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PHYSICAL REVIEW E, VOLUME 65, 021709

Symmetry breaking and interaction of colloidal particles in nematic liquid crystals 1

B. I. Lev,1 S. B. Chernyshuk,2 P. M. Tomchuk,2 and H. Yokoyama1

Yokoyama Nano-structured Liquid Crystal Project, Japan Science and Technology Corporation, Tsukuba Research Consortium, 5-9-9 Tokodai, Tsukuba, Ibaraki 300-2635, Japan 2 Department of Theoretical Physics, Institute of Physics, NAS Ukraine, Prospekt Nauki 46, Kyiv 02022, Ukraine 共Received 21 July 2001; revised manuscript received 13 September 2001; published 25 January 2002兲 We propose a general approach to the description of the long-ranged elastic interaction in the nematic colloids, based on the symmetry breaking of the director field. The type of the far-field interaction between particles immersed in a nematic host is determined by the way the symmetry is broken in the near-field region around the colloidal particle. This is caused both by the particle’s shape and the anchoring at the surface. If the director field near the particle has a set of three symmetry planes, the far-field interaction falls off as d ⫺5 with d being the distance between particles. If one symmetry plane is absent, a dipolar moment perpendicular to it is allowed and yields dipole-dipole interactions, which decays as d ⫺3 . If both the horizontal and vertical mirror symmetries are broken 共it is equivalent to the case when the nonzero torque moment is applied to the particle by the nematic liquid crystal兲, the particles are shown to attract each other following the Coulomb law. We propose a simple method for the experimental observation of this Coulomb attraction. The behavior of colloid particles in curved director fields is analyzed. Quadrupolar particles with planar anchoring are shown to be attracted toward the regions with high splay deformations, while quadrupoles with homeotropic anchoring are depleted from such regions. When there are many colloidal particles in the nematic solvent, the distortions of the director from all of them are overlapped and lead to the exponential screening in the elastic pair interaction potential. This is a many-body interaction effect. This screening is essential in the real dense colloid systems, such as ferronematics—suspensions of magnetic cylindrical grains in the nematic liquid crystal. External magnetic field induces an elastic Yukawa attraction between them. We apply this attraction to the explanation of the cellular texture in magnetically doped liquid crystals. DOI: 10.1103/PhysRevE.65.021709

PACS number共s兲: 61.30.Gd, 82.70.Dd

I. INTRODUCTION

Any interaction is connected with some symmetry breaking. Liquid crystals are materials that break their continuous symmetry under the action of weak external influences. Orientational distortions are easily created by external fields and boundary conditions. Another way to break continuous symmetry in liquid crystals is to introduce a particle of distinct substance in the liquid crystal host. It distorts orientational order of the liquid crystal over distances much larger than the size of the particle. When the regions of the broken continuous symmetry around each of the two particles are overlapped, it leads to interaction between them. A suspension of many particles efficiently breaks the continuous symmetry of the initial liquid crystal and thus could give rise to an entirely different class of composite materials. Properties of such liquid crystal 共LC兲 composites are defined by the collective behavior of the immersed particles mutually interacting via the elastic deformations of the orientational ordering of the liquid crystal. Colloid suspensions in nematic liquid crystals has attracted considerable attention during the last few years 关1– 8兴. The Frank elastic interactions between colloidal particles lead to non-trivial behaviors with the formation of a variety of novel ordered or disordered structures. So far observed are linear chains of small water droplets in an aligned liquid crystal or in large nematic drops 关3,4兴, highly ordered arrays of silicon oil droplet chains in a nematic host 关7兴, and two-dimensional 共2D兲 hexagonal lattices of glycerene droplets in a nematic cell with hybrid boundary conditions 关8兴. In 1063-651X/2002/65共2兲/021709共14兲/$20.00

all these cases, colloidal particles have an effect of distorting the director field n(r) around them and interact elastically because of overlapping of these distortions. Anchoring conditions on the surface of droplets and the global geometry of the LC matrix have a great impact on the elastic interparticle interactions and on the collective formation of structures. Up to now, there are two theoretical approaches to the description of colloid interactions in the nematic liquid crystal. The first one deals with spherical droplets that have strong anchoring strength on the surface 关9,11,12兴. A droplet with strong planar anchoring creates a pair of topological defects, known as boojums. A droplet with strong homeotropic boundary conditions, on the other hand, creates an equatorial disclination ring or a hyperbolic hedgehog 共i.e., characteristic shape in a photo current-time plot兲 as a companion for the radial hedgehog on the surface of the droplet. Using the variational techniques and an electrostatic analogy, Lubensky et al. 关9兴 obtained an approximate director distribution near the droplet with homeotropic boundary conditions, as well as the long-range pair interaction potential between the droplets. It has both the dipole-dipole and the quadrupole-quadrupole components. The dipole-dipole interaction explains the formation of the chains, which are aligned along the director in the nematic host. The second approach was proposed in 关1兴, where the authors have examined the case of weak anchoring strength for particles of general shape. They have found analytically the pair interaction potential, taking into account the different Frank moduli and have expressed the potential in terms of tensors characterizing the shape of the particle. These ten-

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©2002 The American Physical Society

LEV, CHERNYSHUK, TOMCHUK, AND YOKOYAMA

PHYSICAL REVIEW E 65 021709

sors dictate the symmetry of the particle shape and hence the nature of the resulting long-range interaction potential. Their results are in agreement with the quadrupole-quadrupole potential for spheres 关9,12兴 and with the dipole-dipole interaction of Lopatnikov and Namiot 关13兴 for asymmetric cylinders. In this paper we argue that the long-range interaction potential between colloidal particles in nematics is determined by symmetry breaking of the director field in the vicinity of the particles. This symmetry breaking is caused by two reasons: the shape of the particle and the anchoring strength. In the case of weak anchoring it is determined primarily by the form of the particle. In the opposite case the most important is the anchoring strength. In order to universally describe all these phenomena, we introduce the concept of the coat around the particle. The coat embraces all the accompanying topological defects, while it has the same symmetry as the resultant director field near the particle. The director distribution outside the coat undergoes only smooth variations and does not contain any topological defects. In the case of weak anchoring the coat coincides with the particle itself. We note that the same concept has been introduced in 关14兴, where the authors treated the director distribution around the spherical droplet using an electrostatic analogy. They have found its size, or the ‘‘correlation length’’ from the surface, on which the director changes discontinuously passing through the topological defects. The estimates give the value of about 1 ␮m for the correlation length. This is in qualitative agreement with the result of Lubensky l⫽0.26R 0 for the distance between the surface of the droplet and the hyperbolic hedgehog 关9兴, when the radius R 0 of the droplet is about 10 ␮m. We argue that the long-range interaction between particles is determined by the symmetry of the coat and is expressed through tensor characteristics of it. When the director distribution in the vicinity of the particle has three symmetry planes, the pairwise interaction potential falls off as d ⫺5 , with d being the distance between particles. When one symmetry plane is broken, a dipole moment perpendicular to it arises and it leads to the dipoledipole interaction between the particles. When the coat has only one vertical symmetry plane 共the director n without particles is aligned along the vertical axis兲 or when it does not have symmetry planes at all, then the leading interaction at far distance is the Coulomb attraction law. We propose a simple method to observe this attraction. A similar effect was described in the 2D case of smectic-C films where misalignment of a dipolar structure with the far field induces a 2D Coulomb-like charge on the particle 关10兴. In our 3D case we find deformational Coulomb charge in terms of geometrical parameters, anchoring energy, and orientation of the particle. In all previous papers concerned with colloidal particles in liquid crystals, the elastic interaction potential has been obtained as the result of the overlapping of director distortions around the pair of particles. The influence of the director distortions from others particles has not been taken into account yet. So the influence of the particle’s concentration on the elastic pair potential has not been considered. We take into consideration this effect and show that in dense colloids it leads to the exponential screening of the potential at far

distances, which are larger than the average distance between colloidal particles. The real colloid system, where this effect could be observable, is a suspension of long magnetic particles in the nematic, which is called ferronematic. It was examined in the experiment of Chen and Amer 关2兴, where they have observed the appearance of the cellular texture upon the critical external magnetic field. We show that this can be explained via the elastic screened Coulomb attraction between particles, which leads to their collapse. It is produced by the breaking of both horizontal and vertical mirror symmetries with help of the external magnetic field, perpendicular to particles. At the end of the paper we explore the case of nonuniform director field, the movement of particles that is produced by global director distribution. The plan of the paper is as follows. In Sec. II we show how symmetry breaking is connected with the director distribution around the spherical particle. In Sec. III we perceive the dipole-dipole interaction as the result of the mirror symmetry breaking in one plane. In Sec. IV we obtain the Coulomb attraction as the result of symmetry breaking in two planes—horizontal and vertical. In Sec. V we show the result of the screening of the pair interaction in dense colloids. In Sec. VI we conclude our results. In Appendix A we analyze in detail the appearance of the screening, as the result of interference of far-field transverse components n x and n y of the director, which come from all particles. In Appendix B we find the energy of an arbitrary particle in the curved director field. II. SYMMETRY AND THE DIRECTOR DISTRIBUTION

A nematic liquid crystal is an anisotropic fluid in which long molecules have the same average orientation specified by the unit vector n called the director. In the undistorted state the nematic has a spatially uniform orientation n0 , and we consider here that it is parallel to the z axis 关 n0 ⫽(0,0,1) 兴 . The case of global nonuniform director distribution is investigated in the Appendix B, where the expression for the energy of an arbitrary particle in the curved director field is found. Immersed particles distort the uniform orientation of the director in the bulk. The source of bulk director deformations is the preferential orientation imposed at the surface of particles in such a way that the nematic molecules lie either normally or tangentially to it. The phenomenological anchoring energy at the surface of particles can be written in the Rapini-Papoular form

F s⫽

兺p W

冖

ds 关 ␯共 s兲 •n共 s兲兴 2 ,

共1兲

where W is the anchoring energy coefficient. For the homeotropic anchoring W⬍0 and for the planar one W⬎0. Summation should be taken over all particles in the liquid crystal. Bulk energy of the spatial distortions of the director field, which is called the Frank energy, is written in the form

021709-2

SYMMETRY BREAKING AND INTERACTION OF . . .


