Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation
The mean field approximation is formulated within the framework of the density field theory to study the properties of a Maier-Saupe nematogenic fluid near a hard wall. The density and the order parameter profiles are obtained using the analytical expressions derived in the linearized mean field app...
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| Опубліковано в: : | Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation / M. Holovko, T. Patsahan, I. Kravtsiv, D. di Caprio // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13608: 1–11. — Бібліогр.: 27 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860014031863021568 |
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| author | Holovko, M. Patsahan, T. Kravtsiv, I. di Caprio, D. |
| author_facet | Holovko, M. Patsahan, T. Kravtsiv, I. di Caprio, D. |
| citation_txt | Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation / M. Holovko, T. Patsahan, I. Kravtsiv, D. di Caprio // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13608: 1–11. — Бібліогр.: 27 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | The mean field approximation is formulated within the framework of the density field theory to study the properties of a Maier-Saupe nematogenic fluid near a hard wall. The density and the order parameter profiles are obtained using the analytical expressions derived in the linearized mean field approximation. The temperature dependencies of the contact values of the density and order parameter profiles are analyzed in detail. To estimate a validity of the applied approximations, the obtained theoretical results are compared with the original computer simulation data.
В рамках теорiї поля густини сформульовано наближення середнього поля для дослiдження властивостей
нематогенного плину Майєра-Заупе бiля твердої поверхнi. У лiнеаризованому наближеннi середнього
поля розраховано аналiтичнi вирази для профiлю густини та профiлю параметра порядку. Детально проаналiзовано залежнiсть контактних значень профiлiв густини та параметра порядку вiд температури. Для
оцiнки застосовностi використаних наближень проведено порiвняння отриманих теоретичних результатiв з оригiнальними даними комп’ютерного моделювання.
|
| first_indexed | 2025-12-07T16:43:54Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 1, 13608: 1–11
DOI: 10.5488/CMP.19.13608
http://www.icmp.lviv.ua/journal
Maier-Saupe nematogenic fluid with isotropic
Yukawa repulsion at a hard wall:
Mean field approximation
M. Holovko1 , T. Patsahan1∗, I. Kravtsiv1, D. di Caprio2
1 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
2 Institut de Recherche de Chimie Paris, CNRS — Chimie ParisTech,
11 rue Pierre et Marie Curie, 75005 Paris, France
Received December 31, 2015, in final form January 12, 2016
The mean field approximation is formulated within the framework of the density field theory to study the prop-
erties of a Maier-Saupe nematogenic fluid near a hard wall. The density and the order parameter profiles are
obtained using the analytical expressions derived in the linearized mean field approximation. The temperature
dependencies of the contact values of the density and order parameter profiles are analyzed in detail. To esti-
mate a validity of the applied approximations, the obtained theoretical results are compared with the original
computer simulation data.
Key words: Maier-Saupe nematogenic fluid, field theory, interface, hard wall, contact theorem,
Yukawa potential
PACS: 61.30.Gd, 68.08.-p, 61.30.Hn
1. Introduction
It is a great pleasure and a big honor for us to contribute this paper to the festschrift dedicated to
Professor Stefan Sokołowski. Stefan is a well-known expert on the modelling of physico-chemical proper-
ties of complex fluids such as chemically reacting fluids in disordered porous media [1, 2], studies of the
isotropic-nematic phase transition in confined nematic fluids [3], modelling of the properties of fluids in
pores with walls decorated with tethered polymer brushes [4], treatment of phase behavior in confined
functional colloids [5] and many other complex fluids in complex confinements.
In this paper, we study the influence of the surface on a nematic fluid near a hard wall. Due to ori-
entational ordering, such systems have many unique properties which are very important in the display
industry. The anchoring phenomena are among them, according to which the surface induces a specific
orientation of the nematic director with respect to the surface [6]. In order to understand the connec-
tion between the anchoring phenomena and the interaction between nematic molecules and the surface,
there the Henderson-Abraham-Barker (HAB) approach previously developed for isotropic fluids at the
wall [7] has been employed [8, 9]. In this approach, the distribution of the fluid near the wall is described
by the wall-particle Ornstein-Zernike (OZ) equation with the fluid distribution function in the bulk calcu-
lated in the mean spherical approximation (MSA) [10]. Using this approach, it is possible to evaluate the
role of an orientation-dependent interaction of nematic molecules with the surface in the anchoring phe-
nomena. However, in the MSA, the HAB approach does not correctly take into account the contribution
from long-range intermolecular interactions. As a result, it does not satisfy an exact relation known as
the contact theorem, which was formulated in [11, 12] for isotropic fluids near a wall, and recently in [13]
it was reformulated for anisotropic fluids near a hard wall. According to this theorem, the contact value
∗E-mail: tarpa@icmp.lviv.ua
©M. Holovko, T. Patsahan, I. Kravtsiv, D. di Caprio, 2016 13608-1
http://dx.doi.org/10.5488/CMP.19.13608
http://www.icmp.lviv.ua/journal
M. Holovko et al.
of the density profile of a fluid near a hard wall is determined by the pressure of the fluid in the bulk. In
[13], the contact theorem was also formulated for the order parameter profile.
