Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections
The scaled particle theory (SPT) approximation is applied for the study of the influence of a porous medium on the isotropic-nematic transition in a hard spherocylinder fluid. Two new approaches are developed in order to improve the description in the case of small lengths of spherocylinders. In o...
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Інститут фізики конденсованих систем НАН України
2018
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| Цитувати: | Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections / M.F. Holovko, V.I. Shmotolokha // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13602: 1–13. — Бібліогр.: 36 назв. — англ. |
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irk-123456789-1570372019-06-20T01:28:02Z Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections Holovko, M.F. Shmotolokha, V.I. The scaled particle theory (SPT) approximation is applied for the study of the influence of a porous medium on the isotropic-nematic transition in a hard spherocylinder fluid. Two new approaches are developed in order to improve the description in the case of small lengths of spherocylinders. In one of them, the so-called SPT-CS-PL approach, the Carnahan-Starling (CS) correction is introduced to improve the description of thermodynamic properties of the fluid, while the Parsons-Lee (PL) correction is introduced to improve the orientational ordering. The second approach, the so-called SPT-PL approach, is connected with generalization of the PL theory to anisotropic fluids in disordered porous media. The phase diagram is obtained from the bifurcation analysis of a nonlinear integral equation for the singlet distribution function and from the thermodynamic equilibrium conditions. The results obtained are compared with computer simulation data. Both ways and both approaches considerably improve the description in the case of spherocylinder fluids with smaller spherocylinder lengths. We did not find any significant differences between the results of the two developed approaches. We found that the bifurcation analysis slightly overestimates and the thermodynamical analysis underestimates the predictions of the computer simulation data. A porous medium shifts the phase diagram to smaller densities of the fluid and does not change the type of the transition. Теорiя масштабної частинки (ТМЧ) застосовується для вивчення впливу пористого середовища на iзотропно-нематичний перехiд у плинi твердих сфероцилiндрiв. Розроблено два новi пiдходи для покращення опису сфероцилiндрiв невеликої довжини. В одному з них, так званому пiдходi ТМЧ-КС-ПЛ, вводиться поправка Карнагана-Старлiнга (КС) для покращення опису термодинамiчних властивостей плину, тодi як поправка Парсонса-Лi (ПЛ) покращує опис орiєнтацiйного впорядкування. Другий пiдхiд, так званий пiдхiд ТМЧ-ПЛ, пов’язаний з узагальненням теорiї Парсонса-Лi для анiзотропних рiдин у невпорядкованих пористих середовищах. Фазова дiаграма отримана з бiфуркацiйного аналiзу нелiнiйного iнтегрального рiвняння для одночастинкової функцiї розподiлу та умови термодинамiчної рiвноваги. Отриманi данi порiвнюються з даними комп’ютерних симуляцiй. Обидва шляхи i обидва пiдходи iстотно покращують опис системи сфероцилiндричного плину у випадку малих довжин сфероцилiндра. Ми не знайшли iстотної рiзницi в результатах в обох розроблених пiдходах. Ми виявили, що бiфуркацiйний аналiз трохи переоцiнює, а термодинамiчний аналiз недооцiнює передбачення, отриманi з комп’ютерних симуляцiй. Пористе середовище зсуває фазову дiаграму в бiк менших густин плину i не змiнює тип переходу. 2018 Article Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections / M.F. Holovko, V.I. Shmotolokha // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13602: 1–13. — Бібліогр.: 36 назв. — англ. 1607-324X PACS: 61.20.Gy, 61.43.Gy DOI:10.5488/CMP.21.13602 arXiv:1803.11419 http://dspace.nbuv.gov.ua/handle/123456789/157037 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 |
The scaled particle theory (SPT) approximation is applied for the study of the influence of a porous medium on
the isotropic-nematic transition in a hard spherocylinder fluid. Two new approaches are developed in order to
improve the description in the case of small lengths of spherocylinders. In one of them, the so-called SPT-CS-PL
approach, the Carnahan-Starling (CS) correction is introduced to improve the description of thermodynamic
properties of the fluid, while the Parsons-Lee (PL) correction is introduced to improve the orientational ordering. The second approach, the so-called SPT-PL approach, is connected with generalization of the PL theory to
anisotropic fluids in disordered porous media. The phase diagram is obtained from the bifurcation analysis
of a nonlinear integral equation for the singlet distribution function and from the thermodynamic equilibrium
conditions. The results obtained are compared with computer simulation data. Both ways and both approaches
considerably improve the description in the case of spherocylinder fluids with smaller spherocylinder lengths.
We did not find any significant differences between the results of the two developed approaches. We found
that the bifurcation analysis slightly overestimates and the thermodynamical analysis underestimates the predictions of the computer simulation data. A porous medium shifts the phase diagram to smaller densities of the
fluid and does not change the type of the transition. |
| format |
Article |
| author |
Holovko, M.F. Shmotolokha, V.I. |
| spellingShingle |
Holovko, M.F. Shmotolokha, V.I. Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections Condensed Matter Physics |
| author_facet |
Holovko, M.F. Shmotolokha, V.I. |
| author_sort |
Holovko, M.F. |
| title |
Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections |
| title_short |
Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections |
| title_full |
Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections |
| title_fullStr |
Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections |
| title_full_unstemmed |
Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections |
| title_sort |
scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: carnahan-starling and parsons-lee corrections |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157037 |
| citation_txt |
Scaled particle theory for a hard spherocylinder fluid in a disordered porous medium: Carnahan-Starling and Parsons-Lee corrections / M.F. Holovko, V.I. Shmotolokha // Condensed Matter Physics. — 2018. — Т. 21, № 1. — С. 13602: 1–13. — Бібліогр.: 36 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovkomf scaledparticletheoryforahardspherocylinderfluidinadisorderedporousmediumcarnahanstarlingandparsonsleecorrections AT shmotolokhavi scaledparticletheoryforahardspherocylinderfluidinadisorderedporousmediumcarnahanstarlingandparsonsleecorrections |
| first_indexed |
2025-07-14T09:22:42Z |
| last_indexed |
2025-07-14T09:22:42Z |
| _version_ |
1837613672553250816 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 1, 13602: 1–13
DOI: 10.5488/CMP.21.13602
http://www.icmp.lviv.ua/journal
Scaled particle theory for a hard spherocylinder fluid
in a disordered porous medium: Carnahan-Starling
and Parsons-Lee corrections
M.F. Holovko, V.I. Shmotolokha
Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
Received February 1, 2018
The scaled particle theory (SPT) approximation is applied for the study of the influence of a porous medium on
the isotropic-nematic transition in a hard spherocylinder fluid. Two new approaches are developed in order to
improve the description in the case of small lengths of spherocylinders. In one of them, the so-called SPT-CS-PL
approach, the Carnahan-Starling (CS) correction is introduced to improve the description of thermodynamic
properties of the fluid, while the Parsons-Lee (PL) correction is introduced to improve the orientational order-
ing. The second approach, the so-called SPT-PL approach, is connected with generalization of the PL theory to
anisotropic fluids in disordered porous media. The phase diagram is obtained from the bifurcation analysis
of a nonlinear integral equation for the singlet distribution function and from the thermodynamic equilibrium
conditions. The results obtained are compared with computer simulation data. Both ways and both approaches
considerably improve the description in the case of spherocylinder fluids with smaller spherocylinder lengths.
