Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description
The scaled particle theory is developed for the description of thermodynamical properties of a mixture of hard spheres and hard spherocylinders. Analytical expressions for free energy, pressure and chemical potentials are derived. From the minimization of free energy, a nonlinear integral equation...
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| Zitieren: | Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description / M.F. Holovko, M.V. Hvozd // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43501: 1–11. — Бібліогр.: 46 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1570232025-02-09T14:30:23Z Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description Iзотропно-нематичний перехiд в сумiшi твердих сфер та твердих сфероцилiндрiв: застосування теорiї масштабної частинки Holovko, M.F. Hvozd, M.V. The scaled particle theory is developed for the description of thermodynamical properties of a mixture of hard spheres and hard spherocylinders. Analytical expressions for free energy, pressure and chemical potentials are derived. From the minimization of free energy, a nonlinear integral equation for the orientational singlet distribution function is formulated. An isotropic-nematic phase transition in this mixture is investigated from the bifurcation analysis of this equation. It is shown that with an increase of concentration of hard spheres, the total packing fraction of a mixture on phase boundaries slightly increases. The obtained results are compared with computer simulations data. Для опису термодинамiчних властивостей сумiшi твердих сфер та твердих сфероцилiндрiв розвинуто теорiю масштабної частинки. Отримано аналiтичнi вирази для вiльної енергiї, тиску та хiмiчних потенцiалiв. Мiнiмiзацiєю вiльної енергiї сформульовано нелiнiйне iнтегральне рiвняння для орiєнтацiйної унарної функцiї розподiлу. З бiфуркацiйного аналiзу цього рiвняння дослiджено iзотропно-нематичний фазовий перехiд в данiй сумiшi. Показано, що при збiльшеннi концентрацiї твердих сфер загальний коефiцiєнт упаковки сумiшi на границях фаз злегка зростає. Представлено порiвняння отриманих результатiв з даними комп’ютерного моделювання. This project has received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734276, and from the State Fund For Fundamental Research (project N F73/26-2017). 2017 Article Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description / M.F. Holovko, M.V. Hvozd // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43501: 1–11. — Бібліогр.: 46 назв. — англ. 1607-324X PACS: 51.30.+i, 64.70.Md DOI:10.5488/CMP.20.43501 arXiv:1712.05330 https://nasplib.isofts.kiev.ua/handle/123456789/157023 en Condensed Matter Physics application/pdf 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 |
The scaled particle theory is developed for the description of thermodynamical properties of a mixture of hard
spheres and hard spherocylinders. Analytical expressions for free energy, pressure and chemical potentials
are derived. From the minimization of free energy, a nonlinear integral equation for the orientational singlet
distribution function is formulated. An isotropic-nematic phase transition in this mixture is investigated from
the bifurcation analysis of this equation. It is shown that with an increase of concentration of hard spheres, the
total packing fraction of a mixture on phase boundaries slightly increases. The obtained results are compared
with computer simulations data. |
| format |
Article |
| author |
Holovko, M.F. Hvozd, M.V. |
| spellingShingle |
Holovko, M.F. Hvozd, M.V. Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description Condensed Matter Physics |
| author_facet |
Holovko, M.F. Hvozd, M.V. |
| author_sort |
Holovko, M.F. |
| title |
Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description |
| title_short |
Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description |
| title_full |
Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description |
| title_fullStr |
Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description |
| title_full_unstemmed |
Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description |
| title_sort |
isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2017 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/157023 |
| citation_txt |
Isotropic-nematic transition in a mixture of hard spheres and hard spherocylinders: scaled particle theory description / M.F. Holovko, M.V. Hvozd // Condensed Matter Physics. — 2017. — Т. 20, № 4. — С. 43501: 1–11. — Бібліогр.: 46 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovkomf isotropicnematictransitioninamixtureofhardspheresandhardspherocylindersscaledparticletheorydescription AT hvozdmv isotropicnematictransitioninamixtureofhardspheresandhardspherocylindersscaledparticletheorydescription AT holovkomf izotropnonematičnijperehidvsumišitverdihsfertatverdihsferocilindrivzastosuvannâteoriímasštabnoíčastinki AT hvozdmv izotropnonematičnijperehidvsumišitverdihsfertatverdihsferocilindrivzastosuvannâteoriímasštabnoíčastinki |
| first_indexed |
2025-11-26T20:28:50Z |
| last_indexed |
2025-11-26T20:28:50Z |
| _version_ |
1849886176131940352 |
| fulltext |
Condensed Matter Physics, 2017, Vol. 20, No 4, 43501: 1–11
DOI: 10.5488/CMP.20.43501
http://www.icmp.lviv.ua/journal
Isotropic-nematic transition in a mixture of hard
spheres and hard spherocylinders: scaled particle
theory description
M.F. Holovko, M.V. Hvozd
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
Received July 27, 2017
The scaled particle theory is developed for the description of thermodynamical properties of a mixture of hard
spheres and hard spherocylinders. Analytical expressions for free energy, pressure and chemical potentials
are derived. From the minimization of free energy, a nonlinear integral equation for the orientational singlet
distribution function is formulated. An isotropic-nematic phase transition in this mixture is investigated from
the bifurcation analysis of this equation. It is shown that with an increase of concentration of hard spheres, the
total packing fraction of a mixture on phase boundaries slightly increases. The obtained results are compared
with computer simulations data.