An alternative way of describing fluids at a hard wall was developed within the framework of the
density field theory. In this theory, the contributions from the mean field and from fluctuations are sep-
arated. The theory was successfully applied to ionic fluids at a hard wall [14–17] and to simple fluids
with Yukawa-type interactions near a hard wall [18, 19]. It was shown that the mean field treatment of a
Yukawa fluid at a hard wall reduces to the solution of a non-linear differential equation for the density
profile, while the treatment of fluctuations reduces to the OZ equation with the Riemann boundary con-
dition [20]. The density field theory was applied to the description of bulk properties of a nematic fluid
in [21, 22]. The application of the density field theory to the description of a nematic fluid at a hard wall
was initiated in [23].
In this paper, we use the mean field approximation to investigate the effect of a hard wall on the
properties of a nematic fluid.We demonstrate the principal difference between the behavior of the order
parameter profile obtained in the mean field approximation and its linearized version. In order to check
the validity of both approaches we compare the obtained theoretical results with computer simulations
data.
2. Theory
In this paper, we consider Maier-Saupe (MS) nematogenic fluid model [24, 25] as one of the simplest
models that accounts for the isotropic-nematic phase transition. For simplification, we consider a fluid of
point uniaxial nematogens interacting through the pair potential
ν(r12,Ω1Ω2) = ν0(r12)+ν2(r12)P2(cosθ12), (1)
where the first term ν0(r12) = (A0/r12)exp(−α0r12) describes isotropic repulsion and the second term
with ν2(r12) = (A2/r12) exp(−α2r12) describes anisotropic attraction between particles (A0 > 0, A2 < 0),
r12 denotes the distance between particles 1 and 2, Ω =
(
θ,φ
)
are orientations of particles, P2(cosθ12) =
(3cos2θ12 −1)/2 is the second order Legendre polynomial of the relative orientation θ12.
It is necessary, in numerical calculations, to cut-off the potential ν(r12,Ω1Ω2) at some finite distance
and due to this in expression (1), ν0(r ) and ν2(r ) are replaced by ν̃i (r )= νi (r ) for r É rc and ν̃i (r )= 0 for
r > rc, where i = 0, 2 and rc is the cut-off radius.
Within the field-theoretical formalism, the Hamiltonian is a functional of the density field and can be
written as a sum of the entropic and the interaction terms
βH [ρ(r,Ω)] =
∫
ρ(r,Ω)
{
ln
[
ρ(r,Ω)ΛRΛ
3
T
]
−1
}
drdΩ
+
β
2
∫
ν(r12,Ω1Ω2)ρ(r1,Ω1)ρ(r2,Ω2)dr1dr2dΩ1dΩ2, (2)
where β = 1/kBT is the inverse temperature, dΩ = (1/4π)sinθdθdφ is the normalized angle element,
ρ(r,Ω) is particle density per angle such that
∫
ρ(r,Ω)dΩ= ρ(r), ΛT is the thermal de Broglie wavelength
of the molecules, the quantity Λ
−1
R is the rotational partition function for a single molecule [26].
2.1. Mean field approximation
In this paper, we restrict our consideration to the mean field (MF) approximation which is the lowest
order approximation for the partition function. In the canonical formalism, it corresponds to fixing the
Lagrange parameter λ such that the following relation is true for the singlet distribution function
δβH [ρ(r,Ω)]
δρ(r,Ω)
∣
∣
∣
ρMF
=λ. (3)
As a result, we have
ρ(r1,Ω1) = ρbulk(Ω1)exp
{
−β
∫
ν(r12,Ω1Ω2)
[
ρ(r2,Ω2)−ρbulk(Ω2)
]
dr2dΩ2
}
, (4)
13608-2
Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall
where
ρbulk(Ω) = ρb
exp
[
−(κ2
2Sb/α2
2)P2(cosθ)
]
1
∫
0
dcosθexp
[
−(κ2
2Sb/α2
2)P2(cosθ)
]
(5)
is the singlet distribution function for the bulk nematic fluid in the MF approximation, defined within the
framework of the Maier-Saupe theory [24, 25], κ2
2 = 4πρbβA2, ρb is the bulk value of the fluid density,
Sb = (1/ρb)
