Colloidal nematostatics
We give a review of the theory of large distance colloidal interaction via the nematic director eld. The new area of nematic colloidal systems (or nematic emulsions) has been guided by the analogy between the colloidal nematostatics and electrostatics. The elastic charge density representation of t...
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| Опубліковано в: : | Condensed Matter Physics |
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| Дата: | 2010 |
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Інститут фізики конденсованих систем НАН України
2010
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| Цитувати: | Colloidal nematostatics / V.M. Pergamenshchik, V.A. Uzunova // Condensed Matter Physics. — 2010. — Т. 13, № 3. — С. 33602:1-29. — Бібліогр.: 71 назв. — англ. |
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| author | Pergamenshchik, V.M. Uzunova, V.A. |
| author_facet | Pergamenshchik, V.M. Uzunova, V.A. |
| citation_txt | Colloidal nematostatics / V.M. Pergamenshchik, V.A. Uzunova // Condensed Matter Physics. — 2010. — Т. 13, № 3. — С. 33602:1-29. — Бібліогр.: 71 назв. — англ. |
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| description | We give a review of the theory of large distance colloidal interaction via the nematic director eld. The new area of nematic colloidal systems (or nematic emulsions) has been guided by the analogy between the colloidal nematostatics and electrostatics. The elastic charge density representation of the colloidal nematostatics [V.M. Pergamenshchik, V.O. Uzunova, Eur. Phys. J. E, 2007, 23, 161; Phys. Rev. E, 2007, 76, 011707] develops this analogy at the level of charge density and Coulomb interaction. The analogy is shown to lie in common mathematics based on the solutions of Laplace equation. However, the 3d colloidal nematostatics substantially differs from electrostatics both in its mathematical structure and physical implications. The elastic charge is a vector fully determined by the torque exerted upon colloid, the role of Gauss' theorem is played by conservation of the torque components. Elastic multipoles consist of two tensors (dyads). Formulas for the elastic multipoles, the Coulomb-like, dipole-dipole, and quadrupole-quadrupole pair interaction potentials are derived and illustrated by particular examples. Based on the tensorial structure, we list possible types of elastic dipoles and quadrupoles. An elastic dipole is characterized by its isotropic strength, anisotropy, chirality, and its longitudinal component. An elastic quadrupole can be uniaxial and biaxial. Relation between the multipole type and its symmetry is discussed, sketches of some types of multipoles are given. Using the mirror image method of electrostatics as a guiding idea, we develop the mirror image method in nematostatics for arbitrary director tilt at the wall. The method is applied to the charge-wall and dipole-wall interaction.
Розглянуто теорію взаємодії колоїдних частинок через поле нематичного директора на великих відстанях. Нова область фізики нематичних емульсій багато в чому завдячує ідеї про те, що їх фізика є подібною до електростатики. Взаємодія через поле директора є далекосяжною та має багато інших властивостей взаємодії між електричним ди- та квадруполями. Представлення густини пружного заряду в колоїдній нематостатиці розвиває цю аналогію на рівні густини заряду та закону Колона. Показано, що аналогія полягає в подібності математики, яка в обох теоріях спирається на розв'язки рівняння Лапласа. Однак у 3-х вимірах колоїдна нематостатика суттєво відрізняється від електростатики як своєю математичною структурою, так і її фізичними наслідками. Пружний заряд є вектором, який повністю визначається моментом зовнішніх сил, прикладених до колоїда, а роль Гаусової теореми відіграє закон збереження компонент пружного моменту сил. Пружні мультиполі складаються з 2-х тензорів (диад). Виведено формули для пружних мультиполів і потенціалів пружної парної кулонівської, диполь-дипольної та квадруполь-квадрупольної взаємодії. Розглянуто прості приклади, що її ілюструють. Виходячи з тензорної структури, знайдено різні можливі типи пружних ди- і квадруполів. Показано, що пружний диполь характеризується його ізотропною силою, анізотропією, кіральністю та певною поздовжньою компонентою. Пружний квадруполь може бути одно- та двовісним (біаксіальним). Розглянуто зв'язок між типом пружного мультиполя та його симетрією, деякі типи мультиполів проілюстровано графічно. Спираючись на ідею методу електростатичних відображень, розвинено метод дзеркальних відображень в нематостатиці для довільного нахилу директора на стінці, яка обмежує нематичний рідинний кристал. Цей метод застосовано для розв'язання задачі взаємодії пружного заряду та диполя зі стінкою.
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Condensed Matter Physics 2010, Vol. 13, No 3, 33602: 1–29
http://www.icmp.lviv.ua/journal
Colloidal nematostatics
V.M. Pergamenshchik1,2, V.A. Uzunova2
1 Korea University, Display&Semiconductor Physics, Jochiwon-eup, Yeongi-gun,
Chungnam 339–700, South Korea
2 Institute of Physics, prospect Nauki, 46, Kiev 03039, Ukraine
Received January 24, 2010
We give a review of the theory of large distance colloidal interaction via the nematic director field. The new
area of nematic colloidal systems (or nematic emulsions) has been guided by the analogy between the col-
loidal nematostatics and electrostatics. The elastic charge density representation of the colloidal nematostatics
[V.M. Pergamenshchik, V.O. Uzunova, Eur. Phys. J. E, 2007, 23, 161; Phys. Rev. E, 2007, 76, 011707] de-
velops this analogy at the level of charge density and Coulomb interaction. The analogy is shown to lie in
common mathematics based on the solutions of Laplace equation. However, the 3d colloidal nematostatics
substantially differs from electrostatics both in its mathematical structure and physical implications. The elastic
charge is a vector fully determined by the torque exerted upon colloid, the role of Gauss’ theorem is played
by conservation of the torque components. Elastic multipoles consist of two tensors (dyads). Formulas for the
elastic multipoles, the Coulomb-like, dipole-dipole, and quadrupole-quadrupole pair interaction potentials are
derived and illustrated by particular examples. Based on the tensorial structure, we list possible types of elastic
dipoles and quadrupoles. An elastic dipole is characterized by its isotropic strength, anisotropy, chirality, and
its longitudinal component. An elastic quadrupole can be uniaxial and biaxial. Relation between the multipole
type and its symmetry is discussed, sketches of some types of multipoles are given. Using the mirror image
method of electrostatics as a guiding idea, we develop the mirror image method in nematostatics for arbitrary
director tilt at the wall. The method is applied to the charge-wall and dipole-wall interaction.
Key words: nematic colloid, electrostatic analogy, elastic charge and multipoles
PACS: 61.30.Dk, 61.30.Jf, 82.70Dd, 01.55.+b
1. The intrigue of nematic colloids: similarity to the electrostatics, its origin,
and how it is developed in this article.
Isotropic liquids doped with colloidal particles or just colloids are the classical many-particle
systems which, over many decades , have been studied using the methods of molecular and sta-
tistical physics, electrolyte theory, physical chemistry, and so on [1, 2]. Colloidal particles can be
of different physical and chemical origin, but in all cases their interaction via isotropic liquid is of
a short range [3], even if the colloids are charged since their electric field is rapidly screened by a
numerous couterions. The nematic colloids, that have become widely known just over the last 15
years, are fundamentally different: interaction of colloids mediated by distortions of the nematic
director is of a long range. That is why nematic colloids are usually compared not with standard
isotropic colloids, but with a system of electric charges. The similarity with the electrostatics has
always been a powerful, if not dominating, factor of the development of the field of nematic colloids.
But what is the origin of this similarity?
The interaction of particles in a nematic liquid crystal is determined by the elastic free energy
(FE) of the director distortions induced by the surfaces of colloids. The director field in the prox-
imity of colloids is governed by the nonlinear Euler-Lagrange equations which are impossible to
solve analytically even for a single particle. However, outside these nonlinear near zones, the dis-
tortions of the uniform director become weak and take the form of a two-component vector which
satisfies the linear Laplace equation. As the scalar electrostatic potential is also described by the
Laplace equation, there arises an analogy between the electrostatics and colloidal interaction via
the director field. This analogy has been playing a very important heuristic role in the develop-
c© V.M. Pergamenshchik, V.A. Uzunova 33602-1
http://www.icmp.lviv.ua/journal
V.M. Pergamenshchik, V.A. Uzunova
ment of the new area of nematic colloids. In particular, it has allowed for theoretical prediction
and experimental confirmation of the existence of special colloidal particles whose interaction in
a nematic liquid crystal, by analogy with electric dipoles and quadrupoles, scales as the distance
to the power minus three and minus five. The theory of nematic colloids is, however, by far not
so complete and perfect as electrostatics. In particular, the electrostatic charge, its density, and
the Coulomb law, which are fundamental for electrostatics, have not had direct analogues in the
physics of nematic colloids. For this reason the analogy has not been developed to the level of the
Coulomb interaction. Our main goal was 1) to develop the analogy between the two-component
vector colloidal nematostatics and scalar electrostatics to the level of the Coulomb interaction,
namely, to find elastic charge (elastic monopole), its density, elastic multipoles (sections 3.1–3.3),
and analogues of the Coulomb and multipole pairwise interaction (sections 3.4–3.6); 2) to find
limits of this theory and, in particular, to establish when the pairwise approach is applicable to
colloids in a nematic liquid crystal which is a field system with an infinite number of degrees of
freedom (section 3.5); 3) give a classification of all possible colloids, i. e., elastic charges, dipoles,
and quadrupoles, that would be based on a solid ground of mathematics (section 4); 4) demonstrate
the intercolloid interaction by representing examples (section 4.4); 5) develop the image method for
solving boundary problems of the colloidal nematostatics (section 5); 6) speculate about possible
many-body colloidal systems, e. g., elastic plasma (conclusion); 7) to give a brief overview of the
most important (from our point of view) experimental and theoretical results which determined
the fast progress of the area (introduction).
Now it is in order to justify our approach. Being aware of substantial mathematical differences
between the scalar electrostatics and vector nematostatics, we tried to avoid a priori physical analo-
gies. Instead, electrostatics was employed as an example of mathematically perfect and complete
theory, which tells us that the colloidal nematostatics must be based on the mathematical theory
of the Laplace equation. The main element of this theory is the Green function techniques, and
there is no reason (and, actually, no choice) for seeking a different approach. For it gives certain
upper limit: those problems that cannot be solved analytically or, so to say, reasonably well by
using simple approximations, in electrostatics, for the same reason cannot be solved in the newly
born colloidal nematostatics. And vise versa, the maximum success in all problems of colloidal
nematostatics can be achieved by the Green function method. The important exclusion is the in-
teraction of micrometer-size colloids at small separations, when nonlinearity of the equations of
nematostatics allows for no analogy with the linear electrostatics. Colloids in such systems form
very complicated (and nice!) structures associated with colloidal crystals and their fragments (see
recent reviews [4, 5]), and always accompanied by whimsically alongated defects. These systems
can be addressed only numerically (see, e. g., [6, 7]).
2. Introduction
Particles immersed in a nematic liquid crystal interact via the director field n which mediates
the distortions induced thereby. The fact that this interaction is of a long range and resembles
the electrostatic interaction was first brought to the broad attention by Brochard and de Gennes
[8] almost 40 years ago and later by Lopatnikov and Namiot [9]. Over the last 15 years the study
of this director-mediated interaction has developed into a new rapidly growing branch of the
physics of liquid crystals, i. e., the physics of nematic colloidal systems or nematic emulsions [10].
Depending on the size and nature of colloids, these systems can be very different. The recent
development of the field has been inspired mostly by a great diversity of a few- and many-body
ordering phenomena in microemulsions with micrometer and submicrometer size colloids [4–7, 11–
18]. However, there is also an important class of nematic nanoemulsions with molecular size colloids
(dopants) and supramolecular size colloids. One example is a nematic liquid crystal doped with
chiral molecules [19] that induce the well-known macroscopic cholesteric ordering [19]. Moreover,
nowadays an intensive investigation of thermodynamics [20–23] and physical chemistry [24] of a
nematic liquid crystal doped with a great variety of different solute molecules and micelles [25] is
under way. A very special example comprises dye-doped nematic liquid crystals: under the action
33602-2
Colloidal nematostatics
of light some of them demonstrate spectacular orientational Jánossy effect [26] which is possible
only in an anisotropic media. It has been recently shown that the supramolecular aggregates [27],
interacting via the director field, and other supramolecular complexes, including dye molecules
[28], can also play the role of colloids and facilitate the light-induced orientational effect in these
“supra”-nanoemulsions. One more class comprises nematic ferroemulsions [29]. Colloids of different
origin (e. g., inclusions and defects, soft and rigid) and their interaction have also been studied in
different smectic LC phases and their free standing films and membranes (see, e. g., [30–34] and
review [35]).
