Improved ansatz for the direct correlation function in dilute nematic colloids
We present an improved analytical ansatz for the colloid-nematic direct correlation function. This ansatz is more accurate than our earlier version, and yields numerical results that are very close to the true mean spherical approximation for dilute nematic colloids. Furthermore, the improved ansatz...
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| Zitieren: | Improved ansatz for the direct correlation function in dilute nematic colloids / T.G. Sokolovska, R.O. Sokolovskii, G.N. Patey // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 407-414. — Бібліогр.: 15 назв. — англ. |
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irk-123456789-1187052017-06-01T03:05:53Z Improved ansatz for the direct correlation function in dilute nematic colloids Sokolovska, T.G. Sokolovskii, R.O. Patey, G.N. We present an improved analytical ansatz for the colloid-nematic direct correlation function. This ansatz is more accurate than our earlier version, and yields numerical results that are very close to the true mean spherical approximation for dilute nematic colloids. Furthermore, the improved ansatz is valid for external fields of any strength. We examine the zero-field behavior of the colloid-colloid potential of mean force in light of the improved ansatz, and show that at zero field and large separations this function decays as R⁻⁵ and grows as ∑⁶, where ∑ is the colloidal diameter. These dependencies are consistent with our earlier conclusions based on a less accurate version of the ansatz. As with the original ansatz, the improved version remains analytical and can be readily applied to a broad range of physically interesting systems. These include patterned and nonspherical colloids, colloids trapped at interfaces, and nematic fluids in confined geometries, e.g. in droplets. Ми представляємо покращений аналiтичний анзац для прямої кореляцiйної функцiї колоїд-нематик. Цей анзац є точнiшим нiж його попередня версiя, отримана нами, i дає числовi результати для розведених нематичних колоїдiв, що є дуже близькими до тих, якi отримуються в рамках справжнього середньо-сферичного наближення. Крiм того, покращений анзац є справедливим для випадку зовнiшнiх полiв довiльної сили. Ми вивчаємо поведiнку потенцiалу середньої сили колоїд-колоїд в нульовому полi, використовуючи покращений анзац i показуємо, що в нульовому полi i на великих вiддалях ця функцiя загасає як R⁻⁵ i росте як ∑⁶, де ∑ є дiаметром колоїда. Ця залежнiсть узгоджується з нашими попереднiми обчисленнями, якi базуються на менш точнiй версiї анзацу. Як i в попередньому випадку, покращена версiя залишається аналiтичною i може бути легко застосована до широкого класу фiзично цiкавих систем, якi включають колоїди з довiльним розподiлом сурфактантiв на поверхнi i несферичнi колоїди, колоїди при мiжфазнiй границi i нематичнi плини в обмежених геометрiях, наприклад в краплинах. 2007 Article Improved ansatz for the direct correlation function in dilute nematic colloids / T.G. Sokolovska, R.O. Sokolovskii, G.N. Patey // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 407-414. — Бібліогр.: 15 назв. — англ. 1607-324X PACS: 61.30.Cz, 82.70.Dd, 61.30.Gd DOI:10.5488/CMP.10.3.407 http://dspace.nbuv.gov.ua/handle/123456789/118705 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We present an improved analytical ansatz for the colloid-nematic direct correlation function. This ansatz is more accurate than our earlier version, and yields numerical results that are very close to the true mean spherical approximation for dilute nematic colloids. Furthermore, the improved ansatz is valid for external fields of any strength. We examine the zero-field behavior of the colloid-colloid potential of mean force in light of the improved ansatz, and show that at zero field and large separations this function decays as R⁻⁵ and grows as ∑⁶, where ∑ is the colloidal diameter. These dependencies are consistent with our earlier conclusions based on a less accurate version of the ansatz. As with the original ansatz, the improved version remains analytical and can be readily applied to a broad range of physically interesting systems. These include patterned and nonspherical colloids, colloids trapped at interfaces, and nematic fluids in confined geometries, e.g. in droplets. |
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Sokolovska, T.G. Sokolovskii, R.O. Patey, G.N. |
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Sokolovska, T.G. Sokolovskii, R.O. Patey, G.N. Improved ansatz for the direct correlation function in dilute nematic colloids Condensed Matter Physics |
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Sokolovska, T.G. Sokolovskii, R.O. Patey, G.N. |