∫1
0 P2(cosθ)ρbulk(Ω)dcosθ is the bulk value of the orientational order parameter.
After integration with respect to orientation Ω2, we obtain
ρ(r1,Ω1n ,Ωwn)
ρbulk(Ω1)
= exp
{
−
[
V0(r1,Ωwn )−V b
0
]
−
1
p
5
∑
m
Y2m(Ω1n )
[
V2m(r1,Ωwn)−V b
2m
]
}
, (6)
whereΩwn denotes the angle between the nematic director and the surface, and themean field potentials
V0(r1,Ωwn) =β
∫
ν0(r12)ρ(r2,Ωwn )dr2, (7)
V2m(r1,Ωwn) =β
∫
ν2(r12)S2m(r2,Ωwn)dr2. (8)
The bulk values of these potentials are V b
0 = κ2
0/α2
0, V b
20 = κ2
2Sb/α2
2, V b
2m = 0 for m , 0, where κ2
0 =
4πρbβA0,
ρ(r,Ωwn ) =
∫
ρ(r,Ω1n ,Ωwn )dΩ1n (9)
is the density profile. The quantities
S2m(r,Ωwn) =
1
p
5
∫
ρ(r,Ω1n ,Ωwn )Y2m(Ω1n)dΩ1n = ρ(r,Ωwn)S∗
2m(r,Ωwn), (10)
where S∗
2m(r,Ωwn) are the order parameter profiles. Far from the wall we have S∗
20(r,Ωwn) → Sb,
S∗
2m(r,Ωwn) → 0 for m , 0. Simple calculations show that, in order to take into account the cut-off ra-
dius rc, one should substitute the quantities κ2
i
by κ̃2
i
such that
κ̃2
i = 4πρβ
rc
∫
0
νi (r )r 2dr = κ2
i
[
1−exp(−αi rc)−αi rc exp(−αi rc)
]
. (11)
2.2. Linearized MF approximation
The gradient of equation (6) gives
1
ρ(r,Ω1n ,Ωwn )
∇∇∇ρ(r,Ω1n ,Ωwn ) = E0(r,Ωwn)+
1
p
5
∑
m
Y2m(Ω1n)E2m(r,Ωwn), (12)
where we define an equivalent of the electric field as
E0(r,Ωwn) ≡−∇∇∇V0(r,Ωwn), E2m (r,Ωwn)≡−∇∇∇V2m(r,Ωwn). (13)
According to the properties of the Yukawa potential we can write
(
△−α2
0
)
V0(r,Ωwn) =−4πβA0ρ(r,Ωwn), (14)
(
△−α2
2
)
V2m(r,Ωwn) =−4πβA2S2m(r,Ωwn). (15)
Due to translational invariance parallel to the wall, the functions considered depend only on the distance
z to the wall. Equations (10)–(15) make a set of six differential equations for the unknown functions
13608-3
M. Holovko et al.
ρ(r,Ω1n ,Ωwn), S2m(r,Ωwn), E0(r,Ωwn), E2m(r,Ωwn), V0(r,Ωwn), V2m(r,Ωwn). We note that in the case
when the director is oriented perpendicularly to the wall, Ωwn = 0, the singlet distribution function is
axially symmetric. Consequently, the equations considered will retain only the terms with m = 0. In this
paper, we will restrict our further investigation to this special case.
As was shown in [23], the differential equations obtained can be solved analytically in the linear
approximation for the expression (6)
ρ′(z,Ω) = [E0(z)+E20(z)P2(cosθ)]ρbulk(Ω), (16)
where the prime denotes derivative by z.
The resulting solutions of the linearized profile are as follows:
ρ(z)
ρb
= 1−
λ2
0 −α2
2 −
1
5κ
2
2
(
〈Y 2
20〉Ω− 〈Y20〉2
Ω
)
κ2
2 Sb
B1 e−λ0z
−
λ2
2 −α2
2 −
1
5
κ2
2
(
〈Y 2
20〉Ω− 〈Y20〉2
Ω
)
κ2
2 Sb
B2 e−λ2z , (17)
S20(z)
ρb Sb
= 1−
(
λ2
0 −α2
2
)
κ2
2 Sb
B1 e−λ0z −
(
λ2
2 −α2
2
)
κ2
2 Sb
B2 e−λ2z , (18)
where
B1 =
κ2
2 Sb
2
(
λ2
0 −λ2
2
)
[
−
κ2
0
α2
0
+
λ2
2 −α2
2 − (κ2
2/5)〈Y 2
20〉Ω
α2
2
]
, B2 =−
κ2
2 Sb
2α2
2
−B1, (19)
〈Y k
20〉Ω = (1/ρb)
1
∫
0
Y k
20(Ω)ρbulk(Ω)dcosθ. (20)
Parameters λ0 and λ2
λ2
0,2 =
1
2
{
κ2
0 +α2
0 +κ2
2〈P
2
2 (cosθ)〉+α2
2 ±
√
[
κ2
0 +α2
0 −κ2
2〈P
2
2 (cosθ)〉−α2
2
]2 +4κ2
0κ
2
2 S2
b
}
(21)
are identical to the parameters found in the bulk phase when Gaussian fluctuations are taken into ac-
count [22] and characterize a decay of the isotropic repulsive and the anisotropic attractive interactions,
respectively.
Hereafter, the approach presented in this subsection is referred to as the linearized mean field (LMF)
approximation. The expressions for ρ(z) and S∗
20(z) obtained within this approximation correspond to
the case of infinite cut-off radius rc →∞.
2.3. Contact theorem
As it was shown in [13], the density and order parameter profiles satisfy some exact relations known
as the contact theorems. According to these relations in the absence of wall-particle interactions, the
contact values of the density profile ρ(z = 0) and of the order parameter profile S20(z = 0) do not depend
on the angle Ωwn and are equal to
ρ(z = 0) =β
∫
dΩ1nP (Ω1n) =βP, (22)
S20(z = 0) =β
∫
dΩ1nP2(cosθ)P (Ω1n ), (23)
where P is the bulk pressure and P (Ω1n) can be treated as the bulk partial pressure for molecules with a
given orientation Ω1n .
