Bonded Boojum-Colloids in Nematic Liquid Crystals - Langmuir (ACS

Subscriber access provided by University of South Dakota

Article

Bonded boojum-colloids in nematic liquid crystals Zahra Eskandari, Nuno M Silvestre, and Margarida Telo De Gama Langmuir, Just Accepted Manuscript • DOI: 10.1021/la4017195 • Publication Date (Web): 16 Jul 2013 Downloaded from http://pubs.acs.org on July 17, 2013

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Langmuir is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Bonded boojum-colloids in nematic liquid crystals Zahra Eskandari,†,¶ Nuno M Silvestre,∗,† and Margarida M Telo da Gama† Centro de Física Teórica e Computacional, Universidade de Lisboa, Avenida Professor Gama Pinto 2, PT-1649-003 Lisboa, Portugal., and Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, PT-1749-016 Lisboa, Portugal. E-mail: [email protected]

Abstract We investigate bonded boojum-colloids in nematic liquid crystals, configurations where two colloids with planar degenerate anchoring are double-bonded through line defects connecting their surfaces. This bonded structure promotes the formation of linear chains aligned with the nematic director. We show that the bonded configuration is the global minimum in systems that favour twist deformations. In addition, we investigate the influence of confinement on the stability of bonded boojum-colloids. Although the unbonded colloid configuration, where the colloids bundle at oblique angles, is favoured by confinement the bonded configuration is again the global minimum for liquid crystals with sufficiently small twist elastic constants.

Introduction In the past few years, nematic colloids have been proposed as a new route for bottom up assembly of photonic crystals 1 – periodic dielectric structures designed to control and manipulate light. In ∗ To

whom correspondence should be addressed Centro de Física Teórica e Computacional, Universidade de Lisboa, Avenida Professor Gama Pinto 2, PT-1649003 Lisboa, Portugal. ‡ Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, PT-1749-016 Lisboa, Portugal. ¶ Department of Physics, Sharif University of Technology, P.O. Box: 11155-9161, Tehran, Iran. †

1 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

this context, the unique mechanical properties of the liquid crystal (LC) host enables the assembly of a large variety of colloidal crystals 2–8 that, combined with the LC’s electro-optical response, are promising candidates for photonic devices. The assembly of colloidal structures in LCs depends, in particular, on the aligning properties (anchoring) of the surfaces of the colloidal particles and of the confining geometry. 9 In general, these give rise to deformations and to localized frustration of the LC orientational field which can induce the nucleation of topological defects. 10 Colloidal particles in LCs interact predominantly through long-range anisotropic forces, 9,11 with a well defined symmetry determined by the distribution of the defects. 12–14 Typically, the long-range colloidal interactions are either dipolar or quadrupolar and are effective up to several particle diameters. The symmetry of the longrange interaction profile is responsible for the self-assembly of specific structures such as linear chains, 6,8,15,16 periodic lattices, 6–8 anisotropic clusters 17 and cellular-like structures, 18 which, in turn, are stabilized by the presence of the topological defects. Supra-micrometer particles (≥ 1µ m) with perpendicular (homeotropic) anchoring induce pointlike hedgehog defects. 19 The far-field distortions produced by such particles have dipolar symmetry and the resulting pairwise colloidal interaction potential is dipolar decaying with distance d between the particles as d −3 . 12 Under confinement, 20 or through external electric fields, 21 these supra-micrometer particles stabilize equatorial Saturn-ring defects that are typical of submicrometer particles (≤ 1µ m) with homeotropic anchoring. 22 Particles with Saturn-rings induce elastic quadrupolar distortions and their pair potential is quadrupolar decaying with d −5 . 12 The interaction of LC colloids can be altered by confinement. In particular, it influences the long-range interaction changing, the asymptotic power law behaviour to an exponential decay as the thickness of the nematic cell decreases. 23 Elastic quadrupolar distortions are obtained also for colloidal particles with parallel (planar) anchoring. Such particles induce the nucleation of two antipodal surface defects, commonly known as boojums. 19 We refer to these particles as boojum-colloids. The core of boojums in nematic LCs can be of three types: single core, double core, or split core. 24 The single-core boojum is a point-