We did not find any significant differences between the results of the two developed approaches. We found
that the bifurcation analysis slightly overestimates and the thermodynamical analysis underestimates the pre-
dictions of the computer simulation data. A porous medium shifts the phase diagram to smaller densities of the
fluid and does not change the type of the transition.
Key words: hard spherocylinder fluid, porous material, scaled particle theory, isotropic-nematic transition,
Parsons-Lee theory, Carnahan-Starling correction
PACS: 61.20.Gy, 61.43.Gy
1. Introduction
A hard spherocylinder fluid is one of the simplest popular models widely used for the description of
isotropic-nematic phase transitions in the theory of liquid crystals [1]. During this transition the fluid
separates into two phases with two different densities. The phasewith the lower density is the isotropic one
and in this phase the distribution of molecular orientations is uniform. The other phase with the higher
density is the nematic one and in this phase molecular orientations are strongly ordered. This phase
separation was first explained by Onsager [2] nearly seventy years ago as a result of competition between
the orientational entropy that favours disorder and the entropy effect associated with the orientational-
dependent excluded volume of spherocylinder-like particles that favours order. Onsager’s treatment of the
isotropic-nematic transition was given for a very specific model of a hard spherocylinder fluid in which
the length of spherocylinder L1 → ∞ and the diameter D1 → 0 in such a way that the non-dimensional
density of the fluid c1 =
1
4πρ1L2
1 D1 is fixed, where ρ1 =
N1
V , N1 is the number of spherocylinders, V is
the volume of the system. The Onsager theory is based on the low-density expansion of the free energy
functional truncated at the second virial coefficient level. The result obtained for such a model in this
description is exact [1].
The application of the scaled particle theory (SPT) previously developed for a hard-sphere fluid [3, 4]
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.
13602-1
https://doi.org/10.5488/CMP.21.13602
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
M.F. Holovko, V.I. Shmotolokha
provides an efficient approximate way to incorporate the higher order contributions neglected in the
Onsager theory. As a result, it was possible to generalize the Onsager theory for the description of a more
realistic model of the hard spherocylinder fluid with a finite value of the length of spherocylinder L1
and a nonzero value of the diameter D1 [5–7]. We note that for thermodynamic properties of a hard
sphere fluid, the SPT produces the same result as the Percus-Yevick theory [8, 9]. A basic defect of such
a description is known to appear at higher densities where the theory needs some improvement, such
as a semi-empirical Carnahan-Starling (CS) correction [10, 11]. Recently the SPT was applied for the
description of isotropic-nematic phase transition in a mixture of hard spheres and hard spherocylinders.
By comparison with the corresponding computer simulation data [12–14] it was shown that the accuracy
of the SPT description reduces with a decreasing length L1 of spherocylinder. Such a bad accuracy of SPT
description can be improved by CS correction. An alternative to SPT way of improvement of the Onsager
theory is the Parsons-Lee (PL) approach [15–17] which is based on the mapping of the properties of the
hard spherocylinder fluid to those of the hard sphere fluid. The application of the CS theory to the hard
sphere system in the PL theory leads to a correct description of the isotropic-nematic transition in the
hard spherocylinder fluid even at small lengths L1 of spherocylinders [16].
During the last decade the scaled particle theory was extended to generalize for the description of a
hard sphere fluid in disordered porous media [18–24]. The obtained results were generalized for the fluid
of hard convex body particles in disordered porous media [25] and was used for the study of the influence
of porous media on the isotropic-nematic transition in a hard spherocylinder fluid in disordered porous
media [26, 27]. It was shown that a porous medium shifts the isotropic-nematic phase transition to smaller
fluid densities. However, similar to the bulk case, the accuracy of the developed SPT description reduces
with a decreasing spherocylinder length. In this paper, in order to improve the SPT description of a hard
spherocylinder fluid in disordered porous media we will introduce two types of corrections. The first
one is the CS correction which improves the description at higher densities of the fluid. The second one
corrects the description of the orientational ordering in a hard spherocylinder fluid at higher densities.
This correction is formulated by comparison of the constants in the integral equation for the singlet
distribution function of hard spherocylinders in the SPT approach and in the PL theory. In this paper, the
CS and PL corrections constitute the improvement of the SPT description of a hard spherocylinder fluid
in disordered porous media. In parallel to the SPT approach, we also consider the generalization of the
PL theory for a hard spherocylinder fluid in disordered porous media. It is shown that both approaches
provide a correct description of the isotropic-nematic phase transition in a hard spherocylinder fluid in
disordered porous media including the hard spherocylinder fluids with small lengths of spherocylinders.