Key words: hard sphere/hard spherocylinder mixture, isotropic-nematic transition, scaled particle theory
PACS: 51.30.+i, 64.70.Md
1. Introduction
A hard spherocylinder fluid is one of the simplest models widely used for the description of an
isotropic-nematic phase transition in liquid crystals [1, 2]. The first treatment of isotropic-nematic
transition in a hard-spherocylinder fluid was performed by Onsager about seventy years ago [3]. The
Onsager theory can be considered as the density functional theory in which a low-density expansion
of the free energy functional is truncated at the level of second virial coefficient. The equilibrium
state is determined by the functional variation of free energy with respect to the orientational distribution
function. It was shown [3] that such a treatment is exact in the limit of infinitely thin rodswhen L →∞ and
D→ 0, but L2D is fixed, where L and D are the length and the diameter of spherocylinders, respectively.
It was shown that besides an isotropic-nematic transition, the Onsager theory describes a nematic-smectic
transition in a hard-spherocylinder fluid, which appears at higher densities [4]. The application of the
scaled particle theory (SPT) [5–9] provides an efficient approximate way to incorporate the higher-order
contributions neglected in the Onsager theory. An alternative way of improving the Onsager theory is the
Parsons-Lee (PL) approach [10–12], which is based on the mapping of the properties of a spherocylinder
fluid to those of the hard-sphere model. The SPT theory was also extended for the description of a
hard-spherocylinder fluid in random porous media [13, 14].
During the last decades the approaches developed for a hard-spherocylinder fluid have been gener-
alized for the description of mixtures of hard anisotropic particles. In such systems, new phases were
observed and their properties were richer and more complicated than those for the one-component case
[4, 15–25]. The simplest example of such multi-component systems of hard anisotropic particles is a
binary mixture of hard spheres and hard spherocylinders, for the description of which the corresponding
approaches have been proposed. Among them there are the Onsager theory [12, 16, 18, 26] and the
Parsons-Lee approach [22, 24, 25] in the one-fluid and many-fluid approximations for a hard-spheres
mixture and computer simulations [17, 20, 22, 24, 25].
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
43501-1
https://doi.org/10.5488/CMP.20.43501
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
M.F. Holovko, M.V. Hvozd
In this paper we present a development of the scaled particle theory for the description of a binary
mixture of hard spheres and hard spherocylinders. We derive expressions for the chemical potentials of
hard sphere and hard spherocylinder components. From the minimization of free energy, a non-linear
integral equation for the orientational distribution function is obtained. From the bifurcation analysis
of this integral equation, an isotropic-nematic phase diagram of a binary mixture of hard spheres and
hard spherocylinders is analysed and discussed. The results of the presented approach are numerically
compared with some computer simulation data.
The paper is arranged as follows. The theoretical part is presented in section 2. The discussion of
the obtained results and the comparison to computer simulations data are given in section 3. Finally, we
draw some conclusions in the last section.
