PHYSICAL REVIEW E 89, 012509 (2014)

Interaction of small spherical particles in confined cholesteric liquid crystals B. I. Lev,1,2 Jun-ichi Fukuda,2 O. M. Tovkach,1 and S. B. Chernyshuk3 1

Bogolyubov Institute for Theoretical Physics, NAS of Ukraine, Metrologichna 14-b, Kyiv 03680, Ukraine 2 Nanosystem Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba 305-8568, Japan 3 Institute of Physics, NAS of Ukraine, Prospekt Nauky 46, Kyiv 03650, Ukraine (Received 8 November 2013; published 29 January 2014) The theory of the elastic interaction of spherical colloidal particles immersed into a confined cholesteric liquid crystal is proposed. The case of weak anchoring on the particle surfaces is considered. We derive a general expression for the energy of the interaction between small spherical particles (with diameter much smaller than the cholesteric pitch) suspended in a cholesteric confined by two parallel planes. The resulting form of the interaction energy has a more complex spatial pattern and energy versus distance dependence than that in nematic colloids. The absence of translational symmetry related to helical periodicity and local nematic ordering in cholesteric liquid crystals manifest themselves in the complex nature of the interaction maps. DOI: 10.1103/PhysRevE.89.012509

PACS number(s): 61.30.Dk, 82.70.Dd

I. INTRODUCTION

Colloidal suspensions in liquid crystal hosts constitute a new class of composite materials with unique properties, and the physics of these systems has attracted a tremendous amount of interest in recent years [1,2]. Depending on the relative magnitude of the particle size and the anchoring extrapolation length at the surface of every particle, there appears to be a rich variety of superstructures which cannot be observed in typical colloids [3]. These superstructures are a striking manifestation of long-range elastic interactions which do not occur in isotropic media [4]. Colloidal particles distort the liquid crystal ordering in their vicinity and, due to the longrange nature of the distortion, extend their influence over long distances. Since the form of superstructures made by colloidal particles in a liquid crystal is strongly influenced by these elasticity-mediated interactions, understanding them is quite important to predict and control properties and behaviors of liquid crystal colloids and materials derived from them. Traditionally, the main efforts have been focused on nematic hosts. In such media, the colloidal interactions result in different structures such as linear [5,6] and inclined [5,7–9] chains. Particles at a nematic-air interface as well as quasi-two-dimensional colloids in thin nematic cells form a diversity of two-dimensional (2D) crystals [10–15]. Moreover, the authors of [16] recently reported experimental observation of a 3D crystal structure in a system of particles with a dipole configuration of the director field in their vicinity. A theoretical understanding of the matter in bulk nematic liquid crystals has a formal analogy with classical electrostatics. Colloidal particles break continuous symmetry of the director field and cause its distortions, which may be accompanied by topological defects [4,5,17]. Despite such a complex director field distribution, every particle can be effectively presented as a pointlike source of the deformations of monopole, dipole, and quadrupole type [17–19]. This fact became a starting point for a number of approaches toward the theory of nematic liquid crystal colloids [17,18]. Reference [17] gives analytical quantitative results that have been proven experimentally in bulk nematic liquid crystals. Later a proposal was made [20–25] to extend the method of [17] on 1539-3755/2014/89(1)/012509(11)

confined nematics. Using that approach, the elastic interactions between colloidal particles were found in the nematic cell and near one wall with either planar or homeotropic boundary conditions. The proposed theory [20,21] is in good agreement with experimental data for the confinement effect in the case of spheres placed in a homeotropic cell [26] and with the confinement effect in a planar cell with different cell thicknesses [27]. Obviously, as a host fluid, one can choose other types of liquid crystals. The wide variety of liquid-crystal phases would be of great importance because they might provide more possibilities for the formation of colloidal superstructures. Moreover, it is interesting also from a fundamental point of view to know how colloidal particles interact in media with different types of ordering. For example, Ref. [28] reported on the formation of an unusual solid by stabilizing a network of linear defects under tension in the ideal layered structures of a cholesteric liquid crystal. Later, the interaction between particles immersed in cholesteric and smectic liquid crystals was studied in [29–31]. The authors of [32] found that the colloidal stabilization of defect structures in cholesterics between different states typically involves consecutive transformations that preserve the nonsingular nature of defect cores. In addition, they presented reconfigurable particle-defects interaction, which enables patterning of colloidal particles via manipulations with defects and vice versa. In this paper, we focus on a theoretical investigation of cholesteric liquid crystals. The case of small spherical particles in a bulk cholesteric liquid crystal has been theoretically examined in [33]. In practice however, a liquid crystal ought to be confined by some surfaces, and their influence cannot be neglected a priori. For example, recent articles [29–31] reported nontrivial director distribution around the particles suspended in a twisted and cholesteric cell. Interaction between such particles depends crucially on their size and the thickness of the cell. In the case of cholesteric liquid crystals, the cholesteric pitch serves as an additional characteristic length. Therefore, it is interesting to examine how this new length scale would affect the interaction found for nematic suspensions. For this purpose, we apply the method developed in [20,21,23,25] for nematic hosts. Extending that approach, we find an analytical form of the interaction energy in the case of weak anchoring on