The paper is arranged as follows. In section 2 we give a brief review of the application of the
SPT approach for a hard spherocylinder fluid in disordered porous media. An improvement of the SPT
description with the CS and the PL corrections is presented in section 3. In section 4 generalization of
the PL theory for a hard spherocylinder fluid in disordered porous media is presented. The results and
discussion are presented in section 5. We conclude in section 6.
2. SPT for hard spherocylinder fluids in disordered porous media
In this section we present a short review of the SPT for hard spherocylinder fluids in disordered
porous media. The basic idea of the SPT lies in the insertion of an additional hard spherocylinder with
the scaling diameter Ds and the scaling length Ls into a fluid in such a way that
Ds = λsD1 , Ls = αsL1 , (2.1)
where D1 and L1 are the diameter and the length of fluid spherocylinder, respectively. In the presence
of porous media, the excess of chemical potential for the small scaled particle in a spherocylinder fluid
confined in a matrix can be written in the form [26]
βµexs = − ln p0(αs, λs) − ln
{
1 −
η1
V1p0(αs, λs)
[
π
6
D3
1(1 + λs)
3 +
π
4
D2
1 L1(1 + λs)2(1 + αs)
13602-2
Scaled particle theory for a spherocylinder fluid in a porous medium
+
π
4
D1L2
1(1 + λs)αs
∫ ∫
f (Ω1) f (Ω2) sin γ(Ω1,Ω2)dΩ1dΩ2
]}
, (2.2)
where β = 1
kT , k is the Boltzmann constant, T is temperature, η1 = ρ1V1 is the fluid packing fraction, ρ1
is the fluid density,V1 is the volume of spherocylinder; p0(αs, λs) is the probability to find a cavity created
by a scale particle in the empty matrix and is defined by the excess of a chemical potential µ0
s of the scale
particle in the limit of the infinite dilution of a fluid; Ω = (ϑ, ϕ) is the orientation of particles defined by
the angles ϑ and ϕ; dΩ = 1
4π sin ϑdϑdϕ is the normalized angle element; γ(Ω1,Ω2) is an angle between
orientational vectors of two molecules; f (Ω) is the singlet orientational distribution function normalized
in such a way that ∫
f (Ω)dΩ = 1. (2.3)
For a large scale particle, the excess of chemical potential is given by a thermodynamic expression that
can be presented in the form:
βµexs = w(αs, λs) + βPVs/p0(λs, αs), (2.4)
where P is the pressure of the fluid, Vs is the volume of the scaled particle, the multiplier 1/p0(λs, αs)
appears due to an excluded volume confined by matrix particles and can be considered as a probability
to find a cavity created by a scaled particle in the absence of fluid particles. The probability p0(λs, αs) is
directly related to two different types of porosity introduced by us in [10, 12, 14, 26].
The first one corresponds to the geometrical porosity
φ0 = p0(αs = λs = 0), (2.5)
characterizing the free volume for a fluid. The second type of porosity corresponds to the case λs = αs = 1
and leads to the thermodynamic porosity
φ = p0(αs = λs = 1) = exp(−βµ0
1), (2.6)
defined by the excess chemical potential of fluid particles µ0
1 in the limit of infinite dilution. It characterizes
the adsorption of a fluid in the empty matrix. According to the ansatz of the SPT [5–7, 26] w(λs, αs) can
be presented in the form:
w(λs, αs) = w00 + w10λs + w01αs + w11αsλs +
w20λ
2
s
2
, (2.7)
where the coefficients of this expansion can be found from the continuity of the excess chemical potential
given in (2.2) and (2.4), as well as from the corresponding derivatives ∂µexs /∂λs, ∂µexs /∂αs, ∂2µexs /∂αs∂λs
and ∂2µexs /∂λ
2
s . As a result, one derives the coefficients as follows:
w00 = − ln (1 − η1/φ0) , (2.8)
w10 =
η1/φ0
1 − η1/φ0
(
6γ1
3γ1 − 1
−
p′0λ
φ0
)
, (2.9)
w01 =
η1/φ0
1 − η1/φ0
[
3(γ1 − 1)
3γ1 − 1
+
3(γ1 − 1)2
3γ1 − 1
τ( f ) −
p′0α
φ0
]
, (2.10)
w11 =
η1/φ0
1 − η1/φ0
[
6(γ1 − 1)
3γ1 − 1
+
3(γ1 − 1)2τ( f )
3γ1 − 1
−
p′′0αλ
φ0
+ 2
p′0αp′0λ
φ2
0
−
