0.1
0.2
0.3 H = 7d
H = 5d
density of ends of arms
-3 -2 -1 0 1 2 3
0.00
0.03
0.06
0.09
H = 3d
density of centers
Figure 6. Total fluid local density, local density of the ends of arms and local
density of the central beads of the THS/SLM fluid in a slit formed by two hard
walls fixed at three different separations: H = 3d (stars); H = 5d (circles) and
H = 7d (solid line). Particle number density in the slit defined as the number of
all beads devided by the slit volume is fixed at 0.44 in all cases.
surface coverage provided by the fluid of spherical molecules is positive at all bulk
conditions, while in the case of the SLM fluid Γ could be negative, especially, at low
bulk densities, i.e., low bulk pressures. However, in all cases Γ grows if the bulk fluid
becomes denser. This is consistent with the density distributions, particularly, with
the depletion effect observed in the behavior of local density in figure 4. We expect
that Γ calculated within the WOZ/APY theory works better at low densities. At
higher densities the deviations are quite possible. It follows from the fact that the
hard-sphere/hard-wall system behaves in that manner.
Local density in a slit formed by two walls. Similarly to the single-wall
case, the computer simulation offers a valuable insight into the local ordering of the
star-like molecules adsorbed into a space confined by two unmovable surfaces. For
simplicity the surfaces are modeled as the hard walls impenetrable to the centers of
both the central and the arm beads and only entropic effect of the confinement is
considered. Canonical MC simulations for the slit geometry were performed at con-
ditions when the total number density of beads in each pore is maintained the same.
Since simulation cell is not connected with the reservoir, this setup corresponds to
the case when bulk conditions (pressure or density) are not fixed. Although similar
simulation setup is used very often in the literature [11], the laboratory measure-
ments are performed usually under constant pressure conditions.
Three sets of information regarding the total fluid local density distribution,
local ordering of the ends of the arms and the central beads of the SLM fluid have
been extracted from these simulations. Figure 6 shows the trends in local density
distributions of the THS/SLM fluid in a slit-like confinement at the different gap
thickness. First of all, we see that total density in the slit behaves in a different
manner than near a single wall. Namely, the wall density constantly increases when
the slit becomes narrower that could reflect the flattening of the star-like aggregates
against the wall and the layering is more pronounced. When the slit is narrowed
up to three bead diameters, the number density of particles at the center of the
239
Yu.Duda, A.Trokhymchuk
gap increased substantially. Since transformations in the density distributions of the
ends of arms and the central beads are not so evident, it seems that the changes in
the total density are due to the flexible nature of the star arms. In particular, for
all considered separations between the slit walls, the ends of arms are enhanced at
the walls while the central beads occupy the middle region of the pores.
3.3. Films formed from a star-like molecule fluid
In this subsection we show that properties of fluid films formed from the SLM
aggregates could be studied within the FSA/SLMmodeling as well. The local density
distributions in the films composed of spherical molecules have been investigated by
many authors and are now well understood both theoretically and experimentally
[1]. In particular, it has been shown that well-defined layering in the films formed
from simple fluids is the main reason for disjoining pressure and solvation force
oscillations vs the film thickness. Here our aim is to obtain insight into the changes
in the behavior of macroscopic properties of thin films due to the presence of SLM
aggregates.
Theoretical modeling of the fluid film confinement. Now let us consider
the surfaces of the slit pore that are not fixed and could move. To model such a case,
we can imagine a huge reservoir filled with the SLM fluid having a particle number
density, ρ. Into one part of this reservoir we immersed two planar surfaces that
are infinite in area and are separated by the distance H . The material in between
the surfaces will form the fluid film and is in equilibria with the reservoir fluid. To
describe the interaction of the SLM species with a confinement, we assume that the
surfaces of the immersed substrates are structureless walls located at z = 0 and
z = H . The surfaces can attract or repel the adsorbed SLM particles. To model
such an interaction we used the following fluid-wall potential:
UF(z,H) =
{
∞, if z 6 0 or z > H
φ(z) + φ(H − z), if 0 < z < H
, (3.8)
where z is the distance between the center of fluid particle and the film surface in the
direction perpendicular to the surface. Such a form for the potential UF(z,H) implies
that the bead center can be located on the surface either at z = 0 or z = H and the
actual film thickness is H + d. The function φ(z) is modeled by the Lennard-Jones
(9,3) potential:
φ(z) = ǫW
[
(zW
z
)9
− α
(zW
z
)3
]
, (3.9)
where parameters ǫW and zW have the same meaning as for usual (12,6) Lennard-
Jones potential, and switching parameter α = 0 for repulsive film surfaces and α = 1
for attractive film surfaces.