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Sokolovska, T.G. |
| title |
Improved ansatz for the direct correlation function in dilute nematic colloids |
| title_short |
Improved ansatz for the direct correlation function in dilute nematic colloids |
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Improved ansatz for the direct correlation function in dilute nematic colloids |
| title_fullStr |
Improved ansatz for the direct correlation function in dilute nematic colloids |
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Improved ansatz for the direct correlation function in dilute nematic colloids |
| title_sort |
improved ansatz for the direct correlation function in dilute nematic colloids |
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Інститут фізики конденсованих систем НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/118705 |
| citation_txt |
Improved ansatz for the direct correlation function in dilute nematic colloids / T.G. Sokolovska, R.O. Sokolovskii, G.N. Patey // Condensed Matter Physics. — 2007. — Т. 10, № 3(51). — С. 407-414. — Бібліогр.: 15 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT sokolovskatg improvedansatzforthedirectcorrelationfunctionindilutenematiccolloids AT sokolovskiiro improvedansatzforthedirectcorrelationfunctionindilutenematiccolloids AT pateygn improvedansatzforthedirectcorrelationfunctionindilutenematiccolloids |
| first_indexed |
2025-07-08T14:29:32Z |
| last_indexed |
2025-07-08T14:29:32Z |
| _version_ |
1837089405694640128 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 3(51), pp. 407–414
Improved ansatz for the direct correlation function in
dilute nematic colloids
T.G.Sokolovska1,2, R.O.Sokolovskii1,2, G.N.Patey1
1 Department of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
2 Institute for Condensed Matter Physics, 1 Svientsitskii Str., Lviv 79011, Ukraine
Received May 25, 2007, in final form September 4, 2007
We present an improved analytical ansatz for the colloid-nematic direct correlation function. This ansatz is
more accurate than our earlier version, and yields numerical results that are very close to the true mean
spherical approximation for dilute nematic colloids. Furthermore, the improved ansatz is valid for external
fields of any strength. We examine the zero-field behavior of the colloid-colloid potential of mean force in
light of the improved ansatz, and show that at zero field and large separations this function decays as R
−5
and grows as Σ
6, where Σ is the colloidal diameter. These dependencies are consistent with our earlier
conclusions based on a less accurate version of the ansatz. As with the original ansatz, the improved version
remains analytical and can be readily applied to a broad range of physically interesting systems. These include
patterned and nonspherical colloids, colloids trapped at interfaces, and nematic fluids in confined geometries,
e.g. in droplets.
Key words: colloid, liquid crystal, nematic, potential of the mean force
PACS: 61.30.Cz, 82.70.Dd, 61.30.Gd
1. Introduction to molecular modelling and correlations in nematic colloids
Nematic colloids consist of colloidal particles suspended in nematic liquid crystal. These are
of much current interest because the colloidal particles show a variety of structures that can be
altered by varying physical factors such as the strength and direction of external fields [1,2]. Nematic
colloids are intriguing from both fundamental and practical perspectives. All molecular interactions
in the system are short-ranged, but colloidal particles effectively interact as oriented multipoles
at long distances. In a way these systems are the inverse of fluids of charged particles, where the
interaction potential is long-ranged but the correlations are short-ranged due to screening. We
believe that the origin of such electrostatic-like behavior is the presence of long-range correlations
in the bulk. In [3] we applied the Ornstein-Zernike (OZ) equation to nematic colloids to show
how the electrostatic analogy asymptotically appears in the molecular theory of correlations. Our
theory starts at the level of particle interactions modeled in a physically relevant way, and employs
the Ornstein-Zernike and Lovett equations (OZL) for pair correlation functions and single-particle
distributions. The general solution of these equations gives a description of long-range and short-
range effects in nematic colloids, and takes account of orientational and density fluctuations. The
approach can be equally applied to concentrated and dilute mixtures for colloidal particles of any
size ranging from nanometers to micrometers. After the solution is obtained, the corresponding
thermodynamics is available via exact relations of statistical mechanics [4,5].