3(γ1 − 1) + 3(γ1 − 1)2τ( f )
3γ1 − 1
p′0λ
φ0
−
6γ1
3γ1 − 1
p′0α
φ0
]
+
(
η1/φ0
1 − η1/φ0
)2 (
6γ1
3γ1 − 1
−
p′0λ
φ0
) [
3(γ1 − 1)
3γ1 − 1
+
3(γ1 − 1)2τ( f )
3γ1 − 1
−
p′0α
φ0
]
, (2.11)
13602-3
M.F. Holovko, V.I. Shmotolokha
w20 =
η1/φ0
1 − η1/φ0
[
6 (1 + γ1)
3γ1 − 1
−
12γ1
3γ1 − 1
p′0λ
φ0
+ 2
( p′0λ
φ0
)2
−
p′′0λλ
φ0
]
+
(
η1/φ0
1 − η1/φ0
)2 (
6γ1
3γ1 − 1
−
p′0λ
φ0
)2
, (2.12)
where
γ1 = 1 +
L1
D1
, (2.13)
τ( f ) =
4
π
∫ ∫
f (Ω1) f (Ω2) sin γ(Ω1,Ω2)dΩ1dΩ2. (2.14)
Setting αs = λs = 1 in the equation (2.4) leads to the expression
β
(
µex1 − µ
0
1
)
= − ln (1 − η1/φ0) + A
(
τ( f )
) η1/φ0
1 − η1/φ0
+ B
(
τ( f )
) (η1/φ0)
2
(1 − η1/φ0)2
, (2.15)
where the coefficients A
(
τ( f )
)
and B
(
τ( f )
)
define the porous medium structure and the expressions for
them are as follows:
A
(
τ( f )
)
= 6 +
6 (γ1 − 1)2 τ( f )
3γ1 − 1
−
p′0λ
φ0
[
4 +
3 (γ1 − 1)2 τ( f )
3γ1 − 1
]
−
p′0α
φ0
(
1 +
6γ1
3γ1 − 1
)
−
p′′0αλ
φ0
−
1
2
p′′0λλ
φ0
+ 2
p′0αp′0λ
φ2
0
+
( p′0λ
φ0
)2
, (2.16)
B
(
τ( f )
)
=
(
6γ1
3γ1 − 1
−
p′0λ
φ0
) [
3 (2γ1 − 1)
3γ1 − 1
+
3 (γ1 − 1)2 τ( f )
3γ1 − 1
−
p′0α
φ0
−
1
2
p′0λ
φ0
]
, (2.17)
where p′0λ =
∂p0(αs,λs)
∂λs
, p′0α =
∂p0(αs,λs)
∂αs
, p′′0αλ =
∂2p0(αs,λs)
∂αs∂λs
, p′′0λλ =
∂2p0(αs,λs)
∂λ2
s
are the corresponding
derivatives at α = λ = 0. Using the Gibbs-Duhem equation
(
∂P
∂ρ1
)
T
= ρ1
(
∂µ1
∂ρ1
)
T
, which relates the
pressure P of a fluid to its total chemical potential µ1 = ln(η1) + µ0
1 + µex1 one derives the fluid
compressibility in the form
β
(
∂P
∂ρ1
)
T
=
1
(1 − η1/φ)
+
[
1 + A
(
τ( f )
) ] η1/φ0
(1 − η1/φ) (1 − η1/φ0)
+
[
A
(
τ( f )
)
+ 2B
(
τ( f )
) ] (η1/φ0)
2
(1 − η1/φ) (1 − η1/φ0)
2 + 2B
(
τ( f )
) (η1/φ0)
3
(1 − η1/φ) (1 − η1/φ0)
3 . (2.18)
From expression (2.18) it is possible to obtain the chemical expression and the pressure of the fluid
in SPT2 approach [10, 12, 26]. The expression (2.18) at higher densities has two divergences, which
appear at η1 = φ and η1 = φ0, respectively. Since φ < φ0, the divergence at η1 = φ occurs at lower
densities than the second one and, therefore, should be removed. Different corrections and improvements
of SPT2 approach were proposed in [10–12, 14, 26]. The first corrections were considered in [10] where
based on SPT2, four different approximations were proposed. The best one is the SPT2b approximation
which was derived replacing φ by φ0 everywhere in (2.18) except the first term. However, this term has
a divergence at η1 = φ and due to this, some other approximations were proposed in [11, 12, 14, 26].
One of them called SPT2b1 can be obtained from the expression for the chemical potential in SPT2b
approach by removing the divergence at η1 = φ through the expansion of the logarithmic term in the
SPT2b expression for the chemical potential as follows:
− ln(1 − η1/φ) ≈ − ln(1 − η1/φ) +
η1(φ0 − φ)
φ0φ(1 − η1/φ0)
. (2.19)
13602-4
Scaled particle theory for a spherocylinder fluid in a porous medium
Therefore, one obtains the expressions for the chemical potential and pressure within the SPT2b1
approximation as follows:[
β
(
µex1 − µ
0
1
) ]SPT2b1
= σ( f ) − ln(1 − η1/φ0) +
[
1 + A
(
τ( f )
) ] η1/φ0
1 − η1/φ0
+
η1(φ0 − φ)
φ0φ(1 − η1/φ0)
+
1
2
[
A
(
τ( f )
)
+ 2B
(
τ( f )
) ] (η1/φ0)
2
(1 − η1/φ0)2
+
2
3
B
(
τ( f )
) (η1/φ0)
3
(1 − η1/φ0)3
, (2.20)
(
βP
ρ1
)SPT2b1
=
1
1 − η1/φ0
φ0
φ
+
(
φ0
φ
− 1
)
φ0
η1
ln(1 − η1/φ0)
+
A
(
τ( f )
)
2
η1/φ0
(1 − η1/φ0)2
+
2B
(
τ( f )
)
3
(η1/φ0)
2
(1 − η1/φ0)3
, (2.21)
where
σ( f ) =
∫
f (Ω) ln f (Ω)dΩ (2.22)
is the entropic term.
Some other approximations which include the third type of porosity φ∗ defined by the maximum
value of packing fraction of a fluid in a porous media are analyzed in [11, 12, 14]. However, in this paper
we restrict our consideration to the SPT2b1 approximation which is quite accurate at small, intermediate
and higher fluid densities.