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the surfaces of spherical particles smaller than the pitch. This interaction energy depends sensitively on the relative position of two particles and can be nonmonotonic in its landscape. This nontrivial energy landscape might lead to new colloidal superstructures. II. GENERAL APPROACH A. Bulk free energy

The bulk free energy of a cholesteric liquid crystal within the one constant approximation can be written as  K Fbulk = dV [(∇ · n)2 + (n · ∇ × n + q)2 2 + (n × ∇ × n)2 ],

(1)

where K is the Frank elastic constant and q is the wave number of the cholesteric pitch. This functional is minimized by the following director distribution n0 (r) = (cos qz, sin qz,0) with pitch direction along the z axis. The sign of q may be either positive or negative. For definiteness hereinafter we suppose q > 0. To describe the deviations of the director from its ground state n0 (r), which appear in colloidal systems, we can introduce such variables u(r) and v(r) that in the general case n(r) = (cos(qz + u) cos v, sin(qz + u) cos v, sin v). Then in terms of u(r) and v(r), Fbulk is as follows:   K Fbulk = dV (∇v)2 + cos2 v[(∇u)2 −q 2 ] 2  ∂v ∂v +q cos(qz+u)− sin(qz+u) ∂y ∂x   ∂u ∂u cos(qz+u)+ sin(qz+u) , + cos v sin v ∂x ∂y

where ν is the unit normal at the point s on the surface of the pth particle. In the case of homeotropic anchoring, the anchoring strength W (s) is negative. For planar anchoring, it is positive. Due to the anchoring on its surface, every particle gives rise to the distortions of the liquid crystal ordering. These distortions, in turn, depend on the particle shape as well as on the anchoring strength. In this paper, we restrict ourselves to the case of small distortions. This approximation is valid if the anchoring is weak, i.e., WKR  1, where R is the characteristic size of the particle. Typically W ∼ 10−5 J/m2 , K ∼ 10−11 J/m, and the condition can be satisfied for particles of the order of the cholesteric pitch (∼10−7 m). We will require them instead to be much smaller as it allows us to simplify the calculations. For particles larger than the pitch, the deformations of the director field in their vicinity are not small and obey nonlinear equations. In such a case, we can resort to the concept of a coat described in detail in [20,21,23,25] for nematic liquid crystal colloids. A coat is an area that encloses the particle and contains all the large (nonlinear) deformations inside. Outside the coat, the deformations can be considered as small. Therefore, hereinafter we suppose that the director deviations from its ground state n0 are small, i.e., n(r) = n0 + δn(r), δn = (−u sin qz, u cos qz, v), and |δn|  1. Under these conditions, the director field n(r) is defined and continuous throughout the system, even within the particles. Then the relevant terms in Fs are those that are first order in δn. Thus, implying the particle size, one can use the gradient expansion of the director field around the center of mass of the pth particle rp and rewrite the surface energy as

ds W (s){ν(s) · n0 (s)}{ν(s) · δn(s)} Fs  2 p

(2) For small u(r) and v(r), the bulk free energy can be reduced to   K Fbulk = dV (∇v)2 +(∇u)2 +q 2 v 2 2      ∂v ∂u ∂v ∂u v− u cos qz+q v− u sin qz , +q ∂x ∂x ∂y ∂y (3) where we require the deviations u and v to vanish at the confining walls. This bulk free energy is identical to those of [24,25], but here it is written in real space. Unlike the previous studies, we do not use the Fourier presentation of the free energy in this paper. Note that if q = 0, expression (3) completely reproduces the bulk free energy of a nematic liquid crystal [20,21]. B. Surface energy as a source of deformations

In the colloidal system, the interactions between particle surfaces and liquid crystal molecules must also be taken into account . The energy of these interactions is given by the well known Rapini-Popular form

(4) Fs = ds W (s)[ν(s) · n(s)]2 .