FIG. 1. Director configurations around the water droplet in the nematic. 共a兲 Saturn-ring configuration with quadrupolar symmetry. 共b兲 Nonequatorial disclination ring breaks mirror symmetry in the horizontal plane and induces dipolar moment as a measure of skewness. 共c兲 Ground state of the system is the pair of the radial and hyperbolic hedgehogs, which have bigger dipolar moment.

F b⫽

1 2

冕

d 3 r 兵 K 11共 div n兲 2 ⫹K 22共 n•rotn兲 2 ⫹K 33共 n⫻rotn兲 2 其 .

radial and hyperbolic hedgehogs break mirror symmetry in the horizontal plane, while the equatorial disclination ring 共Saturn ring兲 retains it. Saturn-ring configuration has quadrupole symmetry, which is reduced to dipolar symmetry when the disclination ring is shifted above or below the equator. Authors of 关17兴 have shown by Monte Carlo simulations that the configuration with a hyperbolic hedgehog has lower energy than the Saturn ring. It has been confirmed in 关9兴 with help of the dipole Ansatz that though the equatorial ring has some metastability, its energy is higher than that of the dipole. So on this example we see that strong anchoring on the surface breaks mirror symmetry, though the shape of the particle remains spherical. For weak anchoring or smaller particle size 共less than 1 ␮m兲 it is not, and the quadrupole symmetry configuration remains 关18兴 just as the droplet itself has. The breaking of the symmetry in the near-field region, which is achieved either by the anchoring strength or by the particle’s shape, leads to the different solutions in the farfield region. At far distances from the particle, the director field n(r) tends to be uniform n0 ⫽(0,0,1) and can be written in the form n⫽(n x ,n y ,1). In the one-constant approximation, the Frank free energy is given by F b ⫽ 12 K

d 3 r 兵共 “n x 兲 2 ⫹ 共 “n y 兲 2 其 .

共3兲

The equilibrium equations are the Laplace equations for the transverse components n ␮ ( ␮ ⫽x,y), ⌬n ␮ ⫽0.

共2兲

共We do not take into account here surface K24 and K13 terms, inasmuch as they can give a correction only for shortrange director deformations, but in this paper we take into account only long-range deformations and their contribution to the pair interaction energy兲. In order to find possible director configurations one should solve Euler-Lagrange 共EL兲 equations from the minimization of the Frank free energy with taking into account the boundary conditions, which are found from the minimization of bulk and surface energies. But a situation can arise where there are several director distributions with different symmetry, which satisfy both EL equations in the bulk and boundary conditions at the surface. A typical example is possible director configurations around the water droplet with strong homeotropic anchoring, shown in Fig. 1. In both cases the director lies perpendicular to the sphere and it is equivalent to the radial hedgehog in the center. It has topological charge, which is equal to unity. The uniform director distribution far from the droplet has zero topological charge and so there should be another topological defect near the droplet to compensate for the hedgehog in the center. In Fig. 1共a兲 the droplet creates a ⫺ 21 disclination ring on the equator at a distance l⫽1.08R 0 from the center and in Fig. 1共c兲 it has a point hyperbolic hedgehog 关9兴. In Fig. 1共b兲 the intermediate configuration is shown, which is a nonequatorial disclination ring. Obviously, the director configurations have different symmetry. The nonequatorial disclination ring and the pair of

冕

共4兲

At large distances r it can be expanded in multipoles, n ␮⫽

A ␮ p ␮ •r c ␮i j r i r j ⫹¯ . ⫹ 3 ⫹ r r r5

共5兲

It is clearly seen, that transverse components can be treated as two components of the electric field potential and particles are multipolar sources, similar to antennas 关19兴. The first term is connected with the ‘‘charge,’’ the second with the dipole moment, and the last term is connected with the quadrupole moment. The three items in Eq. 共5兲 represent different broken symmetries of the director field around particles and are responsible for three different interaction laws between particles, as we show below. The first term exists when the director distribution does not have any plane of symmetry at all or it has only one vertical plane of symmetry. It appears when the particle in its vicinity breaks mirror symmetry in a horizontal plane and in one vertical plane. In other terms, it exists, when there is a nonzero torque moment ⌫ acting on the particle due the nematic 关20兴. In the absence of ⌫ it is absent. The second term represents broken symmetry in one plane and the dipole moment p is the measure of skewness, so that more the removal of the disclination ring from the equator, more the magnitude of the dipole moment. The last term exists in any case, because it has the same quadrupole symmetry as the director has.

021709-3



The multipole expansion is valid only in the region where nonlinearities can be neglected. For particles with strong anchoring it is the far region, because of strong director deformations in the near region. But for particles with weak anchoring, distortions are small elsewhere and the multipole expansion is applicable in the near region too. In general, lesser the anchoring strength, smaller is the size of the region where the multipole expansion is inapplicable The amplitudes of this expansion in the strong anchoring case can be found from the asymptotic either of exact solutions or of different variational Ansa¨tze that correctly describe the director field in the near region. This has been done in 关9兴 for spherical particles with homeotropic anchoring with help of electrostatic and dipole Ansa¨tze. In the weak anchoring case, amplitudes in Eq. 共5兲 are found directly from the boundary conditions for the linearized EL equations, as has been done in 关18兴. III. MIRROR SYMMETRY BREAKING AND DIPOLEDIPOLE INTERACTION

In this section we intend to clarify the appearance of the dipole moment p with the breaking of the mirror symmetry of the director field in one plane in the vicinity of the surface, and to represent p as the measure of the skewness. In the paper 关9兴 dipolar and quadrupole moments are expressed as p␮ ⫽(p•n0 )e␮ and c ␮i j ⫽c(n 0i e ␮j ⫹n 0 j e ␮i ), where e␮ are the vectors pointing in the ␮ ⫽x,y direction. The authors have found the interaction potential in the far region between the water droplets with hyperbolic hedgehogs, when the dipolar moments lies parallel to the director. Their result is found in the one-constant approximation (K 11⫽K 22⫽K 33 ⫽K) and it is written as U 共 R兲 ⫽4 ␲ K 关 p z p z⬘ V pp 共 R兲 ⫹ 94 cc ⬘ V cc 共 R兲 ⫹ 23 共 c p ⬘ ⫺c ⬘ p 兲 V pc 共 R兲兴 , V pp 共 R兲 ⫽ V cc 共 R兲 ⫽

1 关 1⫺3 cos2 共 ␪ 兲兴 , R3

1 关 9⫺90 cos2 共 ␪ 兲 ⫹105 cos4 共 ␪ 兲兴 , R5

V pc 共 R兲 ⫽

cos共 ␪ 兲关 15 cos2 共 ␪ 兲 ⫺9 兴 , R4

共6兲

where ␪ is the angle between the separation vector R and n0 ; p z ,p z⬘ ,c,c ⬘ are dipolar and quadrupole moments, respectively, at the positions r and r⬘ . In order to find them, it is necessary to compare the multipole expansion 共5兲 with the asymptote of the Ansa¨tze that describe the director field in the near region. From the variational Ansatz it has been found that p z ⫽2.04a 2 , c⫽⫺1.08a 3 , a being the droplet’s radius. The last term in Eq. 共6兲 is absent for equal droplets, and only the dipole-dipole V pp (R) and the quadrupolequadrupole V cc (R) potentials remain. In the paper 关1兴 the authors suggested an approach that enables to find the interaction potential for particles of ordi-

FIG. 2. Coats that contain all topological defects inside. Anchoring coupling W c of the coat depends on the point of the surface. 共a兲 Quadrupolar coat around the Saturn-ring disclination. 共b兲 Dipolar coat with broken mirror symmetry contains radial and hyperbolic hedgehogs.

nary shape with weak anchoring at the surface. It is valid for the different Frank constants and so exceeds the bounds of the electrostatic analogy. In the weak anchoring case there are no topological defects and the director deformations ␦ n are small everywhere so that the multipole expansion is valid at the particle surface too. That is why unknown amplitudes in Eq. 共5兲 can be expressed through tensor characteristics of the particle’s surface and orientation. In the case of strong anchoring, topological defects arise in the near region, but outside, the director deformations ␦ n are small. Therefore we can confine the particle and topological defects within the region called coat. This region contains all strong deformations of the director field. Outside the coat director deformations are small, ␦ nⰆ1. The size of the coat has been estimated in 关14兴 with the help of the electrostatic analogy and it was shown to be a few micrometers from the particle’s surface. It is in a qualitative agreement with the result of 关9兴 l⫽0.26R 0 for the distance between the droplet’s surface and the hyperbolic hedgehog when the radius R 0 of the droplet is about 10 ␮m. The symmetry of the coat is equivalent to the broken symmetry of the director in the vicinity. For example, a droplet with an equatorial disclination ring 共Saturn-ring configuration兲 could be put into a coat that has a horizontal symmetry plane 关see Fig. 2共a兲兴, and a water droplet with a companion hyperbolic hedgehog could be confined into the coat without a horizontal symmetry plane 关see Fig. 2共b兲兴. The anchoring energy on the surface of the coat is determined as the interaction energy between the nematic molecules over the surface and the molecules under the surface of the coat. Phenomenologically it can be written in the Rapini-Papoular form, but it is natural that the anchoring strength W c becomes dependent on the point s on the surface of the coat. The shape and the surface distribution W c (s) determine the symmetry of the coat, which is identical to the symmetry of the real director distribution in the near region. Obviously, it is necessary to know the real director field in every point and shape of the coat exactly to find W c (s), which is hardly achieved. Instead of finding exact solutions of the EL equations, we show that the problem can be effectively solved in terms of some unknown tensors that