2. Theory
We consider a two-component hard convex body (HCB) fluid consisting of hard spheres (HS) and hard
spherocylinders (HSC). In order to characterize HCB particles we use three functionals: the volume V ,
the surface area S and the mean curvature r taken with the factor 1/4π. For hard spheres of the radius R1,
these functionals are as follows:
V1 =
4
3
πR3
1 , S1 = 4πR2
1 , r1 = R1 (2.1)
and for the hard spherocylinders of the radius R2 and of the length L2
V2 = πR2
2 L2 +
4
3
πR3
2 , S2 = 2πR2L2 + 4πR2
2 , r2 =
1
4
L2 + R2. (2.2)
A basic idea of the SPT approach is an insertion of an additional particle of a variable size, e.g.,
a scaled particle, into a fluid. Adding a scaled hard-sphere particle into our system we use the scaling
parameter λs. Therefore, the volume V1s, the surface area S1s and the mean curvature r1s are modified
according to the following relations
V1s = λ
3
sV1 , S1s = λ2
s S1 , r1s = λsr1. (2.3)
When we add a scaled hard-spherocylinder particle into a fluid with the scaling radius R2s and the scaling
length L2s, in addition to the scaling parameter λs, we introduce the scaling parameter αs. Therefore, the
radius R2s and the length L2s are defined as [5, 6]
R2s = λsR2 , L2s = αsL2. (2.4)
As a result, the volume V2s, the surface area S2s and the mean curvature r2s of the scaled spherocylinder
are equal to
V2s = πR2
2 L2αsλ
2
s +
4
3
πR3
2λ
3
s , S2s = 2πR2L2αsλs + 4πR2
2λ
2
s , r2s =
1
4
L2αs + R2λs. (2.5)
The procedure of insertion of a scaled particle into a HCB fluid is equivalent to the creation of a
corresponding cavity. This cavity is free of centers of any other fluid particles. The key point of the SPT
theory is a consideration of the excess chemical potential of a scaled particle µexs [27–33]. The work
needed to create such a cavity is equal to µexs .
For a small scaledHS andHSC particles, the excess chemical potentials can bewritten in the following
form [29]
βµex1s(λs) = − ln
[
1 − η1(1 + λs)3 − η2
(
1 +
r1sS2
V2
+
r2S1s
V2
+
V1s
V2
)]
, (2.6)
βµex2s(αs, λs) = − ln
[
1 − η1
(
1 +
r2sS1
V1
+
r1S2s
V1
+
V2s
V1
)
− η2
(
1 +
r2sS2
V2
+
r2S2s
V2
+
V2s
V2
)]
, (2.7)
43501-2
Isotropic-nematic transition in a HS/HSC mixture
where β = 1/kBT , kB is the Boltzmann constant, T is the temperature, η1 = ρ1V1 is the HS fluid packing
fraction, ρ1 is the HS fluid density, V1 is the volume of a HS particle; η2 = ρ2V2 is the HSC fluid
packing fraction, ρ2 is the HSC fluid density, V2 is the volume of a HSC particle. It should be noted that
equation (2.7) is written for an isotropic case.
After substituting equations (2.1)–(2.5) into equations (2.6)–(2.7) and having generalized equa-
tion (2.7) for the anisotropic case, the chemical potentials of the HS and HSC scaled particles can be
presented as follows:
βµex1s(λs) = − ln
{
1 − η1(1 + λs)3 − η2
[
1 +
1
k1
6γ2
3γ2 − 1
λs +
1
k2
1
3(γ2 + 1)
3γ2 − 1
λ2
s +
1
k3
1
2
3γ2 − 1
λ3
s
]}
, (2.8)
βµex2s(αs, λs) = − ln
(
1 − η1
[
3
4
s1αs (1 + k1λs)
2 + (1 + k1λs)
3
]
− η2
{
1 +
3(γ2 − 1)
3γ2 − 1
[1 + (γ2 − 1)τ( f )]αs +
6γ2
3γ2 − 1
λs +
6(γ2 − 1)
3γ2 − 1
[
1 +
1
2
(γ2 − 1)τ( f )
]
αsλs
+
3(γ2 + 1)
3γ2 − 1
λ2
s +
3(γ2 − 1)
3γ2 − 1
αsλ
2
s +
2
3γ2 − 1
λ3
s
})
, (2.9)
where k1, s1 and γ2 are equal to
k1 =
R2
R1
, s1 =
L2
R1
, γ2 = 1 +
L2
2R2
, (2.10)
and
τ( f ) =
4
π
∫
f (Ω1) f (Ω2) sin γ(Ω1,Ω2)dΩ1dΩ2. (2.11)
Here, Ω = (ϑ, ϕ) denotes the orientation of HSC particles and it is defined by the angles ϑ and ϕ, where
dΩ = 1
4π sin ϑdϑdϕ is the normalized angle element, γ(Ω1,Ω2) is the angle between orientational vectors
of two molecules, f (Ω) is the singlet orientation distribution function normalized in such a way that∫
f (Ω)dΩ = 1. (2.12)
f (Ω) is defined herein below from the minimization of the free energy of a considered mixture.
For a large scaled particle, the excess chemical potential is equal to the work needed to create a
macroscopic cavity within the fluid and it is given by a thermodynamic expression. For a scaled hard
sphere particle, it can be presented as follows:
βµex1s = w(λs) + βPV1s , (2.13)
where P is the pressure of a fluid and V1s is the volume of a scaled HS particle. Similarly, for a scaled
HSC particle, we have
βµex2s = w(αs, λs) + βPV2s , (2.14)
where P and V2s are the pressure of a fluid and the volume of a scaled HSC particle, respectively.