Page 2 of 26

Page 3 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

like index 1 defect with azimuthal symmetry, and is stable at weak anchoring, high temperatures, and small particle sizes. Split cores have two index 1/2 surface point-like defects connected by a 1/2 bulk disclination line. They are favoured at strong anchoring, low temperatures, and for large particles. The double-core boojum is an intermediate structure with broken azimuthal symmetry. The split-core boojum structure is similar to that found on spherical particles in cholesteric liquid crystals. 25,26 The size of the handles, however, is controlled by the nematic correlation length rather than by the cholesteric pitch. As a consequence, the handles in nematic colloids are comparable to the colloidal size only in the limit of nanometer sized particles. By contrast, in cholesterics the size of handles is set by the cholesteric pitch and is comparable to the colloidal size when the latter is twice the particle diameter. 26 For larger particles, the defect lines form twisted disclination pairs that twist around the particles. 26 The interaction of boojum-colloids has been recently studied numerically. 24,27 In agreement with experiment, 28 both studies show that the far-field distortions and the resulting asymptotic pair potential between particles with degenerate planar anchoring are of quadrupolar type. Moreover, the studies revealed that there is substantial rearrangements of the surface defects at short distances. The rearrangements of the defects are not only positional, as reported in, 27 but also configurational. 24 The most striking feature appears when the colloids are oriented along the far-field director. In this case, the inner single-core defects interact directly with each other giving rise to a repulsion. However, if the colloids are brought together, e.g. with optical tweezers, 29,30 the inner boojums undergo a configurational transition to split-core boojums, accompanied by a positional shift that turns the repulsion into an (radial) attraction. As was shown by experiments, 28 this configuration is unstable as the inner defects, displaced from their initial position, induce an angular force that makes the two colloidal particles bundle with an orientation oblique (∼ 30o ) to the farfield director. More recently, it was shown 24 that bringing the colloidal particles closer together along the direction of the far-field, induces a second structural transition, to a configuration where the split-core boojums on each colloid lock-in forming a double-bond defect between the colloidal particles, on the scale of the bulk correlation length, similar to that observed on the colloidal scale


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 26

for spherical colloids in cholesterics. 25,26,31 In cholesterics, two interaction regions were observed at a low diameter to pitch ratio. In addition to a non-bonded interaction at large particle separations, a defect mediated bonded configuration was also reported between closely spaced particles. The latter was claimed not to be observed in the presence of nematics. 31 However, recently this configuration was found for colloids in nematics at very close distances. 24 Here we investigate the stability of the bonded boojum-colloids configuration in nematics, which will, as in cholesterics, promote the formation of linear chains aligned along the nematic director. The paper is organized as follows: In the next section we introduce the Landau-de Gennes free energy functional, which we minimize numerically. In Sec. 3, we study the stability of the bonded configuration in the bulk. In particular, we show that this configuration is a global minimum for systems where twist deformations are energetically favourable. Linear chains in nematic liquid crystals have been observed only for dipolar 32 and distinct quadrupolar particles. 33 Our findings show that such colloidal structures are also possible for similar quadrupoles. In Sec. 4, we consider how confinement affects the stability of the bonded configuration. Finally, in Sec. 5 we present our conclusions.

Model On the mesoscale, nematic liquid crystals are modelled by the coarse-grained Landau-de Gennes (LdG) free energy, written in terms of the tensor order parameter

Qi j =

B Q (3ni n j − δi j ) + (li l j − mi m j ). 2 2

(1)

The Q-tensor is a traceless symmetric 3 × 3 matrix that, in general, can be written as a function of the degree of (uniaxial) order Q, of the biaxial order B, and of the director field n and two other orthogonal vectors l and m. 4 ACS Paragon Plus Environment

Page 5 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

The mean field LdG free energy is taken as:

F=

Z

n 2 o dV aTrQ2 − bTrQ3 + c TrQ2 Ω Z L1 L2 ∂k Qi j ∂k Q ji + ∂i Qik ∂ j Qk j dV + 2 2 Ω ( ) Z 2 2 3Q ˜ −Q ˜ ⊥ )2 + TrQ ˜2− b + W Tr(Q 2 ∂Ω

(2)