The equilibrium single-particle distribution and pair correlation functions for nematic colloids
are found by solving the OZ relationship,
hαβ(1, 2) = cαβ(1, 2) +
∑
γ=C,N
∫
cαγ(1, 3)ργ(3)hγβ(3, 2)d(3), (1)
combined with the exact Lovett equation [6]. In equation (1), cαβ and hαβ are the direct and
total correlation functions, ρN(3), ρC(3) are the density distributions of the nematic (N) and
c© T.G.Sokolovska, R.O.Sokolovskii, G.N.Patey 407
T.G.Sokolovska, R.O.Sokolovskii, G.N.Patey
colloidal (C) components, and α, β represent C or N . Also, for a nematogen, the label 1 denotes
the coordinates (R1, ω̂1), and for a spherical colloid 1 ≡ (R1). The functions in equation (1)
describe all structural properties of the mixture at nanoscale and microscale levels. The exact OZ
equation must be solved in conjunction with another approximate expression relating the total
and direct correlation functions, such as the Percus-Yevick, hypernetted chain (HNC) or mean
spherical approximation (MSA) closures. A closure is the only approximation that is necessary in
the integral equation theory.
For a dilute (ρC → 0) nematic colloid, the OZ equation takes the form,
hαβ(1, 2) = cαβ(1, 2) +
∫
cαN (1, 3)ρN(3)hNβ(3, 2)d(3). (2)
The physical meaning of all functions in equation (2) is clear; ρN(ω̂) gives the orientational distri-
bution in a pure nematic, whereas, ρN(1)[1 + hNC(1, 2)] gives the complete distribution of nematic
fluid about a colloidal particle. The latter function takes into account all changes at a given point
R1 induced by the colloidal particle at R2; these include changes in the local density and in the
orientational distribution of the nematic fluid. The colloid-colloid potential of mean force at the
HNC level is conveniently given by
φCC(1, 2) =
[
cCC(1, 2) − hCC(1, 2)
]
kBT + vCC(1, 2) , (3)
where vCC(1, 2) is the direct pair interaction between the colloidal particles.
We consider a model [7–9] consisting of a system of uniaxial particles [nematogens (N)] inter-
acting through a pair potential taken to be the sum of a hard-sphere interaction (sphere diameter σ)
and an anisotropic part defined by
v(1, 2) = v2(R12)P2(ω̂1 · ω̂2), (4)
where P2(ω̂1 · ω̂2) is the second-order Legendre polynomial, R12 is the center-center distance, the
unit vector ω̂i denotes the orientation of particle i, and
v2(R12) = −AN(zNσ)2
exp(−zNR12)
R12/σ
. (5)
Here, AN and zN are the energy and the length parameters characterizing the interaction. The
nematogen interaction with the external field is given by
vN(1) = −W
√
5P2(ω̂1 · n̂), W > 0, (6)
where W is the field strength. This interaction orders the bulk director n̂ parallel to the field. The
OZL equations for this model have been solved analytically in the MSA [7,8].
The model colloidal particles (C) are taken to be hard spheres of diameter Σ. Van der Waals
or other direct colloid-colloid interactions could easily be included through the vCC(1, 2) term in
equation (3). The interaction of nematogens with the surface of a colloidal particle is modeled as
vNC(1, 2) =
{
∞, if s12 < σ/2,
−AC exp[−zC(s12 − σ/2)]P2(ω̂1 · ŝ12), if s12 > σ/2,
(7)
where s12 is a vector connecting the nearest point of the surface of colloid 2 with the center of
nematogen 1, and ŝ12 = s12/s12. Note that positive and negative values of AC favour, respectively,
perpendicular and parallel orientations of nematogen molecules with respect to the surface. For
zCσ = 1 the nematogen-colloid interaction range is of the order of the nematogen “length”. The
strength of the nematogen-colloid interaction is determined by AC, and zC, and will vary for
different surfactants. The magnitude of AC depends on the surfactant concentration on the colloidal
surfaces.