According to the ansatz of SPT [14, 27–33] w(λs) and w(αs, λs) can be written in the form of
expansions:
w(λs) = w0 + w1λs +
1
2
w2λ
2
s , (2.15)
w(αs, λs) = w00 + w01αs + w10λs + w11αsλs +
1
2
w20λ
2
s . (2.16)
We can derive the coefficients of these expansions from the continuity of µex1s, µ
ex
2s and their corresponding
derivatives ∂µex1s/∂λs, ∂
2µex1s/∂λ
2
s at λs = 0 for a scaled HS particle; ∂µex2s/∂αs, ∂µ
ex
2s/∂λs, ∂
2µex2s/∂αs∂λs,
∂2µex2s/∂λ
2
s at αs = λs = 0 for a scaled HSC particle.
43501-3
M.F. Holovko, M.V. Hvozd
Therefore, for a scaled HS particle, we obtain:
w0 = − ln(1 − η),
w1 = 3
η1
1 − η
+
1
k1
6γ2
3γ2 − 1
η2
1 − η
,
w2 = 6
η1
1 − η
+
1
k2
1
6(γ2 + 1)
3γ2 − 1
η2
1 − η
+
1
(1 − η)2
(
3η1 +
1
k1
6γ2
3γ2 − 1
η2
)2
, (2.17)
where η = η1 + η2 is the total packing fraction of the mixture considered. For a scaled HSC particle, we
find:
w00 = − ln(1 − η),
w01 =
3
4
s1
η1
1 − η
+
[
3(γ2 − 1)
3γ2 − 1
+
3(γ2 − 1)2τ( f )
3γ2 − 1
]
η2
1 − η
,
w10 = 3k1
η1
1 − η
+
6γ2
3γ1 − 1
η2
1 − η
,
w11 =
3
2
k1s1
η1
1 − η
+
[
6(γ2 − 1)
3γ2 − 1
+
3(γ2 − 1)2τ( f )
3γ2 − 1
]
η2
1 − η
+
{
3
4
s1
η1
1 − η
+
[
3(γ2 − 1)
3γ2 − 1
+
3(γ2 − 1)2τ( f )
3γ2 − 1
]
η2
1 − η
} (
3k1
η1
1 − η
+
6γ2
3γ2 − 1
η2
1 − η
)
,
w20 = 6k2
1
η1
1 − η
+
6(γ2 + 1)
3γ1 − 1
η2
1 − η
+
1
(1 − η)2
(
3k1η1 +
6γ2
3γ2 − 1
η2
)2
. (2.18)
After setting λs = 1 in equation (2.15) and αs = λs = 1 in equation (2.16), the HS and HSC scaled
particles become of the same sizes as HS and HSC particles of a fluid, respectively. It makes it possible
to find the relation between the pressure and the excess chemical potentials µex1 and µex2 of a fluid. The
total chemical potentials for HS and HSC particles in a HS/HSC mixture are as follows:
βµ1 = ln(ρ1Λ
3
1) + βµ
ex
1 , (2.19)
βµ2 = ln(ρ2Λ
3
2Λ2R) + βµ
ex
2 , (2.20)
where Λ1 and Λ2 are the fluid thermal wavelengths of the HS and HSC components, respectively; Λ−1
2R is
the rotational partition function of a single HSC molecule [34]. Then, we can write expressions for the
total chemical potentials as follows:
βµ1 = ln(ρ1Λ
3
1) − ln(1 − η) + a1
η
1 − η
+ b1
η2
(1 − η)2
+ βP
η1
ρ1
, (2.21)
βµ2 = ln(ρ2Λ
3
2Λ2R) − ln(1 − η) + a2
η
1 − η
+ b2
η2
(1 − η)2
+ βP
η2
ρ2
, (2.22)
where the coefficients a1, a2, b1 and b2 are:
a1 = 6
η1
η
+
[
1
k1
6γ2
3γ2 − 1
+
1
k2
1
3(γ2 + 1)
3γ2 − 1
]
η2
η
,
b1 =
1
2
(
3
η1
η
+
1
k1
6γ2
3γ2 − 1
η2
η
)2
(2.23)
and
a2
(
τ( f )
)
=
[
3
4
s1(1 + 2k1) + 3k1(1 + k1)
]
η1
η
+
[
6 +
6(γ2 − 1)2τ( f )
3γ2 − 1
]
η2
η
,
43501-4
Isotropic-nematic transition in a HS/HSC mixture
b2
(
τ( f )
)
=
{(
3
4
s1 +
3
2
k1
)
η1
η
+
[
3(2γ2 − 1)
3γ2 − 1
+
3(γ2 − 1)2τ( f )
3γ2 − 1
]
η2
η
} (
3k1
η1
η
+
6γ2
3γ1 − 1
η2
η
)
. (2.24)
Therefore, we have two equations (2.21) and (2.22), and each of them contains two unknown quantities:
one of the chemical potentials and the pressure. In the case of an one-component fluid, we can eliminate
one of these unknowns, βµ1 (βµ2) or P, using equation (2.21) or equation (2.22) and the Gibbs-Duhem
relation. In our case, the Gibbs-Duhem equation has the form
∂(βP)
∂ρ
=
2∑
α=1
ρα
∂(βµα)
∂ρ
. (2.25)
Expressions for the chemical potentials can be obtained according to the recent paper [35], in which
an expression for the pressure is obtained from equation (2.25), and an expression for the free energy is
obtained from an integration over the total density ρ. From the differentiation of the free energy with
respect to ρ1 and ρ2, we derive expressions for the chemical potentials µ1 and µ2.