This ordering diversity owes its existence to the long range, electrostatic-like interaction via
the director field. The rapid development of the field of nematic emulsions has been greatly guided
by this important similarity between the nematic emulsions and electrostatics. The distance de-
pendence of the director-mediated interaction on the colloid separation r can be similar to that
for the interaction between electric dipoles and quadrupoles [9–12, 36–44] which has been verified
experimentally [45–51]. Particle trapping techniques [45] have been used to test this analogy and
demonstrate experimentally 1/r3 [46, 47] and 1/r5 [48, 49] interactions of a colloid pair, which
were naturally interpreted, respectively, as an elastic dipole-dipole and quadrupole-quadrupole in-
teractions. The deep analogy between the two very different areas of physics not only provides a
useful theoretical tool for the study of the new field of anisotropic colloidal systems, but has been
perceived as the fact of a fundamental physical significance.
Mathematically, the similarity between the director-mediated and electrostatic interaction lies
in the massless nature of the both theories and the Coulomb-like behavior of the Green functions
which derives from it. A massive term in the fundamental functional (e. g., Hamiltonian, FE) has
the form of a square of the order parameter describing this system. The theory of electrostatic
field ϕ is massless as the term mϕ2/2 is absent in the energy functional. It is this property that
gives rise to the Coulomb 1/r potential: a finite massive term would have made it short range, i. e.,
exp(−mr)/r. In turn, the elastic theory of nematic liquid crystals describes the field of the director
n which is a unit vector. The massless nature of the director field follows from its definition: the
energy term quadratic in n is trivial as n2 = 1 [19]. As a result, the elastic theory allows the
director components that far from the distortion source behave as 1/r and its higher powers which
is formally similar to the potential of an electric charge, electric dipole, and so on.
The two systems, however, also have fundamental differences. Electrostatic potential is a scalar
described by the linear Laplace (or Poisson) equation. It is the linearity that underlies the def-
inition of the electric charge and its density as the source of electric field. At the same time, n
is a vector field described by a linear equation (in the one constant approximation) only in 2 di-
mensions. Because of this linearity, the deformation source can be straightforwardly established:
core of a point defect plays the role of a charge in 2 dimensions [19, 52–54]. But in 3 dimensions
the field n is described by highly nonlinear equations [19] and point defects cannot be linearly
connected with the distortions of n they induce [10, 42]. In principle, solution to these nonlinear
equations with appropriate boundary conditions determines the director field produced by any
source. However, in most cases an analytical solution to this problem is not known and can be
found only numerically. Some physically interesting sources, such as a sphere imposing normal
director orientation on its surface or defects, are indeed very strong: they produce large magnitude
highly nonlinear distortions in their close vicinity that are not tractable in terms of electrostatic
analogy [10, 42]. If, however, the total topological charge of the source is zero, the deformations
are decreasing functions of the distance r from the source so that far enough they become suffi-
ciently weak to allow for a linear description and thus for electrostatic analogy. Then, the theory
of a large distance colloidal interaction via the director field, which allows for certain analogy to
electrostatic multipole interaction and which we call colloidal nematostatics, consists of two quite
different problems. The first problem is to find the form of interaction between different types of
colloids that are expectedly elastic analogues of the electric multipoles. The second problem is to
express such elastic multipoles in terms of the colloid’s shape and anchoring, which implies making
a connection between the asymptotic electrostatic-like far director field and its nonlinear source.
The fundamental difficulty in solving these problems is that the colloidal nematostatics has no a
33602-3
V.M. Pergamenshchik, V.A. Uzunova
priori analogue of the Coulomb law. Owing to the Coulomb law, the electrostatic interaction can
always be written down in the form of interacting pairs of point-like charges. By contrast, nematic
liquid crystal is a field system which has an infinite number of degrees of freedom and, generally
speaking, cannot be reduced to a system of interacting pairs. Moreover, in the limit of zero size, a
particle is known to produce no director deformations [55]. Hence, nematostatics should consider
particles of a finite size.
Since, however, the mathematical treatment of point sources is much easier and more fami-
liar from electrostatics, the theory of interaction in nematic emulsions has been first developed
phenomenologically for point-like colloids [10, 40, 42–44]. The phenomenological energy density,
describing a director-mediated 1/r5 interaction of point-like sources and thus interpreted as the
elastic quadrupole-quadrupole interaction, was proposed by Ramaswamy et al. [40]. Later Luben-
sky et al. [42] further developed this approach to incorporate both the interaction of point-like
elastic quadrupoles and the 1/r3 interaction, interpreted as the interaction of point-like elastic di-
poles. The connection of these point-like dipoles and quadrupoles with particular finite-size colloids
was established by semi-numerical finding the equivalent multipoles that produce the same far field
effect as the real colloids. Namely, the authors of [10, 42] showed how to find the equivalent elastic
dipole and quadrupole of a sphere with the strong normal boundary condition. The director in the
proximity of the sphere is governed by strongly nonlinear equations and is indeed very complicated:
the near field contains an accompanying hyperbolic hedgehog or disclination ring [36]. This near
field was qualitatively described by an ansatz field based on an expected (visual) similarity with
the real director distribution [10, 39, 42]; from this ansatz, the authors determined the dipolar and
quadrupolar asymptotics as functions of the sphere radius and elastic constants.
However, these elastic dipoles and quadrupoles were of a very particular type and did not reflect
the general symmetry of nematostatics. Later, the tensorial structure of colloidal interaction was
demonstrated by Lev and Tomchuk [43] and Lev et al. [44] directly from the general nematic FE.
In this theory, a point-like source was characterized by symmetry of the deformation coat in its
close vicinity. It was assumed that the symmetry of the director field in this bulk coating can be
represented by distribution of normals to some enclosing surface separating the nonlinear near zone
from the linear far zone. The obtained pair interaction potential contained all powers of 1/r : the
1/r term was interpreted as the Coulomb interaction, 1/r3 as the dipole-dipole interaction, and so
on. These potentials had the form of a contraction of certain tensors which, in principle, could be
associated with the tensorial characteristics of elastic multipoles.
In particular, the authors of [44] introduced a scalar elastic charge and discussed a Coulomb-like
attraction. Its origin, however, remained undetermined. On the one hand, based on the well-known
result [19] that the source of the 1/r director behavior is a mechanical torque exerted on the colloid,
Lev et al. emphasized that the Coulomb-like attraction can be induced by the vector of external
torque Γ [44]. On the other hand, however, it was suggested that, in general, the Coulomb term
appears when the elastic coat has neither horizontal nor vertical mirror symmetry. But, if the
particle is asymmetric, this is the case without any external torque.
A vector of elastic charge and second rank tensor of elastic dipole were introduced by Fournier
as far back as 1993 [56]. For some reason, probably because the multipole interaction was not
considered there, this approach remained overlooked1 and did not receive a further development.
We see that the analogy between the electrostatics and nematostatics has already been greatly
developed, but it is not complete when it comes to the 1/r interaction. Indeed, the ultimate source
of electrostatic 1/r potential and multipole moments is the electric charge density whose analog
has not been established in the nematostatics. In a series of papers [57–60] we developed the
analogy between electrostatics and nematostatics to the level of charge density. In the derived
elastic charge density representation, the director components on a sphere, separating the near
and far zones, play the role of surface elastic charge density which determines the far field and the
elastic multipole moments, section 3. Inasmuch as the director has two independent components,
in the elastic density representation there are two densities and two multipole tensors (dyad) in any
order. The two-component elastic charge is expressed via the vector of external mechanical torque
1We are thankful to I.I. Smaljukh for referring us to [56].
33602-4
Colloidal nematostatics
applied on the particle, and the elastic Coulomb-like coupling between two “charged” particles is
found to be proportional to the scalar product of the two external torques. Consequently, the elastic
Coulomb interaction can be both attractive and repulsive depending on the relative orientation of
the torques: “parallel torques” attract each other whereas “antiparallel torques” repel each other,
section 3.6. The developed real space Green function method allowed us to obtain a regular FE
expansion in a power series of a parameter (colloid size)/(distance between colloids) and consider
the general status of the pairwise approximation to the nematic emulsions, section 3.5. The pairwise
approach has its limits: the pairwise interaction potential in a nematic emulsion can be uniquely
determined and justified only in the leading order in the small parameter (colloid size)/(distance
between colloids) since the next order contains irreducible three-body terms.
Based on the tensorial structure of an elastic dipole and quadrupole, we list all possible types of
elastic dipolar particles and important examples of quadrupolar particles, section 4. A connection
is discussed between different types of particles and the symmetry of the director field they induce.
The proposed classification is illustrated by sketches of some dipolar and quadrupolar colloids.
Dipolar colloids are characterized by their isotropic strength, anisotropy, chirality, and some two-
component vector along the undistorted director (longitudinal component). In particular, we show
that a general asymmetric colloid is a chiral elastic dipole. The chirality is eliminated by any
symmetry plane, so that quadrupoles cannot be chiral. A particle with two vertical and a horizontal
mirror symmetry planes is quadrupolar as all of the four dipolar components vanish. A quadropole
can be uniaxial and biaxial. An azimuthal anchoring on the colloid surface can make an otherwise
quadrupolar particle a dipole if its easy direction field on the colloid surface has a helicoidal
component. In section 5, the theory is applied to the colloid-surface interaction by developing the
mirror image method of nematostatics. The method is applied to the charge-wall and dipole-wall
interaction.
3. Elastic charge density representation of the colloidal nematostatics
3.1. Linearized director distortions and their source
Consider a 3-d director field n(r), uniform and parallel to the z-axis at infinity, n∞ = (0, 0, 1),
and distorted in a finite number N of particle-like (compact) domains. We refer to such a deforma-
tion domain as particle though the distortion therein can be induced by surface of a real particle,
by topological defects with zero total topological charge [10, 42], or by an external field dying out
outside the domain area. Generally, the deformations at the domain center are strong and satisfy
nonlinear equations, but sufficiently far they become weak, figure 1. Here the perturbation n⊥ to
n∞ is small, |n⊥| � 1, pure transverse, n⊥ = (nx, ny, 0), and satisfies linear equations. Enclose
Figure 1. The particle like distortion domain with the nonlinear core (darkened). At the enclosing
spherical surface S the director component normal to the unperturbed uniform director n∞ is
small, whereas at the nonlinear core it can be large. i is the image of the point r in S.
33602-5
V.M. Pergamenshchik, V.A. Uzunova
the i-th domain by a spherical surface Si of radius ai such that outside it |n⊥| � 1. At the same
time, the sphere can be small as compared to the interparticle distances, and the radius-to-distance
ratio is a small parameter. The spheres divide the space into the area Vin inside and the area Vout
outside them. In Vin the theory is nonlinear, intractable in standard terms (maybe experimentally
inaccessible), and the state of i-th particle will be represented by the director distribution n⊥(s)
on the enclosing sphere Si, s ∈ Si; a priori this n⊥(s) on each sphere depends on the positions
and orientations of all particles in the system. The state of the total system will be represented by
the director distribution on the total surface BN =
⋃N
i=1Si of all N spheres (N -sphere boundary).
This is well justified both physically and mathematically. As discussed above, the physical point
of view is that the information about the particle that can be considered known is the director
distribution close to its surface. The mathematical justification, given below, is that the above
boundary condition determines a unique solution to the Laplace equation.
A sphere has the following unique advantages: firstly, it does not introduce any symmetry
element so that the symmetry associated with the i-th source is fully determined by the distribution
of n⊥ on the i-th sphere; secondly, it enables us to calculate Green functions which seems to be a
task impossible for other surfaces. The final results do not depend on the choice of the surface. In
particular, the formulas of this section are valid if i-th particle is enclosed by an arbitrary smooth
surface S̃i.
It is implied in what follows that index t takes values x or y, Greek indices run over x, y and
z, index i stands for the particle number, summation over the repeating indices is performed, and
vectors s and s′ refer to a point on the surface Si. For simplicity, we omit the divergence FE terms
and work in the one constant approximation [19]. Then the distortion FE functional for the area
Vout outside all the enclosing spheres is given by
F{n} =
K
2
∫
(∇nt · ∇nt)d
3Vout . (1)
The transverse director component nt satisfies the Euler-Lagrange equations which for functi-
onal (1) coincides with the Laplace equation 4nt = 0. Here we assume that the perturbation nt
is vanishing at infinity (a sample surface at finite distance will be considered later on). Integrating
(1) by parts in the context of this assumption and the equation 4nt = 0, the FE in Vout reduces
to the integral over the total surface BN of all spheres:
FN,out =
K
2
∫
BN
nt(ν · ∇)ntd
2s, (2)
where ν = ν(s) is the unit outer normal to the surfaceBN at s, figure 1. It is known [65] that solution
nt(r) to the Laplace equation in the region Vout with the inner boundary BN , that vanishes at
infinity, is uniquely determined by the director distribution nt on BN . The formula which expresses
the director in Vout via the director on the surface reads [65]
nt(r) = −
∫
BN
nt(s) (ν(s)·∇s)G(r, s) d2s, (3)
where G(r, r′) is the Green function of the Laplace equation [65] which satisfies the following
boundary condition:
GN (r, s) = 0, s ∈ BN , s = ∞. (4)
The Green function is the sum G(r, r′) = (4π)−1
[
|r − r′|−1 + I(r, r′)
]
where I(r, r′) is a nonsin-
gular function found from the Laplace equation with the boundary condition (4)2. We see that, in
the linearized theory, the surface director component nt(s) plays the role of a source of the director
component nt(r) in the outer area. While the field equation (3) remains the same for any choice
of the enclosing surfaces S ′
i provided the boundary condition GN = 0 is fulfilled on
⋃N
i=1S
′
i, the
2The function I is nonsingular in the area outside the surface BN , but it can have singularities inside BN .