For micron and submicron colloids the solution of equation (2) can be greatly simplified for
model (4–7). For this case we have suggested an ansatz, namely that the direct correlation function
408
Improved ansatz for the direct correlation function in dilute nematic colloids
cCN(1, 2) can be taken from the wall-nematic solution that has been obtained for any orientation
of the wall with respect to the field [9,10]. It is important that the solution was obtained subject to
well defined boundary conditions at infinity: the director parallel to the field. This ansatz is very
convenient because within the MSA the wall-nematic direct correlation function can be obtained
in explicit form [11]. This permits explicit microscopic calculations of the density and orientational
profiles [12], as well as effective interactions (potentials of mean force) between pairs of colloidal
particles [13]. If the director n̂ is fixed parallel to the z-axis, the total correlation functions in the
MSA are found in the form of spherical harmonic expansions,
hNN(R12, ω̂1, ω̂2) =
∑
ll′=0,2
∑
m
hNN
ll′m(R12)Ylm(ω̂1)Y
∗
l′m(ω̂2), (8)
hCN(R12, ω̂2) =
∑
ll′=0,2
∑
m
hCN
ll′m(R12)Ylm(R̂12)Y
∗
l′m(ω̂2). (9)
If all correlation lengths are finite and smaller than the colloidal radius, the ansatz gives a very
good approximation. A problem arises if some correlation length is larger than Σ. The correlation
length of hNN
22|m|=1(R12) in the nematic phase depends on the external field W [3],
ξ =
√
K
WρNS23
√
5
, (10)
where the elastic constant K and order parameter S2 are not sensitive to the influence of small
fields. This means that in the case of zero field the correlation length is infinite. It was shown
in [7] that these critical (Goldstone or elastic) correlations decrease at long distances as 1/R12.
Moreover, they generate long-range asymptotes in hCN(R12, ω̂2) harmonics, in particular those
that are coupled with hNN
22|m|=1(R12) through equation (2). Physically this means that a single
colloidal particle induces long-range distortions in the distribution of surrounding nematic. Using
the ansatz for the colloid-nematic direct correlation function we obtain [3] the following asymptote
for the total correlation function at zero field and Σ � σ,
hCN(1, 2)
R→∞−→ 1
10
hWN
221 (s = σ/2)
BzC
Σ3
R3
[
Y21(R̂12)Y
∗
21(ω̂2) + c.c.
]
, (11)
where hWN
221 (s = σ/2) is the contact value of the corresponding harmonic of the wall-nematic correla-
tion function, and B can be expressed through the bulk elastic constant K and order parameter S2,
B2 =
〈
|Y21|2
〉
ω
βK
15ρNS2
2
. (12)
The notation 〈. . . 〉ω indicates
∫
fN(ω̂)(. . . )dω̂, and fN(ω̂1) = ρN(1)/ρN is a single-particle dis-
tribution function in the bulk. Finally, at the HNC level the asymptote of the effective colloidal
interaction was found in [3] to be
βφCC′
(R)
R→∞−→ 8π
15
hWN
221 (s = σ/2)
zC
hW ′N
221 (s = σ/2)
zC′
ρN
〈
|Y21|2
〉
ω
Σ3Σ′3
R5
Y40(R̂). (13)
As mentioned above, the ansatz is reliable when ξ < Σ, but may give inaccurate results at
small fields, where ξ > Σ. We investigated the reliability of the ansatz in [11]. The test calculations
were based on verification of an exact consequence of the impenetrability of hard cores, namely
hCN(R12, ω̂2) = −1 if R12 < Σ+σ
2 . Thus, the exact condition implies that
hCN
220(R12) = hCN
200(R12) = hCN
020(R12) = hCN
22m(R12) = 0, hCN
000(R12) = −1, (14)
for R12 < (Σ + σ)/2. We found that since harmonics with |m| 6= 1 are not coupled with elastic
bulk harmonics and are relatively short-ranged, for these harmonics our ansatz essentially satisfies
409
T.G.Sokolovska, R.O.Sokolovskii, G.N.Patey
the core condition for colloidal sizes larger than 50σ for all system parameters considered. This is
not the case for the harmonics hCN
22±1(R12). The problematic harmonics calculated with the ansatz
are plotted in figure 1 for two colloid sizes, Σ = 50σ and 500σ, and small, W = 10−4kBT , and zero
external fields. The colloidal length parameter zCσ = 0.2. The remaining model parameters are the
same for all numerical calculations presented in this paper: zNσ = 1, AN = kBT , η = πρσ3/6 = 0.35
and AC/AN = 2. One can see that the condition (14) essentially holds at W = 10−4kBT for the
larger particle, but not for Σ = 50σ. Note that the correlation length equals 161.5σ at the given
field. In the zero field case, where ξ is infinite, the core condition is not well satisfied for colloidal
particles of any size.