13608-4
Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall
In the MF approximation for the model under consideration, relations (22) and (23) give
ρ(0+)
ρb
= 1+
κ2
0
2α2
0
+
κ2
2
2α2
2
S2
b, (24)
S20(0+)
Sbρb
= 1+
κ2
0
2α2
0
+
κ2
2
2α2
2
〈P 2
2 (cosθ)〉. (25)
These relations are used as boundary conditions in the solution of differential equations of the LMF
approximation. In order to take into account the cutoff distance rc in relations (24)–(25), we should change
κ2
i
to κ̃2
i
given by equation (11). However, we should note the principal difference between the exact
results (22)–(23) and results (24)–(25) of the MF approximation. In paper [21], an invariant
α2
0
2κ2
0
V 2
0 (z,Ωwn)−
1
2κ2
0
E 2
0(z,Ωwn)+
∑
m
[
α2
2
2κ2
2
V 2
2m(z,Ωwn )−
1
2κ2
2
E 2
2m(z,Ωwn)
]
(26)
was found which was used to prove the contact theorem for the MF density profile of a nematogenic fluid
in form (24). However, no similar proof of the contact theorem for the MF order parameter profile in
form (25) exists.
3. Numerical calculation details
In order to verify the theoretical approaches presented in the previous section, a series of numerical
calculations were carried out. To this end, for the model pair potential (1), the following parameters were
chosen: A0/|A2| = 3.0 and α0/α2 = 1.6. It should be noted that all quantities marked by a star in our pa-
per are considered as non-dimensional. For instance, all distances are reduced as r ∗ = rα2 or z∗ = zα2,
densities are reduced as ρ∗ = ρ/α3
2 and temperature as T ∗ = kBT /(|A2|α2). The potential (1) is character-
ized by a rather soft repulsive part (isotropic contribution) and a small attractive part dependent on the
relative orientation between a pair of fluid particles (anisotropic contribution). It is worth noting that the
considered fluid is mostly a repulsive one, and a small attractive contribution affects mainly an orienta-
tional properties of the fluid, at least at the conditions used in our study. In particular, we consider the
fluid at the density ρ∗
b
= 1.0, and the temperature interval T ∗ = 0.5−3.5 is chosen. We have found that, at
these temperatures, the considered fluid is beyond its vapour-liquid phase transition region. Therefore,
only a nematic-isotropic phase transition can be expected in our case.
To obtain a numerical solution of the integral equation (6) used in the MF approximation, the Picard
iterative method was applied. The problem was considered in the cylindrical coordinates, which are set
along z-axis normal to the wall surface. The integrations over z were performed using trapezoidal rule
with a step ∆z = 0.02/α2 , while all integrations over r were done analytically. A step of integration over
cos(θ) was chosen as 0.0025. To take into account the confinement, we consider a fluid between two hard
walls at a distance Lz = 36/α2 to each other. Two cut-off radii rc = 6.0/α2 and 12.0/α2 are used in our
study. The chosen distance Lz appears to be sufficient to get a bulk-like region of a fluid in the middle
between the two walls. The presence of the hard walls is introduced by the boundary conditions ρ(z) = 0
if z < 0 or z > Lz . Equation (6) is solved in combination with equation (10) leading to density ρ(z) and
order parameter S∗
20(z) profiles. The precision of this solution expressed in terms of standard deviation
is 10−5.
The bulk order parameter Sb is used both in the MF and LMF calculations. To obtain this quantity,
the integral equation (5) was applied. It was also solved numerically by the iterative method, but here
with a precision 10−12 and with a step of integration over cos(θ) taken equal to 0.00125. The cut-off radii
rc = 6.0/α2 and 12.0/α2 were used to obtain Sb as well.
We compare the numerical results calculated from MF and LMF approaches with Monte-Carlo (MC)
simulation results. For this purpose, a series of MC simulations in canonical ensemble [27] were per-
formed to obtain Sb, ρ(z) and S∗
20(z). A system of Np fluid particles interacting with the pair potential (1)
were placed into a rectangular box of a size Lx×Ly×Lz , where Lx = Ly = 24/α2 and Lz = 4rc if rc = 6.0/α2
and Lz = 3rc if rc = 12.0/α2 . For Lz taken in our simulations, a bulk-like region in the middle of the box
13608-5
M. Holovko et al.
is observable up to the temperature T ∗ = 2.2. Since the bulk fluid density is set to ρ∗
b
= 1.0, a number
of fluid particles Np is set to 13824 for the case of rc = 6.0/α2 and 20736 for the case of rc = 12.0/α2 .
The simulations were carried out with the periodical boundary conditions applied in three dimensions
in the case of a bulk fluid, and in X and Y dimensions in the case of a fluid between two hard walls.
Each simulation procedure was performed at a constant temperature and volume and consisted of three
stages: 1) equilibration of a fluid in a strong field applied along Z -axis to give the fluid particles a pref-
erential orientation, which was normal to the wall surfaces; 2) equilibration of the system obtained at
the previous stage with the field switched off; 3) production of the necessary characteristics for a system
obtained at the second stage with the field switched off. A criterium defining an equilibrated system is a
stabilization of the total order parameter in the system, i.e., when the order parameter fluctuates around
one average value during an essential number of MC steps, usually it was over 20 000 steps at least. It
should be noted that in our simulations, one MC step corresponds to Np trial translational or rotational
movements.