In order to use equation (2.25) and to get one equation containing only one unknown instead of
equations (2.21) and (2.22), we take the derivatives with respect to the total fluid density ρ =
∑2
α=1 ρα
on the both sides of equations (2.21) and (2.22) keeping the fluid composition constant: xα = ρα/ρ,
α = 1, 2. Hence, we derive
∂(βµ1)
∂ρ
=
1
ρ
[
1 +
η
1 − η
+ a1
η
(1 − η)2
+ 2b1
η2
(1 − η)3
]
+
4
3
πR3
1
∂(βP)
∂ρ
, (2.26)
∂(βµ2)
∂ρ
=
1
ρ
[
1 +
η
1 − η
+ a2
η
(1 − η)2
+ 2b2
η2
(1 − η)3
]
+
(
πR2
2 L2 +
4
3
πR3
2
)
∂(βP)
∂ρ
. (2.27)
The combination of equations (2.25) and (2.26)–(2.27) makes it possible to write an expression for the
fluid compressibility. Taking into account that
∑
α xα = 1, we obtain
∂(βP)
∂ρ
=
1
1 − η
+ (1 + A)
η
(1 − η)2
+ (A + 2B)
η2
(1 − η)3
+ 2B
η3
(1 − η)4
, (2.28)
where
A =
2∑
α=1
xαaα , B =
2∑
α=1
xαbα . (2.29)
From the integration of equation (2.28) over the total density ρ at a constant concentration, we find
βP
ρ
= 1 +
η
1 − η
+
A
2
η
(1 − η)2
+
2B
3
η2
(1 − η)3
. (2.30)
Now, we calculate the Helmholtz free energy, which is related to the pressure as
βF
V
= ρ
ρ∫
0
dρ′
1
ρ′
(
βP
ρ′
)
. (2.31)
We carry out this integration at fixed concentrations xα, where α = 1, 2. Thus, the final expression for
the free energy is
βF
V
=
βFid
V
+ ρ
[
− ln(1 − η) +
A
2
η
1 − η
+
B
3
η2
(1 − η)2
]
, (2.32)
where Fid is the ideal gas contribution to the Helmholtz free energy of a mixture:
βFid
V
= ρ1
[
ln(Λ3
1ρ1) − 1
]
+ ρ2
[
ln(Λ3
2ρ2) − 1
]
+ ρ2σ( f ). (2.33)
43501-5
M.F. Holovko, M.V. Hvozd
Here, σ( f ) is the entropic term defined as
σ( f ) =
∫
f (Ω) ln f (Ω)dΩ. (2.34)
The singlet orientational distribution function f (Ω) can be obtained from the minimization of free
energy with respect to variations of this distribution. This procedure leads to a nonlinear integral equation
ln f (Ω1) + λ + C
∫
f (Ω′) sin γ(Ω1Ω
′)dΩ′ = 0, (2.35)
where
C =
η2
1 − η
[
3(γ2 − 1)2
3γ2 − 1
+
1
1 − η
(γ2 − 1)2
3γ2 − 1
(
3k1η1 +
6γ2
3γ2 − 1
η2
)]
. (2.36)
The constant λ is defined from the normalization condition equation (2.12).