33602-6
Colloidal nematostatics
solution I(r, r′) to this boundary condition does depend on S ′
i and greatly simplifies for spheres
Si.
By substituting equation (3) in equation (2) the FE in Vout acquires the following important
form:
FN,out =
K
2
N∑
i=1
N∑
i′=1
∫
Si
∫
Si′
nt(s)UN (s, s′)nt(s
′)d2sd2s′, (5)
where
UN (s, s′) = − (ν(s′)·∇s
′) (ν(s)·∇s)GN (s′, s). (6)
These two formulas form the starting point of our approach. They describe an interaction of
the surface elements nt(s)d
2s and nt(s
′)d2s′ with the potential UN (s, s′). This potential, however,
depends on coordinates of all particles by virtue of the boundary condition onGN which vanishes on
all the spheres, not just on the two spheres represented by s and s′. Therefore, generally speaking,
UN describes a many-particle interaction in accordance with the fact that a field system does not
reduce to a finite number of particle-like sources with a pairwise interaction. In what follows we
obtain the function I(r, r′) for different numbers of particles, show that nx(s) and ny(s) determine
surface densities of what can be considered as a two-component elastic charge, and establish the
conditions when formula (6) indeed describes a pairwise interaction of particles imbedded in a
nematic medium.
3.1.1. Incorporation of real surfaces and finite anchoring
Here, for a moment, we make a diversion from our main line to demonstrate how the anchoring
effect on real surfaces can be incorporated in our approach. Consideration of finite anchoring effects
in colloidal nematostatics requires incorporation of a finite-size source, and theories, based on the
approximation of point-like sources, encounter difficulties. For instance, the treatment of finite
anchoring in [61, 62], based on the approach [43, 44, 63], required an introduction of an artificial
director field inside the colloids’ material. In our approach, the anchoring effects can be easily
included.
Above, for simplicity, the sample surface at finite separations from the colloids, as well as a
finite anchoring on real surfaces were disregarded. The above formulas (2)–(5), however, provide
a natural basis for straightforward incorporation of these surface effects. Indeed, these formulas
reduce the equilibrium elastic FE to the surface integral. But so is the anchoring energy, which
thus should be just added to the elastic FE.
If the real surface of the i-th particle is S̃i and the sample surface is S0, the total anchoring
energy Fa has the form
Fa =
N∑
i=1
∫
S̃i
fi,a(n⊥(s))d2s+
∫
S0
f0,a(n⊥(s))d2s, (7)
where fi,a and f0,a are the correspondent anchoring energy densities. Assuming that the director
field everywhere (even just outside S̃i) is described by the Laplace equation, the sum of the bulk
FE (2) and the anchoring energy (7) takes the form of the surface integral which is a functional
of the director field on the surfaces S̃i and S0, FN = FN{director field on all of the real surfaces}.
This functional is minimized by the standard boundary conditions to the bulk Laplace equation:
K(ν · ∇)nt + ∂fi,a(n⊥)/∂nt = 0, s ∈ S̃i ,
K(ν · ∇)nt + ∂f0,a(n⊥)/∂nt = 0, s ∈ S0 . (8)
Substituting here the solution (3) of the Laplace equation and using the definition (6) of the
potential UN , one finally obtains following equations for the director on the i-th colloid surface S̃i
33602-7
V.M. Pergamenshchik, V.A. Uzunova
and on the sample surface S0:
K
N∑
i=1
∫
S̃i
UN (s, s′)nt(s
′)d2s′ +K
∫
S0
UN(s, s′)nt(s
′)d2s′ + ∂fi,a(n⊥(s))/∂nt = 0, s ∈ S̃i ,
K
N∑
i=1
∫
S̃i
UN(s, s′)nt(s
′)d2s′ +K
∫
S0
UN (s, s′)nt(s
′)d2s′ + ∂f0,a(n⊥(s))/∂nt = 0, s ∈ S0 . (9)
The Green function which determines the potential UN via formula (6) now satisfies the zero
boundary condition on all of the real surfaces, i. e.,
GN (r, s) = 0, s ∈
⋃N
i=1
S̃i
⋃
S0 . (10)
Equations (9) are self-consistent integral equations showing that the perturbation of the field
at each point of any surface is superposition of the perturbations from all points of all the surfaces.
These equations are exact provided the bulk director can be determined from the Laplace equation.
Note that system (9) considerably simplifies if the director is fixed on some S̃k of S̃i. In this case the
equations for the surfaces S̃k are trivial and disappear from the system (9), while the contributions
from S̃k to other surfaces reduce to the fields of correspondent (known) elastic multipoles.
Thus, the problem of colloidal interaction in all cases, be it with fixed or soft boundary condi-
tions, is reduced to finding the appropriate Green function of the Laplace equation. We are again
to stress that the Green function method is the most direct and mathematically most justified and
developed way to address the problems of colloidal nematostatics. And, actually, it is this way that
makes the essence of the electrostatic analogy (in particular, Jackson’s “Classical Electrodynamics”
[64] can be as useful as it has been in solving electrostatic problems).
Actual experiments with colloids in a nematic liquid crystal are performed in confined ge-
ometries where the bounding surface gives rise to important or even dominating effects such as
exponential screening of the 1/r5 interaction at distances larger than the cell thickness [18]. As in
electrostatics, these effects can be addressed by introducing image-multipoles behind the confining
surface [59–63]. The image method for finite-size colloids [59, 60] is presented in section 3.4. Till
then we consider colloids far from confining surfaces.
3.2. Elastic charge density and multipoles of a single particle
Consider first a single particle enclosed by a sphere Si of radius ai and set the coordinate origin
at its center; all other particles are assumed to be very far. In this case the director field is the
unperturbed equilibrium field n
(i)
t (r) of a single i-th particle which is indicated by the subscript i.
The one particle Green function G
(i)
1 subject to the boundary condition G
(i)
1 (r, s) = 0 for s ∈ Si is
known exactly [65, 66]:
G
(i)
1 (r, r′) =
1
4π
(
1
|r − r′| −
ai
r
1
|i(r) − r′|
)
, (11)
where i(r) = (ai/r)
2r is location of the image of the point r in the spherical surface which has to
be taken with the coefficient ai/r, figure 1. The normal surface derivative of the Green function is
obtained in the form [57, 65]
(ν(s)·∇s)G
(i)
1 (r, s) = − 1
4π
r2 − a2
i
ai|r − s|3 . (12)
Substituting (12) in (3) gives the outer Poisson integral which is exact formula for the director
induced outside the single sphere by the director components on its surface, i. e.,
n
(i)
t (r) =
1
4π
∫
Si
n
(i)
t (s)
r2 − a2
i
ai|r − s|3 d2s. (13)
33602-8
Colloidal nematostatics
Let us introduce the surface density of the transverse director field
σ
(i)
t (s) = n
(i)
t (s)/a2
i . (14)
Then expansion of (13) in inverse powers of the distance r from the origin yields
n
(i)
t (r) =
q
(i)
t
r
+ 3
(d
(i)
t · r)
r3
+ 5
(Q
(i)
t : r : r)
r5
+ · · · , (15)
where
q
(i)
t =
ai
4π
∫
Si
σ
(i)
t d2s, (16)
d
(i)
t,α = − a2
i
4π
∫
Si
σ
(i)
t ναd2s, (17)
Q
(i)
t,aβ =
a3
i
8π
∫
Si
σ
(i)
t (3νανβ − δαβ)d2s (18)
(as usual, Q
(i)
t : r : r = Q
(i)
t,αβrαrβ). The form of (14)–(18) prompts the following interpretation.
q
(i)
t is the t-th component of the elastic charge and σ
(i)
t (s) is its surface density at point s on
the i-th sphere. The vector d
(i)
t and tensor Q
(i)
t are the t-th dipole and quadrupole moments
determined in the standard way by the surface charge density σ
(i)
t on the i-th sphere. As σ
(i)
x and
σ
(i)
y are two separate sources, the dyad, they determine not only the x and y director components
outside the single i-th particle, (13), but also two independent tensors, the tensor dyad, for each
multipole moment, i. e., q
(i)
x and q
(i)
y , d
(i)
x and d
(i)
y ,Q
(i)
x and Q
(i)
y , and so on. These multipoles are
uniquely determined quantities that characterize the director in the far zone of a single i-th particle
unperturbed by other particles. Along with the director n
(i)
t (r) and its large r asymptotics, given by
the r.h.s. of (15), the multipole moments do not depend on the choice of the enclosing surface. Our
choice of a sphere surface is just one possibility motivated by its simplicity: as indicated above, it
enables one to obtain the formulas for multipole moments in the explicit form given in this section.
In particular, below it enables us to express the elastic charge and Coulomb-like interaction solely
in terms of the mechanical torques exerted on the colloids.
3.3. Torque balance, Gauss’ theorem, and elastic charge in 3 dimensions
The fundamental physical quantity of electric charge is purely phenomenological and should
be postulated in the theory of elementary particles. By contrast, nematostatics of the director
field n allows for introduction of two different charges. Owing to the linearity of the one constant
approximation in 2d, the deformation source can be straightforwardly established: core of a point
defect plays the role of a charge in 2d [19, 52–54]. The integral, expressing the topological invariant,
is independent of the integration contour. This property plays the role analogous to Gauss’ theorem
in electrostatics while the invariant itself plays the role of a conserved charge. As a result, the 2d
nematostatics is similar to the 2d electrostatics with its logarithmic potential: disclinations with
topological charges of the same sign repel one another and those with topological charges of opposite
sign attract one another.
In 3d, however, the analogy between topological defects and charge is completely lost. In 3d,
the field n is described by highly nonlinear equations [19] so that point defects, though remain
topological invariants, cannot be linearly connected with the distortions of n they induce [10, 42].
Here the deformation source is the director distribution in a domain of the size ∼ a [57, 58], figure 1.
We seek the elastic analog of charge, following de Gennes’ idea outlined in [19].
A colloidal system can be subject to external electromagnetic fields. If colloids have a permanent
dipole (electric or magnetic) or finite polarizability, these fields exert a mechanical torque on
33602-9
V.M. Pergamenshchik, V.A. Uzunova
colloids. Due to its elasticity, a nematic liquid crystal transfers mechanical torque. A torque, exerted
on a particle, induces director deformations in the ambient medium. Nonzero deformations result
in a nontrivial torque balance in the director field. It is this torque balance that is described by
the standard Euler-Lagrange equations minimizing the FE functional. The torque balance implies
two torques at any spatial point. Namely, the balance at a given point shows that an elastic torque
is applied at both sides of any virtual surface passing through this point, and that these two
torques have equal magnitudes and opposite signs [19]. Thus, a torque on a particle, located within
a closed surface S, is transferred from inside S to outside S, the total torque being conserved.
It means that certain torque density, integrated over an arbitrary closed surface, is equal to the
total torque applied inside the surface irrespective of its form. In the static equilibrium, only a
transverse external torque Γ⊥ = (Γx,Γy, 0) can be exerted on a particle; a longitudinal torque Γz
cannot be balanced as a rotation about the z-axis does not change the elastic energy, Γz = 0. In
the one-constant approximation, the integral, which expresses the conservation and transfer of a
torque Γ⊥, is of the form [19]
Γt = K
∫
S
εαtρ
(
rρ
∂nγ
∂rβ
∂nγ
∂rα
+ nρ
∂nα
∂rβ
)
dSβ , (19)
where K is the elastic constant, εαtρ is the absolute antisymmetric tensor, all indices but t run over
x, y, z, and summation over the repeated indices is implied. The integral in the r.h.s. of (19) does
not depend on the choice of enclosing surface S, and the equality (19) reminds us Gauss’ theorem
with Γt in place of the electric charge. To further justify this connection one notices that integral
(19) over a remote surface S vanishes for any term in the expansion (15) except for the first one.