021709-4



characterize the surface of the coat. For this purpose we introduce the surface energy in the form F cs⫽

冖

兺p

ds W c 共 s兲关 ␯共 s兲 •n共 s兲兴 2

共7兲

instead of Eq. 共1兲. Such substitution allows us to consider the director deformations ␦ n small everywhere in the space. All defects are hidden now inside the coat. This makes it possible to find the interaction potential between the coats. We can apply the results of the paper 关1兴 because ␦ nⰆ1 far and wide. It was the only physical assumption under which the results are valid. According to the 关1兴 the interaction potential between the two coats separated by the vector R in the general case is written as U 共 R兲 ⫽⫺

⫻

⫻

1 p Aˆ p Aˆ ⬘ 8 ␲ m,m ⬘ ⫽1,2,3 m m ⬘ ␮ ⫽1,2

兺

再

兺

冑K ␮␮

共冑

⫹ 共 ⫺1 兲 ␮

Q m,m ⬘

储

冑

K 33R⬜2 ⫹K ␮␮ R 2储 ⫺ K ␮␮ R 2储兲 2 K 33R⬜2 ⫹K ␮␮ R 2储

冑

R⬜ ⫻n0 , R⬜

r2 ⫽

R⬜ , R⬜

r3 ⫽n0 ,

冎

.

R⬜ ⫽n0 ⫻R.

␥ klmn ⫽

冖

冖

d ␴ W c 共 s兲 ␯ k 共 s兲 ␯ l 共 s兲 ,

d ␴ W c 共 s兲 ␯ k 共 s兲 ␯ l 共 s兲 ␳ m 共 s兲 ,

d ␴ W c 共 s兲 ␯ k 共 s兲 ␯ l 共 s兲 ␳ m 共 s兲 ␳ n 共 s兲 .

where the summation on the repeating indices is made. Now we consider the case of ␣ 3m ⫽0 and below we show circumstances under which this is not true. Now we want to examine thoroughly the second term, which represents the dipole-dipole interaction. It takes the form U dd ⫽ ␤ 3ms ␤ 3ms ⬘

Superscript p means that in operator Aˆ mp we need to substitute “⫽ ⳵ / ⳵ rp . Here ␣ lm , ␤ lms , ␥ lmst are tensor characteristics of the coat, which contain all information about its symmetry. Let ␳ be the vector pointing from the center of mass of the particle to the point s at the surface of the coat and ␯ be the unit normal to the surface at this point. Then tensors are expressed as

共11兲

1 , 4␲R 共13兲

共9兲

Aˆ m ⫽ 共 kl •n0 兲关 ␣ lm ⫹ ␤ lms 共 ks •“ 兲 ⫹ ␥ lmst 共 ks •“ 兲共 kt •“ 兲兴 . 共10兲

␤ klm ⫽2

␣ 3m ␣ 3m 1 ⫹ ␤ 3ms ␤ 3ms ⬘ 共 ks •“ 兲共 ks ⬘ •“ 兲 4␲R 4␲R

共8兲

where (k1 ,k2 ,k3 ) is the local basis rigidly bound with each particle. Operators Aˆ m are defined as

冖

⫹

⫺ ␥ 3mst ␥ 3ms ⬘ t ⬘ 共 ks •“ 兲共 ks ⬘ •“ 兲共 kt •“ 兲共 kt ⬘ •“ 兲

(⫾) Q m,m ⬘ ⫽(r1 •km )(r1 •km ⬘ )⫾(r2 •km )(r2 •km ⬘ ),

␣ kl ⫽2

共12兲

It can be easily shown that Q m,m ⬘ ⫽0 for m, m ⬘ ⫽3, and for m,m ⬘ ⫽1,2. Using this allows us to write the expression 共12兲 in the form

R⬜2

In this expression R 储 and R⬜ are parallel and perpendicular to the undeformed director n0 components of R⫽rp ⫺rp ⬘ , r1 ⫽

兺

U 共 R兲 ⫽⫺

⫺

冑K 33R⬜2 ⫹K ␮␮ R 2

冉冊 ⫹

Q m,m ⬘ 1 p U 共 R兲 ⫽⫺ Aˆ mp Aˆ m⬘⬘ . 8 ␲ m,m ⬘ ⫽1,2,3 R ⫹ Q m,m ⬘ ⫽ ␦ m,m ⬘ ,

1

⫹

Q m,m ⬘

The integration is over the surface of the coat. The symmetry of these tensors contains all data about broken symmetry of the director field in the vicinity of particles and defines distinctive features of the interaction potential in the far region. Its magnitude can be treated like variational parameters for the concordance with experimental data or it can be evaluated from the comparison with long-range asymptotic of Ansa¨tze solutions. For instance, let us consider the comparison of Eq. 共8兲 with the Lubensky potential 共6兲. For this purpose we consider the one-constant approximation, and Eq. 共8兲 results in

3 共 ks •R兲共 ks ⬘ •R兲 ⫺ ␦ ss ⬘ . R3

共14兲

Symmetry of the tensor ␤ reflects symmetry of the director and defines a dipole type of the interaction. Let us consider components ␤ 31s ⫽2养d ␴ W c (s) ␯ 1 ␯ 3 ␳ s . If the director distribution has a horizontal symmetry plane, then for every point ␯ the mirror image exists, for which ␯ 3 changes sign whereas ␯ 1 , ␳ 1 , ␳ 2 remain the same. Therefore ␤ 311⫽ ␤ 312 ⫽0. Besides, if there is symmetry plane YZ, then ␤ 313⫽0, else ␤ 313⫽0. Similar, if there is symmetry plane XZ, then ␤ 323⫽0, else ␤ 323⫽0. For example, banana-shaped particles with the symmetry plane XZ have ␤ 313⫽0, and thus induce elastic dipole-dipole interaction as the result of the director distortions. So we come to the important conclusion that if the director distribution near the particle has three perpendicular planes 共one of which is horizontal兲, then all ␤ 3ms are zero and only quadrupole interaction remains, which dies off as R ⫺5 , as is clearly seen from Eq. 共13兲. If one symmetry plane in the director distribution is broken, then a dipole moment in the perpendicular direction arises. If the horizontal symmetry plane is broken, as in the case of the water droplets in nematics 关4兴, the nonzero components of the tensor ␤ are ␤ 311⫽ ␤ 322 , and the dipolar part takes the form

021709-5

LEV, CHERNYSHUK, TOMCHUK, AND YOKOYAMA 2 ␤ 311 U dd ⫽ 关 1⫺3 cos2 共 ␪ 兲兴 4 ␲ KR 3


共15兲

with cos ␪⫽n0 R/R. This coincides with the first term in Eq. 共6兲, and we clearly see that ␤ 311 is proportional to the z component of the dipolar moment p z ,

cos ␸⫽(k1 •er ), ␥ 3131⫽ ␲ r 3 LW c /4. They repel in the X direction and attract in the Y direction. If the coat has the axis of symmetry C 4 then ␥ 3131 and ␥ 3232 are equal and the quadrupole interaction takes the form U qq⫽

␤ 311⫽⫺4 ␲ Kp z . 共We choose the minus sign in order to agree on the results on the behavior of particles in the curved director field with 关9兴, see Appendix B兲. To demonstrate the appearance of the dipolar moment as the result of the breaking of the mirror symmetry, we consider a model system—a sphere with nonequatorial thread, on which anchoring strength is infinite. Surface anchoring on the sphere can be written as W c 共 ␪ 兲 ⫽W 0 ⫹W 1 共 cos ␪ 兲 ␦ 共 cos ␪ ⫺cos ␪ th兲 . W 0 is the constant anchoring on the surface and the second term is the anchoring on the thread. Then integration on the sphere surface gives

␤ 311⫽2

冖

d ␴ W c 共 s兲 ␯ 1 ␯ 3 ␳ 1 ⫽2 ␲ R 3 W 1 共 x th兲 x th共 1⫺x 2th兲 ,

where x th⫽cos(␪th). Inasmuch as W 1 (0)⫽0, we conclude that

␤ 311⬃cos共 ␪ th兲 when ␪ th⬇ ␲ /2. It becomes zero, when the thread lies on the equator and changes its sign, when the thread is shifted above or below the equator, so that it really behaves similar to the dipolar moment. When the mirror symmetry is not broken, all components of the tensor ␤ are zero and the last term remains, which represents a quadrupole-quadrupole interaction. In this case nonzero components of the tensor ␥ that are included in the interaction are ␥ 3131 , and ␥ 3232 . In general, they are different if the particle does not have an axis of symmetry C 4 共for instance, for the parallelepipeds with different sides兲. In this case the general quadrupole-quadrupole interaction potential is U qq⫽⫺

2 3 ␥ 3131

2 ␲ KR 5

关 1⫺5 cos2 ␪ ⫺5 共 k1 •er 兲 2

⫹35 cos2 ␪ 共 k1 •er 兲 2 兴 ⫺

2 3 ␥ 3232

2 ␲ KR 5

关 1⫺5 cos2 ␪

⫺5 共 k2 •er 兲 ⫹35 cos ␪ 共 k2 •er 兲兴 , 2

2

2

U qq,horizont⫽

2 3 ␥ 3131

2 ␲ KR 5

2 ␲ KR 5

共 3⫺30 cos2 ␪ ⫹35 cos4 ␪ 兲 .