The bulk terms a, b and c are material-dependent. a = ao (T − T ∗ ) is assumed to depend linearly on the temperature T , where T ∗ is the supercooling temperature of the isotropic phase. L1 and L2 are the LdG elastic parameters, which can be related to the Frank-Oseen elastic constants by K1 = K3 = 9Q2b (L1 + L2 /2) /2 and K2 = 9Q2b L1 /2, where K1 , K2 , and K3 are, respectively, the splay, twist and bend elastic constants. Clearly, using only two elastic constants in the LdG model does not uniquely define the Frank-Oseen elastic constants. In general, the elastic constants are different, but in some cases the difference is small and this simplified model is deemed adequate to describe the relevant phenomenology. The planar (degenerate) anchoring on the surface of the colloidal particles is modelled by the Fournier-Galatola surface energy. 34 The quadratic term favours tangential orientation of the director field n, and the quartic term guarantees the existence of a min p imum for the surface scalar order parameter equal to its bulk value Qb = (b/8c) 1 + 1 − 8τ /9 ,

where τ = 24ao (T − T ∗ )c/b2 is the reduced temperature.

˜ ˜⊥ ˜ Q˜ i j = Qi j + 31 Qb δi j and Q˜ ⊥ i j is the projection of Qi j on the surface, Qi j = (δik − νi νk )Qkl (δl j −

νl ν j ), where ν is the normal to the surface. Typical values of the parameters for 5CB are ao = 0.044 × 106 J K/m3 , b = 0.816 × 106 J/m3, c = 0.45 × 106 J/m3 , L1 = 6 × 10−12 J/m, L2 = 12 × 10−12 J/m and T ∗ = 307 K. We have taken a(T ) = 0.01 × 106 J/m3 , which corresponds to the temperature T = 307.2K, just below the isotropic-nematic transition temperature, TIN = 308.5K. Assuming strong degenerate anchoring we take W = 10−3 J/m2 . We have minimized numerically the LdG free energy Eq. (2) using Finite Elements Methods 5 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

with Adaptive Meshing, 24 coupled to the Conjugated Gradient Method, for each particle configuration. Calculations were performed in a cubic box of linear size 2 µ m and for spherical particles of radius R = 0.1 µ m. At the outer boundary the nematic is assumed to be uniformly aligned. The free energy for each configuration is calculated with a precision of . 0.04%, on unstructured meshes with tetrahedral elements of sizes ranging from 0.001 µ m – on regions with strong elastic deformations – to 2 µ m. The size of the particles in this study is smaller than the typical size of particles in experiments. Nonetheless, we believe that this does not change the resulting phenomenology.

On the stability of bonded boojum-colloids We start by addressing the stability of the bonded configuration. Such configuration typically appears when the boojum-colloids are relatively close (. 2.4R) and approach each other along a path parallel to the global orientation of the nematic director field (θ = 0). At long range, two boojumcolloids interact as two quadrupoles and thus, along that particular direction they repel. However, as these colloidal particles are brought together, this repulsion turns into an attraction induced by the positional change of the inner defects, 24,27 as shown in Figure 1a. In such configuration the force acting on the colloidal particles has a non-zero angular component that induces an oblique configuration (θ ≃ 27o ). As these particles are brought even closer, along θ = 0, their inner defects undergo a structural transition, where the two inner defects detach one of its ends from the surface of the corresponding colloidal particle and re-attach to the surface of the opposite boojum-colloid, see Figure 1b, thus creating a double-bond. We will call this a bonded boojum-colloids (BBC) configuration. The BBC configuration cannot be self-assembled, requiring some type of assisted assembly. Our results, discussed below, indicate that the stable configuration at large and intermediate distances is the unbonded one, which becomes metastable at short distances, where the BBC is the global minimum. The nematic between the particles at near contact, brought together along the far-field direction, has to be melted using laser tweezers, in order for the colloids to assemble in the BBC configuration.