=2



ds W (s){ν(s) · [n0 (rp ) + (ρ · ∇)n0 (rp )

p

+ (1/2)(ρ · ∇)(ρ · ∇)n0 (rp ) + · · · ]} × {ν(s) · [δn(rp ) + (ρ · ∇)δn(rp ) + (1/2)(ρ · ∇)(ρ · ∇)δn(rp ) + · · · ]}.

(5)

In particular, the expansion of the ground state n0 (s) ≈ n0 (rp ) + (ρ · ∇)n0 (rp ) + (1/2)(ρ · ∇)(ρ · ∇)n0 (rp ) + · · · is valid only for ρq  1 so that we consider only small particles as compared to the pitch in this paper. Neglecting the O(ν 2 ρ 3 ) terms (the physical meaning of this assumption will be discussed in the next section), we can represent Fs in the following compact form: pi δni (rp ), (6) Fs = A p

pi is determined by the particle shape where the operator A [18,19,23,25,33] and the director ground state, ip = α ik (kk · n0 ) + β ikl kk (kl · ∇)n0 A

p

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+ γ iklm kk (kl · ∇)(km · ∇)n0 + β ikl (kk · n0 )(kl · ∇) + 2γ iklm kk (kl · ∇)n0 (km · ∇) + γ iklm (kk · n0 )(kl · ∇)(km · ∇),

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with ki being the versors of the coordinate system associated with each particle. αil ,βilm ,γilmn are tensor characteristics of the surface (or coat) which contain all information about the symmetry of the director distribution around the particle [18,19], αkl = 2 dσ Wc (s)ν k (s)νl (s), βklm = 2 dσ Wc (s)νk (s)νl (s)ρm (s), and γklmn = dσ Wc (s)νk (s)νl (s)ρm (s)ρn (s), with ρ being a vector pointing from the center of mass of the particle to point s at the surface. Such a treatment of the surface energy represents a powerful technique as it allows us to extract the far-field interaction between the colloids of arbitrary shape to any order. Hence, the surface energy of a cholesteric liquid crystal can be written as  p1 δn1 + A p2 δn2 + A p3 δn3 Fs = Fsource = A =



p

p1 u sin qz −A

 p2 u cos qz + A p3 v . +A

(8)

p

p1 u sin qz + A p2 u cos qz and p u = −A Introducing operators B p3 v, one can rewrite (8) in a simpler form, p v = A C p u(rp ) + C p v(rp )}. {B (9) Fs = Fsource =

director field distortion is the sum of distortions caused by every single particle. Substituting solutions (11) into the free energy and implying the superposition principle, we arrive at   the fact that Fbulk + Fs = p>p Up,p + p Up , where Up is the self-energy of the pth particle and Up,p is the energy of the interaction between the pth and p th particles [20,21,23,25]. The self-energy can be written in the following form, which allows us to exclude the divergent terms: Up = −

1 p C p Hv (rp ,rp )], [Bp Bp Hu (rp ,rp ) + C 8π K

(12)

1 and Hv (r,r ) = where Hu (r,r ) = Gu (r,r ) − |r−r | 1  Gv (r,r ) − |r−r | . The subject of the present paper is the energy of the interaction between the pth and p th particles [20,21,23,25],

Up,p = −

1 p C p Gv (rp ,rp )]. [Bp Bp Gu (rp ,rp ) + C 4π K (13)

This expression is a general representation of the pair interaction energy in cholesteric liquid crystal colloids.

p

Now we can use the method proposed in Refs. [18–21,23,25] to derive the energy of the interaction between the particles immersed in a confined cholesteric liquid crystal. C. Energy of the pair interactions

The distortion profile that minimizes the elastic energy in the presence of the particles can be determined from the following Euler-Lagrange equations: δ(Fbulk + Fs ) δ(Fbulk + Fs ) = = 0. δu(r) δv(r)

(10)