4. Results and discussion
4.1. Bulk fluid
The order parameter of the bulk MS fluid is calculated at temperatures in the range of T ∗ = 0.5−3.5
using the MF approximation with the different cut-off radii (figure 1). We have checked at which r ∗
c
the results tend to the case of r ∗
c → ∞ and have tested what the effect of the cut-off radius in general
is. This information is valuable to make a comparison with the computer simulations in which the cut-
off radius is used. As can be seen in the case of r ∗
c = 6.0, the MF approximation leads to the values of
the order parameter lower than in the cases of r ∗
c = 12.0 and r ∗
c → ∞ for the whole stable nematic
region up to the critical region of the nematic-isotropic phase transition, which appears at Sb < 0.443 [22]
(figure 1, left-hand panel). Also, it is observed that the results for r ∗
c = 12.0 totally coincide with the case
of r ∗
c →∞ (figure 1, right-hand panel). The values of the order parameter obtained from the simulations
are systematically lower than in the MF approximations (figure 1, triangle symbols).
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
MF (r*
c )
MF (r*
c = 6.0)
MC (r*
c = 6.0)S b
T*
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
MF (r*
c )
MF (r*
c = 12.0)
MC (r*
c= 12.0)S b
T*
Figure 1. Temperature dependence of the order parameter Sb of a bulk MS fluid. The results obtained in
the MF approximation as well as with the use of the Monte-Carlo (MC) simulation method.
4.2. Density and order parameter profiles
A series of density profiles are calculated for the MS fluid near a hard wall at different temperatures
in the range T ∗ = 0.5−3.5. For this purpose, the LMF and MF approximations are applied and compared
with the corresponding simulation results. In figure 2, we present two selected results at temperatures
T ∗ = 1.3 and 2.0. The cut-off radius r ∗
c = 12.0 is used in the MF approximation and in the computer
13608-6
Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall
0 1 2 3 4 5
0.5
1.0
1.5
2.0
2.5
3.0
LMF (r*
c )
MF (r*
c = 12.0)
MC (r*
c = 12.0)
(z
)
z*
T* = 1.3
0.0 0.1 0.2 0.3 0.4
0.5
1.0
1.5
2.0
2.5
0 1 2 3 4 5
0.5
1.0
1.5
2.0
2.5
3.0
LMF (r*
c )
MF (r*
c = 12.0)
MC (r*
c = 12.0)
(z
)
z*
T* = 2.0
0.0 0.1 0.2 0.3 0.4
0.5
1.0
1.5
2.0
2.5
Figure 2. Density profiles of the MS fluid near a hard wall obtained in the LMF and MF approximations
as well as with the use of the Monte-Carlo (MC) simulation method. The results were obtained at temper-
ature T∗ = 1.3 (left-hand panel) and T∗ = 2.0 (right-hand panel).
simulations. As has been shown above, this cut-off should be long enough to give results comparable to
the case r ∗
c →∞. It is observed that all presented profiles have the same qualitative behavior, i.e., they
have a distinct highmaximum at the contact with the wall, then aminimum appears around z∗ = 0.5−1.0
and at z∗ > 5 one can see a convergence of ρ∗(z) to ρ∗
b
when the bulk-like region is reached. We note that
the contact values obtained in the MF and LMF practically coincide— the cut off correction is negligible,
while for ρ∗(0) obtained from the simulations, we observe some small difference. The contact value of
the obtained density profiles will be discussed more in detail later on. For this moment, we focus on ρ∗(z)
at distances z∗ > 0 and on comparison of the theoretical approaches with the computer simulations. One
can see that at small distances z∗, the LMF approximation agrees with the simulations better than theMF,
while the MF approximation describes ρ∗(z) better at distances z∗ around the minimum of ρ∗(z) and
larger. Moreover, at all temperatures considered in our study, the z∗-position of the minimum of ρ∗(z) is
very close in the MF approximation to that in the computer simulations, while in the LMF approximation,
the z∗-position of the minimum is notably shifted towards the higher values.
A different situation is observed for the order parameter profile S∗
20(z) of the MS fluid near a hard
wall, which is presented in figure 3 as a normalized quantity S̃(z) = S∗
20(z)/Sb. We take such a presenta-
tion because of an essential deviation of the theory from the simulations results for bulk order parame-
0 2 4 6 8 10
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
LMF (r*
c )
MF (r*
c = 12.0)
MC (r*
c = 12.0)
S 20
(z
)
z*
T* = 1.3
~
0 2 4 6 8 10
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
LMF (r*
c )
MF (r*
c = 12.0)
MC (r*
c = 12.0)
S 20
(z
)
z*
T* = 2.0
~
Figure 3. Profiles of the normalized order parameter S∗
20(z)/Sb of the MS fluid near a hard wall obtained
in the LMF and MF approximations and with the use of the MC simulation method. The results were
obtained at temperature T∗ = 1.3 (left-hand panel) and 2.0 (right-hand panel).
13608-7
M. Holovko et al.
ters (figure 1). Using the normalization we can consider all profiles on the same plot. Nevertheless, even
with this normalization, one can observe an important quantitative and qualitative difference between
not only the theory and simulations, but also between the MF and LMF approximations (figure 3). In the
LMF approximation, a minimum of S̃(z) is found at z∗ = 0.54 and z∗ = 0.58 at T ∗ = 1.3 and T ∗ = 2.0,
respectively, while in the MF approximation no minimum has been noticed at all. At the same time, the
simulation results do not give any evidence of a S̃(z) minimum either. In this context, the MF results are
similar to those obtained from simulations. However, all the considered approaches give completely dif-
ferent contact values of S̃(z). The contact value of the order parameter obtained from the simulation is
the lowest one, then an essentially larger S̃(z) is given by the MF approximation and the largest value of
S̃(0) close to 1.0 is calculated from the LMF approximation. In the bulk region (far enough from the wall),
all the profiles converge to 1.0 as expected. The only difference is found in the rates of this convergence,
which is lower in the MF approximation than in the LMF approximation, and in the simulations it is the
lowest one. Also, it is worth noting that the rate of convergence of S∗
20(z) to Sb [i.e., S̃(z) to 1.0] decreases
with the temperature.