Substituting nt = qt/r in (19) and integrating over a sphere gives Γy = −4πKqx, Γx = 4πKqy, or
qt =
1
4πK
[Γ× n∞]t . (20)
The above consideration clearly shows that the Coulomb director field dying out as 1/r can
be induced by a colloid if and only if there is an external mechanical torque upon this colloid. In
particular, a colloid that has no symmetry planes does not induce the 1/r field if there is no external
torque (such a colloid is a chiral elastic dipole whose field is dying out as 1/r2, see section 4.2).
Thus, the tentative conclusion is that, in 3d, the role of Gauss’ theorem and a charge is played,
respectively, by the conservation of an elastic torque transferred via the director field, and by two
transverse components of an external torque exerted on a particle [59, 60]. This is fully justified
by calculating the Coulomb-like interaction in the elastic charge density representation [57, 58].
3.4. The two particle Green function, point-to-point potential and its expansion in a power
series of 1/r
Now consider two particles. In contrast to the single sphere case, the Green function G2(r, r
′)
for two spheres cannot be found in the form of a sum of the Coulomb source and a finite number
of its point-like images. In [57] we obtained the lowest order terms of expansion of the Green
function in a power series of the small radius-to-distance ratio by considering a few successive
images of the Coulomb source in two spheres. Here we briefly outline the method and give the
main results. Figure 2 shows the two spheres S1 and S2 of radii a1 and a2, centered at points o1
and o2 and representing two particles. The radius vectors of the points o1, o2, r, s1 . . . are denoted
by the correspondent bold symbols. The Green function G2(r, r
′) to the Laplace equation, subject
to the boundary condition G2(r, r
′) = 0 for any r′ ∈ B2 = S1
⋃
S2, is of the form
G2(r, r
′) = (4π)−1[1/|r− r′|+I(r, r′)], (21)
where I(r, r′) is a field of unknown “image” source that should compensate the field of the Coulomb
source 1/|r − r′| on the surface B2 [65]. The vectors i1 and i2 represent the first-order images of
r, respectively, in S1 and S2; the vectors i12 and i21 are the second-order images of the point r:
33602-10
Colloidal nematostatics
r
o 1 o 2
i1
i2
i1,1i2,1
s1
S 1 S 2
s2
Figure 2. The two spheres enclosing particles 1 and 2.
i12 is the image of the point i1 in sphere S2, and i21 is the image of the point i2 in sphere S1. The
function I is the sum of all of the image sources. The reflection process can be continued, but we
do not need higher images as they result in terms of higher order in the small parameter a/R than
the leading order. For r close to the surface S2, up to terms∼ O[(a/R)2], the Green function can
be written as
4πG2(r, s1) =
(
1
|r− s1|
− a1
|r − o1|
1
|i1 − s1|
)
− a2
|r − o2|
(
1
|i2 − s1|
− a1
|i2 − o1|
1
|i21 − s1|
)
. (22)
As seen from general formula (5), the pair interaction potential is the sum U2(s1, s2)+U2(s2, s1)
where U2 is determined by (6) for N = 2. Finally, the pairwise interaction potential is calculated
from the Green function as
U(s2, s1) = −a2
1a
2
2 [ν(s1)·∇s1
(ν(s2)·∇s2
)G2(s1, s2) + (1 � 2)] , (23)
where, for later convenience, we introduced the factor a2
1a
2
2. Calculation of (23) with the Green
function (22) gives the potential in the following form [57, 58]:
U(s2, s1) = −a1a2
2π
(R·(s2 − s1))
|s2 − s1|3
, s1 ∈ S1, s2 ∈ S2 , (24)
where R = o1 − o2. Now we introduce the unit vector u = R/R in the separation direction, the
vectors (directed inward the spheres) of unit outer normals ν1(s1) and ν2(s2), respectively, to the
surface S1 at point s1 and to S2 at point s2, and expand (24) in a power series of the inverse
separation 1/R. The result can be converted to the following important formula for the pairwise
point-to-point potential [57, 58]:
2π
a1a2
U(s2, s1) = − 1
R
− 2
a1(ν1 · u) − a2(ν2 · u)
R2
− 3
a1a2[(ν1 · ν2) − 3(ν1 · u)(ν2 · u)]
R3
+ · · ·
− 15a2
1a
2
2
4R5
[35(ν1 · u)2(ν2 · u)2 − 5(ν1 · u)2 − 5(ν2 · u)2 + 2(ν1 · ν2)2
− 20(ν1 · ν2)(ν1 · u)(ν2 · u) + 1] + · · · (25)
The four terms shown in this expansion give rise, respectively, to the Coulomb, charge-dipole,
dipole-dipole, and quadrupole-quadrupole colloid-colloid interaction potentials which will be de-
rived below.
The arguments s1 and s2 of the potential (25) belong to different surfaces. For completeness, we
give here the point-to-point potential for two points s1 and s2 lying on the same sphere S with the
radius a. This potential enters equations (9) and the colloid self-energy is introduced below. This
potential is equal to −a4 limr−→s
′(ν(r) · ∇r)(ν(s) · ∇s)G1(r, s) where G1 is given in equation (11).
The result is
U(s2, s1) =
−a4
2π|s1 − s2|3
, s1 ∈ S, s2 ∈ S.
33602-11
V.M. Pergamenshchik, V.A. Uzunova
3.5. Interaction of two particles at large separations, the pairwise approximation and its
limits
Now consider the energy of two particles of the previous section, figure 2. The director di-
stribution on Si, i = 1, 2, is now perturbed by another particle. In terms of the elastic charge
density (14) this means that the unperturbed single particle charge density σ
(i)
t (s) changed,
σ
(i)
t (s) → σ
(i)
t (s) + δσ
(i)
t (s). Then the total elastic FE F1&2 of the two particles, obtained from the
general many-particle FE (5) for N = 2, is the sum of the following terms:
F1&2 = F1{σ(1)
t } + F2{σ(2)
t } + F12{σ(1)
t σ
(2)
t } + ∆F1&2 . (26)
The first three terms are independent of the perturbation δσ. The first term is the self-energy of
the unperturbed particle 1 which does not depend on the state of particle 2. Similarly, the second
term does not depend on particle 1, so that these two terms do not contribute to interparticle
interaction.
The third term in (26) is the pairwise interaction FE given by the formula
F12 =
K
2
∫
S1
∫
S2
σ
(1)
t (s1)σ
(2)
t (s2)U2(s1, s2)d
2s2d
2s1 , (27)
where U2 is defined by (25). The last term in the FE (26) is the sum
∆F1&2 = ∆F12{σ(1)
t } + ∆F21{σ(2)
t } + ∆12{δσt} (28)
of the following corrections. ∆F12 is the correction to the self-energy F1 because of the presence
of particle 2, and ∆F21 is the correction to F2 because of the presence of particle 1. Though the
sum ∆F12+ ∆F21 appears as correction to the self-FE, it represents interaction as it depends on
the locations of both particles. The term ∆12{δσt} is the total FE contribution due to the final
perturbation δσ(i) of the single-particle fields of both particles. We also need the correction due to
nonlinearity of the director field. This term, which has the form [42]
Fan =
K
8
∫
Vout
dV
(
∇n2
⊥
)2
, (29)
is not present in the FE (5) of linearized field and should be added individually. The above F1&2
(26) totally disregards nonlinearity of the exact equations of nematostatics neglecting terms ∼
n2
⊥
= n2
x + n2
y. In this concern, the natural question arises as to how small should nt be on the
integration spheres Si in the formulas (14)–(18) for the multipoles that Fan be indeed sufficiently
small.
We have shown in [57] that in the system of similar multipoles (only charges, only dipoles, or
only quadrupoles), the term F12 (27) is the principal contribution while all the correction terms in
the sum (28) are higher by an order in the small parameter a/R, i. e.,
∆F12 + ∆F21 + ∆12 ∼ (a/R)F12 � F12 . (30)
The anharmonic correction Fan can be neglected as compared to the leading term F12 if Fan .
(a/R)F12, which gives
〈n⊥〉 .
√
a/R, (31)
where 〈n⊥〉 is the average magnitude of the transverse director on the integration sphere: 〈n⊥〉 ∼√
〈ntnt〉. This is a rather weak restriction, and nt on the integration sphere in equations (14)–(16)
can be quite appreciable. The inequalities (30) and (31) show that the leading order pair interaction
potential is F12 given in equation (27). Moreover, it was shown in [57] that the interaction between
colloids via the nematic director field is pairwise only in the leading order because already in the
next order there appear irreducible three-particle interaction terms.
33602-12
Colloidal nematostatics
Now the following important conclusions can be drawn as to the restrictions of the pairwise
approach in the case of particles bearing similar elastic multipoles. First, if the particle size and
interparticle distance are of the same scale, the omitted terms make the pairwise approximation
inapplicable, and one is left with the original field description alone. Therefore, the ratio (a/R)
should be small. Second, the pairwise approach is valid with the accuracy up to terms ∼ (a/R)F12.
Third, the perturbations of the individual multipole moments, induced by other particles and de-
scribed by the functional ∆{δσ} ∼ (a/R)F12, can be adopted within the pairwise approach only
in a system of two particles, because only in this case the three-body terms do not appear in
this order. This means that the pairwise term F12 describes the interaction between, respectively,
the unperturbed elastic charges, dipoles, and quadrupoles, which are fully determined by the un-
perturbed director distributions of individual particles. Since, as explained above, these multipole
moments are uniquely determined characteristics of an individual particle, the pairwise interaction
is also uniquely determined by equations (27). In particular, it is independent of our choice of the
enclosing surfaces.
It is known that correlation between two particles via the third one statistically occurs in the
second virial coefficient of the thermodynamic virial FE expansion. In our case, the correlations
are of static origin and are much stronger: they give rise to FE corrections in the order a/R
whereas the statistical mechanical correlations appear in the order (a/R)3 (the small parameter of
thermodynamics is the packing fraction ∝ (a/R)3).
3.6. Interaction of two similar elastic multipoles: the Coulomb, dipole-dipole, and
quadrupole-quadrupole potentials
Here we consider the leading interaction term F12 which, at the same time, is the only pairwise
term, by substituting expansion (25) in (27). Consider contribution of the first term in (25) to F12.
Using the definition of elastic charge qt (16) we obtain [57, 58]
UCoulomb = −4πK
q
(1)
t · q(2)t
R
, (32)
which is obviously the Coulomb interaction. We see that, in contrast to the electrostatic Coulomb
law and 2d nematostatics, charge components with similar signs attract while those with different
signs repel each other. The relation (20) allows us to express the Coulomb term via the transverse
components Γ⊥ = (Γx,Γy) of the torques exerted upon the particles:
UCoulomb = − (Γ
(1)
⊥
· Γ(2)
⊥
)
4πKR
. (33)
This formula shows that the Coulomb attraction is induced when (Γ
(1)
⊥
· Γ
(2)
⊥
) > 0 while, if
(Γ
(1)
⊥
· Γ(2)
⊥
) < 0, the torques induce a Coulomb repulsion: parallel torques attract while antipar-
allel torques repel each other. An important conclusion from the formula (33) is that the elastic
Coulomb-like interaction can be induced only by torques upon the interacting particles and does
not directly depend on their form and anchoring [57, 58]. In particular, breaking the horizontal and
vertical mirror symmetry of the director field, induced by asymmetric particles, is insufficient for
the Coulomb interaction. Calculation of an external torque requires much less information about
the source than that contained in the distribution of anchoring strength and easy axes over the
colloid surface. For instance, if the anchoring is very strong, then the torque that can be exerted
on the colloid by a given (electric or magnetic) field, is determined by the vector of permanent
dipole (electric or magnetic) or the polarizability tensor of the particle, and no further information
about the anchoring is needed. If the anchoring is weak, then it determines the maximum value of
the applied torque.
If the external torques are not applied, the interaction F12 can be expressed solely in terms of
multipoles of the particles. Using definition (17) of the elastic dipole, the potential (27) calculated
33602-13
V.M. Pergamenshchik, V.A. Uzunova
with the third term of the expansion (25) is obtained in the following form:
Udd = −12πK
(
d
(1)
t · d(2)
t
)
− 3
(
d
(1)
t · u
)(
d
(2)
t · u
)
R3
, (34)
which can naturally be interpreted as the elastic dipole-dipole interaction.
Using the definition (18) of the elastic quadrupole, the potential (27) calculated with the fourth
term of the expansion (25) gives the elastic quadrupole-quadrupole potential:
UQQ = −20πK
3R5
[
2
(
Q
(1)
t : Q
(2)
t
)
+ 35
(
Q
(1)
t : u : u
) (
Q
(2)
t : u : u
)
− 20
(
Q
(1)
t : u : Q
(2)
t : u
)]
.