Local density distribution in the thick film. To calculate the local density
distribution of the SLM fluid under the film confinement, similarly to the single-wall
case, the associative version of the HAB integral equation is applied [39]. However,
since the application of Perram procedure [36] is limited, at present, to the single-wall
240
Simplified approach to the star-like molecule fluids
-4 -2 0 2 4
0.0
0.5
1.0
1.5
2.0
z/d
fluid-repulsive surfaces
film thickness H = 8d
-4 -2 0 2
0.0
0.5
1.0
1.5
2.0
z/d
fluid-attractive surfaces
film thickness H = 8d
Figure 7. Normalized local density distribution of the FSA/SLM fluid in a film
formed by fluid-attractive and fluid-repulsive confining surfaces.
case only, we used the direct iteration solution of the set of WOZ/APY equations
in a form:
yαWj0 (z,H) = δj0 +
∑
lm
∑
β,γ
∫
dr′cαγjl (r
′)σγβ
lmh
βW
m0
(|r− r′|, H), (3.10)
but with the particle-wall interaction defined according to equation (3.8). The func-
tions, hαW
i0 (z,H) and yαWi0 (z,H), are the fluid-surface partial pair and cavity corre-
lation functions, respectively. The total local density distribution in a film, ρ F(z,H),
is calculated as follows:
ρF(z,H) = ργ(z,H)
[
y0W
00
(z,H) +
1
σΓ
Γ
∑
i,α
σα
i y
Γ−α,W
Γ−i,0 (z,H)
]
, (3.11)
where γ(z,H) is the Boltzmann factor for the fluid-surface repulsion.
Some representative results for the local density distribution of the SLM fluid in
films with attractive and repulsive surfaces are plotted in figure 7 as a function of
separation from the surface. The film thickness is fixed at eight bead diameters. The
local density is a convenient quantitative measure of the self-arrangement of the film
fluid. Quite tall and narrow peaks that are observed in the immediate vicinity of
the film surfaces reflect a well localized surface layer for spherical molecules. These
peaks almost disappear for SLM fluids, especially in the case of the repulsive fluid-
surface interaction. As the distance from the film surfaces increases, the magnitudes
of the peaks decrease while their widths increase. There is a well-defined homoge-
neous region with the bulk density in the middle of the film that covers about one
particle diameter for a fluid of spherical molecules but extends to approximately two
diameters for the SLM fluid. This picture is nearly independent of the fluid-surface
interaction. All this allows us to conclude that SLM fluid in the film tends to be
separated from the film surfaces and becomes homogeneous faster because of the
diminished influence of the confinement.
241
Yu.Duda, A.Trokhymchuk
1 2 3 4 5
-1
0
1
2
surface separation, H/d
HS
SLM
surface separation, H/d
fluid-repulsive surfaces
1 2 3 4 5
-1
0
1
2
HS
SLM
disjoining pressure, bd2P(H)
fluid-attractive surfaces
Figure 8. Structural disjoining pressure exerted by a film formed from the
FSA/SLM fluid (thick solid line) and HS fluid (thin solid line) confined between
two fluid-attractive and two fluid-repulsive surfaces.