Before discussing the contact values obtained in our study we will try to understand the behavior of
the order parameter profile S∗
20(z) and how it relates to the density profile ρ∗(z). First of all, we have
noticed that the order parameter of fluid particles next to the wall (z∗ → 0) is smaller than the bulk order
parameter Sb. However, the density profiles (figure 2) for the same z∗ is higher than the bulk density.
From the knowledge of the bulk, for which the higher density leads to the higher order parameter, one
can expect that the order parameter near the wall is also higher than Sb. Therefore, we encounter a con-
tradiction, since both the MF approximations and simulation predict the values of S∗
20(z) smaller than Sb
everywhere except the bulk-like region where S∗
20(z) is equal to Sb (figure 3). Apparently we are dealing
with two competing effects: densification of fluid particles near the wall, which should increase S∗
20(z) at
small z∗ and the absence of fluid particles beyond the wall, which should decrease S∗
20(z). Following the
obtained results, one can conclude that the latter effect is more essential for the system considered in our
study.
4.3. Contact values
The density and order parameter profiles of a MS fluid near a hard wall are used to calculate the cor-
responding contact values ρ∗(0) and S∗
20(0) as functions of the temperature. These contact values can be
calculated from the expressions of the contact theorem (CT) (22) and (23), which in the MF approximation
reduce to the relations (24) and (25), respectively. In the LMF approximation, the contact values ρ∗(0) and
S∗
20(0) satisfy the relations (24) and (25) automatically due to the boundary conditions applied in the so-
lution of the corresponding differential equation. In the MF approximation, the situation is different for
ρ∗(0) and S∗
20(0). As it was shown in [23], the contact value for the density profile ρ∗(0) satisfies the rela-
tion (24), although it has not been proven that the contact value of the order parameter S∗
20(0) obtained
in the MF approximation should satisfy the relation (25).
First we consider the temperature dependencies of the contact value of density profiles obtained
with the different cut-off radii r ∗
c = 6.0 and 12.0 (figure 4). As can be seen, the CT leads to the same
result as the MF approximation. A small difference between the LMF and the MF approximation (or the
CT) appears when the cut-off radius is equal to r ∗
c = 6.0 (figure 4, left-hand panel). If the cut-off radius
is increased to r ∗
c = 12.0, equivalent results are obtained in all of the approximations (figure 4, right-
hand panel). At the same time, the agreement between the theoretical approaches and the simulations is
mostly qualitative. We would like to draw attention to the non-monotonous behavior of the contact value
ρ∗(0) with a distinct minimum observed both in the simulations and in the theoretical predictions. To
understand this interesting effect, we should analyze the temperature dependence of the density contact
value more in detail.
As it is seen in figure 4, at low temperatures, ρ∗(0) is large and lowers as the temperature increases.
This is related to the reduction of the repulsive contribution of the pair potential (1). Since in our model
a soft repulsive interaction is used, it becomes weaker if the temperature increases. It should be noted
that the attractive part of the pair potential in our model is rather small and mainly affects orientational
properties of fluid particles. At the same time, a relative orientation can indirectly strengthen the fluid-
fluid repulsion contribution by reducing the attractive interaction term. Therefore, at some point in the
13608-8
Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall
0.5 1.0 1.5 2.0 2.5 3.0 3.5
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
LMF (r*
c )
CT (r*
c = 6.0)
MF (r*
c = 6.0)
MC (r*
c = 6.0)
(0
)
T*
0.5 1.0 1.5 2.0 2.5 3.0 3.5
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
4.2
LMF (r*
c )
CT (r*
c = 12.0)
MF (r*
c = 12.0)
MC (r*
c = 12.0)
(0
)
T*
Figure 4. Contact value of the density profile as a function of the temperature obtained by different ap-
proaches. Two cut-off radii are used: r∗c = 6.0 (left panel) and r∗c = 12.0 (right-hand panel).
nematic phase, when the order parameter remains sufficiently small, the repulsion becomes stronger
causing an increase of the contact value ρ∗(0). This effect is observed in figure 4 at temperature T ∗ =
1.885, where ρ∗(0) reaches its minimum and starts to increase rapidly until the fluid becomes totally
isotropic [S∗
20(z) = 0]. In the isotropic phase, the fluid particles are totally orientationally disordered, and
a further temperature increase leads to the weakening of the repulsion. Thus, a continuous decrease of
ρ∗(0) is obtained at high temperatures. It should be noted that the explanation presented here concerns
solely, models with a pair potential consisting of a soft-core term combined with a Maier-Saupe attractive
potential. In the case of a hard-core type of repulsive interaction, the temperature dependence can be
opposite and the effect of orientational ordering can be completely different.