(35)
Thus, we derived the pairwise interaction F12 for two interacting charges, dipoles, and quadrupo-
les. The natural relation between the definitions (14)–(18) and the interaction potentials (32)–(35)
supports our idea of the elastic charge density.
Below we will consider the geometrical structure of different elastic multipoles, present a math-
ematical classification of possible dipoles and quadrupoles, illustrate it by sketching some repre-
sentative colloid shapes, and consider important examples of the multipole interaction.
4. Tensorial structure and classification of different elastic dipoles
4.1. Dyads of elastic multipoles and their transformation in the intrinsic (a la isotopic)
space
The definition (14)–(18) and interaction potentials (32)–(35) suggest that there are two scalars
of elastic charge, two vectors of elastic dipole, two tensors of elastic quadrupoles, i. e., all of the
elastic multipoles are dyads. A question arises when this interpretation is correct. Imagine that
the director distribution at the source, nx and ny, changes but this change has nothing to do with
a rotation about the z-axis, e. g., colloids change alignment at their surfaces dynamically. Then,
the new components after the change, n′
x and n′
y, have no connection with their old values, nx and
ny, via a tensorial transform of the local (i. e., individual for each colloid) reference frame in the
real x, y-space. Similarly, the new values of dt,α (Qt,αβ) have no connection with their old values
via a local tensorial transform in the real x, y-space. Such a change can be considered as a local
transform in some intrinsic space with the coordinates nx and ny or dx and dy (Qx,αβ and Qy,αβ),
which is individual for each colloid and reminiscent of the isotopic space of nuclear physics. Thus,
under such an “intrinsic” transformation, nx and ny do not transform as components of a spatial
vector. Then qt is not a spatial vector but rather can be viewed as two independent charges, a
dyad; similarly, dt,α (Qt,αβ) is not a spatial tensor but a dyad of two independent vectors dx,α and
dy,α (tensors Qx,αβ and Qy,αβ). Another obstacle in viewing the objects dt,α and Qt,αβ as tensors
is that t runs over 1 and 2 whereas α and β run over 1, 2, and 3. Nevertheless we need to rotate an
elastic multipole about the z-axis in the real space to a new reference frame. To this end, in the
next section we make it a tensor in the dimension 2 + 1 = 3.
4.2. Elastic dipoles in the 2+1 and 2 dimensions
Since the rotation about a homogeneous director n∞, which is along the z-axis, does not alter
the FE of a particle, all the elastic multipoles should be determined up to an arbitrary azimuthal
angular variable φ. Let us consider the subscript α = t′ with the values 1 and 2 and with the
value α = 3 separately. The indices taking values 1 and 2 will be denoted by t or t′. Then qt and
dt,3 are transformed as a 2d vector, and dtt′ is transformed as a 2nd rank tensor.
The 2d vector (16) of elastic charge can be formally imbedded into the 2 + 1 = 3 dimensional
space by defining the 3d elastic charge vector
q = (qx, qy, 0) =
1
4πK
[Γ × n∞] (36)
33602-14
Colloidal nematostatics
where Γ = (Γx,Γy, 0) is the torque vector. Though “vector of charge” does sound unusual, the
tensorial structure of a vector = first rank tensor of elastic charge is quite simple and absolutely
clear.
An elastic dipole is a more complex quantity. Let us imbed dt,β (17) into the 2 + 1 dimensions.
To this end, we introduce a second rank 3 × 3 tensor dα,β with the matrix ||dα,β || whose first and
second rows coincide, respectively, with the components of dx and dy,i. e.,
||dα,β || =
dx,x dx,y d3,x
dy,x dy,y d3,y
0 0 0
=
dx
dy
0
. (37)
In turn, the first two rows can be decomposed into the second rank 2 × 2 tensor D and the 2d
vector d3, i. e., (
dx
dy
)
= (D⊕ d3) , (38)
D = ||dt,t′ || =
(
dx,x dx,y
dy,x dy,y
)
, (39)
d3 = (d3,x, d3,y). (40)
Consider D and d3 individually and find their general φ-dependent form induced by a rotation of
the particle about the z-axis.
A general 2nd rank tensor D is a sum of its symmetric and antisymmetric parts. In the proper
reference frame O0, D can be reduced to a special form D0 with the diagonal symmetric part, i. e.,
D0 =
(
D11 0
0 D22
)
+ C
(
0 1
−1 0
)
. (41)
The antisymmetric part is invariant with respect to rotations about the z-axis and has the same
form and coefficient C in O0 and in any other reference frame. A symmetric 2nd rank tensor can
be decomposed into an isotropic part ∝ δij and anisotropic traceless part. In particular, D0 (41)
(i. e., D in the particular reference frame O0) can be written as
D0=d
(
1 0
0 1
)
+ ∆
(
1 0
0 −1
)
+ C
(
0 1
−1 0
)
, (42)
where d = (D11 + D22)/2, ∆ = (D11 − D22)/2. Now we want to find the form of the tensor D
in an arbitrary reference frame Oφ that can be obtained from O0 by rotating it by an angle φ
anticlockwise. Such a rotation induces the following transformation of an arbitrary vector r =
(r1, r2, r3): r → r′ = (r′1, r
′
2, r
′
3), where r′α = Rαβ′rβ′ and the matrix ||Rαβ′ || is of the form
||Rαβ || =
cosφ sinφ 0
− sinφ cosφ 0
0 0 1
. (43)
The 2nd rank tensor D0 is transformed to the reference frame Oφ as Dtt′ = RtsRt′s′D0ss′ . In the
context of (42) and (43) this finally gives
D =d
(
1 0
0 1
)
+ ∆
(
cos 2φ − sin 2φ
− sin 2φ − cos 2φ
)
+ C
(
0 1
−1 0
)
. (44)
One can easily verify that d3 transforms as a 2d vector. If d3 is equal to d
(0)
3 = (d
(0)
3,x, d
(0)
3,y) in
the reference frame O0, then d3,t = Rtt′d
(0)
3,t′ . Thus, the dyad of dipole “vectors” in an arbitrary
reference frame Oφ has the form
dx = (d+ ∆ cos 2φ, − ∆ sin 2φ+ C, d
(0)
3,x cosφ+ d
(0)
3,y sinφ), (45)
dy = (−∆ sin 2φ− C, d− ∆ cos 2φ, d
(0)
3,y cosφ− d
(0)
3,x sinφ).
33602-15
V.M. Pergamenshchik, V.A. Uzunova
The angle φ can also be viewed as a rotation angle of the particle itself in the fixed reference
frame O0 where the dipole has the diagonal form D0 (44). Then φ is the angle of a clockwise
particle rotation from its special diagonal state D(φ = 0) = D0 to an arbitrary state D: after
this rotation the form of the dipole tensor D(φ) will be presented by formula (45). Note that
formula (17) provides the following estimates for the magnitude of the coefficients d and ∆.
The coefficient d describes an isotropic source, which in our case is uniaxial and produce no spiral
component (no chirality): d ∼ a2 〈|nt|〉 where 〈|nt|〉 is the average of the absolute value of the
transverse director component over the enclosing sphere of radius a. The coefficient ∆ describes
an anisotropy related to biaxiality of a nonchiral source: ∆ ∼ a2(〈|nx|〉 − 〈|ny|〉). Since d and
∆ contribute to the transverse components of the dyad (45) normal to the unperturbed director
n∞, we will refer to them as uniaxial and biaxial transverse dipole strength, respectively. The
antisymmetric tensor with the coefficient C describes chiral strength of a source particle. Our idea
of chiral source is similar to that by de Gennes [19] and Fournier [56], but it is expressed directly in
terms of the director components (see section 4.3.1 below). The vector d3 is a longitudinal dipole
since it consists of the components of the dipole dyad (45) along n∞.
The formulas (38) and (42) provide a mathematical basis for classifying all the possible types
of dipolar colloids. Thus, in general, an elastic dipole can be described by the three coefficients d,
∆, C, and the vector d3: the isotropic dipole strength, anisotropy, chirality, and the longitudinal
component. Now we discuss a connection between d, ∆, C, d3 and the symmetry of source particle
(which throughout this article implies symmetry of the equilibrium single-particle director field in
the vicinity of the particle determined by distribution of the polar and azimuthal anchoring over
the particle surface).
Asymmetric particle (general chiral dipole). In the case of a general asymmetric particle, all the
three coefficients of the tensor D (44) and the vector d3 are nonzero. In contrast to the assertion
of [44] where the elastic charge was associated with the total absence of mirror reflection planes,
such a particle is a general chiral elastic dipole rather than an elastic charge.
Figure 3. Three nonchiral dipoles inducing the director fields with symmetries exemplified by
conventional forms: a) uniaxial dipole is isotropic (circular cone), b) biaxial dipole is anisotropic,
it is flattened from one side (elliptical cone); c) longitudinal dipole with a horizontal mirror plane
(vertical hemisphere). The azimuthal easy directions of the surface anchoring of the particles
have no helicoidal component.
The C∞v symmetry (uniaxial particle) (see [67] for symmetry notations). An important partic-
ular case of a colloid is a particle with the symmetry C∞v , i. e., infinite order rotational symmetry
about the z-axis plus mirror symmetry with respect to any “vertical” plane passing through the
z-axis. Such a particle is uniaxial, i. e., isotropic, ∆ = 0, and nonchiral, C = 0; its dipole is trans-
verse, d3 = 0. The only nonzero coefficient is d. Here we call such a particle uniaxial; examples of
uniaxial particles are sketched in figure 3.
The C∞ symmetry (chiral helicoid). An azimuthally symmetric particle with the symmetry C∞
without reflection planes (e. g., a helicoid, figure 4 (a), (b), (c)) is a chiral source with nonzero
C, but with ∆ = d3 = 0. From the physical point of view, the mirror reflection planes of a C∞v
33602-16
Colloidal nematostatics
uniaxial particle can be eliminated, and, at the same time, the chirality can be induced, by an
azimuthal anchoring with the helicoidal alignment of its easy axes, figure 4 (a), (b), (c). In general,
C is eliminated by any single symmetry plane (horizontal or vertical); both d and ∆ are eliminated
by a horizontal mirror plane; whereas d3 is eliminated by two vertical mirror symmetry planes,
but can be nonzero for one vertical and one horizontal reflection planes, figure 3 (d). For instance,
the well-known bent-core molecules with their planes along the director have a nonzero d3.
Figure 4. Three chiral dipoles, a), b), and c), and an onion-like quadrupole, d). The helicoidal
field of the azimuthal anchoring easy directions on the surfaces of the dipoles makes them chiral.
The removal of the helicoidal easy directions on the surface of sphere c) causes the disappearance
of all the dipolar coefficients, and the colloid d) is a uniaxial onion-like bipolar quadrupole.
The C1v symmetry (general biaxial particle), C2v symmetry (biaxial particle). A C1v particle
with a single vertical reflection plane has no chirality, C = 0, but the vector d3 is not generally zero.
Its tensor D is symmetric and has finite anisotropy ∆ 6= 0. We refer to such particles as general
biaxial. The C2v symmetry with two mutually perpendicular vertical reflection planes passing
through the z-axis eliminates the vector d3, d3 = 0, and particles with this symmetry constitute
an important particular case of anisotropic colloids which we call (just) biaxial, figure 3 (b). A
uniaxial particle is circular in its horizontal cross-sections. A general biaxial particle is biaxial in
the horizontal cross-sections, its director distribution in the x, y planes has mirror symmetry with
respect to one of the two principal axes, e. g., it is an isosceles triangle. A just biaxial particle
is more symmetric: its director distribution in the x, y-cross-sections has mirror symmetry with
respect to both principal axes, e. g., the field in x, y-cross-sections is elliptic. Thus, under rotation
of a biaxial particle about the z-axis its elastic dipole transforms as a 2nd rank 2 × 2 tensor (44)
with the isotropic fraction of magnitude d, anisotropic fraction of magnitude ∆, and C = 0, in
which φ is the angle of the anticlockwise rotation of the particle from the diagonal dipolar state.
Very recently we have described possible symmetry operations and point groups of elastic
dipoles. These unpublished results can be summarized as follows. There are four pure types char-
acterized by a single nonzero coefficient, the four dipolar singlets, and eight mixed types: single
dipolar quintet, single quartet, two triplets, and four doublets. The idea of a single electrostatic
dipole is thus left far behind: the nematostatic dipoles are much more numerous and versatile.
A dipolar particle cannot have three mutually perpendicular planes of symmetry as these would
make it an elastic quadrupole.