Disjoining pressure. An important equilibrium property of any fluid confined
to the film is the pressure, PN(H), exerted by confined fluid on the inner side of
confining surfaces in the direction perpendicular to the surfaces. It can be calculated
from the local density distribution in the film, ρF(z,H), as follows [41]:
PN(H) = −
∫ H/2
0
∂UF(z,H)
∂z
ρF(z,H)dz. (3.12)
The pressure PN(H), measured relative to the bulk pressure, PB, defines the so-called
disjoining pressure:
Π(H) = PN (H)− PB. (3.13)
The bulk pressure, PB, is one of the parameters that determine the thermodynamic
state of the SLM fluid in a bulk phase. It can be calculated independently or can
be identified as a limiting value of the normal pressure acting between film surfaces
at infinite separation. To satisfy the self-consistency of the numerical procedure we
used the second possibility, i.e. we assume PB ≡ PN(H → ∞).
Figure 8 shows the results of the WOZ/APY theory calculations of the disjoining
pressure vs film thickness for the FSA/SLM fluid confined by attractive and repulsive
surfaces. Disjoining pressure oscillates between positive and negative values and
causes the confined SLM fluid to climb or spread on film surfaces. The differences in
the amplitude, periodicity and decay of oscillations for confined HS fluid or confined
FSA/SLM aggregates are evident. In reality this will influence the stability of the
films formed from these two fluids, since the zeros of the disjoining pressure (surface
separations at which the force exerted by the film fluid on the inner surfaces equals
the bulk pressure on the outer side of the surfaces) are related to the stable or
unstable thickness of fluid films.
Structural forces due to the SLM aggregates. To understand the qual-
itative trends in the film stability initiated by the architecture of the FSA/SLM
aggregates we applied the approach known in colloid science as the Derjaguin ap-
proximation [1]. According to Derjaguin, the work per unit area, W (H), to bring
242
Simplified approach to the star-like molecule fluids
1 2 3 4 5
-4
-2
0
2
4
6
surface separation, H/d
HS
SLM
fluid-repulsive surfaces
1 2 3 4 5
-4
-2
0
2
4
6
HS SLM
surface separation, H/d
energy per unit area, bd2W(H)
fluid-attractive surfaces
Figure 9. Structural energy per unit area between two fluid-attractive and two
fluid-repulsive surfaces immersed into the FSA/SLM fluid (thick solid line) and
HS fluid (thin solid line).
two film surfaces from infinity to the separation, H , can be calculated as
W (H) =
∫
∞
H
Π(H ′)dH ′. (3.14)
There is no direct interaction between two film surfaces and the energy, W (H), is
due to the structuring of the film fluid only, i.e. due to the so-called structural forces
[42]. Since complexes in the FSA/SLM fluid are formed by tangent spheres, the
extraction from W (H) the similar quantity calculated for the HS monomers give us
the information about the contribution into the film interaction of the energy induced
by the shape of the SLM formations. Proceeding in this way, we found out [39] that
the contribution due to the SLM shape is of the same order as the HS monomer result
itself, however, it has an opposite sign and results in a decrease in the magnitude,
shift in the phase, and increase in the periodicity of the oscillations of the force
exerted by the SLM fluid film. The energy law observed in a HS fluid is modified
switching to the SLM fluid (see figure 9). In general, the structural interaction energy
is a damped oscillatory function of the film thickness, but with a periodicity different
from that determined by the bead diameter. In contrast, the hard-sphere monomers
are able to pack more efficiently and, due to this, fill in the space between the
macrosurfaces down to very thickened films: range of oscillations observed in figure 9
is up to four particle diameters and the periodicity is approximately equal to the
monomer diameter. The oscillations in the film formed from SLM fluid are weaker
and periodicity increases to about two bead diameters. The structural interaction
between film surfaces due to the SLM fluid at small film thickness (of the order of
1.5 bead diameter) is prevailed by the repulsive forces in the case when fluid-surface
interaction is attractive, and promotes an attraction between the film surfaces when
the fluid-surface interaction is repulsive. When the film thickness increases, the films
formed from the SLM fluid stimulate attraction for both kinds of surfaces, though
for the fluid attractive surfaces this effect is more pronounced.