In contrast to the case of nematics, in the cholesteric free ∂u energy the coupling between u and v is present (terms like ∂x v, ∂v u, etc.). These off-diagonal terms make rigorous treatment ∂x of Fbulk very difficult. Similar to our previous work [33], for the first step we neglect the off-diagonal terms and obtain the interaction potential. Then we will show under which conditions this neglect is justified (see Sec. IV). After such an assumption, the system of differential equations (10) splits into two separate linear equations for u(r) and v(r). Solutions to these equations can be written via appropriate Green functions,  1 p δ(r − rp )Gu (r − r ), dr B u(r) = − 4π K p (11)  1    dr Cp δ(r − rp )Gv (r − r ), v(r) = − 4π K p where Gu (r,r )=−4π δ(r − r ), Gv (r,r ) − q 2 Gv (r,r ) = −4π δ(r − r ), and both Gu and Gv vanish at the confining surfaces. Due to the linearity of the Euler-Lagrange equations, we can use the superposition principle for the system of many colloidal particles. That is, the resulting

III. SPHERICAL PARTICLES IN A CHOLESTERIC CELL

To illustrate the peculiarities of the interaction arising from (13), let us consider the case of cholesteric liquid crystal confined by two parallel planes (see Fig. 1). Introducing a coordinate system in such a way that the planes are orthogonal to the z axis and placed at z = 0 and z = L, where L is the cell thickness, we can use the following representation of the appropriate Green functions [20–22]:   ∞ ∞ nπ nπ z 4 im(φ−φ  ) nπ z sin Im ρ< Gu = e sin L n=1 m=−∞ L L L   nπ × Km ρ> , (14) L Gv =

∞ ∞ nπ z 4 im(φ−φ  ) nπ z sin e sin L n=1 m=−∞ L L       π π 2 2 2 2 × Im n + N ρ< Km n + N ρ> , (15) L L

where Im (x) and Km (x) are modified Bessel functions √ of the mth √ kind, ρ> ( − ρ < | = (x − x)2 + (y  − y)2 , and on the angle θ between ρ and the rubbing direction. Thus, for definiteness, hereafter we can assume that the pth particle is located at (x,y,z) and the p th at (0,0,z ). However, the general expression (20) is still difficult to analyze. Therefore, we consider some simpler cases below. But first we should remark that although functions (14) and (15) are well-defined in the case of equal ρ> and ρ< , they give infinite values of the interaction energy in that case. If ρ > = ρ < , then ρ = 0 and, consequently, U → ∞ (see the Appendix for the explicit form of the derivatives). Therefore, the interaction between the particles located along the z axis is beyond the present scope. Another form of the Green functions should be applied to that problem [34]. A. Interaction in the middle of the cell (z = z  =

L ) 2

Let us assume that both particles are located in the middle of the cell, i.e., z = z = L2 . Then, as follows from (20), the quadrupole-quadrupole interaction for this configuration takes the form   2 ∂ 4 Gu ∂ 4 Gv 2k−1  2 ∂ Gv (21) + + 4q UQQ = −4π KQQ ∂x∂x  ∂y∂y  ∂y∂y  ∂z∂z ∂x∂x  in a (2k − 1)π cell, where k ∈ Z+ , or 2k = −4π KQQ UQQ



∂ 4 Gu ∂ 2 Gv ∂ 4 Gv + + 4q 2     ∂x∂x ∂y∂y ∂x∂x ∂z∂z ∂y∂y 

 (22)

in a 2kπ cell, where we used the fact that cos nπ sin nπ = 0. 2 2 As can be seen from Fig. 2, the maps of the interaction are quite similar to those in a planar nematic cell. We recall briefly that in the planar nematic cell, spherical particles repel everywhere along the rubbing direction as well as along the direction perpendicular to it. We have a similar picture in cholesteric cells, though the zone borders, especially at short distances, differ from those in the nematic case. It should be noted that the maps plotted in Fig. 2 differ considerably from those obtained in [31] via numerical computations. We suppose that the reason for such a mismatch is rooted in the nontrivial distribution of the director field around the particle with strong anchoring on its surface [29,30]. Interaction between such particles requires further theoretical investigations. UQQ as a function of the interparticle distance in the plane perpendicular to the pitch direction, i.e., in the xy plane, is plotted in Fig. 3. It clearly shows that the confinement effect observed in nematics [26] appears in cholesterics as well. Obviously, screening of the interaction at√ρ  L follows from the asymptotic behavior of modified Bessel functions at large values of argument Km (x) ∝ exp[−x]/ x. Since the explicit forms of the Green functions are completely dictated by the geometry of the confining walls, screening does not depend on the particles shapes and the liquid crystal type. It is a necessary consequence of the presence of the bounding surfaces. It is worth noting that in cholesterics, the confinement effect appears at larger distances as compared to nematics. Figure 3 reflects one more interesting feature of UQQ in cholesterics. At short distances, the interaction between the particles with quadrupole distribution of the director field in their vicinity does not behave as 1/ρ 5 exactly. Different symmetry of the cholesteric ground state as compared to nematics manifests itself in the existence of additional “slower” terms in the energy. B. Interaction between the particles with z =