4.3. Elastic quadrupoles in the 2+1 dimensions
It turns out that if the dipolar dyad of a colloid is zero its elastic quadrupole Qt,αβ (18) is
a much less complex quantity than an elastic dipole. The reason is that a quadrupole particle is
highly symmetric. For instance, two vertical and a horizontal reflection symmetry planes eliminate
all the four dipolar coefficients and make the particle a quadrupole. Consider such a particle. In
the proper reference frame O0 with the x-and y axes lying in the vertical symmetry planes, the
33602-17
V.M. Pergamenshchik, V.A. Uzunova
dyad of quadrupole tensors has the following form:
||Qx,αβ|| =
0 0 Qx
0 0 0
Qx 0 0
, ||Qy,αβ|| =
0 0 0
0 0 Qy
0 Qy 0
. (46)
To imbed this dyad into the 2+1=3 dimensional space, one can take ||Q3,αβ || = 0 (similarly to the
zero third line in the matrix (37)). Each tensor above is symmetric by the definition of Qt,αβ and
depends on a single number, Qx or Qy. If the particle is axially symmetric, then Qx = Qy = Q
and we call it a uniaxial quadrupole. The well-known examples of such particles are the Saturn
ring quadrupole [36, 39, 42], induced by a sphere with normal boundary condition, and an onion-
like bipolar quadrupole, induced by a sphere with tangential surface director field aligned along
the meridians of the sphere, figure 4 (d). If Qx 6= Qy, particle is a biaxial quadrupole. A biaxial
quadrupole is anisotropic in the horizontal cross-sections and can be characterized by the isotropic
coefficient Q = (Qx +Qy)/2 and anisotropy ∆ = (Qx −Qy)/2. Choosing the x-axis of the proper
reference frame O0 along the longer symmetry axis of the director near field (which is equivalent to
|Qx| > |Qy|), we can make both coefficients of the same sign, Q∆ > 0. For instance, an anisotropic
modification of the Saturn-ring and onion quadrupoles would be obtained by flattening in the y-
direction. Then, for anisotropic Saturn ring quadrupole, Q > 0 and ∆ > 0; whereas for anisotropic
onion quadrupole, Q < 0 and ∆ < 0. Obviously, ∆ = 0 for a uniaxial quadrupole.
Under anticlockwise rotation (43) of the reference frame by the angle φ from O0 the components
(46) transform as the third rank tensors with Q3,α,β = 0, i. e., Q′
t,αβ = Rtt′Rαα′Rββ′Qt,α′β′ .
Performing this calculation gives the dyad (46) in the reference frame Oφ:
||Qx,αβ|| =
0 0 Q+ ∆ cos 2φ
0 0 −∆ sin 2φ
Q+ ∆ cos 2φ −∆ sin 2φ 0
,
||Qy,αβ|| =
0 0 −∆ sin 2φ
0 0 Q− ∆ cos 2φ
−∆ sin 2φ Q− ∆ cos 2φ 0
. (47)
For φ = 0, formulas (47) go over into (46). For a uniaxial quadrupole with ∆ = 0, the tensors
Qt,α,β are invariant and (47) remain of the form (46) in any reference frame.
The azimuthal anchoring plays an exclusive role for elastic quadrupoles. Indeed, the reflection
planes can be eliminated by a helicoidal component of the (tangential) easy axes of azimuthal
anchoring, which can make the otherwise quadrupolar particle an elastic dipole. For instance, an
onion quadrupole is indeed a quadrupole if the easy directions on its surface are along the meridians
connecting the upper and lower poles (boojums). If, however, the field of tangential easy axes is
twisted and has a helicoidal component, the onion-like particle becomes a chiral dipole, 4c. This
dipole has all the coefficients zero but C which illustrates that C is indeed the chirality. Another
interesting example is a particle made of two cones of the same dimensions but with opposite sense
of chirality, whose bases are attached to each other along the horizontal symmetry plane. Such
a particle is not chiral (due to the symmetry plane), but rather an elastic dipole with nonzero
longitudinal component d3.
To conclude this section, we note that, depending on its anchoring, a particle of the same shape
can be a dipole or quadrupole. For instance, an equilateral triangle-shaped platelett is a dipole if
its plane is parallel to n∞ (planar anchoring), but it is a quadrupole if the plane is normal to n∞
(normal anchoring).
4.3.1. The electrostatic analogy and tensorial multipole structure of [56]
A vector of elastic charge similar to our qt and the 3×3 tensor of dipole moment were introduced
by Fournier in [56]. His analysis of the elastic dipole [56] and our analysis presented above share
certain common ideas, too. However, the difference is substantial. Using the techniques proposed
by de Gennes, Fournier also introduces the chirality, the isotropic and anisotropic terms, but
33602-18
Colloidal nematostatics
rather for the rotation vector than for the director components themselves. The two approaches
also differ in the choice of the analogy between electrostatics and nematostatics. In our approach
the transverse director components nx and ny are considered to be an analogue of electrostatic
potential ϕ [39, 40, 43, 44, 57, 58]. The fundamental reason is that nt enters the FE functional
the same way ϕ enters the electrostatic energy. Then, similar to ϕ, which is the canonical variable
for the electrostatic field theory, the vector (nx, ny) is the canonical variable for the nematic field
theory. Another way, chosen in [56], is to seek analogy between the director and the electric field,
i. e., between nt and the gradients of ϕ. The analogy can be established between the director field
induced by an elastic dipole and the electric field of an electric charge (both scale as 1/r2). Of
course, technically this can be justified, but such an analogy is no longer direct which results in
certain mismatch. In particular, the elastic charge has no electrostatic counterpart (electric charge is
already connected with the elastic dipole). We believe that the direct analogy [39, 40, 43, 44, 57, 58]
between nt and ϕ is more natural, deep, and far going than the indirect analogy chosen in [56]
which becomes obvious when the multipole interaction is considered.
4.4. Particular examples of the elastic dipole-dipole and quadrupole-quadrupole interac-
tion
Now we illustrate the general interaction potential by important examples. First we substitute
particular dipole dyads (45) to the general dipole-dipole interaction potential (34) to obtain its
particular form for some dipolar particles. To this end, we choose a separation vector R to lie in
the xz-plane as shown in figure 5, introduce the polar angle θ made by the unit separation vector
u and the z-axis, and the azimuthal angle φi, i = 1, 2, made by the actual x-axis and the x-axis
of the proper reference frame O
(i)
0 of the i-th particle, figure 5. Then, the separation vector lies in
the xz plane, u = (sin θ, 0, cos θ), and the dipole diad of the i-th particle is given by formula (45)
in which all the parameters are indicated by index i.
Figure 5. Two biaxial dipolar colloids shown in the vertical zx-plane (left) and horizontal xy-
plane (right). The polar angle θ and the azimuthal angles φ1 and φ2 made by the longer symmetry
axes of particles 1 and 2 and the zx-plane.
The i-th C∞v uniaxial dipole is characterized by ∆ = 0, d
(i)
x = (di, 0, 0), d
(i)
y = (0, di, 0). The
interaction potential of two uniaxial particles is of the form
U1d−1d =
12πKd1d2
R3
(1 − 3 cos2 θ), (48)
which, in particular, describes the interaction of topological dipoles [42].
The i-th C∞ uniaxial chiral dipole is characterized by ∆ = 0, d
(i)
x = (di, Ci, 0), d
(i)
y =
(−Ci, di, 0). The interaction potential of two uniaxial chiral particles is of the form
U1dC−1dC =
12πK(d1d2 + C1C2)
R3
(1 − 3 cos2 θ). (49)
33602-19
V.M. Pergamenshchik, V.A. Uzunova
This formula is incapable of demonstrating the full effect of chirality which can be described only if
the restriction of uniform unperturbed director is removed. Then the effect of chirality is expected
to be a spiral cholesteric-like rotation of the director field.
The dipole dyad of an i-th biaxial dipole is obtained from (45) for C(i) = d
(i)
3,t = 0: d
(i)
x =
(di +∆i cos 2φi,−∆i sin 2φi, 0), d
(i)
y = (−∆i sin 2φi, di −∆i cos 2φi, 0). The interaction potential of
two biaxial dipoles has the form
U2d−2d =
12πK
R3
[
(d1d2 + ∆1∆2 cos(2φ1 − 2φ2))(1 − 3 cos2 θ)
+3 sin2 θ(d2∆1 cos 2φ1 + d1∆2 cos 2φ2)
]
. (50)
Now we substitute quadrupole diad (47) to the general quadrupole-quadrupole interaction
potential (35) to obtain its particular form for uniaxial and biaxial quadrupolar particles. The i-th
quadrupole is determined by coefficients Qi and ∆i, (47). Two biaxial quadrupoles interact via the
potential
U2Q−2Q =
80πK
9R5
[(Q1Q2 + ∆1∆2 cos(2φ1 − 2φ2))(9 − 90 cos2 θ + 105 cos4 θ)
− 5(Q2∆1 cos 2φ1 +Q1∆2 cos 2φ2) sin2 θ(21 cos2 θ − 3)]. (51)
The potential for two uniaxial quadrupoles is obtained by setting ∆i = 0, which gives
U1Q−1Q =
80πK
9R5
Q1Q2[9 − 90 cos2 θ + 105 cos4 θ]. (52)
This formula differs from that derived for Saturn ring quadrupoles in [42] only by coefficient.
General formulas for the interaction potentials (32)–(35) and their particular forms derived in
this Subsection solve the first problem of colloidal nematostatics formulated in the Introduction:
they provide the form of interaction between different types of colloids. However, these formulas
can be used if the material parameters of the multipoles are known. The problem of finding these
parameters in terms of the shape of the colloid and anchoring is entirely different from the first
one and can be addressed only numerically. Nevertheless, formulas (17) and (18) greatly simplify
the calculation of a dipole moment. For instance, the coefficient d for the topological dipole and
quadrupole [42] mentioned in this review more than once, was calculated by using ansatz (29) of
[42]. Although this anzatz describes the near field only qualitatively, integrating it over the sphere,
enclosing the pair, closely reproduces the result which was obtained in [42] by tailoring the fields
in the far and near zones [57, 58]. Notice, that equation (17) was used in [16] to estimate the dipole
moment of liquid droplets trapped at a nematic-air interface.
5. The mirror image method of colloidal nematostatics
The above formulas can be used to solve boundary problems similar to those of electrostatics.
The simplest boundary problem is the interaction of an elastic multipole with a plane surface
bounding the nematic sample and imposing a fixed homogeneous director alignment. Here we
consider this problem for the elastic charge and dipole. The field of colloidal emulsions deals with
macroscopic distances, and here we are interested in macroscopic colloid-wall separations of an order
of micrometers. Until recently, the problem of calculating a mechanical torque on a particle near a
nematic liquid crystal surface and its energy has been specified by a particular shape, anchoring,
and orientation of the particle, and thus could be addressed only numerically [68, 69]. The theory
[57, 58] developed above considerably simplifies the problem. The specific parameters of a particle
now enter the problem via its multipole moment, and the problem reduces to the interaction
between a wall and an elastic multipole. Now this problem can be solved analytically in a universal
form by means of an image method specific to nematostatics. Using the mirror image method of
electrostatics as a guiding idea, we developed a mirror image method in nematostatics for arbitrary
33602-20
Colloidal nematostatics
director tilt at the wall [59, 60]. A wall is shown to induce a repulsive tilt-dependent 1/R4 force
on an elastic dipole and its reorientation to the minimum energy alignment. The external torque,
however, induces the elastic charge in this colloid and triggers switching to the 1/R2 repulsion.
The calculations demonstrate that the dyadic nature of an elastic dipole is essential.
Consider a single particle at distance h from a plane surface of a nematic liquid crystal sample
(wall). We assume that the anchoring of the wall is strong, the director alignment in the sample
far from the particle is homogeneous and parallel to the z-axis, n∞ = (0, 0, 1), but the angle θ it
makes to the surface normal is arbitrary, figure 6; θ = π/2 and θ = 0 correspond to the planar and
homeotropic surface director alignment, respectively. To justify the linearized theory, h is assumed
to be large compared to the particle size ∼ a. As the director on the wall is fixed, the boundary
condition consists of two equations
nt(rwall) = 0, t = x, y, (53)
which have to be satisfied for any point rwall of the wall. This boundary condition (53) can be
satisfied by placing an image-particle with the multipole moment of the same order on the other
side of the wall.
Figure 6. Geometry of the problem considered. A wall (bold line), a particle at O1, and its
image at O2. The z-axis, which is along the unperturbed director n∞, makes the angle θ with
the normal to the wall. The x-axis lies in the figure plane whereas the y-axis is normal to it and
directed from the reader. The perturbation nt of n∞ vanishes on the wall: nt = 0, t = x, y. The
director field outside the dashed spheres, representing a finite size of the particle and its image,
obeys linear equations.