From the thermodynamic relationship
βF
V
= βµ1ρ1 − βP, (2.23)
one can obtain an expression for the free energy. Within the SPT2b1 approximation, the free energy of a
confined fluid is as follows:(
βF
N
)SPT2b1
= σ( f ) + ln
η1
φ
− 1 − ln(1 − η1/φ0) +
(
1 −
φ0
φ
) [
1 +
φ0
η1
ln(1 − η1/φ0)
]
+
A
(
τ( f )
)
2
η1/φ0
1 − η1/φ0
+
B
(
τ( f )
)
3
(
η1/φ0
1 − η1/φ0
)2
. (2.24)
The singlet orientational distribution function f (Ω) can be obtained from the minimization of the free
energy with respect to variations of this distribution. This procedure leads to the nonlinear integral
equation
ln f (Ω1) + λ +
8
π
C
∫
f (Ω′) sin γ(Ω1Ω
′)dΩ′ = 0, (2.25)
where
CSPT2b1 =
η1/φ0
1 − η1/φ0
[
3(γ1 − 1)2
3γ1 − 1
(
1 −
p′0λ
2φ0
)
+
η1/φ0
(1 − η1/φ0)
(γ1 − 1)2
3γ1 − 1
(
6γ1
3γ1 − 1
−
p′0λ
φ0
)]
. (2.26)
3. Carnahan-Starling and Parsons-Lee corrections
As it was already noted at the beginning of this paper, the SPT approach is not accurate enough for
higher fluid densities as the length of spherocylinders decreases and the CS correction [11] should be
taken into account. The CS correction is generalized for the presence of a porous media. We present the
equation of state in the following form:(
βP
ρ1
)SPT2b1-CS
=
(
βP
ρ1
)SPT2b1
+
(
β∆P
ρ1
)CS
, (3.1)
13602-5
M.F. Holovko, V.I. Shmotolokha
where (βP/ρ1)
SPT2b1 is given by equation (2.21), (β∆P/ρ1)
CS is the CS correction which we present in
the form (
β∆P
ρ1
)CS
= −
(η1/φ0)
3
(1 − η1/φ0)
3 . (3.2)
We present the chemical potentials in a similar form
(βµ1)
SPT2b1-CS = (βµ1)
SPT2b1 + (β∆µ1)
CS, (3.3)
where the correction (∆µ1)
CS can be obtained from the Gibbs-Duhem equation
(β∆µ1)
CS = β
η1∫
0
dη1
η1
(
∂∆P
∂ρ1
)CS
. (3.4)
As a result,
(β∆µ1)
CS = ln(1 − η1/φ0) +
η1/φ0
1 − η1/φ0
−
1
2
(η1/φ0)
2
(1 − η1/φ0)2
−
(η1/φ0)
3
(1 − η1/φ0)3
. (3.5)
The free energy can also be presented in the form
(
βF
N1
)SPT2b1-CS
=
(
βF
N1
)SPT2b1
+
(
βF
N1
)CS
, (3.6)
where the first term (βF/N1)
SPT2b1 is given by equation (2.24) and the second term can be found from
thermodynamic relation (2.23)(
β∆F
N1
)CS
= ln(1 − η1/φ0) +
η1/φ0
1 − η1/φ0
−
1
2
(η1/φ0)
2
(1 − η1/φ0)
2 . (3.7)
However, the considered CS correction improves only thermodynamic properties and does not modify
the description of orientational ordering which is described by the integral equation (2.25) for the singlet
distribution function f (Ω). In order to improve the description of orientational ordering we shouldmodify
the parameterC given by the expression (2.26). This parameter has two terms. The first term appears from
the coefficient A
(
τ( f )
)
and the second one appears from the coefficient B
(
τ( f )
)
in the expression (2.24)
for the free energy. By simple comparison of parameter C in the SPT approach and the PL theory for the
bulk case we can see that the first termwhich appears from the coefficient A
(
τ( f )
)
in SPT2b1 theory is the
same as in the PL approach. Although there are some differences in the second term which appears from
the coefficient B
(
τ( f )
)
, it is possible to have practically the same result for the description of isotropic-
nematic transition from SPT and PL approaches if we introduce some parameter δ as a multiplier near
the term with τ( f ) in coefficient B
(
τ( f )
)
. After generalization of this result for the hard spherocylinder
fluid in disordered porous media, we can rewrite the expression for B
(
τ( f )
)
in the following form
B
(
τ( f )
)
=
(
6γ1
3γ1 − 1
−
p′0λ
φ0
) [
3 (2γ1 − 1)
3γ1 − 1
+
3 (γ1 − 1)2 δτ( f )
3γ1 − 1
−
p′0α
φ0
−
1
2
p′0λ
φ0
]
. (3.8)
Using the Parsons-Lee approach in the framework of Onsager investigation for sufficiently long sphero-
cylinders we determined that δ = 3/8 [16]. As a result, we can present the constant C in the form
CCS-PL =
η1/φ0
1 − η1/φ0
[
3(γ1 − 1)2
3γ1 − 1
(
1 −
p′0λ
2φ0
)
+
η1/φ0
(1 − η1/φ0)
δ
(γ1 − 1)2
3γ1 − 1
(
6γ1
3γ1 − 1
−
p′0λ
φ0
)]
. (3.9)
13602-6
Scaled particle theory for a spherocylinder fluid in a porous medium
4. Generalization of the Parsons-Lee theory for the hard spherocylinder
fluid in disordered porous media
In this section we generalize the PL theory for the case of hard spherocylinder fluid in disordered
porous media. In [16] in the framework of the functional scaling concept, a direct generalization of the CS
equation for the free energy of hard sphere fluid for a nematic fluid was constructed. Following [16], in
accordance with (2.25) and (3.7), a generalized expression for the hard spherocylinder fluid in disordered
porous media can be written as(
βF
N1
)PL
= ln
(
η1
φ
)
− 1 + σ( f ) +
{ (
1 −
φ0
φ
) [
1 +
φ0
η1
ln(1 − φ0/η1)
]
+
(
1 +
A
2
)
η1/φ0
1 − η1/φ0
+
(
B
3
−
1
2
) (
η1/φ0
1 − η1/φ0
)2 } [
1 +
3
4
(γ1 − 1)2
3γ1 − 1
τ( f )
]
, (4.1)
where
A = 6 + 4
p′0λ
φ0
+
( p′0λ
φ0
)2
−
1
2
p′′0λ
φ0
, (4.2)
B =
1
2
(
3 −
p′0λ
φ0
)2
. (4.3)
For the pressure and the chemical potential we will respectively have(
βP
ρ1
)PL
= 1 +
{
φ0
φ
η1/φ0
1 − η1/φ0
+
(
φ0
φ
− 1
) [
1 +
φ0
η1
ln(1 − η1/φ0)
]
+
A
2
η1/φ0
(1 − η1/φ0)2
+
2B
3
(η1/φ0)
2
(1 − η1/φ0)3
−
(η1/φ0)
3
(1 − η1/φ0)3
} [
1 +
3
4
(γ1 − 1)2
3γ1 − 1
τ( f )
]
, (4.4)
(βµ1)
PL = ln
(
η1
φ
)
+ σ( f ) +
{(
1 +
φ0
φ
+ A
)
η1/φ0
1 − η1/φ0
+
[
1
2
(A − 1) + B
]
(η1/φ0)
2
(1 − η1/φ0)2
+
(
2B
3
− 1
)
(η1/φ0)
3
(1 − η1/φ0)3
} [
1 +
3
4
(γ1 − 1)2
3γ1 − 1
τ( f )
]
. (4.5)
After minimization of the free energy, in the considered approach we obtain an integral equation for the
singlet orientational distribution function in the form (2.25), in which, however, the constant C has the
following form
CPL =
6
π
(γ1 − 1)2
3γ1 − 1
{ (
1 −
φ0
φ
) [
1 +
φ0
η1
ln(1 − η1/φ0)
] (
1 +
A
2
)
η1/φ0
1 − η1/φ0
+
(
B
3
−
1
2
) (
η1/φ0
1 − η1/φ0
)2 }
. (4.6)
5. Results and discussions
We will illustrate the developed approaches for the hard spherocylinder fluid in a hard sphere matrix.