It coincides with the second term of Eq. 共6兲. From the comparison with it, we find along with the sign the connection with the quadrupole moment c of the coat,

␥ 1313⫽c

4␲K &. 3

Then for particles with planar anchoring W⬎0, ␥ 1313 ⬎0, c⬎0 and for particles with homeotropic anchoring ␥ 1313⬍0, c⬍0, this is in an agreement with the Ansatz result for the quadrupole moment of the water droplet in the nematic 关9兴. IV. THE COULOMB ATTRACTION AS THE RESULT OF THE BREAKING OF HORIZONTAL AND VERTICAL MIRROR SYMMETRY

Expression 共13兲 shows that the Coulomb attraction is present in the general case. Let us consider the nature of this term. It is determined by the components ␣ 13 and ␣ 23 , ␣ 13 ⫽2养d ␴ W c (s) ␯ 1 (s) ␯ 3 (s). It is obvious that, if there is a horizontal symmetry plane, then both components are zero. The presence of the symmetry plane XZ makes ␣ 23⫽0 and YZ makes ␣ 13⫽0. So we come to the significant conclusion: the breaking of both horizontal and vertical mirror symmetry leads to ␣ 13⫽0 or ␣ 23⫽0 and thus to the Coulomb attraction 共17兲

U⫽⫺Q 2 /R,

where the role of the charge is played by the geometrical 2 2 ⫹ ␣ 23 )/4␲ K. factor Q⫽ 冑( ␣ 13 We can calculate the charge, for instance, for the long cylinder (LⰇd), which makes the tilt angle ␪ with the z axis and lies in the YZ plane 共Fig. 3兲. For such a cylinder ␣ 13 ⫽0, ␣ 23⫽0. Let O ⬘ be the coordinate basis (XY ⬘ Z ⬘ ), which is turned by the angle ␲ /2⫺ ␪ with respect to the axis X. In this basis ␯⬘ ⫽(cos ␸,0,sin ␸). In the 共X,Y,Z兲 basis ␯ i ⫽A i j ␯ ⬘j , where the rotation matrix

冋

共16兲

where er ⫽R/R. We see that potential depends not only on the angle ␪ but also on the azimuthal angle ␸. For the cylinders that lie parallel to the axis Y, ␥ 3232⫽0. Then the interaction potential, for instance, for the cylinders that lie in the horizontal plane ( ␪ ⫽ ␲ /2) becomes

2 3 ␥ 3131

1

Ai j⫽ 0 0

0

0

sin ␪

⫺cos ␪

cos ␪

sin ␪

册

so that ␯ 1 ⫽cos ␸, ␯ 2 ⫽⫺cos ␪ sin ␸, ␯ 3 ⫽⫺sin ␪ sin ␸. Then ␣ 23⫽2养d ␴ W ␯ 2 ␯ 3 ⫽⫺dL ␲ W cos ␪ sin ␪ 共we integrate only on the side area兲 and the charge is

共 5 cos2 ␸ ⫺1 兲 ,

Q⫽ 021709-6

dL 兩 W sin 2 ␪ 兩 4

冑

␲ . K

共18兲



the director distortions around the pair of particles. In principle, however, the rest of particles should also have an influence on the interaction potential. The presence of particle 3 affects the interaction between particles 1 and 2. This effect should not be significant on small distances, but if the distance between particles 1 and 2 is large, so that there are many particles with numbers 3,4,5... located between them, then their collective action, i.e., deformations from them can greatly influence on the interaction potential between particles 1 and 2. In a previous paper 关22兴, we have examined this case and shown that the deformations from all particles lead to the exponential screening of the pair interaction potential. Physically it can be explained following the argument by Brochard and de Gennes 关21兴. Let us consider cylindrical particles, labeled by a unit vector u, and ␪ is the angle between n0 and u. It produces a director in the far region, n(r)⫽n0 ⫹ ␦ n(r), where ␦ n(r) ⫽( ␦ nx , ␦ ny ,0). It is convenient to put ␦ n(r)⫽ ␻(r)⫻n0 , where ⌬ ␻ x ⫽⌬ ␻ y ⫽0. Since ␻ z is arbitrary, we may put ⌬ ␻ z ⫽0. Then 1 1 ␻共 r兲 ⫽ 肀 ⫹ ␹:ⵜ ⫹¯ . r r

FIG. 3. Orientation of the cylindrical particle.

When the cylinder lies in the XY plane ( ␪ ⫽ ␲ /2) or YZ plane ( ␪ ⫽0) then Q⫽0, as follows from general symmetry considerations. The maximum charge occurs when ␪ ⫽ ␲ /4. Such an inclined position of the cylinder is not of course profitable energetically. In the ground state the cylinders make angles ␪ ⫽0 or ␪ ⫽ ␲ /2 as functions of the ratio Wd/K 关24兴. But if they have magnetic moment, the external magnetic field can fix the tilt angle ␪ ⫽0, ␲ /2 and cause the elastic Coulomb attraction between them. Such a situation is realized, for example, in the suspension of ferromagnetic particles in the nematic, which is called ferronematic 关21兴, in the presence of the external magnetic field. This Coulomb attraction can be found experimentally also between the liquid drops in the nematics in the presence of the inclined electric field. In Ref. 关23兴 it was shown that isotropic drops suspended in the nematic phase 共5CB and MBBA兲 are deformed into an elliptical shape along the field. If the direction of the electric field is neither parallel nor perpendicular to the director, then the drops should deform parallel to it, so that both horizontal and vertical mirror symmetry is broken. This must lead to the elastic Coulomb attraction between the drops. As well, we consider that Coulomb attraction must play an important role in the 2D hexagonal lattice of the glycerol droplets on the nematic surface 关8兴. Hybrid boundary conditions break rotational symmetry of the director field around droplets and thus should induce Coulomb attraction between them. We are currently planning an experiment for checking of this idea.

We are interested mainly in the first item. In a nematic states n and ⫺n are identical, therefore 肀 must be an even function of n0 . It must be also an even function of u. The most general vector constructed from u and n0 and even in each of them is

肀 ⫽l 共 cos ␪ 兲 n0 ⫻u, where l(x) is an odd function of x. This leads us to

␦ n共 r兲 ⫽

l 共 cos ␪ 兲 u⬜ . r

Note that the coefficient of 1/r vanishes when u is parallel to n0 and also when u is normal to n0 关since l(0)⫽0兴. This coincides with our symmetry considerations made above. Let us define l 1 ⫽l(cos ␪). In 关21兴 it is shown that the distortion energy is

␦ F⫽4 ␲ Kl 1 sin ␪ ␦ ␪

共20兲

from which it is seen that if the parallel orientation of the particle satisfies the minimum energy, then l 1 should be positive. If we have a suspension of identical particles with positions rp , then the distortion ␦ n(r) from all of them is given by the following equation:

␦ n共 r兲 ⫽ 兺 p

V. SPILLOVER OF THE TWO-PARTICLE INTERACTION

In all previous papers concerning colloidal particles in liquid crystals, the elastic interaction potential between particles has been obtained as the result of the overlapping of

共19兲

l1 关 u ⫺ ␦ n共 rp 兲兴 . 兩 r⫺rp 兩 p⬜

The presence of ␦ n(rp ) means that the particle p creates no long deformations if it is aligned parallel to the director n0 ⫹ ␦ n(rp ).

021709-7



Let us now go to the continuum approach, substituting 兺 →c 兰 dV, c being the concentration of particles c⫽N/V. After acting with the operator ⵜ 2 on both sides

␦ n共 r兲 ⫽cl 1

冕

dr⬘

1 e ⫺ ⑀ expt兩 r⫺r⬘ 兩 u⬜ 共 r⬘ 兲 , 兩 r⫺r⬘ 兩

2 ␰ expt ⫽4 ␲ cl 1 ⬎0 .

Thus the effect of each particle is screened out at dis⫺1 . It is obvious that this screening tances larger than ␰ expt should be manifested also in the interaction potential. More precise analysis shows 共see Appendix A兲 that in dense colloids the interaction potential instead of Eq. 共8兲 takes the form 1 p p ⬘ expt ˆ Aˆ 关 I 共 r ⫺r 兲 ⫹I expt 共 rp ⫺rp 兲兴 , A U pp ⬘ ⫽⫺ 2ll ⬘ ⬘ 8 ␲ m m ⬘ 1ll ⬘ p p ⬘

expt

I ␮ mm ⬘ 共 R兲 ⫽

1

冑K ␮␮ K 33 ⫻

⫹

W cⰆ 3 共21兲

⫺

exp关 ⫺ ␰ ␮ 冑共 K ␮␮ /K 33兲 R 2储 ⫹R⬜2 兴

冑共 K ␮␮ /K 33兲 R 2 ⫹R⬜2 储

⫹

冑

兵 exp关 ⫺ ␰ ␮ 冑共 K ␮␮ /K 33兲 R 储兴

⫺exp关 ⫺ ␰ ␮ 冑共 K ␮␮ /K 33兲 R 2储 ⫹R⬜2 兴其 ,

␰ ␮ ⫽ 冑c 共 a 11⫹a 22兲 /2K ␮␮ ,

共22兲

where ␰ ␮ are inverse screening lengths ( ␮ ⫽1,2) and c is the concentration of particles, a 11 and a 22 are the corresponding components of the tensor a ␮ ␯ ⫽ ␣ lm 关 k l ␮ k m ␯ ⫺k l 3 k m 3 ␦ ␮ ␯ 兴 . In the one-constant approximation K ␮␮ ⫽K 33⫽K it becomes dependent only on the scalar of the vector R, ⫹