Local density distribution and film stability. To understand why the SLM
fluid tends to eliminate the structural force oscillations, we show in figure 10 the
243
Yu.Duda, A.Trokhymchuk
0
1
2
3
4
fluid-attractive surfaces
H = 1.1d
0
1
2
3 H=2.1d
-2 -1 0 1 2
0
1
2
3
H = 3.1d
0
1
2
3
4
fluid-repulsive surfaces
H = 1.4d
0
1
2
3 H = 2.4d
-2 -1 0 1 2
0
1
2
3
H = 3.4d
Figure 10. Local density distribution in the films formed from the FSA/SLM
fluid (thick solid line) and HS fluid (thin solid line) between two attractive and
two repulsive surfaces. Vertical thick lines indicate the position of film surfaces.
local density distributions across the films of the thicknesses determined from the
energy law discussed in figure 9. Particularly, the film configurations (i.e., film thick-
nesses) that we chose to analyze in this figure roughly correspond to the film energy
minima in the case of the film formed from the HS monomers. We see that a re-
pulsive structural force between two fluid attractive surfaces at small separations
between them originates from the presence of the surface layer of the SLM fluid and
extends till the separations of the order of the thickness of this layer. For the surface
separations intermediate between one and two bead diameters, the film fluid under-
goes a local restructuring and the attraction between confining surfaces dominates.
The existence of layering near film surfaces can be identified from the oscillations
in a local density. This is a case of the film formed from spherical monomers. Two
density maxima at a small film thickness, that we observe in the case of the SLM
film, can be treated as a consequence of the spliting of a rather thick surface layer of
the star aggregates; this layer remains disordered or amorphous. The local density
in this film is always higher than the corresponding bulk density, and the force ex-
erted by the film fluid on the film surfaces is repulsive. For the results discussed in
figure 10 such a disordered first layer of the SLM film is extended until around three
bead diameters. At this thickness the local density distribution starts to exhibit a
relatively deep failure, that causes attraction between film boundaries. At a larger
film thickness, the density distribution behaves as for the thick film discussed in
figure 7.
244
Simplified approach to the star-like molecule fluids
4. Conclusions
In the present paper we discussed an application of the associative integral equa-
tion theory to the four-site model of associating fluids in order to study the properties
of the star-like molecule fluids. We exploited the fact that the ideal network approx-
imation, consisting in the neglect in the diagrammatic analysis of the intramolecular
diagrams representing the formation of the loops and of the crossing rings, allows one
to treat the problem analytically. Particularly, the analytical solution of the associa-
tive OZ integral equation within the associative PY closure, which we abbreviated
as WOZ/APY theory, is available in literature [22–27].
There are at least two approaches to the star-like molecule fluids that are based
on an associative hard-sphere model. One of them is discussed in the present issue
by Kalyuzhnyi and Holovko [43]. These authors are using an analytical solution
of the polymer Percus-Yevick approximation for the multicomponent mixture of
associating hard spheres forming the star-like molecules in the limit of complete
association. In contrast, in our contribution to describe the structure of the star
fluid we have used the version of the four-site model of the associative fluid called
as FSA/SLM model [40]. Despite its simple pair potential, the FSA/SLM model
predicts a similar anomalous behavior for the structure factor [26] that is analogous
to that observed by Likos et al. [2,3,12], i.e. the first peak decreases with increas-
ing density while the second peak grows. To test and to justify our approach, we
provide readers with comparison of the radial distribution functions and structure
factors of the FSA/SLM model with MC simulation data for the tangent hard-sphere
(THS/SLM) fluid model. Inhomogeneous star-like molecule fluids have been consid-
ered as well. Particularly, the density profiles of the star-like molecule fluids near a
single wall and in the slit-like pore have been analyzed. Moreover, the FSA/SLM
model within the framework of the WOZ/APY theory and associative version of the
HAB approach was employed to probe the effect of the presence of star-like forma-
tions on the interaction between two macrosurfaces immersed into the SLM fluid.
In many cases we found that the obtained results qualitatively reproduce important
features reported from laboratory measurements.