First, we specify the geometrical and the probe particle porosities [26]. The geometrical porosity φ0 in
this case has the form
φ0 = 1 − η0 , (5.1)
13602-7
M.F. Holovko, V.I. Shmotolokha
where η0 = ρ0V0, ρ0 =
N0
V , N0 is the number of matrix particles, V0 =
1
6πD3
0 is the volume of a matrix
particle, V is the total volume of the system, D0 is the diameter of matrix hard spheres.
Using the SPT, the following expression for the probe particle porosity is derived [26]
φ = (1 − η0) exp
{
−
η0
1 − η0
D1
D0
[
3
2
(γ1 + 1) + 3γ1
D1
D0
]
−
η2
0
(1 − η0)2
9
2
γ1
D2
1
D2
0
−
η3
0
(1 − η0)3
(3γ1 − 1)
1
2
D3
1
D3
0
(
1 + η0 + η
2
0
)}
. (5.2)
The probability to find a small scaled spherocylinder in an empty matrix is
p0(αs, λs) = 1 − η0
1
V0
π
2
[
1
3
(D0 + λsD1)
3 +
1
2
αsL1(D0 + λsD1)
2
]
. (5.3)
Hereupon we can find the derivatives needed for the description of thermodynamic properties of a
confined fluid:
p′0λ = −3
D1
D0
η0 , p′0α = −
3
2
η0
L1
D0
, p′′0αλ = −3η0
L1
D0
D1
D0
, p′′0λλ = −6η0
D2
1
D2
0
. (5.4)
Now we apply the theory presented in the previous section for investigation of the isotropic-nematic
phase transition in a hard spherocylinder fluid confined in a matrix formed by a disordered hard sphere.
We start this study from the bifurcation analysis of the integral equation (2.25) for the singlet distribution
function f (Ω). This equation has the same form as the corresponding equation obtained by Onsager [2]
for the hard spherocylinder fluid in the limit of L1 →∞, D1 → 0 while the dimensional density of fluid
c1 =
1
4πρ1L2
1 D1 is fixed. In the Onsager limit
C → c1 =
1
4
πρ1L2
1 D1. (5.5)
From the bifurcation analysis of the integral equation (2.25) it was found that this equation has two
characteristic points Ci and Cn [28], which define the range of stability of a considered system. The first
point Ci corresponds to the highest value of a possible density of a stable isotropic state and the second
point Cn corresponds to the lowest value of a possible density of a stable nematic state. For the Onsager
model from the solution of the coexistence equations, the values of the density of coexisting isotropic
and nematic phases were obtained [29–31]
ci = 3.289, cn = 4.192. (5.6)
For the finite values of L1 and D1 we can put
Ci = 3.289, Cn = 4.192. (5.7)
For the constantC in this paperwe have three different approximations. In the SPT2b1C is given by the
expression (2.26), for CS-PL approximation C is given by the expression (3.9) and for PL approximation
C is given by the expression (4.6). The values (5.7) for C define the isotropic-nematic phase diagram for
a hard spherocylinder fluid in disordered porous media depending on the ratio L1/D1 = γ1 − 1 and the
parameter of the matrix, namely η0 = 1 − φ0 and the ratio D1/D0. To be more specific, we will fix the
last ratio by putting D0 = L1. As a result, D1
D0
= 1/(γ1 − 1).
At the beginning we will demonstrate how the developed approaches describe the isotropic-nematic
coexistence curves for a hard spherocylinder fluid in the bulk case. In figure 1 we present the dependence
of the density η1 on the parameter γ1 along the isotropic-nematic coexistence curves obtained from the
bifurcation analysis in SPT2b1, CS-PL and PL approximations. For comparison the computer simulation
results taken from [32, 33] are presented as well. As we can see all three approximations for large
enough values of γ1 give the same results in good agreement with the simulation data. However, starting
13602-8
Scaled particle theory for a spherocylinder fluid in a porous medium
5 10 15 20 25 30 35 40 45 50 55 60 65
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
4 6 8 10 12 14 16
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Figure 1. (Colour online) Isotropic-nematic coexistence diagram in the bulk case for a hard spherocylinder
fluid in the plane of the packing fraction of the fluid η1 versus parameter γ1. The results presented are
obtained from the bifurcation analysis of the integral equation (2.25) in different approximations for the
constant C: red dashed line denotes SPT2b1, green solid line denotes PL, brown dotted line denotes
CS-PL, down pointed green triangles 5 are the results of the simulations [32], left pointed triangles C
are GDI simulation results [33] and star points ? are the GEMC simulation results taken from [33].
from γ1 near γ1 ≈ 30 there is a deviation of the SPT2b1 approximation from general tendency and the
computer simulation data. This deviation increases with a decreasing parameter γ1 and at γ1 smaller than
20 it leads to incorrect results. Two other approximations, namely the CS-PL and PL ones, reproduce
more correctly the general tendency of the dependence of coexistence curves on γ1. We do not observe
differences between the CS-PL and PL approximations. However, at small γ1, the results predicted from
the bifurcation analysis slightly overestimate the jump of the density at the phase transition.