U pp ⬘ ⫽⫺

Q m,m ⬘ 8␲K

ˆ mp Aˆ p ⬘ A m⬘

冋

册

exp共 ⫺ ␰ 兩 rp ⫺rp ⬘ 兩兲 , 兩 rp ⫺rp ⬘ 兩

K

冑cS c

关 Q m,m ⬘ ⫹ 共 ⫺1 兲 ␮ ⫹1 Q m,m ⬘ 兴

⫺ 2 共 ⫺1 兲 ␮ Q m,m ⬘ 2 K ␮␮ K 33 ␰ ␮ R⬜

the particle itself. In this case tensor a ␮ ␯ is identical to zero. Really ␣ lm ⫽ ␣ ␦ lm and k l ␮ k l ␯ ⫽ ␦ ␮ ␯ so a ␮ ␯ ⬅0 for small spherical particles. This means that no screening takes place. It arises only for the anisotropic coats. The physical meaning of this elastic screening can be easily understood from the following example: two people badly hear one another in the crowded room. The more people between them the worse they hear one another. Similarly with particles, the formula 共22兲 is true when the screening length is bigger than the average distance between particles, ␰ ⫺1 Ⰷ 具 l 典 ⫽1/3冑c. From here we can write the condition on the anchoring strength of the coat under which the given approach is applicable,

共23兲

where ␰ ⫽ 冑c(a 11⫹a 22)/2K. It is clearly seen that the presence of the macroscopic concentration c of particles leads to the screening of the pair interaction potential with the screening length ␰ ⫺1 ⬇ 冑K/WcS 共W is absolute value here, not depending on the sign兲, S area of the particle. This screening takes place both for the homeotropic and for the planar anchoring. Concentration here is included in the inverse screening length ␰ only, so that the limit c→0 makes ␰ ⫽0 and gives back the unscreened result of Eq. 共12兲. The presence of the tensor a ␮ ␯ in the screening lengths indicates that screening has an anisotropic nature, i.e., it exists only for the anisotropic coats. For the small spherical particles 共less than 1 ␮m兲 without any topological defects, the coat coincides with

.

共24兲

VI. EXPLANATION OF THE CELLULAR TEXTURE IN FERRONEMATICS

In 1970, Brochard and de Gennes proposed doping the liquid crystal matrix with ferromagnetic particles to make possible the coupling of the liquid crystal molecular orientation to weak external fields 关21兴. The authors treated such a system theoretically and predicted Freédericksz effect in the weak magnetic fields H⬃10 G. So the doped matrix exhibits collective orientational distortion in the weak magnetic fields. Also they predicted segregation effects, i.e., the smooth change of particle concentration c(R) from point to point by applying the magnetic field, so that the concentration increases in the center of the cell. In a paper 关2兴 authors observed experimentally the collective behavior in the MBBA doped with magnetic particles, manifested as a longranged uniform distortion of the molecular orientation of the entire sample upon application of weak magnetic fields H ⬍1 G. In that experiment, particles were coated with DMOAP, which provides homeotropic anchoring on its surfaces, so that the magnetic particles lie perpendicular to the nematic director in the absence of the magnetic field. On reaching the field H⬃30 G, the experiment shows 关2兴 that the uniform orientational distortion is replaced by a new field-induced cellular texture with the cells having dimensions of the order of tens of micrometers. So at the critical concentration in the center, magnetic particles clump into aggregates. This clumping had no explanation because the magnetic dipole-dipole interaction is much smaller than the interaction with the external magnetic field. Indeed, the magnetic moment ␮ ⫽M s v induces interaction E dd⫽ ␮ 2 /R 3 , where R is average distance between particles R ⫺3 ⬃c ⬃1010 cm3 . So E dd⬃4⫻10⫺15 erg. Energy of interaction with external magnetic field H⬃10 G is E H ⫽ ␮ H⬃3 ⫻10⫺12 erg and E H ⰇE dd . We explain this field induced cellular texture by the clumping of particles, which is caused by the elastic deformations of the director, i.e., by the elastic interaction between particles. The magnetic field rotates particles and induces Coulomb attraction between them according to Eq. 共17兲, which is screened by the collective action of neighbors,

021709-8



so that the leading interaction potential becomes a Yukawa attraction, U⫽⫺

Q2 exp共 ⫺ ␰ R 兲 . R

共25兲

Now we can find the conditions of the spinodal decomposition in the system of elastically interacting particles. In the mean-field approximation, the free energy of such a system is written in the form F⫽

kT v

冕

f 共 R兲 ln f 共 R兲 dV⫹

1 2v2

冕

f 共 R兲 f 共 R⫹r兲

⫻U 共 r兲 dR dr,

共26兲

where f (R) is the volume fraction of particles, f ⫽c v , v being the volume of the particle. We need to find the conditions under which the homogeneous distribution becomes unstable. The concentration becomes f (R)⫽ f 0 ⫹ ␦ f (R), f 0 is the ground volume fraction. We make a series expansion f 共 R⫹r兲 ⬇ f 共 R兲 ⫹ 共 r“ 兲 f 共 R兲 ⫹ 21 共 r“ 兲 2 f 共 R兲 . Then we have F⫺F 0 ⫽ 12 N⫽2kT/ v ⫹

1 v2

冕

⬁

R0

冕

N ␦ f 2 共 R兲 ⫹M 共 “ ␦ f 兲 2 ,

U 共 r 兲 dr,

M ⫽⫺

1 2v2

冕

⬁

R0

U 共 r 兲 r 2 dr. 共27兲

Here R 0 is the size of the particle. Inasmuch as U⬍0, a phase transition occurs for N⬍0. In our case ␰ R 0 Ⰶ1 and we may write N⬇

2kT 4 ␲ e ⫺ , f 0v ␰ 2v 2

M⬇

12␲ e . ␰ 4v 2

tor in the near region and this breaking defines the interaction law between particles in the far region. We take note of the coat, that is, the region that confines all topological defects and strong director deformations and has the same symmetry as the director near the particle. Outside the coat there are no topological defects and the director deformations are small. We show that if the coat has three symmetry planes, then the far-field interaction is of quadrupole-quadrupole type and, in general, depends not only on the angle ␪ between R and n0 , but also on the azimuthal angle, for example, for the cylinders that lie horizontally in the vertical director field. If one symmetry plane is broken, the dipolar moment is induced perpendicular to it and leads to the dipole-dipole interaction between the coats. The typical realization of this situation is the pair of radial and hyperbolic hedgehogs or nonequatorial disclination rings near the water droplet with homeotropic boundary conditions. One more distinctive feature of colloidal particles in nematics is the Coulomb attraction between them, when the nonzero torque moment is applied to them by the nematic. This can be reached by the external field, for example, by the magnetic field in the suspension of magnetic particles, or by the inclined electric field in the nematic emulsions. From symmetry considerations, it is equivalent to the breaking of both the horizontal and vertical mirror symmetries. Collective action of all anisotropic particles leads to the exponential screening of the interaction. In the ferronematics, external magnetic field gives rise to the Yukawa elastic attraction between particles, which is the reason of the threshold clumping and formation of the cellular texture. We are planning now to observe experimentally the Coulomb attraction in the nematic emulsions. In Appendix B, the general expression for the energy of colloid particles in the curved director field is given. The quadrupole particles move toward the high splay deformations if they have planar anchoring and repel from them if they have homeotropic anchoring on their surfaces.

Below the critical point N⬃4 ␲ e/ ␰ 2 v 2 . The length of the most growing instability is 1 l inst⫽ 冑2M /N⬃ ⬃ ␰

冑

K . WcS

共28兲

There are the following parameters in the experiment 关2兴 with cylindrical particles: c⬇1010 cm⫺3 , S⬇2 ␲ rL, the radius of the particle r⬇0.05 ␮ m, the length L⬇0.5 ␮ m, elastic constant K⬃10⫺7 dyn, anchoring energy W ⬃10⫺3 dyn/cm, and we find l inst⬇30 ␮ m. This is in a qualitative agreement with the experimental size of the cells. VII. CONCLUSIONS

In this paper we have examined theoretically elastic interactions in nematic colloids with the help of the general considerations concerning breaking of different symmetries of the director field in the vicinity of colloid particles. This is caused both by the shape of particles and the anchoring strength. Their joint efforts break the symmetry of the direc-

ACKNOWLEDGMENTS

We acknowledge helpful discussion with J. Fukuda, M. Yoneya, K. M. Aoki, and E. Kats. Authors S.B.C. and P.M.T. gratefully acknowledge financial support by the Grant from the Ministry of Education and Science of Ukraine No. 02.07/ 00145. APPENDIX A:

SCREENING EFFECTS IN THE INTERACTION

The total free energy of the system—particles plus the nematic liquid crystal—is sum of the bulk 共2兲 and surface energies 共7兲, F⫽F b ⫹F s .

共A1兲

We do not consider here the distribution entropy part of the free energy, because it does not influence the director distribution and does not influence the finding of the elastic interaction potential between particles. We consider that all

021709-9



topological defects are hidden inside coats, so that the director distortion from the homogenous state n0 is small anywhere, n共 r兲 ⫽n0 ⫹ ␦ n共 r兲 ,

兩 ␦ n兩 Ⰶ1.