Using the expression for the Helmholtz free energy, we calculate the total chemical potentials for the
components of HS and HSC in a mixture. From the relationship
βµα =
∂
∂ρα
(
βF
V
)
, (2.37)
we derive
βµ1 = lnΛ3
1ρ1 − ln(1 − η) +
1
2
η
1 − η
{
a1 + 6
ρ1V1
η
+
ρ2V1
η
[
3
4
s1(1 + 2k1) + 3k1(1 + k1)
] }
+
1
3
η2
(1 − η)2
[
b1 + 3
ρ1V1
η2
(
3η1 +
1
k1
6γ2
3γ2 − 1
η2
)
+
ρ2V1
η2
(
9k1
(
1
2
s1 + k1
)
η1 +
{
3
4
6γ2
3γ2 − 1
s1 + 3k1
[
3 +
3(γ2 − 1)2τ( f )
3γ2 − 1
]}
η2
)]
+ βPV1 (2.38)
for the chemical potential of HS and
βµ2 = lnΛ3
2ρ2 + σ( f ) − ln(1 − η)
+
1
2
η
1 − η
{
a2 +
ρ1V2
η
[
1
k1
6γ2
3γ2 − 1
+
1
2
1
k2
1
6(γ2 + 1)
3γ2 − 1
]
+
ρ2V2
η
[
6 +
6(γ2 − 1)2τ( f )
3γ2 − 1
] }
+
1
3
η2
(1 − η)2
[
b2 +
ρ1V2
η2
1
k1
6γ2
3γ2 − 1
(
3η1 +
1
k1
6γ2
3γ2 − 1
η2
)
+
ρ2V2
η2
( {
3
4
6γ2
3γ2 − 1
s1 + 3k1
[
3 +
3(γ2 − 1)2τ( f )
3γ2 − 1
]}
η1
+
6γ2
3γ2 − 1
[
6(2γ2 − 1)
3γ2 − 1
+
6(γ2 − 1)2τ( f )
3γ2 − 1
]
η2
)]
+ βPV2 (2.39)
for the chemical potential of HSC.
3. Results and discussions
We use the theory presented in the previous section to study the effect of hard spheres on the isotropic-
nematic phase transition in a binary mixture of hard spheres and hard spherocylinders. This investigation
is done within the framework of bifurcation analysis of the integral equation equation (2.35) for the
singlet distribution function f (Ω). It is worth noting that for the first time this equation was obtained by
Onsager [3] for a hard-spherocylinder fluid in the limit L2 → ∞ and R2 → 0, when the dimensionless
density of a spherocylinder fluid c2 =
1
2πρ2L2
2 R2 was fixed. Therefore, in the Onsager limit we have
C → c2 =
1
2
πρ2L2
2 R2. (3.1)
43501-6
Isotropic-nematic transition in a HS/HSC mixture
The result, equation (2.36), for C is the generalization of the SPT result for a HSC fluid for the finite
values of L2 and R2 [9, 36]. In this case,
C →
η2
1 − η2
[
3 (γ2 − 1)2
3γ2 − 1
+
η2
1 − η2
6γ2 (γ2 − 1)2
(3γ2 − 1)2
]
. (3.2)
From the bifurcation analysis of the integral equation equation (2.35) for the singlet distribution function
f (Ω), it was found that this equation has two characteristic points Ci and Cn [37], which defined the
range of stability of a considered mixture. The first point Ci corresponds to the highest possible density
of the stable isotropic state and the second point Cn corresponds the lowest possible density of a stable
nematic state. For the Onsager model, from the minimization of the free energy with respect to the
singlet distribution function f (Ω), and subsequently from the solution of the coexistence equations, the
following values of density of coexisting isotropic and nematic phases were obtained [38–40]:
ci = 3.289, cn = 4.192. (3.3)
In the presence of hard spheres for the Onsager model, we have
C =
c2
1 − η1
. (3.4)
It means that the isotropic-nematic transition in the presence of hard spheres shifts to lower densities of
spherocylinders.
For the binary mixture of hard spheres and hard spherocylinders at the finite value of L2/2R2, we can
put
Ci = 3.289, Cn = 4.192, (3.5)
where Ci and Cn are determined from equation (2.36). The values of Ci and Cn in equation (3.5) define
the isotropic-nematic phase diagram for a HS/HSC mixture depending on the ratios L2/R2 = 2 (γ2 − 1)
and k1 = R2/R1, as well as on the densities of HS and HSC particles, η1 and η2, respectively. We note
that s1 defined by equation (2.10) is not an independent parameter, since
s1 = 2 (γ2 − 1) k1. (3.6)
The packing fraction of hard spheres η1 as a function of the packing fraction of hard spherocylinders η2
for γ2 = 21 along the boundaries of isotropic-nematic phase transition is estimated from equation (3.5)
for a HS/HSC mixture at a fixed ratio k1. As it is seen from figure 1, the presence of hard spheres shifts
the phase transition to lower densities of hard spherocylinders. Moreover, the interfacial region becomes
broader if the size of hard spheres increases (k1 decreases).