L 2

and z  =

L 4

It is also interesting to explore the interaction between the particles located at different distances from the walls. To simplify our task slightly, we set z = L2 and z = L4 . Then for the case of the (2k − 1)π cell, one can easily find   ∂ 4 Gu UQQ (2k − 1)π ∂ 4 Gu − = cos(2k − 1)π sin − 4π KQQ 2 ∂x∂y∂y  ∂y  ∂x∂x  ∂x  ∂y + sin

(2k − 1)π ∂ 4 Gv (2k − 1)π ∂ 4 Gv (2k − 1)π (2k − 1)π sin cos + sin 2 4 ∂y∂y  ∂z∂z 2 4 ∂x  ∂y∂z∂z

+ 2q sin

(2k − 1)π ∂ 3 Gv (2k − 1)π (2k − 1)π ∂ 3 Gv (2k − 1)π cos − 2q sin cos 2 4 ∂y∂y  ∂z 2 4 ∂x∂x  ∂z

− 2q sin

(2k − 1)π ∂ 3 Gv (2k − 1)π ∂ 3 Gv (2k − 1)π (2k − 1)π sin sin − 2q sin   2 4 ∂x∂y ∂z 2 4 ∂x  ∂y∂z

+ 4q 2 sin

(2k − 1)π (2k − 1)π (2k − 1)π ∂ 2 Gv (2k − 1)π ∂ 2 Gv 2 − 4q sin . sin cos 2 4 ∂x∂x  2 4 ∂x∂y  012509-5

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FIG. 2. (Color online) Maps of the quadrupole-quadrupole interaction between particles located at (x,y,L/2) and (0,0,L/2). Arrow lines UQQ < 0. indicate the local direction of the force F = −∇UQQ . Painted regions are repulsion zones, ∂ρ

If the particles are placed in a 2kπ cell, the interaction potential is as follows: −

∂ 4 Gv ∂ 4 Gv ∂ 4 Gu kπ kπ UQQ = cos 2kπ cos kπ + cos kπ cos + cos kπ sin 4π KQQ ∂x∂x  ∂y∂y  2 ∂x∂x  ∂z∂z 2 ∂x∂y  ∂z∂z + 2q cos kπ cos

kπ ∂ 3 Gv kπ ∂ 3 Gv kπ ∂ 3 Gv + 2q cos kπ sin + 2q cos kπ cos 2 ∂x  ∂y∂z 2 ∂x∂y  ∂z 2 ∂y∂y  ∂z

− 2q cos kπ sin

kπ ∂ 3 Gv kπ ∂ 2 Gv kπ ∂ 2 Gv 2 2 + 4q . cos kπ cos − 4q cos kπ sin 2 ∂x∂x  ∂z 2 ∂y∂y  2 ∂x  ∂y

(24)

For such a configuration, the maps of the interaction (see Fig. 4) are much more diverse than those of the previous case. Though an explicit look of every map depends on the value of q, all of them consist of a few separate repulsion zones. This suggests that along the alternating zones of repulsion and attraction, the interaction potential has nonmonotonic behavior. However, UQQ as a function of two variables (x,y) has no local minimum. Consequently, there is no metastable configuration of the particle positions in this case.

C. Interactions in vertical planes

Now let us assume that one particle is located in the middle of the cell, i.e., z = L2 , and the other, unlike the previous cases, can move freely. Then in the case of a π cell, the energy of the interaction as a function of the particle position (x,y,z) is 012509-6

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described by −

  ∂ 4 Gv ∂ 4 Gu ∂ 4 Gu ∂ 4 Gu ∂ 4 Gv UQQ + sin qz = − cos 2qz − sin 2qz − + cos qz          4π KQQ ∂x∂x ∂y∂y ∂x ∂y∂y∂y ∂x∂x∂x ∂y ∂y∂y ∂z∂z ∂x∂y  ∂z∂z + 2q cos qz