While the contact value of the density profile is rather understandable, the behavior of the contact
value of the order parameter profile is not so clear. First of all, as it has been already shown, there is
an essential inconsistency between the MF and LMF approximations (figure 3). There is also a problem if
one compares the contact values S∗
20(0) obtained in theMF and LMF approximations and those calculated
from the CT theorem. As can be seen in figure 5, the MF results significantly differ from those of the LMF
and the CT. A perfect agreement of the LMF and the CT appears due to the definition of the boundary
conditions used in the LMF approximation, which are taken to fit the CT theorem (25). A deviation of the
CT from the LMF is seen only for the case of r ∗
c = 6.0 chosen in the CT (figure 5, left-hand panel). For
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
LMF (r*
c )
CT (r*
c = 6.0)
MF (r*
c = 6.0)
MC (r*
c = 6.0)
S* 20
(0
)
T*
0.5 1.0 1.5 2.0 2.5 3.0 3.5
0.0
0.2
0.4
0.6
0.8
1.0
LMF (rc )
CT (rc = 12.0)
MF (rc = 12.0)
MC (rc = 12.0)
S* 20
(0
)
T*
Figure 5. Contact value of the order parameter profile as a function of the temperature obtained by dif-
ferent approaches. Two cut-off radii are used: r∗c = 6.0 (left-hand panel) and r∗c = 12.0 (right-hand panel).
13608-9
M. Holovko et al.
r ∗
c = 12.0, the contact values of the order parameter of a MS fluid obtained from the CT and the LMF
totally coincide. At the same time, the MF approximation gives much lower S∗
20(0), although it is closer
to the simulation results than the CT and LMF. We assume that the difference between the CT and the
MF approaches is related to a poor description of the bulk partial pressure P (Ω1n). In order to improve
the results obtained from the CT (or the LMF approximation), one should take into account higher order
terms, for instance the Gaussian fluctuations [22]. Also, it should be noted that in contrast to the CT and
LMF approximation, S∗
20(0) obtained from the MF approximation and the simulations decays to zero
before the critical region is reached. Thus, the convex part of S∗
20(0) is absent for them. It means that in
the critical region there is no orientational ordering of fluid particles at a contact with the wall.
5. Conclusions
Within the framework of the density field theory, we have formulated the mean field (MF) approxi-
mation as a starting point for the theoretical description of a nematic fluid at a hard wall. Using the devel-
oped approach we have investigated the density and order parameter profiles of a confined Maier-Saupe
nematogenic fluid with an isotropic Yukawa-like repulsion. Theoretical predictions have been compared
with analytical results obtained within the framework of the linearized mean field (LMF) approxima-
tion [23]. For the density profile, the results obtained in the MF and LMF approximations are in qual-
itative agreement with computer simulations data. For the order parameter profile, the results of the
MF and LMF approximations have qualitatively different behaviors. In the LMF approximation, a min-
imum of the order parameter profile is present, while in the MF approximation no minimum has been
observed. The results of computer simulations do not give any evidence of the existence of a minimum in
the z-dependence of the order parameter profile either. In this context, the MF and computer simulations
results are qualitatively similar. This contradiction in the description of the density and order parameter
profiles within the framework of the LMF approximation is connected with the problem of the respective
contact theorems in forms (24) and (25) which are used as boundary conditions in the solution of the
corresponding system of differential equations. As it was shown in reference [23], the contact value of
the density profile ρ(0) satisfies the relation (24) but the validity of the relation (25) for the contact value
of the order parameter S20(0) is not evident. Moreover, the comparison with the results of computer sim-
ulations in figure 5 shows that relation (25) is probably incorrect. As we can see from figure 5, the LMF
approximation can give a better result if we change the boundary condition for the order parameter pro-
file by correcting the contact value for S(0) obtained within the framework of the MF approximation. We
hope that the considered problem can be better understood by going beyond the MF approximation and
including a contribution from fluctuations.
The temperature dependencies of the contact values of the density and order parameter profiles have
been analyzed more in detail. We have found a non-monotonous behavior of the density contact value
ρ(0) as a function of the temperature with a distinct minimum observed in both theoretical predictions
and computer simulations. This non-monotonous temperature dependence of ρ(0) is explained by the
competition of the soft isotropic repulsive and the soft anisotropic attractive contributions. For the con-
tact value of the order parameter, we have observed a monotonous decrease with an increase of tem-
perature. Both the simulations and the MF results indicate that there is no orientational ordering of fluid
particles at the contact with the wall when the fluid is in the critical region.
The results presented in this paper have been obtained within the framework of the MF approxi-
mation. We note that the agreement between theoretical predictions and computer simulation data is
mostly qualitative. For a better theoretical description one should take fluctuations into account. For the
bulk properties of the model under consideration, the influence of the contribution from fluctuations
was already discussed in reference [22]. It was shown that the singlet distribution function reduces to
the form (5) with a change of κ2
2 to κ2
2 t , where t = 1−βA0α
2
2/(λ0 +λ2). It was shown that the tempera-
ture trend of deviation between the order parameter Sb calculated with fluctuations included and theMF
value is the same as that between computer simulations and the MF value presented in figure 1. In the
next paper we plan to include the contribution from fluctuations in the description of a MS nematogenic
fluid at a hard wall similar to the way it was done for isotropic Yukawa fluids [18, 19].