共A2兲

We can use the Fourier representation for the director in the entire space, thereby considerably simplifying the problem. In the Fourier representation we have:

␦ n共 r兲 ⫽

1 共 2␲ 兲3

冕

d 3 q exp共 ⫺iq•r兲 ␦ n共 q兲 .

共A3兲

We substitute Eq. 共A3兲 in the bulk Frank energy 共2兲 to obtain F b⫽

1 1 2 共 2␲ 兲3

冕

d 3 q 兵 K 11兩 q␦ n共 q兲兩 2 ⫹K 22兩关 n⫻q兴 ␦ n共 q兲兩 2

⫹K 33兩共 n•q兲 ␦ n共 q兲兩 2 其 .

共A4兲

To simplify this expression we choose the special basis e1 ⫽

共 q⬜ ⫻n0 兲 , q⬜

e2 ⫽

q⬜ , q⬜

e3 ⫽n0 ,

q⬜ ⫽n0 ⫻q. 共A5兲

For this basis we have q⫽(q⬜ ,0,q 储 ) ⫽( ␦ n 1 , ␦ n 2 ,0), and Eq. 共A6兲 reduces to F b⫽

1 1 2 共 2␲ 兲3

兺i

冕

␦n

and

d 3 q 兵 K ii q⬜2 ⫹K 33q 2储其兩 ␦ n i 共 q兲兩 2 . 共A6兲

Because we assume that the director varies smoothly from point to point and relation 共A2兲 is true, we can consider the director to have a given value inside the volume of the particle. This assumption is valid if the total volume of the suspended particles is much less than the entire volume of the system, i.e., the volume fraction of particles is small, c v Ⰶ1, where c⫽N/V is the concentration and v is the volume of the particle 共the ‘‘gas’’ approximation兲. For the real system 关2兴, c⫽1010 cm⫺3 , v ⬃10⫺15 cm3 , and c v ⬃10⫺5 , and our assumption is therefore true. The director on the surface can therefore be expressed through the director in the center of mass Rp of the particle and its derivatives,

␦ n共 s兲 ⫽ ␦ n共 Rp 兲 ⫹ 共 ␳•“ 兲 ␦ n共 Rp 兲 ⫹ 21 共 ␳•“ 兲 2 ␦ n共 Rp 兲 ,

• ␯(s) 兴 2 and also include the second powers of the perpendicular director deformations ␦ n x and ␦ n y . We thus write 关 ␯共 s兲 •n共 s兲兴 2 ⫽ 共 ␯•n0 兲 2 ⫹2 共 ␯•n0 兲共 ␯• ␦ n兲 ⫹2 共 ␯•n0 兲共 ␳•“ 兲

⫻共 ␯• ␦ n兲 ⫹2 共 ␯•n0 兲共 ␳•“ 兲 2 共 ␯• ␦ n兲 ⫹ 共 ␯• ␦ n兲 2 ,

where ␯⫽ ␯(s) and ␦ n⫽ ␦ n(Rp ). We note that this expression involves two smallness parameters. The first is the perpendicular component of the director 兩 ␦ n x 兩 , 兩 ␦ n y 兩 ⬃␧, ␦ n 3 ⬃␧ 2 and the second is the ratio of the particle size to the average deformation length l n of the director %⫽ ␳ /l n . In 关1兴, the respective terms that are proportional to ␧, %␧, and % 2 ␧ were taken into account. The expansion in % is equivalent to the multipole expansion in 关9兴. In this paper we also take the last term proportional to ␧ 2 into account. This term is not essential at the distances comparable to the average distance between particles, as we see below, and it can therefore be omitted for the systems considered in 关4,5,9兴, where the concentration of the dispersed particles is small. It becomes essential for dense colloids, where there are too many particles and where the interference of the distortions from all particles is considerable. In 关9兴, 兩 ␦ n ␮ 兩共with ␮ ⫽x,y兲 was shown to fall off as R ⫺2 and R ⫺3 , depending of the dipole or quadrupole symmetry. We thus conclude that ␧⬃% 2 for the third term in Eq. 共A8兲, which has dipole symmetry 共and ␧ 2 Ⰶ%␧ for this case兲 and ␧⬃% 3 (␧ 2 Ⰶ% 2 ␧) for the fourth term with quadrupole symmetry. In any case, taking the last term into account gives only small corrections at the average distances and for a small number of particles. As we see below, it is essential in the collective effect of the screening at large distances, where the concentration of particles is high. We specifically clarify this problem in what follows. For this, we write the scalar products 关 ␯(s)•n(s) 兴 in the local basis (k1 ,k2 ,k3 ) associated with each particle. For ex␯(s)•n(s)⫽ 兺 l⫽1,2,3 ( ␯•kl )( ␦ n•kl )⫽ ␯ l ␦ n ␮ k l ␮ ample, ⫹ ␯ l ␦ n 3 k l 3 , where ␯ l ⫽( ␯•kl ), k l 3 ⫽kl •n0 , and ␦ n 3 ⫽⫺ 12 ( ␦ n 2x ⫹ ␦ n 2y ). The surface energy is then written as F s ⫽F 共s0 兲 ⫹F 共s1 兲 ⫹F 共s2 兲 , F 共s0 兲 ⫽N F 共s1 兲 ⬇

where ␳ is the vector drawn from the center of mass to the point s on the surface. The complete expression for the director on the surface through the director in the center of mass of the particle is therefore given by

冖

F 共s2 兲 ⫽

2

We now fix a coordinate system 共x,y,z兲 where the z axis is parallel to the nondeformed director n0 and x,y are perpendicular to it. This system is firmly fixed in space. We next substitute director field 共A7兲 in the scalar product 关 n(s)

兺p

冖

d ␴ W c 共 s兲关 ␯共 s兲 •n0 兴 2 ,

共A9兲共A10兲

d ␴ W c 共 s 兲 ␯ l ␯ m k l 3 兵 2 ␦ n ␮ k l ␮ ⫹2 共 ␳•“ 兲 ␦ n ␮ k l ␮

⫹ 共 ␳•“ 兲 2 ␦ n ␮ k l ␮ 其 ,

n共 s兲 ⫽n0 ⫹ ␦ n共 Rp 兲 ⫹ 共 ␳•“ 兲 ␦ n共 Rp 兲 ⫹ 共 ␳•“ 兲 ␦ n共 Rp 兲 . 共A7兲 1 2

共A8兲

兺p

冖

共A11兲

d ␴ W c共 s 兲 ␯ l␯ m关 ␦ n ␮␦ n ␯k l␮k m␯

⫺ 共 ␦ n 2x ⫹ ␦ n 2y 兲 k l 3 k m 3 兴 .

共A12兲

Here N is the total number of particles in the whole volume V of the liquid crystal matrix, E s(1) and E s(2) are surface terms linear and quadratic in ␦ n ␮ , respectively. It can be

021709-10



rewritten in terms of the tensors 共11兲 so that the elastic energy F el , i.e., the energy of the deformations of the director field has the form F el⫽F b ⫹F 共s1 兲 ⫹F 共s2 兲 , F 共s1 兲 ⫽

共A14兲

k ⫺ 共 ␦ n 2x ⫹ ␦ n 2y 兲 k l 3 k m 3 兴 .

␮ m␯

共A15兲 The main difference between this approach and 关1兴 is in taking the term in Eq. 共A15兲 into account. It is quadratic in the director deformations and can be regarded as the contribution of all particles to the interference of the distortions. Precisely this term leads to the screening effects. Some of its features can be considered without finding the director. For example, it is clearly seen that it vanishes for the spherical particles. Indeed, ␣ lm ⫽ ␣ ␦ lm and k l ␮ k l ␯ ⫽ ␦ ␮ ␯ for the sphere, and therefore F s(2) ⬅0. For any other shape, Eq. 共A15兲 does not vanish. To describe its effect analytically, we go to the continuum limit in this expression and replace the summation with the integration over the entire space, 兺 p ⇒c 兰 dV, where c⫽N/V is the concentration of particles, F 共s2 兲 ⫽

c 2

冕

dV ã ␮ ␯ ␦ n ␮ 共 x兲 ␦ n ␯ 共 x兲 ,

ã ␮ ␯ ⫽ ␣ lm 关 k l ␮ k m ␯ ⫺k l 3 k m 3 ␦ ␮ ␯ 兴 .

c 2共 2␲ 兲3

冕

F el⫽

1 2共 2␲ 兲3

冋

共A19兲共A20兲

冕

d 3 q V i j 共 q兲 ␦ n i 共 q兲 ␦ n *j 共 q兲 ⫹b i* 共 q兲 ␦ n i 共 q兲共A21兲

V i j 共 q兲 ⫽ 共 K ii q⬜2 ⫹K 33q 2储兲 ␦ i j ⫹ca i j , b i 共 q兲 ⫽

兺p e iq•r a m k m . p

i

共A22兲共A23兲

Here, m⫽1,2,3, i, j⫽1,2, and ␦ n i (q), and k m i are the projections of these vectors on the basis (e1 ,e2 ,e3 ). 1. Director distribution in the doped nematic liquid crystal

Having found the complete expression for the elastic energy of the liquid crystal with particles, we can find the director at any point of the system from the extremum condition

␦ F ⫽V i j 共 q兲 ␦ n i 共 q兲 ⫹b j 共 q兲 ⫽0, ␦ n *j 共 q兲 el ␦ n i 共 q兲 ⫽⫺V ⫺1 i j 共 q兲 b j 共 q兲 .

共A24兲

In matrix form, the last equation becomes d 3 q ã ␮ ␯ ␦ n ␮ 共 q兲 ␦ n ␯* 共 q兲 .