∂ 3 Gv ∂ 3 Gv ∂ 3 Gv ∂ 3 Gv − 2q sin qz − 2q cos qz − 2q sin qz ∂y∂y  ∂z ∂x∂x  ∂z ∂x∂y  ∂z ∂x  ∂y∂z

+ 4q 2 sin qz

∂ 2 Gv ∂ 2 Gv 2 , − 4q cos qz ∂x∂x  ∂x  ∂y

(25)

and in the 2π -cell case, UQQ is as follows: −

  ∂ 4 Gv ∂ 4 Gu ∂ 4 Gu ∂ 4 Gu ∂ 4 Gv UQQ − cos qz = cos 2qz + sin 2qz − − sin qz 4π KQQ ∂x∂x  ∂y∂y  ∂x  ∂y∂y∂y  ∂x∂x∂x  ∂y  ∂x∂x  ∂z∂z ∂x  ∂y∂z∂z − 2q cos qz

∂ 3 Gv ∂ 3 Gv ∂ 3 Gv ∂ 3 Gv − 2q cos qz − 2q sin qz + 2q sin qz     ∂x ∂y∂z ∂x∂y ∂z ∂y∂y ∂z ∂x∂x  ∂z

− 4q 2 cos qz

∂ 2 Gv ∂ 2 Gv 2 + 4q sin qz . ∂y∂y  ∂x∂y 

(26)

Unfortunately, it is not possible to visualize a behavior of UQQ as a function of three variables (x,y,z). Therefore, we consider here the interactions in the xz and yz planes only. As in the preceding subsection, the maps of the interaction depend on the cell types and consist of a few separate attraction-repulsion zones (see Fig. 5). This, in turn, leads to the existence of stationary points in the energy profile. An important distinction of this case is that some of these points (marked by the stars in Fig. 5) correspond to the minima of the energy as a function of variables (x,z) and (y,z), respectively. Such a profile of the interaction should manifest itself in the complex dynamics of motion of the particles suspended in a cholesteric liquid crystal cell. Similar oscillating motion of the particles larger than the cholesteric pitch was observed recently in [31]. IV. OFF-DIAGONAL TERMS

In this section, we consider the off-diagonal coupling in bulk free energy (3). In the general case, rigorous treatment

∂u ∂v of these off-diagonal terms ( ∂x v, ∂x u, etc.) is very difficult. However, their influence on the interparticle interactions can be easily estimated. Indeed, we can consider that part of the free energy as an additional source of the deformations,    ∂v ∂u (2) v− u cos qz Fsource = Kq dr δ(r − rp ) ∂x ∂x p    ∂u ∂v + v− u sin qz . (27) ∂y ∂y

Then substituting “unperturbed” solutions   the p δ(r − rp )Gu (r − r ) u(r) = −(4π K)−1 p dr B and   p δ(r − rp )Gv (r − r ), where v(r) = −(4π K)−1 p dr C Gu (r − r ) and Gv (r − r ) are Green’s functions (14) and (15), respectively, in expression (27), one can find that the off-diagonal part of the free energy gives rise to the following correction to UQQ :     q (2) p Gv (rp ,rp )] p ∂Gu (rp ,rp ) [C cos qz UQQ = B p 8π 2 K ∂xp    ∂Gu (rp ,rp )    [Cp Gv (rp ,rp )] . + sin qzp Bp ∂yp (28)

FIG. 3. (Color online) Confinement effect along the rubbing direction in a π cell. U = −UQQ /(4π KQQ ), z = z = L2 . The dashed line is the appropriate asymptotic U  9(L/ρ)5 + 4π 2 L3 (2qπ/ρ 2 + 2/ρ 3 + q 2 π 2 /ρ) exp[−πρ/L] arising from the Green functions for exp[−q(r−r )] 1 . bulk cholesterics, Gu = |r−r  | , and Gv = |r−r |

Implying the asymptotic √ behavior of modified Bessel functions, Km (x) ∝ exp[−x]/ x if x → ∞, one can easily find that  √  (2)   UQQ  n,m exp[−(n + m)πρ/L]/( nmπρ/L)  ∝   → 0, √ U  QQ n exp[−nπρ/L]/ nπρ/L ρ → ∞. (29) L Thus, the off-diagonal terms have no impact on the interaction at large interparticle distances (larger than the cell thickness L). To shed some light on the case of small distances, let us again assume that both particles are located in the middle of the π cell. As can be concluded from Fig. 6, the influence of the off-diagonal part of the free energy is non-negligible

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FIG. 4. (Color online) Maps of the quadrupole-quadrupole interaction between particles located at (x,y,L/2) and (0,0,L/4). Arrow lines UQQ < 0. indicate the local direction of the force F = −∇UQQ . Painted regions are repulsion zones, ∂ρ

in this case. However, it depends on the relative positions of the particles and may vanish along some directions. For this (2) particular configuration, UQQ = 0 along the x and y axes.