Consider an elastic multipole at a distance h from a wall and its image multipole located at
the mirror point behind the wall at the distance R = 2h from the real particle, figure 6. Variables,
attributed to the particle and its image, will be indicated by index 1 and index 2, respectively.
The director field, which determines the multipoles, is fixed at the spheres S1 and S2 of radius
a enclosing, respectively, the particle and its image. To the order a/R, the transverse director
components nt on the wall are given by the sum [58]
nt(r) = n
(1)
t (r)
(
1 − a
r2
)
+ n
(2)
t (r)
(
1 − a
r1
)
, (54)
where n
(1)
t is the far field (15) of the single particle, r1 = r − o1, r2 = r − o2, where o1 and o2
are the radius vectors of the two centers. Equation (54) explicitly shows nonadditivity of the fields
produced by the two particles: nt = n
(1)
t on S1 (r1 = a) and nt = n
(2)
t on S2 (r2 = a); the additivity
nt(r) ∼ n
(1)
t +n
(2)
t takes place only when both a/r1 � 1 and a/r2 � 1, i. e., far from the particles.
We set the reference frame with the onset on the wall between the particles so that the xz-plane
(with z-axis along n∞) is perpendicular to the wall, and the y-axis is normal to the figure plane,
figure 6. Then o1 = (−h sin θ, 0,−h cos θ), o2 = (h sin θ, 0, h cos θ), an arbitrary point of the wall
33602-21
V.M. Pergamenshchik, V.A. Uzunova
rwall = (−z cos θ, y, z sin θ), and
r1 = (−z cos θ + h sin θ, y, z sin θ + h cos θ),
r2 = (−z cos θ − h sin θ, y, z sin θ − h cos θ). (55)
The particle-image separation vector is R = o2 − o1 = Ru, where u = (sin θ, 0, cos θ). In the
context of (54) and the equality r1 = r2, the condition nt = 0 on the wall yields
n
(1)
t (rwall) + n
(2)
t (rwall) = 0, t = x, y. (56)
These equations determine the components of the image multipole. We will find the image charge
and image dipole for arbitrary tilt θ and then, by substituting its components in the interaction
energy (32) or (34), calculate the interaction between the colloid and the wall. Below we consider
a charged, nonchiral C∞v uniaxial, biaxial, and general dipolar particle individually.
5.1. Elastic charge-wall interaction
The charge q
(1)
t can be induced by an external field exerting the torque Γ with the components
Γy = 4πKq
(1)
x and Γx = −4πKq
(1)
y ; the image-charge q
(2)
t is on the other side of the wall at the
distance h from it, figure 6. The director field of a single elastic charge is n
(i)
t = q
(i)
t /ri. Then
from (56) we obtain q
(2)
t = −q(1)t , t = x, y. Thus, the particle and its image have opposite charges
which corresponds to Γ
(2)
⊥
= −Γ
(1)
⊥
, figure 7. Since two opposite elastic charges repel one another,
the elastic charge-wall interaction is repulsive. The repulsion force is obtained from the interaction
Figure 7. Elastic charge at a wall with fixed planar (left sketch) and homeotropic (right sketch)
director alignment. Elastic charge qt induced by an external torque Γ and its image −qt induced
by the image-torque −Γ. The director at the wall remains unperturbed and equal to n∞.
energy (32) of the torques Γ⊥ and −Γ⊥ by differentiating with respect to R at R = 2h, i. e.,
Fq−wall =
Γ2
⊥
16πKh2
. (57)
The result depends on the direction of n∞ and thus on the surface tilt only via the trivial relation
Γ⊥ = Γ − (Γ · n∞).
5.2. Interaction of a uniaxial C∞v dipole with a wall
Now consider the interaction between a wall and an elastic dipole dyad (dx,dy), (45). Consider
first the simplest and practically important case of a uniaxial C∞v particle with the isotropic
nonchiral dipole dyad:
d(1)
x = (d, 0, 0), d(1)
y = (0, d, 0). (58)
33602-22
Colloidal nematostatics
To solve the boundary condition (56), the image-dipole d
(2)
t is chosen in the form
d(2)
x =
(
d(2)
x cosψ, 0,−d(2)
x sinψ
)
,
d(2)
y =
(
0, d(2)
y , 0
)
, (59)
where ψ is the angle that d
(2)
x makes with the x-axis. The director field of a single i-th dipole at
point rwall of the surface is n
(i)
t = 3(d
(i)
t · ri)/r
3
i , (15), and the boundary conditions read
(
d
(1)
t · r1
)
+
(
d
(2)
t · r2
)
= 0, t = x, y. (60)
Substituting (58),(59) and (55) in (60) yields the equations which can be reduced to the form
d sin θ − d(2)
x sin(θ − ψ) = 0,
d cos θ + d(2)
x cos(θ − ψ) = 0,
d+ d(2)
y = 0. (61)
This system is solved by d
(2)
x = d
(2)
y = −d, ψ = 2θ, i. e.,
d(2)
x = (−d cos 2θ, 0, d sin 2θ),
d(2)
y = (0,−d, 0). (62)
Substituting (62) in (48) with u = (sin θ, 0, cos θ) yields
U1d = 12πK
d2(sin2 θ + 2)
R3
. (63)
Differentiating (63) with respect to R at R = 2h, we obtain the force with which the wall repels
a uniaxial dipole:
F1d−wall = πK
9d2(2 + sin2 θ)
4h4
. (64)
Now consider two important particular cases of the planar and homeotropic wall.
For the planar wall θ = π/2, the image has the form d
(2)
x = (d, 0, 0), d
(2)
y = (0,−d, 0), figure 8,
and the repulsive force has the magnitude
F1d−wall,planar = πK
27d2
4h4
. (65)
For the homeotropic wall θ = 0, d
(2)
x = (−d, 0, 0), d
(2)
y = (0,−d, 0), figure 9, and the repulsive
force has the magnitude
F1d−wall,hom = πK
9d2
2h4
. (66)
The force F1d−wall,hom is 1.5 times weaker than F1d−wall,planar. In this concern we would like to
emphasize that the results, obtained above by means of nematostatics of [57, 58], show that even
in the simplest case of a homeotropic and planar wall, the dyadic nature of the elastic dipole is
essential. Indeed, a single dipole component is insufficient to satisfy the two boundary conditions
(53) on the wall. Therefore, phenomenological theories, such as [42], which deal solely with a
uniaxial dipolar particle and describe it by a single coefficient d, cannot be used in the problem of
a particle-wall interaction. Nevertheless, let us compare our result with what could be “naively”
obtained from the phenomenological approach in the simplest cases. When the poles of a single
dipole interchange, the dipole changes its orientation to the inverse one and is described by the
coefficient (−d). In this situation, the image dipole can be chosen only in the form of a single-
component dipole with the same or inverted sign. In the homeotropic case, the image dipole
33602-23
V.M. Pergamenshchik, V.A. Uzunova
Figure 8. Dyad of elastic dipole at a wall with fixed planar director alignment. The dipole (left)
with dx = (d, 0, 0) and dy = (0, d, 0) and the image (right) with d
(2)
x = (d, 0, 0) and d
(2)
y =
(0,−d, 0) shown in two mutually perpendicular planes: xz-plane normal to the wall (upper
sketch) and planes x = −h (left) and x = h (right) parallel to the wall (lower sketch).
Figure 9. Dyad of elastic dipole at a wall with fixed homeotropic director alignment. The
dipole (left) with dx = (d, 0, 0) and dy = (0, d, 0) and the image (right) with d
(2)
x = (−d, 0, 0)
and d
(2)
y = (0,−d, 0) shown in the xz-plane normal to the wall.
d(2) = −d allows us to eliminate the particle-induced perturbation nt on the wall and thus to satisfy
the boundary condition (56). In the planar case, however, the y-equation cannot be satisfied: the
choice d(2) = −d is unacceptable as it would result in attraction which is obviously incorrect,
whereas the choice d(2) = d satisfies only the x-equation (56). Nevertheless, this last choice is the
only possible one in the sense that at least it results in a repulsion from the wall [51]. Then, using
(48) with |d(1)d(2)| = d2, one obtains for the planar (θ = π/2) and homeotropic (θ = 0) cases,
respectively, the following formulas:
F̃1d−wall,planar = πK
9d2
4h4
,
F̃1d−wall,hom = πK
18d2
4h4
. (67)
Apart from the difference in the coefficients, the “naive” prediction is that the repulsion from
a wall with homeotropic anchoring is twice stronger than that from a wall with planar an-
choring, F̃1d−wall,hom/F̃1d−wall,planar = 2. This is in contrast to the above exact result F1d−wall,hom/
F1d−wall,planar = 2/3.
5.2.1. Elastic charge and simple uniaxial elastic dipole in the gravitational field: a sketch
of experiment
The above prediction can be verified in an experiment similar to that recently reported by
Pishnyak et al. [51] who studied the interaction between colloids of the topological dipole type [42]
and a nematic liquid crystal surface with a planar anchoring. In a thick cell with horizontal surfaces,
33602-24
Colloidal nematostatics
the distance h from the lower surface is set in balance between the attraction due to the weight of
the colloid and the elastic dipole-surface repulsion. The ratio hh/hp of the equilibrium distance hh
of a topological dipole from a lower surface with a homeotropic anchoring to the distance hp from
that with a planar anchoring can be estimated experimentally and compared both with the ratio
hh/hp = 4
√
2/3 ' 0.9 obtained above, and with hh/hp = 4
√
2 ' 1. 2, following from the “naive”
prediction (67). Note that the value of the coefficient d in our theory, which does not enter the
ratio hh/hp, is 1/3 of that calculated numerically in [42] (see [57, 58]).
Another interesting effect to observe is “charging” of a colloid, i. e., inducing an elastic charge by
exerting external torque [e. g., by applying an electric (magnetic) field if the particle is ferroelectric
(ferromagnetic)]. A torque on a colloid creates an elastic charge which is repelled from the wall with
a force (57). In the case of a dipole-type colloid, an external torque would switch the h−4 dipole-
wall repulsion to a h−2 charge-wall repulsion which can manifest itself, in particular, in a sharp
increase of the equilibrium distance from the wall. Such an effect was observed by Lapointe et al.
in 2004 [70]. A ferromagnetic Ni wire with the planar anchoring along the long-axis in zero field
was aligned with the director and rested on the lower substrate of a planar nematic cell. A torque
exerted by an external magnetic field turned the wire thereby inducing the elastic distortions of
the director field. As a result, the wire repelled from the substrate and levitated over it3.
5.3. Interaction of a general dipole with a wall
The mirror-image method can be applied to dipolar particles of arbitrary shape. In general case,
dipole dyads of a real particle and its image have all the three components, d
(1)
t = (α
(1)
t , β
(1)
t , γ
(1)
t )
and d
(2)
t = (α
(2)
t , β
(2)
t , γ
(2)
t ). The dyad of the general image-dipole obtained from boundary condi-
tions (56) has the form
d
(2)
t =
(
−α(1)
t cos 2θ + γ
(1)
t sin 2θ, − β
(1)
t , γ
(1)
t cos 2θ + α
(1)
t sin 2θ
)
. (68)
Substituting (68) in (34), we obtain
Ud =
12πK
R3
[
α
(1)
t α
(1)
t
(
1 + sin2 θ
)
+ β
(1)
t β
(1)
t + γ
(1)
t γ
(1)
t
(
1 + cos2 θ
)
− α
(1)
t γ
(1)
t sin2 θ
]
. (69)
Differentiating (63) with respect to R at R = 2h yields the repulsion force of an arbitrary elastic
dipole from the wall, i. e.,
Fd−wall =
9πK
4h4
[
α
(1)
t α
(1)
t
(
1 + sin2 θ
)
+ β
(1)
t β
(1)
t + γ
(1)
t γ
(1)
t
(
1 + cos2 θ
)
− α
(1)
t γ
(1)
t sin2 θ
]
. (70)
It is important to remember that the components of the dipole dyad entering this formula depend
on an arbitrary angle φ, (45). This fact is trivial in a homogeneous space as the particle energy is
degenerate in φ, but the presence of a wall breaks the azimuthal symmetry around the z-axis and
removes the degeneracy. This means that the angle φ changes to minimize the interaction energy
with the wall. In other words, a particle, approaching the wall from a large distance, will turn to
assume some particular orientation with φ = φm. Below we consider an important particular case
described by formula (70) with an explicit φ dependence.