Page 6 of 26

Page 7 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

A more detailed view of the inner regions is shown in Figure 1c and d. The color code gives information on the component perpendicular to the plane of observation. It clearly shows that in the region between the colloidal particles the BBC exhibits a larger region where the nematic director sticks out of the plane. In fact, closer inspection reveals that the director field for the BBC exhibits twist deformations in those regions, which are not seen for the configuration shown in Figure 1a. Although, in cholesterics, twist deformations appear spontaneously, in nematics twist appears as a result of the energetic balance of splay, bend, and twist deformations. In nematic droplets, with tangential (degenerate) anchoring, the elastic anisotropy was found to drive an instability from a uniform bipolar configuration – where the director field is parallel to the meridians crossing two diametrically opposite boojum defects –, into a twisted bipolar configuration. 35 The twisted bipolar configuration is stable when K3 . 2.32(K1 − K2 ) and has been observed for droplets of 8CB, and of 8OCB, dispersed in glycerine. 36 A similar transition – from uniform to twisted bipolar configuration – for isolated colloidal particles, with tangential degenerate anchoring, in nematics, is not likely to occur as it requires nematics with K1 ≥ 2.3K3 + 19.6K2 . 37 For two interacting boojum-particles, however, the combined effect of the curved surfaces is responsible for the emergence of twist deformations near the surfaces, which adds a dipoledipole contribution to the overall pair interaction, 37 leading to the formation of linear chains. This suggests that favouring twist deformations increases the stability of the BBC configuration. Figure 2 shows the pair interaction energy F (2) = F − F (0) − 2F (1) as a function of the relative orientation of the boojum-colloids at a distance d = 2.05R, for different values of the elastic constants L2 /L1 = −1, 0, 2. F is the free energy defined in Eq.2, F (0) is the free energy of a particle-free, uniformly aligned nematic liquid crystal, and F (1) is the free energy of an isolated colloidal particle. For simplicity, we represent only the free energy of the stable configurations and omit the metastable branches. In the two represented quadrants (|θ | ≤ 45o ) F (2) exhibits three minima: two at θ ≃ ±27o – degenerate due to the cylindrical symmetry of the system – corresponding to the unbonded configuration, and another at θ = 0 that corresponds to the BBC configuration.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

For systems with large twist elastic constant L2 = −L1 , the BBC configuration becomes a shallow local minimum with an energetic barrier – measured from the point of coexistence of the two configurations –, smaller that 10kB T . Figure 2 suggests that for sufficiently large values of the twist elastic constant the BBC configuration could become unstable. On the other hand, for systems with small K2 (L2 = 2L1 ) the BBC configuration is the global minimum with a barrier that exceeds 50kB T . This indicates that the BBC configuration is stabilized by the presence of twist deformations. Elastic anisotropy has a role on the stability of the BBC configuration also at larger distances. Figure 3 shows the pair interaction energy, for relative orientation of the colloidal particles parallel to the far field director (θ = 0), as a function of distance for several elastic constants L2 /L1 = −1, 0, 2. The branch corresponding to the unbonded configuration is shown by open symbols, while full symbols correspond to the BBC configuration. The results clearly show that in systems where twist deformations are energetically favourable, the BBC configuration is stable at larger distances. The binding potential of this configuration, measured from the point of coexistence is, for L2 = −L1 , ∼ 55kB T , while for L2 = 2L1 , which corresponds to 5CB, it is ∼ 361kB T . In the nematic, as in the cholesteric, the force between particles, along the far-field direction, is a discontinuous function of the distance, jumping to stronger values at shorter distances. The discontinuity occurs when the defects on neighbouring particles join together, forming the bondedboojum colloidal pairs. Figure 3 also shows that the unbonded configuration is metastable at very short distances. This means that to observe the BBC configuration the particles would have to be placed in the BBC stability region and the liquid crystal between the colloidal particles would have to be heated. In order to check if this twisted configuration is associated with the twist instability described above and, in more detail, in 35–37 we compared the angle that the director field at the surface of the colloidal particles makes with the local meridional plane for each set of elastic constants, but no significant differences were observed. This indicates that the twist is not induced by the elastic anisotropy but rather by the confinement created by the two spherical surfaces. The elastic


Page 8 of 26

Page 9 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

anisotropy simply enhances the stability of the BBC configuration.

On the stability of bonded boojum-colloids under confinement In some experiments, liquid crystal colloids are studied under strong confinement – in cells with a few particle diameters thick. Although it has been shown that the cell thickness can influence the effective interaction of colloidal particles, 23 confinement allows a more precise control of the global liquid crystal orientation. Therefore, it is essential to consider the effect of the confining walls in the stability of the BBC configuration. To investigate this, we start by considering the effect of a single flat wall. To simplify our study we will consider the one-elastic constant approximation (L2 = 0).