We should note that the isotropic-nematic coexistence lines can also be found from the condition of
thermodynamic equilibrium, according to which the isotropic and nematic phases have the same pressure
and the same chemical potential:
P (ηi) = P(ηn), µ (ηi) = µ1(ηn). (5.8)
The coexistence curves obtained from the condition (5.8) for a hard spherocylinder fluid in the
bulk case are presented in figure 2. As it was shown in [28], in the Onsager limit, the results obtained
from the bifurcation analysis and from the condition of thermodynamic equilibrium coincide exactly.
Similar to the bifurcation analysis in the thermodynamic way for large enough values of γ1 all three
approximations give the same results, but with a decreasing γ1 we observed a deviation in SPT2b1
approximation and computer simulation data which leads to incorrect results at small γ1. Again we do
not observe the difference between the CS-PL and PL results. However, at small γ1, contrary to the
bifurcation analysis, the thermodynamic consideration slightly underestimates the value of the density
jump between the isotropic and nematic phases. Nevertheless, comparing figure 1 and figure 2 we can
see that thermodynamic consideration leads to a better agreement with the computer simulation data.
As we have already noted in the Onsager limit all three approximations in the thermodynamic
approach and in the bifurcation analysis reproduce correctly the exact result (5.6). In the presence of a
porous medium for the Onsager model we obtain
ci/φ0 = 3.289, cn/φ0 = 4.192. (5.9)
It means that for the isotropic-nematic phase transition, the presence of a porous medium shifts the
phase diagram to lower densities of a fluid. This effect of the porous medium is illustrated in figure 3
where the dependence of the density of fluid η1 on parameter γ1 along the isotropic-nematic coexistence
curves calculated from thermodynamic equilibrium (5.8) for the hard spherocylinder fluid in a porous
13602-9
M.F. Holovko, V.I. Shmotolokha
5 10 15 20 25 30 35 40 45 50 55 60 65
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
4 6 8 10 12 14 16
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Figure 2. (Colour online) Isotropic-nematic coexistence diagram in the bulk case for a hard spherocylinder
fluid in the plane of the packing fraction of the fluid η1 versus parameter γ1. The results presented
are obtained from the thermodynamics analysis in different approximations: black dotted line denotes
SPT2b1, orange dash-dot-dot line denotes PL, blue dash-dot-dash line denotes CS-PL, down pointed
green triangles 5 are the results of simulations taken from [32], left pointed triangles C are GDI
simulations results from [33] and star points ? are the GEMC simulations results taken from [33].
5 10 15 20 25 30 35 40 45 50 55 60
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.3
D0=L1
4 6 8 10 12 14 16
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Figure 3. (Colour online) The influence of the porous medium on the isotropic-nematic coexistence
diagram for a hard spherocylinder fluid in the plane of the packing fraction of the fluid η1 versus
parameter γ1. The results are calculated from the thermodynamics analysis. The results for a fluid in
the disordered porous medium with the porosity φ0 = 0.7 (η0 = 0.3) are presented by dotted lines. For
comparison purposes the results for the bulk case are also presented. The notations are the same as in
figure 2.
medium with the porosity φ0 = 0.7 (η0 = 0.3) is presented. For comparison, the isotropic-nematic
diagram for the bulk case is also presented.
The influence of porosity φ0 on the coexistence lines of the isotropic-nematic phase transition
calculated from the conditions (5.8) for a hard spherocylinder fluid in coordinates fluid density η1 versus
packing fraction η0 of matrix particles (the porosity φ0 = 1 − η0) is illustrated also in figure 4 and
figure 5 for the cases L1/D1 = 20 and L1/D1 = 5, correspondingly. All the curves are obtained from the
condition of thermodynamic equilibrium (5.8) in CS-PL and PL approximations, correspondingly. As
we can see there are small insignificant differences between the predictions from both approaches. For
13602-10
Scaled particle theory for a spherocylinder fluid in a porous medium
N
I-N
I
L1/D1=20
Figure 4. (Colour online) Coexistence lines of isotropic-nematic phases of a hard spherocylinder fluid
in a hard sphere matrix for L1/D1 = 20 and D0/L1 = 1. Dependencies of the spherocylinder fluid
packing fraction η1 on the matrix packing fraction η0 are presented. The results are obtained from the
thermodynamics analysis with green dotted lines corresponding to the PL approximation and the blue
solid lines corresponding to the CS approximation. The GEMC results taken from [34] are shown as
circles and those taken from [33] are shown as squares and triangles (GDI).
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.225
0.250
0.275
0.300
0.325
0.350
0.375
0.400
I
L1/D1=5
N
I-N
Figure 5. (Colour online) Coexistence lines of isotropic-nematic phases of a hard spherocylinder fluid
in a hard sphere matrix for L1/D1 = 5 and D0/L1 = 1. Dependencies of the spherocylinder fluid
packing fraction η1 on the matrix packing fraction η0 are presented. The results are obtained from the
thermodynamics analysis with green dotted lines corresponding to the PL approximation and the blue
solid lines corresponding to the CS approximation. The GEMC results taken from [32] are shown as red
circles.
comparison, in figure 4 the results of the computer simulations of Schmidt and Dijkstra [34] obtained
by the method of Gibbs ensemble Monte Carlo (GEMC) are also presented. For the bulk case (η0 = 0),
the results of Bolhuis and Frenkel [33] obtained by GEMC method and GEMC combined with modified
Gibbs-Duhem integration (GDI) method are shown as well. As we can see there are some differences
between the computer data from [34] and [33]. Our theoretical prediction is in better agreement with the
data from [33] and correctly reproduces the dependences of η1 on η0 along the coexistence curves. The
computer simulation results from [32] are also presented in figure 5. We see a good correlation between
computer simulation data and theoretical prediction.