Summarizing, we conclude that the considered FSA/SLM model of the star-like
molecule fluid is qualitatively correct and exhibits the main features attributed to
the star fluids. An application of the associative integral equation theory in the
form of WOZ/APY realization to the case of a star-like molecule fluid is quite
successful in the prediction of homogeneous structure ordering. It is noteworthy
that the agreement of the WOZ/APY predictions and MC data is the best for a
high density and the long arms, that is for the case of real star molecule fluids. We
hope that the results and the insight we provided will stimulate both new theoretical
and experimental studies in this field of macromolecular science.
5. Acknowledgement
YD was supported in part by the CONACyT of Mexico under Grant J32211-E.
245
Yu.Duda, A.Trokhymchuk
References
1. Israelachvili J.N. Intermolecular and Surface Forces. New-York, Academic Press, 1992.
2. Likos C.N., Löwen H., Watzlawek M., Abbas B., Jucknischke O., Allgaier J., Richter D.
// Phys. Rev. Lett., 1988, vol. 80, p. 4450.
3. Likos C.N., Löwen H., Poppe A., Willner L., Roovers J., Cubitt B., Richter D. //
Phys. Rev. E, 1998, vol. 58, p. 6299.
4. Lekkerkerker H.N.W., Poon W.C.-K., Pusey P.N., Stroobants A., Warren P.B. //
Europhys. Lett., 1992, vol. 20, p. 559.
5. Roovers J.E., Hadjichristidis N., Fetters L.J. // Macromolecules, 1983, vol. 16, p. 214.
6. Grest G.S., Kremer K., Witten T.A. // Macromolecules, 1987, vol. 20, p. 1376.
7. Grest G.S., Fetters L.J., Huang J.S., Richter D. // Adv. Chem. Phys., 1996, vol. 94,
p. 67.
8. Zifferer G. // Macromol. Theory Simul., 1997, vol. 6, p. 381.
9. Sikorski A., Romiszowski P. // Macromol. Theory Simul., 1999, vol. 8, p. 103.
10. Lipson J.E.G., Whittington S.G., Wilkinson M.K., Martin J.L., Graunt D.S. // J.
Phys. A: Math. Gen., 1985, vol. 18, p. L469.
11. Yethiraj A., Hall C.K. // Macromolecules, 1991, vol. 24, p. 709.
12. Watzlawek M., Löwen H., Likos C.N. // J. Phys.: Condens. Matter, 1998, vol. 10,
p. 8189.
13. MacDowell L.G., Vega C. // J. Chem. Phys., 1998, vol. 109, p. 5681.
14. Yethiraj A., Hall C.K. // J. Chem. Phys., 1991, vol. 94, p. 3943.
15. von Ferber C., Holovatch Y. // Physica A, 1998, vol. 249, p. 327.
16. Alessandrini J.L., Carignano M.A. // Macromolecules, 1992, vol. 25, p. 1157.
17. Cherayil B.J., Bawendi M.G., Miyake A., Freed K.F. // Macromolecules, 1986, vol. 19,
p. 2770.
18. Sear R., Jackson G. // Molec. Phys., 1994, vol. 81, p. 801.
19. Daoud M., Cotton J.P. // J. Phys. (France), 1982, vol. 43, p. 531.
20. Kalyuzhnyi Yu. // Condens. Matter Phys., 1997, No. 10, p. 51.
21. Yethiraj A., Curro G., Rajasekaran J.J. // J. Chem. Phys., 1995, vol. 103, p. 2229.
22. Wertheim M.S. // J. Stat. Phys., 1986, vol. 42, p. 459.
23. Wertheim M.S. // J. Stat. Phys., 1986, vol. 42, p. 477.
24. Chang J., Sandler S. // J. Chem. Phys., 1995, vol. 102, p. 437.
25. Chang J., Sandler S. // J. Chem. Phys., 1995, vol. 103, p. 3196.
26. Vakarin E., Duda Yu., Holovko M.F. // Molec. Phys., 1997, vol. 90, No. 4, p. 611.
27. Vakarin E., Duda Yu., Holovko M.F. // J. Stat. Phys., 1997, vol. 88, p. 1333.
28. Vakarin E., Holovko M.F., Duda Yu. // Molec. Phys., 1997, vol. 91, No. 2, p. 203–214.
29. Duda Yu., Segura C.J., Vakarin E., Holovko M.F., Chapman W. // J. Chem. Phys.,
1998, vol. 108, p. 9168.