Interacting boojum-colloids near a flat wall To calculate the effective interaction energy between two boojum-colloids close to a flat wall, we fixed one particle at a distance D from the wall and rotated the other particle around it in the plane perpendicular to the wall keeping the inter-particle distance d fixed. Figure 4 shows a schematic representation of the system with two particles of radius R near a wall. θ is the polar angle relative to the far-field director. The wall and the far field director are assumed to be along the z-axis and fixed boundary conditions are applied at the wall. In the asymptotic limit, d, D ≫ R, the pair interaction energy of boojum-colloids, labelled 1 and 2, near a flat wall, with (fixed) planar boundary conditions parallel to the far-field director, is given by 38 Fwall = F12 + F11′ + F22′ + 2F12′ , where the interaction between the two particles is F12 = 4π KQ 2

9 − 90 cos2 θ + 105 cos4 θ , d5

(3)

Fαα ′ = 15π KQ 2 /(16hα5 ) is the interaction of particle α with its image α ′ , and the interaction of


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 26

particle α with the image of particle β is F12′ = −4π KQ

2 (2D − d sin θ )

2

105d 2 cos2 θ − 4D2 + d 2 − 4dD sin θ (4D2 + d 2 − 4dD sin θ )

9/2

.

(4)

Here, Q is the quadrupolar moment of the boojum particles, hα is the distance of particle α from the wall – e.g., h1 = D and h2 = D − d sin θ –, and the one-constant approximation is used K1 = K2 = K3 = K. Although the asymptotic theory is only valid for large inter-particle and particle-wall separations, it gives an idea of the symmetries and other relevant effects at play. In Figure 5 we plot the interaction energy of boojum-colloids near a flat wall as given by the asymptotic theory Fwall as a function of the relative orientation between particles θ and their separation d for the reference particle located at distances D/R = 2, 3 from the wall. For θ ≤ 0 the free energy profiles are similar, exhibiting repulsion for θ = 0◦ and θ = 90◦ , and attraction for an oblique angle. It is as if the reference particle blocks the effect of the flat wall. As the particles are brought closer to the wall, the

θ > 0 region of the interaction profile is affected by a repulsive contribution from the wall. Such effect is strong enough to suppress the attraction for (positive) oblique angles, as seen for D = 2R. The asymptotic theory indicates that particles in an unbonded configuration will not stay close to the flat wall, but nothing can be inferred on the effect of the wall on the BBC configuration. Note, however, that if the reference particle is attached to the wall in the process of fabrication, playing the role of a spherical pillar, the asymptotic theory indicates that boojum-colloids self-assemble at such localized protuberances. Our numerical results are in good qualitative agreement with the asymptotic theory, as shown in Figure 6 where we plot the interaction energy as a function of the relative orientation for several interparticle separations d. Again, we have considered the case where the reference particle is fixed at distances D = 3R (Figure 6 top) and D = 2R (Figure 6 bottom) from the wall. For θ ≤ 0, particles are attracted at oblique angles and their equilibrium configuration is reached at contact d = 0 and

θ ≃ −27◦ . As shown in Figure 5, for θ > 0 the wall contributes with a repulsive component that


Page 11 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

affects the minimum located at θ ≃ 27◦ . However, by contrast to the results of the asymptotic theory, this minimum is not completely suppressed for D = 2R. We remind the reader that in the bulk there is cylindrical symmetry around the reference particle. What Figure 6 shows is that the degeneracy at θ ≃ 27◦ is lifted due to the presence of the wall and that the second particle will be pushed, around the reference particle, towards θ ≃ −27◦ on the plane perpendicular to the wall. The presence of the wall also affects the stability of the BBC configuration. In Figure 6 we show only the BBC nearly at contact d = 2.05R. When the reference particle is at a distance D ≥ 3R, the BBC configuration is the global minimum of the free energy, as was shown in Figure 3 for L2 = 0, with a barrier of ∼ 23kB T . As the reference particle is brought closer to the wall, as in Figure 6 at D = 2R, the BBC configuration is no longer the global minimum.