13608-10
Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall
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Нематогенний плин Майєра-Заупе з iзотропним
юкавiвським вiдштовхуванням бiля твердої поверхнi:
наближення середнього поля
М. Головко1 , Т. Пацаган1, I. Кравцiв1, Д. дi Капрiо2
1 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
2 Лабораторiя електрохiмiї, хiмiї поверхонь i енергетичного моделювання, вiддiлення хiмiї вищої
нацiональної школи ПарiТех, вул. П. i М. Кюрi, 11, 75005 Париж, Францiя
В рамках теорiї поля густини сформульовано наближення середнього поля для дослiдження властивостей
нематогенного плину Майєра-Заупе бiля твердої поверхнi. У лiнеаризованому наближеннi середнього
поля розраховано аналiтичнi вирази для профiлю густини та профiлю параметра порядку. Детально про-
аналiзовано залежнiсть контактних значень профiлiв густини та параметра порядку вiд температури. Для
оцiнки застосовностi використаних наближень проведено порiвняння отриманих теоретичних результа-
тiв з оригiнальними даними комп’ютерного моделювання.
Ключовi слова: нематогенний плин Майєра-Заупе, теорiя поля, поверхня, контактна теорема,
потенцiал Юкави
13608-11
http://dx.doi.org/10.1021/jp961443l
http://dx.doi.org/10.1063/1.473042
http://dx.doi.org/10.1103/PhysRevE.59.4161
http://dx.doi.org/10.1063/1.3592562
http://dx.doi.org/10.1021/jp503826p
http://dx.doi.org/10.1088/0034-4885/54/3/002
http://dx.doi.org/10.1080/00268977600101021
http://dx.doi.org/10.1103/PhysRevLett.92.185508
http://dx.doi.org/10.1063/1.1825373
http://dx.doi.org/10.1016/S0167-7322(99)00098-7
http://dx.doi.org/10.1016/S0022-0728(79)80459-3
http://dx.doi.org/10.1063/1.2137707
http://dx.doi.org/10.1063/1.4905239
http://dx.doi.org/10.1080/0026897031000154293
http://dx.doi.org/10.1063/1.476286
http://dx.doi.org/10.1016/j.jelechem.2005.02.008
http://dx.doi.org/10.1021/jp0737395
http://dx.doi.org/10.1080/00268976.2010.547524
http://dx.doi.org/10.1063/1.4921242
http://dx.doi.org/10.5488/CMP.14.33605
http://dx.doi.org/10.1080/00268976.2012.762615
http://dx.doi.org/10.5488/CMP.16.14002
http://dx.doi.org/10.1515/zna-1959-1005
http://dx.doi.org/10.1515/zna-1960-0401
Introduction
Theory
Mean field approximation
Linearized MF approximation
Contact theorem
Numerical calculation details
Results and discussion
Bulk fluid
Density and order parameter profiles
Contact values
Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-155778 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T16:43:54Z |
| publishDate | 2016 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Holovko, M. Patsahan, T. Kravtsiv, I. di Caprio, D. 2019-06-17T12:21:00Z 2019-06-17T12:21:00Z 2016 Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation / M. Holovko, T. Patsahan, I. Kravtsiv, D. di Caprio // Condensed Matter Physics. — 2016. — Т. 19, № 1. — С. 13608: 1–11. — Бібліогр.: 27 назв. — англ. 1607-324X PACS: 61.30.Gd, 68.08.-p, 61.30.Hn DOI:10.5488/CMP.19.13608 arXiv:1603.02186 https://nasplib.isofts.kiev.ua/handle/123456789/155778 The mean field approximation is formulated within the framework of the density field theory to study the properties of a Maier-Saupe nematogenic fluid near a hard wall. The density and the order parameter profiles are obtained using the analytical expressions derived in the linearized mean field approximation. The temperature dependencies of the contact values of the density and order parameter profiles are analyzed in detail. To estimate a validity of the applied approximations, the obtained theoretical results are compared with the original computer simulation data. В рамках теорiї поля густини сформульовано наближення середнього поля для дослiдження властивостей
 нематогенного плину Майєра-Заупе бiля твердої поверхнi. У лiнеаризованому наближеннi середнього
 поля розраховано аналiтичнi вирази для профiлю густини та профiлю параметра порядку. Детально проаналiзовано залежнiсть контактних значень профiлiв густини та параметра порядку вiд температури. Для
 оцiнки застосовностi використаних наближень проведено порiвняння отриманих теоретичних результатiв з оригiнальними даними комп’ютерного моделювання. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation Нематогенний плин Майєра-Заупе з iзотропним юкавiвським вiдштовхуванням бiля твердої поверхнi: наближення середнього поля Article published earlier |
| spellingShingle | Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation Holovko, M. Patsahan, T. Kravtsiv, I. di Caprio, D. |
| title | Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation |
| title_alt | Нематогенний плин Майєра-Заупе з iзотропним юкавiвським вiдштовхуванням бiля твердої поверхнi: наближення середнього поля |
| title_full | Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation |
| title_fullStr | Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation |
| title_full_unstemmed | Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation |
| title_short | Maier-Saupe nematogenic fluid with isotropic Yukawa repulsion at a hard wall: Mean field approximation |
| title_sort | maier-saupe nematogenic fluid with isotropic yukawa repulsion at a hard wall: mean field approximation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155778 |
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