V. RESULTS AND DISCUSSION

We used the method developed in [20,21,23,25] for a theoretical description of colloidal elastic interactions in confined cholesteric liquid crystals. Extending that approach, we derived a general expression for the energy of a pair interaction between small spherical particles (with radii much smaller than the cholesteric pitch) suspended in a cholesteric cell. The interaction energy has a more complex form than in the case of nematic liquid crystal hosts. The absence of translational symmetry related to the helical periodicity and local nematic ordering in cholesteric liquid crystals manifest themselves in the complex nature of the interaction maps. These maps depend crucially on the number of pitches fitted in the cell, especially if the particles are located at different distances from the walls. The maps of the interaction in vertical planes (along the pitch) clearly indicate nontrivial dynamics of motion of the colloids in a cholesteric host.

The common feature of the colloidal interactions in cholesterics and nematics is the confinement effect. In both cases, screening of the interaction at distances larger that the cell thickness L does not depend on the particles shape. The nature of this effect is completely rooted in the geometry of the bounding surfaces. In cholesterics, however, it appears at larger distances. Another interesting peculiarity of the interaction between spherical colloids in cholesterics is the presence of the nonquadrupole terms (i.e., the terms that do not behave as 1/ρ 5 if ρ → 0) in the interaction potential for the particles of quadrupole symmetry. This effect reflects the breaking of the symmetry of the director distribution around the particle in the case of the helical ground state. We hope that this prediction can be observed experimentally. The off-diagonal coupling in the bulk free energy was shown to be of minor importance at interparticle distances larger than the cell thickness. Though its impact is nonnegligible at small ρ, it can be vanishing along some particular directions. Hence, we believe that cholesteric liquid crystals will provide an interesting example of a host fluid for colloidal systems. We therefore suggest that experiments be performed

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FIG. 5. (Color online) Maps of the quadrupole-quadrupole interaction in the xz plane (θ = 0, along the rubbing) and in the yz plane (θ = π/2, perpendicular to the rubbing). One particle is located at (0,0,L/2), the other is at (x,y,z). Arrow lines indicate the local direction ∂U ∂U of the force F = −∇UQQ . Both ∂ρQQ and ∂zQQ are positive within the painted regions. The stars depict minima of the energy as a function of (x,z) and (y,z), respectively. All the plots are symmetrical about the plane z = L/2. In addition, the left plots are symmetrical about the yz plane and the right plots are symmetrical about the xz plane.

to reveal how the forces acting between particles in cholesteric liquid crystals are influenced by the cholesteric structures. A

possible continuation of the present study will be to extend our calculations in the case of particles comparable to the

FIG. 6. (Color online) Impact of the off-diagonal terms on the interaction in a π cell. Particles are located at (x,y,L/2) and (0,0,L/2). (2) /UQQ | as a function of the particle position in the xy plane. (b) Intersection of the surface by plane x = y (θ = π/4). (a) |UQQ 012509-9

LEV, FUKUDA, TOVKACH, AND CHERNYSHUK

PHYSICAL REVIEW E 89, 012509 (2014) ∞ 2π 2 nπ z nπ z ∂ 3 Gv = − sin [K0 (λn ρ) λn n cos  2 ∂x∂x ∂z L n=1 L L

cholesteric pitch. Under such circumstances, the gradient expansion of n0 (s) in (5) is no longer applicable. This may lead to interesting consequences. All numerical calculations in the paper were performed by MATHEMATICA 7.

+ K2 (λn ρ) cos 2θ ],

(A6)

∞ ∂ 3 Gv nπ z 2π 2 nπ z cos [K0 (λn ρ) = − λ n sin n ∂x∂x  ∂z L2 n=1 L L

ACKNOWLEDGMENTS

+ K2 (λn ρ) cos 2θ ],

(A7)

B.I.L and J.F. are grateful to the Japan Society for the Promotion of Science (JSPS) for the financial support that enabled B.I.L. to stay at the National Institute of Advanced Industrial Science and Technology (AIST) and to carry out this work there. J.F. is also supported by JSPS KAKENHI (Grant-in-Aid for Scientific Research), Grant No. 25400437.