30. Baxter R.J. // J. Chem. Phys., 1968, vol. 49, p. 2770.
31. Duda Yu. // J. Chem. Phys., 1998, vol. 109, p. 9015.
32. Wertheim M.S. // J. Math. Phys., 1964, vol. 5, No. 3, p. 643–654.
33. Duda Yu., Millan-Malo B., Pizio O., Henderson D. // J. Phys. Studies, 1997, vol. 1,
p. 45.
34. Dickman R., Hall C.K. // J. Chem. Phys., 1988, vol. 89, p. 3168.
35. Metropolis N.A., Rosenbluth A.W., Rosenbluth M.N., Teller A.H., Teller E. // J.
Chem. Phys., 1953, vol. 21, p. 1087.
246
Simplified approach to the star-like molecule fluids
36. Perram J.W. // Molec. Phys., 1975, vol. 30, p. 1505.
37. Trokhymchuk A., Henderson D., Sokolowski S. // Canadian J. Physics, 1996, vol. 74,
p. 65.
38. Henderson D., Abraham F.F., Barker J.A. // Molec. Phys., 1976, vol. 31, No. 6,
p. 1291.
39. Duda Yu., Henderson D., Trokhymchuk A., Wasan D. // J. Phys. Chem., 1999,
vol. 103, p. 7495.
40. Duda Y., Garcia I., Trokhymchuk A., Henderson D. // Molec. Phys., 2000, vol. 98,
No. 17, p. 1287.
41. Henderson D., Blum L., Lebowitz J.L. // J. Electroanal. Chem., 1979, vol. 102, p. 315.
42. Trokhymchuk A., Henderson D., Nikolov A., Wasan D.T. // Langmuir, 2001, vol. 17,
No. 16, p. 4940.
43. Kalyuzhnyi Yu.V., Holovko M.F. // Condens. Matter Phys., 2002, vol. 5, No. 2, p. 211.
Про один спрощений підхід до опису рідин зіркових
молекул
Ю.Дуда 1,2 , A.Трохимчук 2,3
1 Відділ молекулярної інженерії, Мексиканський інститут нафти,
Мехіко, Мексика
2 Інститут фізики конденсованих систем НАН України,
79011 Львів, вул. Свєнціцького, 1
3 Факультет хімії та біохімії, Університет Брайхем Янґ,
Прово, UT 84602, США
Отримано 29 березня 2002 р.
Запропоновано теоретичний підхід до вивчення широкого спектру
рівноважних властивостей рідин, молекули яких мають форму зірок
(як наприклад полістирен, та ряд кополімерів). Підхід у своїй осно-
ві ґрунтується на аналітичному розв’язку, отриманому в теорії інтег-
ральних рівнянь для так званої асоціативної моделі силових центрів.
Результати та висновки з теоретичного підходу підкріплені порівнян-
нями з комп’ютерним експериментом, проведеним для моделі гнуч-
ких полімерів, що мають форму зірок. Показано, що запропонований
теоретичний підхід працює добре для однорідної фази, де добре від-
творює структурний фактор для відносно довгих кінцівок зірки і ви-
соких густин рідини. Отримані результати відтворюють найважливі-
ші характеристики сольватаційної сили між двома плоскими повер-
хнями, зануреними у розчин з наявними зірковими молекулами, що
спостерігається у лабораторному експерименті.
Ключові слова: зіркові молекули, асоціативна теорія інтегральних
рівнянь, комп’ютерний експеримент
PACS: 05.20.Jj, 61.25.Em
247
248
|