Figure 1. Long-range harmonics of the colloid-nematic total correlation function hCN
221(R) for
colloidal particles with Σ = 50σ and 500σ, at zero external field and at W = 10−4kBT . The
parameter zCσ = 0.2.
Clearly our ansatz requires improvement in the low field (large ξ) limit, and this is the purpose
of the present paper. We show how the ansatz can be much improved and examine the consequences
for the zero-field asymptotes defined by equations (10) and (12).
2. Corrected ansatz for small or zero fields
Numerically, it is possible to correct the ansatz taken from the wall problem by introducing a
single parameter. We scale the analytical expression for harmonics |m| = 1 inside the hard core,
c̄CN
221(R12) ≈ qcWN
221 (s = R12 − Σ/2) for R12 <
Σ + σ
2
, (15)
and choose the coefficient q (0 < q < 1) such as to minimize
∫ (Σ+σ)/2
0
|h̄CN
221(R12)|dR12 . (16)
Note that c̄CN
221(R12) outside the core remains the same and satisfies the MSA closure,
cCN(R12, ω̂2) = −φCN(R12, ω̂2)
kBT
, if R12 >
Σ + σ
2
. (17)
The corrected ansatz appears to be very close to the true MSA solution of equation (2), even at
zero field (see figure 2). We have calculated h̄CN
221(R12) with the corrected ansatz (15) for colloids
with Σ = 500σ and different surface parameters. At zero field h̄CN
221(R12)/hCN
221(R12) ≈ 1.27 for
zCσ = 1, and ≈ 1.23 for zCσ = 0.2 (figure 3). Numerically, we find that the deviation from 5/4 lies
within 2 per cent. Thus for sufficiently large colloidal particles, the asymptote (11) can be easily
410
Improved ansatz for the direct correlation function in dilute nematic colloids
Figure 2. Corrected long-range harmonics of the colloid-nematic total correlation function
h̄CN
221(R) for colloidal particles with Σ = 50σ and 500σ, at zero external field and at W =
10−4kBT . The parameter zCσ = 0.2.
Figure 3. The ratio of the corrected long-range harmonics of the colloid-nematic total correlation
function to the uncorrected case, h̄CN
221(R12)/hCN
221(R12), at zero external field, for Σ = 500σ and
for two ranges of the colloid-nematic interaction, zCσ = 1 and zCσ = 0.2.
corrected at zero field by including a factor of 5/4. In experiments typical colloidal sizes are of
order 103σ.
The factor 5/4 is not accidental. Formally we can take the limit of zero field (ξ → ∞) for the
case of the tilted wall (from equation (17) of [11]). In this case the asymptote of the wall-nematogen
total correlation function tends to a constant value at large distances from the wall,
lim
ξ→∞
hWN(s, ω̂N)
s→∞−→ lim
ξ→∞
hWN
221 (s = σ/2)
BzC
[Y21(ŝ)Y ∗
21(ω̂N) + c.c.] exp[(−s − σ/2)/ξ], (18)
where the unit vector ŝ defines the orientation of the wall normal with respect to the field, and
ω̂N is a molecule orientation.