共A17兲

cos共 ␺ 兲

⫺sin共 ␺ 兲

sin共 ␺ 兲

cos共 ␺ 兲

册

冉

1

⫹e iq•rp a m 关 ␦ n* 共 q兲 •km 兴其 ,

共A18兲

共A25兲

Having found the director field, we substitute Eq. 共A25兲 in Eq. 共A21兲 and obtain the elastic energy of the director deformations in the DNLC, F el⫽⫺

p

册冉冊

2. Elastic energy and the pair interaction potential between particles

.

兺p 2 共 2 ␲ 兲 3 冕 d 3 q 兵 e ⫺iq•r a m* 关 ␦ n共 q兲 •km 兴

冋

冊

1 V 22 ⫺V 12 b 1 共 q兲 ␦ n 1 共 q兲 ⫽⫺ ␦ n 2 共 q兲 D ⫺V 12 V 11 b 2 共 q兲

2 . with D⫽V 11V 22⫺V 12

In the basis (e1 ,e2 ,e3 ) the surface energy becomes F 共s1 兲 ⫽

d 3 q a i j ␦ n i 共 q兲 ␦ n *j 共 q兲 ,

⫹b i 共 q兲 ␦ n * i 共 q兲 ,

共A16兲

The tensor ã ␮ ␯ is here taken in the 共x,y,z兲 coordinate system, which is not convenient. It is much more suitable to write the surface energy and the bulk energy in Eq. 共A6兲 in the same basis (e1 ,e2 ,e3 ). This basis is rotated by the angle ␺ (q) with respect to 共x,y,z兲 around the z axis. In the new basis the director has the components ␦ n⫽( ␦ n 1 , ␦ n 2 ,0) and ␦ n ␮ ⫽␼ ␮ i ␦ n i 共with ␮ ⫽x,y and i⫽1,2兲. The rotation matrix is given by ␼ ␮i⫽

冕

We now add the bulk energy F b and the surface energy F s and find the total energy of the system F total⫽F s(0) ⫹F el with the elastic energy

We thus consider the interference of only long wavelength distortions of the director field. In the Fourier representation, we have F 共s2 兲 ⫽

c 2共 2␲ 兲3

a i j ⫽␼ iT␮ã ␮ ␯ ␼ ␯ j .

兺p 兵 ␣ lm ⫹ ␤ lms共 ks •“ 兲 ⫹ ␥ lmst共 ks •“ 兲

兺p ␣ lm 关 ␦ n ␮ ␦ n ␯ k l

F 共s2 兲 ⫽

共A13兲

⫻共 kt •“ 兲其 ␦ n ␮ k l ␮ k m 3 , F 共s2 兲 ⫽ 21

a m ⫽ 共 k•n0 兲关 ␣ lm ⫹i ␤ lms 共 q•ks 兲 ⫺ ␥ lmst 共 q•ks 兲共 q•kt 兲兴 ,

1 2共 2␲ 兲3

冕

d 3 q V ⫺1 i j 共 q 兲 b i* 共 q 兲 b j 共 q 兲 ⬍0. 共A26兲

The negative sign implies that the total free energy F⫽F s(0) ⫹F el evaluated for solution 共A25兲 is less than the energy F ⫽F s(0) for the undeformed director field n0 . The total energy

021709-11



F el can be represented as the sum of the pair potentials between two particles. Indeed, we introduce the operator Aˆ m , such that Aˆ m e iq•r⫽a m e iq•r,

共A27兲

Aˆ m ⫽ 共 kl •n0 兲关 ␣ lm ⫹ ␤ lms 共 ks •“ 兲 ⫹ ␥ lmst 共 ks •“ 兲共 kt •“ 兲兴 . 共A28兲 The elastic energy F el then takes the form F el⫽ 12 1 ˆ p Aˆ p ⬘ A U pp ⬘ ⫽⫺ 共 2 ␲ 兲3 m m⬘

冕

兺

p,p ⬘

where a( ␪ )⫽ 12 (a ˜ 11⫹a ˜ 22). This approximation makes the (q) diagonal and allows us to take all the propagator V ⫺1 ij integrals analytically. The diagonal approximation is exact only for ␪ ⫽0, when the whole system has the rotational symmetry in the 共x,y兲 plane. The coefficient a( ␪ ) depends on the shape of particles. For example, for the cylinders with rⰆL, we have a( ␪ )⫽ ␲ rLW(2⫺3 sin2 ␪), and for flat 共pancake兲 particles with rⰇh, a( ␪ )⫽2 ␲ r 2 W(1⫺3 cos2 ␪) 共where ␪ is the angle between the normal to the pancake plane and the director兲. In the diagonal approximation, the propagator therefore becomes

共A29兲

U pp ⬘ ,

2 2 ⫺1 ␦ij V ⫺1 i j 共 q 兲 ⫽ 关 K ii q⬜ ⫹K 33q 储 ⫹ca 共 ␪ 兲兴

and the pair potential is given by 3

d qe

iq• 共 rp ⫺rp ⬘ 兲

V ⫺1 i j 共 q 兲 k m i k m ⬘j .

U pp ⬘ ⫽⫺

共A30兲

The expression U pp ⬘ has the meaning of the pair interaction potential between particles p and p ⬘ , which is caused by long-range deformations of the director field. The subscript p indicates that we must substitute “⫽ ⳵ / ⳵ r p in the operator Aˆ mp . This expression is valid for particles of ordinary shape and orientation. It accounts for screening effects that arise from the interference of the director field distortions by all particles. 3. Pair potential in the diagonal approximation: Analytical results

Although expression 共A30兲 is exact, it is too difficult to find the results analytically. In the most general case, the pair potential U(R,⍀) depends on all the three components of the radius vector R⫽rp ⫺rp ⬘ and on three Euler angles ⍀, which determine the orientation of particles in space 共we assume that all particles are oriented in the same way, and therefore all of them have the same Euler angles兲. To take the screening effects into account analytically, we consider particles with rotational symmetry around one axis. For such particles, the pair potential U(R, ␪ ) depends on the angle ␪ between this symmetry axis and the director. If ␪ ⫽0, all particles are parallel to the director. In this case, the entire DNLC has the rotational symmetry around the director n0 and the pair potential U(R⬜ ,R 储 ) depends on the perpendicular and parallel projections of R with respect to n0 . But in the case where ␪ ⫽0 there arises the second preferential direction in the system—the direction in which all particles lie. We project this direction on the plane perpendicular to the director and let s denote the projection, s•n0 ⫽0. In this case, we have potential U⫽U(R⬜ ,R 储 , ␸ , ␪ ), where ␸ is the azimuthal angle between R and s. To obtain analytical results, we average over the angle ␸. For this purpose, we average the tensor a i j in Eq. 共A20兲 over the angle ␺ and diagonalize it. We call this dropping of the off-diagonal terms the diagonal approximation, a i j 共 ␺ , ␪ 兲 ⇒ 具 a i j 典 ␺ ⫽ 21

冋

共A32兲

ã 11⫹a ˜ 22

0

0

ã 11⫹a ˜ 22

册

⫽a 共 ␪ 兲 ␦ i j , 共A31兲

1 p p⬘ ˆ Aˆ 关 I 共 R兲兴 , A 8 ␲ l l ⬘ ll ⬘

I ll ⬘ 共 R兲 ⫽I 1ll ⬘ 共 R兲 ⫹I 2ll ⬘ 共 R兲 ,

共A33兲共A34兲

I 1ll ⬘ 共 R兲 ⫽

冕

关 kl • 共 q⬜ ⫻n0 兲兴关 kl ⬘ • 共 q⬜ ⫻n0 兲兴 d 3 q e iq•R 2 2 , ␲ q⬜ 关 K 11q⬜2 ⫹K 33q 2储 ⫹ca 共 ␪ 兲兴共A35兲

I 2ll ⬘ 共 R兲 ⫽

冕

共 kl •q⬜ 兲共 kl ⬘ •q⬜ 兲 d 3 q e iq•R 2 2 . ␲ q⬜ 关 K 22q⬜2 ⫹K 33q 2储 ⫹ca 共 ␪ 兲兴共A36兲

It is easy to integrate over q in Eqs. 共A35兲 and 共A36兲 using the coordinate system with the basis r1 ⫽

R⬜ ⫻n0 , R⬜

r2 ⫽

R⬜ , R⬜

r3 ⫽n0 ,

R⬜ ⫽n0 ⫻R. 共A37兲

This basis is rotated with respect to that in Eq. 共A5兲 by some angle ␸ about the axis n0 . The quantities q⬜ and q 储 are similar in both bases. We therefore have exp(⫺iq•R) ⫽exp关⫺i(q⬜R⬜ cos ␸⫹q储R储)兴 and the denominators of the fractions involved in Eqs. 共A35兲 and 共A36兲 do not depend on the angle ␸. Integrating over the ␸ gives I ␮ ll ⬘ 共 R兲 ⫽

冕

⬁

0

⫹

dq⬜ q⬜ 兵 Q l,l ⬘ J 0 共 q⬜ R⬜ 兲 ⫺

⫹ 共 ⫺1 兲 ␮ Q l,l ⬘ J 2 共 q⬜ R⬜ 兲其 ⫻

冕

⬁

⫺⬁

dq 储

exp共 ⫺iq 储 R 储兲 , ␲ 关 K ␮␮ q⬜2 ⫹K 33q 2储 ⫹ca 共 ␪ 兲兴共A38兲

(⫾)

where ␮ ⫽1,2 and Q l,l ⬘ ⫽(r1 •kl )(r1 •kl ⬘ )⫾(r2 •kl )(r2 •kl ⬘ ) and J 0 and J 2 are the Bessel functions. In order to take these integrals we must scrutinize thoroughly the function a( ␪ ). As mentioned above a( ␪ ) ⫽ ␲ rLW(2⫺3 sin2 ␪) for cylindrical particles. The case where W⬎0 corresponds to the planar anchoring, and W ⬍0 corresponds to the normal anchoring. For planar anchor-