5.3.1. Interaction of a biaxial dipole with a wall
From (45), dipole dyad of a biaxial particle is obtained as d
(1)
x = (d + ∆ cos 2φ,−∆ sin 2φ, 0),
d
(1)
y = (−∆ sin 2φ, d − ∆ cos 2φ, 0). Identifying from this expression the components of d
(1)
t =
(α
(1)
t , β
(1)
t , γ
(1)
t ) and substituting them in (70) yields the repulsion force
F2d−wall =
9πK
4h4
[(
d2 + ∆2
) (
2 + sin2 θ
)
+ 2∆d cos 2φ sin2 θ
]
. (71)
3We are thankful to P. Pieranski for referring us to the experiment of [70].
33602-25
V.M. Pergamenshchik, V.A. Uzunova
Minimization with respect to φ (the free energy and the force have the same φ-dependence (71))
yields: i) φ is arbitrary for a uniaxial C∞v particle with ∆ = 0, which is obvious since such a
particle is invariant with respect to rotation about the z-axis; ii) φ is arbitrary at the homeotropic
wall, θ = 0, which is obvious since such a wall does not break the symmetry about the z-axis, Fig.
5; iii) for 0 < θ 6 π/2, φ = 0 if ∆ < 0 and φ = π/2 if ∆ > 0; this means that the longer axis of the
x, y-cross-section of the particle turns parallel to the wall which obviously saves the elastic energy.
Particles producing director deformations with two symmetry planes passing through the z-
axis, referred to here as just biaxial, can be exemplified by objects of the following geometries: the
topological dipole of [42] created by an ellipsoid with the normal surface anchoring rather than by
a sphere; an elliptical cone with its long axis along n∞ and planar anchoring at its lateral surface.
In all of these cases the equilibrium orientation of the long axis in the transverse x, y-cross-section
is parallel to the wall, and thus the repulsion from a wall is accompanied by a reorientation about
the homogeneous director. The only exclusion is the case of homeotropic wall where the only effect
is repulsion.
6. Conclusion
The theory developed above deals with two colloids. However, as the director-mediated inter-
action is of a long range, the many-particle effects can dominate in the large scale phase behavior
of nematic colloids. The cholesteric spiral induced by molecular chiral dipoles is the best known
example [19, 56]. The theory consistent with the many-particle effects should adopt a slow spatial
director variation such as the above cholesteric spiral. Such a theory is our goal for the near future,
and the two-particle theory presented above is going to be the ground for building this many-
particle generalization. To conclude this review we would like to bring to the readers’ attention
three known systems in which many-particle effects are or expectedly should be essential.
The first one is a kind of elastic plasma. Dye molecules aggregate in a nematic solvent and
form columnar aggregates [27]. Their long axes make a certain nonzero polar angle to the director
whereas azimuthal angles are distributed isotropically due to thermal randomization. When an
external electric field is applied, the torques exerted on the colloids are also distributed homoge-
neously in the plane normal to the director: two aggregates, whose azimuthal angles differ by π,
experience opposite torques (e. g., one is turned clockwise, another one anticlockwise). In terms of
the elastic charge, this is an elastic plasma: the external torque charge the columnar colloids while
the temperature makes the number of charges with opposite signs equal, i. e., the total charge is
equal to zero. The elastic charges interact via the Coulomb potential (32) which results in a col-
lective macroscopic director reorientation. This effect in [27] was obtained in a phenomenological
way since the potential (32) was at the time unknown. The effect is awaiting for its many-body
consideration in terms of elastic plasma.
Another system is a nematic doped with ferromagnetic colloidal particles [29]. Under the action
of magnetic field such colloids will become elastic charges, too. We expect that the many-body
interaction effects are important to the physics of such ferrocolloids. Interaction effects have not
been considered so far.
The third system is formed by colloids trapped at a nematic-air interface [14–16]. The colloids
form hexagonal and quasi-hexagonal lattices with different periods which can coexist [15, 16] and be
transformed into each other under certain conditions [16]. The lattices are stabilized by the elastic
repulsion between the dipolar colloids and the capillary attraction induced by a vertical force on
the colloids [16]. This force is definitely due to the elasticity of the nematic liquid since stabilization
holds only while the liquid is in its nematic phase and disappears in the isotropic phase. The many-
body effect in the capillary attraction is shown to be by 4 − 5 orders of magnitude stronger than
the pair interaction [71]. At the same time, the many-body effects of the long-range elastic dipolar
repulsion has not been considered.
In our opinion, the next step in the theory of nematic colloids is to incorporate the long-range
many-body colloidal interaction for the purpose of describing spatially inhomogeneous phases.
33602-26
Colloidal nematostatics
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33602-28
Colloidal nematostatics
Колоїдна нематостатика
В.М. Пергаменщик1,2, В.А. Узунова2
1 Корейський унiверситет, Пiвденна Корея
2 Iнститут фiзики, Київ, Україна
Стаття є оглядом теорiї взаємодiї колоїдних частинок через поле нематичного директора на вели-
ких вiдстанях. Нова область фiзики нематичних емульсiй багато в чому завдячує iдеї про те, що
їх фiзика є подiбною до електростатики. Взаємодiя через поле директора є далекосяжною та має
багато iнших властивостей взаємодiї мiж електричним диполями та квадруполями. Представлен-
ня густини пружного заряду в колоїднiй нематостатицi розвиває цю аналогiю на рiвнi густини за-
ряду i закону Колона. Показано, що аналогiя полягає в подiбностi математики, яка в обох теорiях
спирається на розв’язки рiвняння Лапласа. Однак у трьох вимiрах колоїдна нематостатика суттєво
вiдрiзняється вiд електростатики як своєю математичною структурою, так i її фiзичними наслiдка-
ми. Пружний заряд є вектором, який повнiстю визначається моментом зовнiшнiх сил, прикладених
до колоїда, а роль Гаусової теореми вiдiграє закон збереження компонент пружного моменту сил.
Пружнi мультиполi складаються з двох тензорiв (диад). Виведено формули для пружних мультипо-
лiв i потенцiалiв пружної парної кулонiвської, диполь-дипольної i квадруполь-квадрупольної взаємо-
дiї; розглянуто простi приклади, що її iлюструють. Виходячи з тензорної структури, знайдено рiзнi
можливi типи пружних диполiв i квадруполiв. Показано, що пружний диполь характеризується його
iзотропною силою, анiзотропiєю, кiральнiстю, та певною повздовжньою компонентою. Пружний ква-
друполь може бути одновiсним i двовiсним (бiаксiальним). Розглянуто зв’язок мiж типом пружного
мультиполя i його симетрiєю, деякi типи мультиполiв проiлюстровано графiчно. Спираючись на iдею
методу електростатичних вiдображень, розвинуто метод дзеркальних вiдображень в нематостатицi
для довiльного нахилу директора на стiнцi, яка обмежує нематичний рiдинний кристал. Цей метод
застосовано для розв’язання задачi взаємодiї пружного заряду i диполя зi стiнкою.
Ключовi слова: нематичний колоїд, електростатична аналогiя, пружний заряд i мультиполi
33602-29
The intrigue of nematic colloids: similarity to the electrostatics, its origin, and how it is developed in this article.
Introduction
Elastic charge density representation of the colloidal nematostatics
Linearized director distortions and their source
Incorporation of real surfaces and finite anchoring
Elastic charge density and multipoles of a single particle
Torque balance, Gauss' theorem, and elastic charge in 3 dimensions
The two particle Green function, point-to-point potential and its expansion in a power series of 1/r
Interaction of two particles at large separations, the pairwise approximation and its limits
Interaction of two similar elastic multipoles: the Coulomb, dipole-dipole, and quadrupole-quadrupole potentials
Tensorial structure and classification of different elastic dipoles
Dyads of elastic multipoles and their transformation in the intrinsic (a la isotopic) space
Elastic dipoles in the 2+1 and 2 dimensions
Elastic quadrupoles in the 2+1 dimensions
The electrostatic analogy and tensorial multipole structure of [56]
Particular examples of the elastic dipole-dipole and quadrupole-quadrupole interaction
The mirror image method of colloidal nematostatics
Elastic charge-wall interaction
Interaction of a uniaxial Cv dipole with a wall
Elastic charge and simple uniaxial elastic dipole in the gravitational field: a sketch of experiment
Interaction of a general dipole with a wall
Interaction of a biaxial dipole with a wall
Conclusion
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| id | nasplib_isofts_kiev_ua-123456789-32107 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T17:04:06Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Pergamenshchik, V.M. Uzunova, V.A. 2012-04-08T19:10:37Z 2012-04-08T19:10:37Z 2010 Colloidal nematostatics / V.M. Pergamenshchik, V.A. Uzunova // Condensed Matter Physics. — 2010. — Т. 13, № 3. — С. 33602:1-29. — Бібліогр.: 71 назв. — англ. 1607-324X PACS: 61.30.Dk, 61.30.Jf, 82.70Dd, 01.55.+b https://nasplib.isofts.kiev.ua/handle/123456789/32107 We give a review of the theory of large distance colloidal interaction via the nematic director eld. The new area of nematic colloidal systems (or nematic emulsions) has been guided by the analogy between the colloidal nematostatics and electrostatics. The elastic charge density representation of the colloidal nematostatics [V.M. Pergamenshchik, V.O. Uzunova, Eur. Phys. J. E, 2007, 23, 161; Phys. Rev. E, 2007, 76, 011707] develops this analogy at the level of charge density and Coulomb interaction. The analogy is shown to lie in common mathematics based on the solutions of Laplace equation. However, the 3d colloidal nematostatics substantially differs from electrostatics both in its mathematical structure and physical implications. The elastic charge is a vector fully determined by the torque exerted upon colloid, the role of Gauss' theorem is played by conservation of the torque components. Elastic multipoles consist of two tensors (dyads). Formulas for the elastic multipoles, the Coulomb-like, dipole-dipole, and quadrupole-quadrupole pair interaction potentials are derived and illustrated by particular examples. Based on the tensorial structure, we list possible types of elastic dipoles and quadrupoles. An elastic dipole is characterized by its isotropic strength, anisotropy, chirality, and its longitudinal component. An elastic quadrupole can be uniaxial and biaxial. Relation between the multipole type and its symmetry is discussed, sketches of some types of multipoles are given. Using the mirror image method of electrostatics as a guiding idea, we develop the mirror image method in nematostatics for arbitrary director tilt at the wall. The method is applied to the charge-wall and dipole-wall interaction. Розглянуто теорію взаємодії колоїдних частинок через поле нематичного директора на великих відстанях. Нова область фізики нематичних емульсій багато в чому завдячує ідеї про те, що їх фізика є подібною до електростатики. Взаємодія через поле директора є далекосяжною та має багато інших властивостей взаємодії між електричним ди- та квадруполями. Представлення густини пружного заряду в колоїдній нематостатиці розвиває цю аналогію на рівні густини заряду та закону Колона. Показано, що аналогія полягає в подібності математики, яка в обох теоріях спирається на розв'язки рівняння Лапласа. Однак у 3-х вимірах колоїдна нематостатика суттєво відрізняється від електростатики як своєю математичною структурою, так і її фізичними наслідками. Пружний заряд є вектором, який повністю визначається моментом зовнішніх сил, прикладених до колоїда, а роль Гаусової теореми відіграє закон збереження компонент пружного моменту сил. Пружні мультиполі складаються з 2-х тензорів (диад). Виведено формули для пружних мультиполів і потенціалів пружної парної кулонівської, диполь-дипольної та квадруполь-квадрупольної взаємодії. Розглянуто прості приклади, що її ілюструють. Виходячи з тензорної структури, знайдено різні можливі типи пружних ди- і квадруполів. Показано, що пружний диполь характеризується його ізотропною силою, анізотропією, кіральністю та певною поздовжньою компонентою. Пружний квадруполь може бути одно- та двовісним (біаксіальним). Розглянуто зв'язок між типом пружного мультиполя та його симетрією, деякі типи мультиполів проілюстровано графічно. Спираючись на ідею методу електростатичних відображень, розвинено метод дзеркальних відображень в нематостатиці для довільного нахилу директора на стінці, яка обмежує нематичний рідинний кристал. Цей метод застосовано для розв'язання задачі взаємодії пружного заряду та диполя зі стінкою. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Colloidal nematostatics Колоїдна нематостатика Article published earlier |
| spellingShingle | Colloidal nematostatics Pergamenshchik, V.M. Uzunova, V.A. |
| title | Colloidal nematostatics |
| title_alt | Колоїдна нематостатика |
| title_full | Colloidal nematostatics |
| title_fullStr | Colloidal nematostatics |
| title_full_unstemmed | Colloidal nematostatics |
| title_short | Colloidal nematostatics |
| title_sort | colloidal nematostatics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32107 |
| work_keys_str_mv | AT pergamenshchikvm colloidalnematostatics AT uzunovava colloidalnematostatics AT pergamenshchikvm koloídnanematostatika AT uzunovava koloídnanematostatika |