The effect of the size of hard spheres on the isotropic-nematic phase boundaries of the same HS/HCS
mixture, but at a fixed packing fraction η1, is demonstrated in figure 2. One can observe that an increase
of the packing fraction of hard spheres leads to a contraction of interfacial region.
The boundaries of an isotropic-nematic phase transition for the same model are presented in figure 3
using the coordinates η = η1 + η2 and x1 = ρ1/(ρ1 + ρ2). It is seen that an increase of the composition
of spherical particles x1 makes the total packing fraction η slightly higher in comparison with the results
obtained in the coordinates η1 and η2 (see figure 1). In the case of a pure HSC fluid (x1 = 0), the SPT
results are rather close to the computer simulations data taken from [41]. It is noticed that the SPT theory
underestimates the values of ηi and ηn, and the difference from the computer simulations increases with
a decrease of the parameter L2/2R2.
The isotropic-nematic phase transition boundaries for a HS/HSC mixture when γ2 = 6 and k1 = 1
are presented in figure 4 (the left-hand panel) using the coordinates η and x1. A comparison with the
computer simulations data taken from [17] and [25] are also shown in this figure. It is found that the
theoretical prediction of the isotropic line is about ∆η = 0.05 lower than the simulations results, while for
the nematic line it is approximately ∆η = 0.023 lower. On the other hand, qualitatively the effect of the
composition x1 predicted by the theory and the one obtained from the computer simulations are similar.
Hence, in figure 4 (the right-hand panel) we present the modified results of a theoretical prediction,
43501-7
M.F. Holovko, M.V. Hvozd
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Cn
κ1=0.1
Ciη
2
η1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Cn
κ1=0.4
Ciη
2
η1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Cn
κ1=0.7
Ci
η
2
η1
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Cn
κ1=1
Ci
η
2
η1
Figure 1. (Color online) Coexistence lines of isotropic-nematic phases of a HS/HSCmixture for L2/2R2 =
20 presented as a dependence of the packing fraction of HSC particles η2 = ρ2V2 on the packing fraction
of HS particles η1 = ρ1V1 at fixed k1 = R2/R1. The black line below denoted by Ci corresponds to the
isotropic phase, the red line above denoted by Cn corresponds to the nematic phase. The area between
the solid black and dotted red lines corresponds to the region of the coexistence of isotropic and nematic
phases.
0.105
0.11
0.115
0.12
0.125
0.13
0.135
0.14
0.145
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cn η1=0.1
Ci
η
2
κ1
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cn
η1=0.3
Ci
η
2
κ1
0.035
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cn
η1=0.5
Ci
η
2
κ1
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Cn
η1=0.7
Ci
η
2
κ1
Figure 2. (Color online) Coexistence lines of isotropic-nematic phases of a HS/HSCmixture for L2/2R2 =
20 presented as a dependence of the packing fraction of HSC particles η2 = ρ2V2 on the ratio k1 = R2/R1
at the fixed packing fraction of HS particles η1 = ρ1V1. The black line below denoted by Ci corresponds
to the isotropic phase, the red line above denoted by Cn corresponds to the nematic phase. The area
between the solid black and dotted red lines corresponds to the region of the coexistence of isotropic and
nematic phases.
43501-8
Isotropic-nematic transition in a HS/HSC mixture
0.13
0.135
0.14
0.145
0.15
0.155
0.16
0.165
0.17
0.175
0 0.05 0.1 0.15 0.2 0.25
η
x
1
Figure 3. (Color online) Coexistence lines of isotropic-nematic phases of a HS/HSCmixture for L2/2R2 =
20 and k1 = 1 presented as a dependence of the packing fraction of HS/HSC mixture η = η1 + η2 on
the composition of spherical particles x1 = ρ1/(ρ1 + ρ2). For the case of pure HSC system (x1 = 0) the
SPT results are close to those obtained in computer simulations with a use of the modified Gibbs-Duhem
integration procedure (filled black triangle for the isotropic branch and open red triangle for the nematic
branch) [41].