Bonded boojum-colloids between two flat walls As shown in Figure 7, approaching the particles to the wall changes only slightly the distance where the two configurations coexist d ∗ . In fact, an 87.5% change in D produces a 0.52% in d ∗ . This change, although still small, is more pronounced when the colloidal particles, assumed to be at the center of the cell, are under confinement. In this case the interparticle separation for which the two configurations coexist changes by an amount of 1.7%. It is possible that the effects of one or two walls on the stability of the BBC configuration is not conclusive at fixed relative orientation at θ = 0. The presence of a second wall placed symmetrically from the reference particle adds an extra repulsive contribution to the pair interaction directed towards the first wall. This results in a modulation of the interaction energy profile on both sides of the reference particle, that is more pronounced as the cell is made thinner. Figure 8 shows that while for cells of thickness 2D = 8.0R the particles interact without any influence from the confining walls, while for thin cells (2D = 3.0R) the walls suppress the unbonded equilibrium configurations increasing significantly the stability of the BBC configuration. However, the unbonded equilibrium configurations are still possible if the colloidal particles escape in the plane parallel to the confining walls. In fact, on that plane, the oblique configuration (θ ≃ 27◦ ) 11 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 26

is a global minimum. Nevertheless, our numerics show that, even under confinement, in systems that promote twist deformations (L2 = 2L1 ) the BBC is again the global minimum.

Conclusion Recently, a detailed study of the effective colloidal interactions of particles dispersed in nematic liquid crystals, with planar anchoring, revealed a new colloidal configuration where the particles form linear chains aligned parallel to the global orientation of the nematic molecules, and the surfaces of the colloidal particles are connected by two defect lines. 24 In this work, we have carried out a detailed study of the stability of this new configuration, named bonded boojum-colloids (BBC) configuration , both in the bulk and in the presence of confining walls. The BBC configuration induces twist deformations in the inner region of the colloidal particles. These deformations appear due to the confinement created by the curved surfaces of the particles. There is no reason to believe that these twist deformations are related to the elastic anisotropy of the system, as reported for the instability of isolated boojum-colloids. 37 In fact, if this was the case, changing the elastic anisotropy would result in a change of the helical angle around the particles, which was found to be the same for L2 /L1 = −1, 0, 2. We have found, however, that the BBC configuration is more stable in liquid crystals where twist deformations cost less energy, like in cholesterics where such deformations are always present, and a similar double-bonded defect configuration is observed. The stability of the BBC configuration under confinement was also studied. We started by addressing the interaction of boojum-colloids near a single flat wall. We have shown that the wall contributes with a repulsive potential that breaks the cylindrical symmetry of the pair interaction and decreases the stability of the BBC configuration. For real confinement, there is a repulsion from both walls that drives the colloidal particles to the middle of the cell. Moreover, for narrow cells the unbonded configuration at oblique angles is stable only in the plane parallel to the confining walls but, for liquid crystals with small twist elastic constants the BBC configuration is again


Page 13 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

the (global) equilibrium configuration. In addition, we have shown that it is possible to use walls decorated with spherical pillars, with planar anchoring, to localize colloidal particles, at an oblique angle to the wall (∼ 27◦ ). Recent studies 5,33 have shown that it is possible for unlike quadrupoles – boojum-colloids and particles with homeotropic anchoring surrounded by Saturn-ring defects – to self-assemble at right angles relative to the far-field director. This indicates that, if the free colloidal particles, interacting with the spherical pillars, have homeotropic anchoring, the particles will assemble at the structured surface perpendicularly to the wall. Then, by tuning the anchoring of the colloidal particles by photosensitive azobenzene groups 39 it should be possible to switch between a state where the colloid is at an oblique angle with the wall Figure 9(a) and a state where the particle is localized perpendicularly to the wall Figure 9(b). The study of such devices, as well as their optical properties, goes beyond the scope of present paper and is left for future work.

Acknowledgement We gratefully acknowledge financial support from the Portuguese Foundation for Science and Technology (FCT) through Grants Nos. PEstOE/FIS/U10618/2011, PTDC/FIS/098254/2008, and SFRH/BPD/40327/2007 (NMS). We would like to thank Mykola Tasinkevych for fruitful discussions.