13602-11
M.F. Holovko, V.I. Shmotolokha
6. Conclusions
In this paper the scaled particle theory (SPT) is extended for the description of a hard spherocylinder
fluid in a disordered porous medium. We started from the SPT2b1 approach previously developed by
us [21, 22] for a hard sphere fluid in a disordered porous medium and generalized in [26] for a hard
spherocylinder fluid. The theory in this paper is applied for the study of the influence of disordered
porous media on the isotropic-nematic transition in a hard spherocylinder fluid. It is shown that the
accuracy of the SPT2b1 decreases with decreasing lengths of spherocylinders. Two different approaches
are developed in order to improve the SPT2b1 theory. In one of them, the so-called SPT2b1-CS-
PL approach, two corrections are involved. The first one is the Carnahan-Starling correction which
improves SPT description of thermodynamical properties at higher densities of the fluid. The second
one corrects the description of orientational ordering in a hard spherocylinder fluid at higher densities.
The constant of this correction is obtained from the comparison of the integral equation for the singlet
distribution function of a hard spherocylinder fluid in the SPT2b1 and Parsons-Lee (PL) approaches. In
the second approach, the so-called SPT2b1-PL approach, the PL theory [16] is generalized for a hard
spherocylinder fluid in a disordered porous medium. To this end, according to the original PL theory [16],
thermodynamic properties of a hard spherocylinder fluid in a disordered porous medium are mapped
with the thermodynamic properties of a hard sphere fluid in a disordered porous medium in the SPT2b1
approximation [21, 22] with the CS correction considered in this paper.
The phase diagram of a hard spherocylinder fluid in a disordered porous medium is calculated in two
different ways. One of them is connected with the bifurcation analysis of the nonlinear integral equation
for the singlet distribution function obtained from minimization of the free energy of the considered
system. The second way is based on the condition of thermodynamic equilibrium. The obtained results
are compared with the existing computer simulation data [32–34]. It is shown that in both approaches
the original SPT2b1 approximation is not very accurate with the decreasing length of spherocylinders.
The SPT2b1-CS-PL and SPT2b1-PL approximations in the bifurcation analysis and in thermodynamic
way more or less correctly reproduce the coexistence curves with decreasing lengths of spherocylinders.
We do not find a significant difference between the SPT2b1-CS-PL and SPT2b1-PL approximations.
However, the bifurcation analysis slightly overestimates the change of density at the phase transition. In
thermodynamic way, we also did not find the best agreement between theoretical prediction and computer
simulation data for small enough lengths of spherocylinders. In any case, the thermodynamic way is more
preferable for the description of a phase transition. From bifurcation analysis and from thermodynamic
way we show that the porous medium shifts the phase diagram to lower densities of a fluid. Comparison
with computer simulation results is discussed.
The model considered in this paper can be used as the reference system for generalization of the Van
der Waals theory for anisotropic fluids in disordered porous media [26, 27, 35] and for taking different
types of associations [36] into account.
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Теорiя масштабної частинки для системи
сфероцилiндричного плину в невпорядкованому пористому
середовищi: поправки Карнагана-Старлiнга i Парсонса-Лi
М.Ф. Головко, В.I.Шмотолоха
Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
Теорiя масштабної частинки (ТМЧ) застосовується для вивчення впливу пористого середовища на
iзотропно-нематичний перехiд у плинi твердих сфероцилiндрiв. Розроблено два новi пiдходи для покра-
щення опису сфероцилiндрiв невеликої довжини. В одному з них, так званому пiдходi ТМЧ-КС-ПЛ, вводи-
ться поправка Карнагана-Старлiнга (КС) для покращення опису термодинамiчних властивостей плину, то-
дi як поправка Парсонса-Лi (ПЛ) покращує опис орiєнтацiйного впорядкування. Другий пiдхiд, так званий
пiдхiд ТМЧ-ПЛ, пов’язаний з узагальненням теорiї Парсонса-Лi для анiзотропних рiдин у невпорядкова-
них пористих середовищах. Фазова дiаграма отримана з бiфуркацiйного аналiзу нелiнiйного iнтеграль-
ного рiвняння для одночастинкової функцiї розподiлу та умови термодинамiчної рiвноваги. Отриманi
данi порiвнюються з даними комп’ютерних симуляцiй. Обидва шляхи i обидва пiдходи iстотно покращу-
ють опис системи сфероцилiндричного плину у випадку малих довжин сфероцилiндра. Ми не знайшли
iстотної рiзницi в результатах в обох розроблених пiдходах.Ми виявили,що бiфуркацiйний аналiз трохи
переоцiнює, а термодинамiчний аналiз недооцiнює передбачення, отриманi з комп’ютерних симуляцiй.
Пористе середовище зсуває фазову дiаграму в бiк менших густин плину i не змiнює тип переходу.
Ключовi слова: твердий сфероцилiндричний плин, пористий матерiал, теорiя масштабної частинки,
iзотропно-нематичний перехiд, теорiя Парсонса-Лi, поправка Карнагана-Старлiнга
13602-13
https://doi.org/10.1063/1.4923291
https://doi.org/10.1103/PhysRevA.19.1225
https://doi.org/10.1063/1.452811
https://doi.org/10.1063/1.455332
https://doi.org/10.1021/jp809706n
https://doi.org/10.1021/jp9106603
https://doi.org/10.1063/1.3532546
https://doi.org/10.5488/CMP.15.23607
https://doi.org/10.1351/PAC-CON-12-05-06
https://doi.org/10.1021/acs.jpcb.6b02957
https://doi.org/10.5488/CMP.20.33602
https://doi.org/10.1016/j.molliq.2013.05.030
https://doi.org/10.15407/ujpe60.08.0770
https://doi.org/10.1103/PhysRevA.17.2067
https://doi.org/10.1021/ma00139a014
https://doi.org/10.1063/1.447098
https://doi.org/10.1021/ma00065a027
https://doi.org/10.1063/1.471343
https://doi.org/10.1063/1.473404
https://doi.org/10.1063/1.1815294
https://doi.org/10.5488/CMP.18.13607
https://doi.org/10.1021/jz502135f
Introduction
SPT for hard spherocylinder fluids in disordered porous media
Carnahan-Starling and Parsons-Lee corrections
Generalization of the Parsons-Lee theory for the hard spherocylinder fluid in disordered porous media
Results and discussions
Conclusions
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