0.35
0.36
0.37
0.38
0.39
0.4
0.41
0.42
0.43
0.44
0 0.05 0.1 0.15 0.2 0.25
η
x
1
0.35
0.36
0.37
0.38
0.39
0.4
0.41
0.42
0.43
0.44
0 0.05 0.1 0.15 0.2 0.25
η
x
1
Figure 4. (Color online) Coexistence lines of the isotropic and nematic phases of a HS/HSC mixture for
L2/2R2 = 5 and k1 = 1 presented as a dependence of the packing fraction of HS/HSCmixture η = η1+η2
on the composition of spherical particles x1 = ρ1/(ρ1 + ρ2). The computer simulations data taken from
[17] are denoted by filled circles for the isotropic branch and by open circles for the nematic branch. The
computer simulations data taken from [25] are shown as filled squares for the isotropic branch and open
squares for the nematic branch.
where the coexistence lines are shifted up by ∆η = 0.05 for the isotropic phase and by ∆η = 0.023 for
the nematic phase. As one can see, in this case an agreement between theoretical and simulation results
is rather satisfactory.
It is worth noting that the isotropic-nematic coexistence lines can be also obtained from the condition
of thermodynamic equilibrium. According to this condition, the both phases should have the same
pressure and the same chemical potentials:
Pi(ηi, xi) = Pn(ηn, xn), µi,1(ηi, xi) = µn,1(ηn, xn), µi,2(ηi, xi) = µn,2(ηn, xn), (3.7)
where µi,1 (or µi,2) and µn,1 (or µn,2) are the chemical potentials of HS (or HSC) particles in the isotropic
and nematic phases, respectively.
In [37] it was shown that for the Onsager model the results obtained from the bifurcation analysis
and from the thermodynamic calculations coincide exactly. We also observe the same for the mixture of
Onsager spherocylinders and hard spheres. On the other hand, it was found in [14] that for the finite values
of L2/2R2, there is some difference between the results obtained from these two different approaches,
and the difference slightly increases with a decrease of the ratio L2/2R2.
43501-9
M.F. Holovko, M.V. Hvozd
4. Conclusions
In this paper we have generalized the scaled particle theory for the investigation of thermodynamic
properties of a mixture of hard spheres and hard spherocylinders. The expressions for the chemical
potentials of hard spheres and hard spherocylinders are derived from the consideration of a scaled hard
sphere and a scaled hard spherocylinder inserted into a system under study. Analytical expressions for the
free energy and for the pressure of the considered mixture are also obtained. From the minimization of
the free energy, a nonlinear integral equation for the orientational distribution function is obtained. From
the bifurcation analysis of this integral equation, an isotropic-nematic phase transition in a mixture of
hard spheres and hard spherocylinders is investigated. It is shown that the presence of hard spheres shifts
the phase transition to the lower densities of hard spherocylinders. With an increase of the sizes of hard
spheres, the interfacial region is expanded and with an increase of the packing fraction of hard spheres,
the interfacial region decreases. It is also shown that with an increase of concentration of hard spheres, the
total packing fraction of a mixture on the phase boundaries slightly increases in comparison with phase
boundaries in the coordinates of the packing fraction of hard spheres η1 and the packing fraction of hard
spherocylinders. The obtained results are qualitatively in agreement with computer simulations data.
The present work can be extended directly to the presence of disordered porous media. For the present
time, the scaled particle theory for a hard sphere fluid in disordered porous media is quite well developed
[27–29, 33, 42, 43] and has found applications in describing a reference system within the perturbation
theory for fluids with different types of attraction, such as associative [44] and ionic fluids [45, 46]. A
generalization of SPT theory for the description of a hard spherocylinder fluid in disordered porous media
is presented in [13, 14]. Using the obtained results in our separate paper we are going to consider the
generalization of SPT theory for the mixture of hard spheres and hard spherocylinders in random porous
media.
Acknowledgements
This project has received funding from the European Unions Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No 734276, and from the State Fund
For Fundamental Research (project N F73/26-2017).
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рiю масштабної частинки. Отримано аналiтичнi вирази для вiльної енергiї, тиску та хiмiчних потенцiалiв.
Мiнiмiзацiєю вiльної енергiї сформульовано нелiнiйне iнтегральне рiвняння для орiєнтацiйної унарної
функцiї розподiлу. З бiфуркацiйного аналiзу цього рiвняння дослiджено iзотропно-нематичний фазовий
перехiд в данiй сумiшi. Показано, що при збiльшеннi концентрацiї твердих сфер загальний коефiцiєнт
упаковки сумiшi на границях фаз злегка зростає. Представлено порiвняння отриманих результатiв з да-
ними комп’ютерного моделювання.
Ключовi слова: сумiш твердих сфер та твердих сфероцилiндрiв, iзотропно-нематичний перехiд, метод
масштабної частинки
43501-11
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Introduction
Theory
Results and discussions
Conclusions
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