References (1) Muševiˇc, I.; Škarabot, M.; Humar, M. Direct and inverted nematic dispersions for soft matter photonics. J. Phys.: Condens. Matter 2011, 23, 284112. (2) Muševiˇc, I.; Škarabot, M.; U. Tkalec, M. R.; Žumer, S. Two-dimensional nematic colloidal crystals self-assembled by topological defects. Science 2006, 313, 954. (3) Ravnik, M.; Škarabot, M.; Žumer, S.; Tkalec, U.; Poberaj, I.; Babiˇc, D.; Osterman, N.; Muševiˇc, I. Entangled nematic colloidal dimers and wires. Phys. Rev. Lett. 2007, 99, 247801. 13 ACS Paragon Plus Environment

Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 26

(4) Ognysta, U.; Nych, A.; Nazarenko, V.; Škarabot, M.; Muševiˇc, I. Design of 2D Binary Colloidal Crystals in a Nematic Liquid Crystal. Langmuir 2009, 25, 12092–12100. (5) Ognysta, U. M.; Nych, A. B.; Uzunova, V. A.; Pergamenshchik, V. M.; Nazarenko, V. G.; Skarabot, M.; Musevic, I. Square colloidal lattices and pair interaction in a binary system of quadrupolar nematic colloids. Phys. Rev. E 2011, 83, 041709. (6) Völtz, C.; Stannarius, R. Self-organization of isotropic droplets in smectic-C free-standing films. Phys. Rev. E 2004, 70, 061702. (7) Nazarenko, V. G.; Nych, A.; Lev, B. Crystal structure in nematic emulsion. Phys. Rev. Lett. 2001, 87, 075504. (8) Cluzeau, P.; Joly, G.; Nguyen, H. T.; Dolganov, V. K. Formation of Two-Dimensional CrystalLike Structures. JETP Lett. 2002, 75, 482. (9) Stark, H. Physics of colloidal dispersions in nematic liquid crystals. Phys. Rep. 2001, 351, 387–474. (10) Mermin, N. D. The topological theory of defects in ordered media. Rev. Mod. Phys. 1979, 51, 591–648. (11) Tasinkevych, M.; Andrienko, D. Colloidal particles in liquid crystal films and at interfaces. Condens. Matter Phys. 2010, 13, 33603. (12) Lubensky, T. C.; Pettey, D.; Currier, N.; Stark, H. Topological defects and interactions in nematic emulsions. Phys. Rev. E 1998, 57, 610. (13) Lev, B. I.; Tomchuk, P. M. Interaction of foreign macrodroplets in a nematic liquid crystal and induced supermolecular structures. Phys. Rev. E 1999, 59, 591. (14) Pergamenshchik, V. M.; Uzunova, V. A. Colloidal nematostatics. Condens. Matter Phys. 2010, 13, 33602. 14 ACS Paragon Plus Environment

Page 15 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(15) Loudet, J.-C.; Barois, P.; Poulin, P. Colloidal ordering from phase separation in a liquidcrystalline continuous phase. Nature 2000, 407, 611. (16) Cluzeau, P.; Poulin, P.; Joly, G.; Nguyen, H. Interactions between colloidal inclusions in two-dimensional smectic-C∗ films. Phys. Rev. E 2001, 63, 031702. (17) Poulin, P.; Frances, N.; Mondail-Monval, O. Suspension of spherical particles in nematic solutions of disks and rods. Phys. Rev. E 1999, 59, 4384. (18) Meeker, S. P.; Crain, W. P. J.; Terentjev, E. Colloid-liquid-crystal composites: An unusual soft solid. Phys. Rev. E 2000, 61, R6083. (19) Poulin, P.; Weitz, D. Inverted and multiple nematic emulsions. Phys. Rev. E 1998, 57, 626. (20) Škarabot, M.; Ravnik, M.; Žumer, S.; Tkalec, U.; Poberaj, I.; Babiˇc, D.; Osterman, N.; Muševiˇc, I. Interactions of quadrupolar nematic colloids. Phys. Rev. E 2008, 77, 031705. (21) Loudet, J.; Poulin, P. Application of an electric field to colloidal particles suspended in a liquid-crystal solvent. Phys. Rev. Lett. 2001, 87, 165503. (22) Gu, Y. D.; Abbott, N. L. Observation of Saturn-ring defects around solid microspheres in nematic liquid crystals. Phys. Rev. Lett. 2000, 85, 4719. ˇ c, M.; Ravnik, M.; Žumer, S.; Kotar, J.; Babiˇc, D.; Poberaj, I. (23) Vilfan, M.; Osterman, N.; Copiˇ Confinement effect on interparticle potential in nematic colloids. Phys. Rev. Lett. 2008, 101, 237801. (24) Tasinkevych, M.; Silvestre, N. M.; Telo da Gama, M. M. Liquid crystal boojum-colloids. New J. Phys. 2012, 14, 073030. (25) Mackay, F. E.; Denniston, C. Modelling defect-bonded chains produced by colloidal particles in a cholesteric liquid crystal. Europhys. Lett. 2011, 94, 66003.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 26