021709-12



ing the equilibrium state of particles is ␪ ⫽0 and a planar(0) ⫽2 ␲ rLW⬎0, and for normal anchoring the equilibrium state is ␪ ⫽ ␲ /2 and a normal( ␲ /2)⫽⫺ ␲ rLW⬎0. expt We write I ␮ ll ⬘ (R) for the expression I ␮ ll ⬘ (R) and introduce p ␮ ⫽ 冑K ␮␮ /K 33R 储 , s⫽R⬜ , and z ␮ ⫽ 冑ca( ␪ )/K ␮␮ . Afexpt ter the integration over the q 储 , I ␮ ll ⬘ (R) becomes expt

I ␮ ll ⬘ 共 R兲 ⫽

1

冑K ␮␮ K 33

冕

exp共 ⫺p ␮ 冑q⬜2 ⫹z ␮2 兲

⬁

0

dq⬜ q⬜

冑q⬜2 ⫹z ␮2

⫹

⫺

⫻ 兵 Q l,l ⬘ J 0 共 sq⬜ 兲 ⫹ 共 ⫺1 兲 ␮ Q l,l ⬘ J 2 共 sq⬜ 兲其 . 共A39兲 For the Bessel functions, we have the relation 2 ␯ J ␯ 共 x 兲 ⫽xJ ␯ ⫹1 共 x 兲 ⫹xJ ␯ ⫺1 共 x 兲 ,

FIG. 4. Movement of the small particle in the curved director field. Quadrupolar particles with planar anchoring move toward the high splay deformations while those with homeotropic anchoring are repelled from such regions.

which for ␯ ⫽1 gives 2 J 2 共 x 兲 ⫽ J 1 共 x 兲 ⫺J 0 共 x 兲 . x

APPENDIX B:

The corresponding integrals with the functions J 1 (x) and J 0 (x) are given by

冕

⬁

0

dq⬜ q⬜ ⫽

exp共 ⫺p ␮ 冑q⬜2 ⫹z ␮2 兲

冑q⬜2 ⫹z ␮2

exp共 ⫺z ␮ 冑p ␮2 ⫹s 2 兲

冑p ␮2 ⫹s 2

冕

冑

⫽

共A40兲

,

⬁ exp共 ⫺p ␮ q⬜2 ⫹z ␮2 兲 dq⬜ 0 q⬜2 ⫹z ␮2

冑

J 0 共 sq⬜ 兲

expt

J 1 共 sq⬜ 兲

1 关 e ⫺p ␮ z ␮ ⫺exp共 ⫺z ␮ 冑p ␮2 ⫹s 2 兲兴 . sz ␮

1

冑K ␮␮ K 33

再

⫹

⫺

关 Q l,l ⬘ ⫹ 共 ⫺1 兲 ␮ ⫹1 Q l,l ⬘ 兴

⫺

⫹ 共 ⫺1 兲 ␮ Q l,l ⬘

In this appendix we consider the behavior of particles in the curved director field. We suppose that the director field is not homogeneous because of the global boundary conditions, and that the director deformation length is much more than the size of particles. So in this section we do not consider the deformations of the director that are caused by particles themselves. The anchoring strength on the surface of the coat is written as F sc⫽

共A41兲

e ⫺z ␮ 冑

冖

d ␴ W c 共 s兲关 ␯共 s兲 •n共 s兲兴 2 .

共B1兲

Inasmuch as the global deformations scale is large, we can express the director on the surface n(s) through the director in the center of mass of the coat, n共 s兲 ⫽n0 ⫹ 共 ␳•“ 兲 n0 ⫹ 21 共 ␳•“ 兲 2 n0 .

Using these relations we find I ␮ ll ⬘ 共 R兲 ⫽

BEHAVIOR OF PARTICLES IN CURVED DIRECTOR FIELDS

2 p ␮ ⫹s 2

冑p ␮2 ⫹s 2

冎

␳ is the vector from the center of mass to the point s. Here n0 is the local director field, which would have been in the center if there was no coat in that place 共See Fig. 4兲. Then the anchoring energy takes the form F sc⫽F 0 ⫹F 1 ⫹F 2 ,

2 2 2 关 e ⫺p ␮ z ␮ ⫺e ⫺z ␮ 冑p ␮ ⫹s 兴 . 2 s z␮

F 0⫽

共A42兲 The pair interaction potential is then given by 1 p p ⬘ expt ˆ Aˆ 关 I 共 r ⫺r 兲 ⫹I expt 共 rp ⫺rp 兲兴 , A U pp ⬘ ⫽⫺ 2ll ⬘ ⬘ 8 ␲ l l ⬘ 1ll ⬘ p p ⬘ 共A43兲

冖

d ␴ W c 共 s兲共 ␯•n0 兲 2 ,

冖␴ 冖␴

F 1 ⫽2

d W c 共 s兲共 ␯•n0 兲共 ␳•“ 兲共 ␯•n0 兲 ,

F 2⫽

d W c 共 s兲关共 ␯•n0 兲共 ␳•“ 兲 2 共 ␯•n0 兲

⫹ 兵共 ␳•“ 兲共 ␯•n0 兲其 2 兴 .

which is equivalent to Eq. 共21兲. 021709-13



The first item F 0 is responsible for the orientation of the particle with respect to the director, whereas the second and the third items describes the behavior of the whole particle in the curved director. All the scalars can be represented in the basis (k1 ,k2 ,k3 ) as ␯•n0 ⫽( ␯•kl )(n0 •kl )⫽ ␯ l n l , ( ␳•“) ⫽ ␳ s (ks •“)⫽ ␳ s ⳵ s . Then the result is

high splay deformations. This is confirmed experimentally: small water drops gather in the center of the big nematic droplet with homeotropic anchoring on the surface 共near radial hedgehog兲 and assemble near the surface boojums, when the global conditions are planar 关3兴.

F 1⫽ ␤ l␮sn l⳵ sn ␮ ,

共B2兲

The F 2 term for the quadrupole particles with azimuthal symmetry takes the form

F 2 ⫽ ␥ l ␮ st ⳵ s 共 n l ⳵ t n ␮ 兲 .

共B3兲

F 2 ⫽ ␥ 1313共 n•“ 兲 div n.

共B4兲

These are the general expressions that describe movement of particles in the curved director field. The particle will move in the way to minimize the sum of F 1 and F 2 . The first term describes the behavior of the coat with dipolar symmetry in the curved director and the second one describes the behavior of the coat with quadrupole symmetry in the curved director. If the coat has azimuthal symmetry, then nonzero components of the ␤ are ␤ 311⫽ ␤ 322 and in the Cartesian frame 关 n⫽(0,0,1) 兴 , F 1 takes the form F 1 ⫽ ␤ 311n 3 ( ⳵ 1 n 1 ⫹ ⳵ 2 n 2 )⫽⫺4 ␲ Kp z n z div n, which coincides with the result 关9兴 and indicates that dipoles assemble in the places with

We see that particles that do not have a dipole moment 共for example, small spherical particles for which WR/KⰆ1兲 move differently as a function of the sign of the anchoring. For planar anchoring W⬎0 and ␥ 1313⬎0, for homeotropic anchoring ␥ 1313⬍0, so that particles with planar anchoring move toward the places with high splay and particles with homeotropic anchoring are repelled from the regions with high splay. We see that although the sign of the anchoring does not influence the interparticle quadrupole-quadrupole interaction 关see Eq. 共16兲兴, it plays a crucial role in the behavior of particles in the curved director fields.

关1兴 B. I. Lev and P. M. Tomchuk, Phys. Rev. E 59, 591 共1999兲. 关2兴 S. H. Chen and N. M. Amer, Phys. Rev. Lett. 51, 2298 共1983兲. 关3兴 P. Poulin, H. Stark, T. C. Lubensky, and D. A. Weitz, Science 275, 1770 共1997兲. 关4兴 P. Poulin and D. A. Weitz, Phys. Rev. E 57, 626 共1998兲. 关5兴 P. Poulin, V. Cabuil, and D. A. Weitz, Phys. Rev. Lett. 79, 4862 共1997兲. 关6兴 P. Poulin, V. A. Raghunathan, P. Richetti, and D. Roux, J. Phys. I 4, 1557 共1994兲. 关7兴 J. Loudet, P. Barois, and P. Poulin, Nature 共London兲 407, 611 共2000兲. 关8兴 V. Nazarenko, A. Nych, and B. Lev, Phys. Rev. Lett. 87, 075504 共2001兲. 关9兴 T. C. Lubensky, D. Pettey, N. Currier, and H. Stark, Phys. Rev. E 57, 610 共1998兲. 关10兴 D. Pettey, T. C. Lubensky, and D. R. Link, Liq. Cryst. 25, 579 共1998兲. 关11兴 V. A. Raghunathan, P. Richetti, and D. Roux, Langmuir 12, 3789 共1996兲. 关12兴 V. A. Raghunathan et al., Mol. Cryst. Liq. Cryst. Sci. Technol., Sect. A 288, 181 共1996兲. 关13兴 S. L. Lopatnikov and V. A. Namiot, Zh. Eksp. Teor. Fiz. 75, 361 共1978兲. 关14兴 E. D. Belockii, B. I. Lev, and P. M. Tomchuk, Pis’ma Zh.

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