∞ nπ z 2π 2 2 2 nπ z ∂ 4 Gv cos [K0 (λn ρ) = − λ n cos ∂x∂x  ∂z∂z L3 n=1 n L L

APPENDIX: DERIVATIVES OF THE GREEN FUNCTIONS

∂ 2 Gv nπ z 2 2 nπ z sin [K0 (λn ρ) =− λn sin  ∂y∂y L n=1 L L

+ K2 (λn ρ) cos 2θ ],

(A8)



Here ρ = ρ > − ρ < is the interparticle distance in the xy plane, θ is the angle between ρ and the √ x axis (rubnπ π , μ = , λ = n2 + N 2 , and bing direction), q = Nπ n n L L L p (p) d Km (x) = dx p Km (x),

− K2 (λn ρ) cos 2θ ],

∞ ∂ 3 Gv 2π 2 nπ z nπ z = − sin [K0 (λn ρ) λ n cos n ∂y∂y  ∂z L2 n=1 L L



nπ z 2 2 nπ z ∂ 4 Gu sin [A0 + A2 ], = − μ sin ∂x∂x  ∂y∂y  L n=1 n L L

− K2 (λn ρ) cos 2θ ],

where

− K2 (λn ρ) cos 2θ ],

μn  K (μn ρ) cos2 θ − μ2n K0 (μn ρ) sin2 θ ρ 0

∞ ∂ 4 Gv nπ z 2π 2 2 2 nπ z cos [K0 (λn ρ) = − λ n cos ∂y∂y  ∂z∂z L3 n=1 n L L

2 K2 (μn ρ)[cos 2θ + cos 4θ ] ρ2   μn  5 K2 (μn ρ) sin2 2θ − cos2 θ + ρ 2

− K2 (λn ρ) cos 2θ ],

(A3)

n=1

 ×

μ2n

nπ z K2 (λn ρ) sin 2θ, L

(A13)

∞ ∂ 3 Gv ∂ 2 Gv 2π 2 nπz = = 2 λ n cos ∂x  ∂y∂z ∂x∂y  ∂z L n=1 n L

nπ z K2 (λn ρ) sin 2θ, L

× sin

nπ z nπ z sin sin L L 

8 5μn  K (μn ρ) + μ2n K2 (μn ρ) sin 4θ, K2 (μn ρ) − 2 ρ ρ 2 (A4)

(A14)

∞ ∂ 3 Gv ∂ 2 Gv 2π 2 nπ z = = λn n sin     2 ∂x ∂y∂z ∂x∂y ∂z L n=1 L

× cos

nπ z K2 (λn ρ) sin 2θ, L

(A15)

∞ ∂ 4 Gv ∂ 2 Gv 2π 2 2 2 nπz = = λn n cos     3 ∂x ∂y∂z∂z ∂x∂y ∂z∂z L n=1 L



∂ 2 Gv nπ z 2 2 nπ z sin [K0 (λn ρ) =− λn sin  ∂x∂x L n=1 L L + K2 (λn ρ) cos 2θ ],

∂ 2 Gv ∂ 2 Gv 2 2 nπ z = = λ sin   ∂x ∂y ∂x∂y L n=1 n L × sin

 2  ∂2 ∂ Gu ∂ 2 Gu −   ∂x∂y ∂y  ∂y  ∂x ∂x  2  2 ∂ Gu ∂ 2 Gu ∂ − =   ∂x ∂y ∂y∂y ∂x∂x 1 = L

(A12)



− μ2n K2 (μn ρ) cos 2θ sin2 θ,



(A11)

(A2)

and A2 =

(A10)

∞ ∂ 3 Gv 2π 2 nπ z nπ z = − λ n sin cos [K0 (λn ρ) n ∂y∂y  ∂z L2 n=1 L L

(A1)

A0 = −

(A9)

(A5)

012509-10

× cos

nπ z K2 (λn ρ) sin 2θ. L

(A16)

INTERACTION OF SMALL SPHERICAL PARTICLES IN . . .

PHYSICAL REVIEW E 89, 012509 (2014)

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Interaction of small spherical particles in confined cholesteric liquid crystals.

The theory of the elastic interaction of spherical colloidal particles immersed into a confined cholesteric liquid crystal is proposed. The case of we...
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