On the other hand, we can consider the limit of the distribution function around a spherical
colloid [equation (10)] as Σ → ∞. Note that the limit of zero field, ξ → ∞, is taken first. At the
distance s = R−Σ/2 outside the surface, Σ3/R3 tends to 8, since Σ → ∞ and s � Σ. By putting
ŝ ≡ R̂12 and ω̂N ≡ ω̂2 one can see from equation (11) that
lim
ξ→∞
hWN
221 (s) =
5
4
lim
Σ→∞
lim
ξ→∞
hCN
221(s). (19)
In other words, the order in which two limits are taken does matter if one uses the direct correlation
function from the wall problem as an ansatz for spherical colloids. If the asymptote is corrected with
411
T.G.Sokolovska, R.O.Sokolovskii, G.N.Patey
a factor of 5/4 the inconsistency does not appear. Following [3], one can see that any correction of
asymptote (10) with a given factor results in a correction of asymptote (12) with the same factor
squared. Thus, the asymptote of the potential of mean force should be corrected with a factor of
(5/4)2.
If the correlation length ξ is smaller than Σ, one would expect the fluid distribution about a
colloid obtained with the ansatz to coincide with the distribution near the plane wall, at least for
s � Σ. Our numerical analysis for Σ = 500σ, zcσ = 1 and distances up to several 10σ shows
that this is true for βW = 10−4 and ξ = 161.5σ. The picture is different, however, for the smaller
field, βW = 10−6, where ξ ∼ 1615σ is larger than Σ. In this case hWN
221 (s)/hCN
221(s) tends to 5/4 at
s ≈ 10σ.
We are especially interested in verifying the Σ-dependencies of the asymptotes (10) and (12),
particularly because phenomenological theories appear to disagree on this matter. A theory with
imposed weak boundary conditions at colloidal surfaces predicts the interaction of colloids (that
exhibit up-down symmetry) to be proportional to Σ8 [14]. At the same time the theory of Lubensky
et al. [15], based on rigid boundary conditions, predicts that the interactions behave as Σ6. Our
molecular theory does not impose boundary conditions at colloidal surfaces, but instead calculates
them. All information about any Σ-dependence is contained in the direct colloid-nematic correlation
function. In fact the Σ-dependence of asymptote (10) at zero field is defined by the cCN
221 behavior
at small k in Fourier space. The uncorrected ansatz gives the following expansion for large Σ at
zero field [3],
cCN
221(k)
k→0−→ −4π
hWN
221 (s = σ/2)
30zC
B
[
Σ3 + O(Σ2)
]
k2 + O(k4). (20)
To investigate further, we consider the expansion (20) for the corrected ansatz, which takes the
form,
c̄CN
221(k)
k→0−→
[
4π
βAC(−1 + q)
75zC
(Σ/2)4 + O(Σ3)
]
k2 + O(k4). (21)
The first impression is that the coefficient of k2 now has a Σ4-dependence. However, this turns
out not to be true. Numerical calculations of (−1 + q)Σ were done for different Σ, up to 1500σ.
For σ = 2nm, order of size of 5CB molecules, this corresponds to colloids of 3µm in diameter. We
considered various interaction parameters (zCσ = 0.1, 0.2, 1, 2). In all cases the value of (−1+ q)Σ
tends to a constant limit at large Σ (figure 4). In other words, the fact that q 6= 1 corrects
asymptotes (10) and (12) with the factors 5/4 and (5/4)2, respectively, but does not change their
Σ-dependence at zero field. Thus, the MSA results support the conclusion on the Σ-dependence
reached in [15], where rigid boundary conditions were employed. It should be emphasized that in
our model the calculated director distribution in the vicinity of colloids is not coupled rigidly with
normals to the colloidal surface [12]. The full coefficient of k2 in equation (21) divided by Σ3/zC
is plotted in figure 5 for different values of zCσ. One can see that a Σ3-dependence holds in all the
cases for Σ > 500σ.
3. Summary and conclusions
In this paper we have developed an improved ansatz for the colloid-nematogen direct correlation
function. In our previous studies we suggested that a general solution of the Ornstein-Zernike
equation for a nematic fluid in the presence of a wall, arbitrarily oriented with respect to the nematic
director, can be very useful for the investigation of dilute nematic colloids. In fact, we suggested
using the wall-nematogen direct correlation function, obtained explicitly in [11], as an ansatz
for the direct correlation function in nematic colloids. This ansatz permits explicit microscopic
calculations of the density and orientational profiles, as well as potentials of mean force for large
colloidal particles, such as those used in numerous experiments. This ansatz works very well in
the presence of external fields, when all correlation lengths are smaller than the colloidal radius.
At small and zero external fields the results of this ansatz, particularly asymptotes at zero field,
need some correction. The reason for this is that the “elastic” correlation length in the nematic
phase is infinite at zero field. Here we suggest a simple scaling of the ansatz, which gives results
412
Improved ansatz for the direct correlation function in dilute nematic colloids
Figure 4. The values of (−1 + q)Σ for vari-
ous ranges of the colloid-nematic interaction,
zCσ = 0.1, 0.2, 1, 2.
Figure 5. The lim
k→0
c̄CN
221(k)zC/(k2Σ3) at zero
external field as a function of Σ for various
ranges of the colloid-nematic interaction zCσ.
numerically very close to the MSA solution of the OZ equation for nematic colloids in the presence
of arbitrarily weak fields, including the zero field case. Our study of spherical colloids shows that
the improved ansatz corrects asymptotes of the nematic distribution about a colloidal particle and
the colloid-colloid potential of mean force with factors 5/4 and (5/4)2, respectively, but it does not
change the dependence of the asymptotes on the colloidal diameter, or on the interparticle vector.
Finally, the ansatz taken from the wall-nematic problem has some additional advantages. Unlike
a direct solution of the OZ equation for nematic colloids, the same analytical expression can be used
with little adjustments for physically different systems, e.g., nonspherical colloids, colloids with a
nonuniform distribution of surface surfactant (patterned colloids), colloids trapped at interfaces,
as well as nematic droplets.
Acknowledgements
Acknowledge is made to the Donors of the American Chemical Society Petroleum Research
Fund for partial support of this research. The financial support of the Natural Science and Engi-
neering Research Council of Canada is gratefully acknowledged. This research has been enabled
by the use of WestGrid computing resources, which are funded in part by the Canada Foundation
for Innovation, Alberta Innovation and Science, BC Advanced Education, and the participating
research institutions. WestGrid equipment is provided by IBM, Hewlett Packard and SGI.
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Покращений анзац для прямої кореляцiйної функцiї в
розведених нематичних колоїдах
Т.Г. Соколовська1,2, Р.О.Соколовський1,2, Г.Н.Пейтi1
1 Хiмiчний факультет, Унiверситет Британської Кулумбiї, Ванкувер, Канада
2 Iнститут фiзики конденсованих систем НАН України, вул. Свєнцiцького 1, Львiв 79011, Україна
Отримано 25 травня 2007 р., в остаточному виглядi – 4 вересня 2007 р.
Ми представляємо покращений аналiтичний анзац для прямої кореляцiйної функцiї колоїд-нематик.
Цей анзац є точнiшим нiж його попередня версiя, отримана нами, i дає числовi результати для роз-
ведених нематичних колоїдiв, що є дуже близькими до тих, якi отримуються в рамках справжньо-
го середньо-сферичного наближення. Крiм того, покращений анзац є справедливим для випадку
зовнiшнiх полiв довiльної сили. Ми вивчаємо поведiнку потенцiалу середньої сили колоїд-колоїд в
нульовому полi, використовуючи покращений анзац i показуємо, що в нульовому полi i на великих
вiддалях ця функцiя загасає як R
−5 i росте як Σ
6, де Σ є дiаметром колоїда. Ця залежнiсть узго-
джується з нашими попереднiми обчисленнями, якi базуються на менш точнiй версiї анзацу. Як i
в попередньому випадку, покращена версiя залишається аналiтичною i може бути легко застосо-
вана до широкого класу фiзично цiкавих систем, якi включають колоїди з довiльним розподiлом
сурфактантiв на поверхнi i несферичнi колоїди, колоїди при мiжфазнiй границi i нематичнi плини в
обмежених геометрiях, наприклад в краплинах.
Ключовi слова: колоїд, рiдкий кристал, нематик, потенцiал середньої сили
PACS: 61.30.Cz, 82.70.Dd, 61.30.Gd
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