(26) Lintuvuori, J. S.; Stratford, K.; Cates, M. E.; Marenduzzo, D. Colloids in Cholesterics: SizeDependent Defects and Non-Stokesian Microrheology. Phys. Rev. Lett. 2010, 105, 178302. (27) Mozaffari, M. R.; Babadi, M.; Fukuda, J.; Ejtehadi, M. R. Interaction of spherical colloidal particles in nematic media with degenerate planar anchoring. Soft Matter 2011, 7, 1107–1113. (28) Smalyukh, I. I.; Lavrentovich, O. D.; Kuzmin, A. N.; Kachynski, A. V.; Prasad, P. N. Elasticity-mediated self-organization and colloidal interactions of solid spheres with tangential anchoring in a nematic liquid crystal. Phys. Rev. Lett. 2005, 95, 157801. (29) Yada, M.; Yamamoto, J.; Yokoyama, H. Direct observation of anisotropic interparticle forces in nematic colloids with optical tweezers. Phys. Rev. Lett. 2004, 92, 185501. (30) Takahashi, K.; Ichikawa, M.; Kimura, Y. Force between colloidal particles in a nematic liquid crystal studied by optical tweezers. Phys. Rev. E 2008, 77, 020703. (31) Lintuvuori, J. S.; Stratford, K.; Cates, M. E.; Marenduzzo, D. Self-assembly and nonlinear dynamics of dimeric colloidal rotors in cholesterics. Phys. Rev. Lett. 2011, 107, 267802. (32) Škarabot, M.; Ravnik, M.; Žumer, S.; Tkalec, U.; Poberaj, I.; Babiˇc, D.; Osterman, N.; Muševiˇc, I. Two-dimensional dipolar nematic colloidal crystals. Phys. Rev. E 2007, 76, 051406. (33) Eskandari, Z.; Silvestre, N. M.; Tasinkevych, M.; Telo da Gama, M. M. Interactions of distinct quadrupolar nematic colloids. Soft Matter 2012, 8, 10100–10106. (34) Fournier, J. B.; Galatola, P. Modeling planar degenerate wetting and anchoring in nematic liquid crystals. Europhys. Lett. 2005, 72, 403–409. (35) Williams, R. D. Two Transitions in tangentially anchored nematic droplets. J. Phys. A: Math. Gen. 1986, 19, 3211–3222. (36) Lavrentovich, O. D.; Sergan, V. V. Parity-breaking phase transition in tangentially anchored nematic drops. Il Nuovo Cimento D 1990, 12, 1219–1222. 16 ACS Paragon Plus Environment

Page 17 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(37) Uzunova, V. A.; Pergamenshchik, V. M. Chiral dipole induced by azimuthal anchoring on the surface of a planar elastic quadrupole. Phys. Rev. E 2011, 84, 031702. (38) Chernyshuk, S. B.; Lev, B. I. Theory of elastic interaction of colloidal particles in nematic liquid crystals near one wall and in the nematic cell. Phys. Rev. E 2011, 84, 011707. (39) Chandran, S.; Mondiot, F.; Mondain-Monval, O.; Loudet, J. C. Photonic Control of Surface Anchoring on Solid Colloids Dispersed in Liquid Crystals. Langmuir 2011, 27, 15185– 15198.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(a)

(b)

(c)

(d)

Page 18 of 26

Figure 1: Equilibrium configurations of boojum-colloids at close distance and aligned with the far field director n∞ . a) Boojum-colloid configuration for interparticle distance d = 2.2R, where the inner defects slide on the surface of the corresponding colloidal particles in opposite directions, resulting in an angular force that induces an equilibrium configuration at an oblique angle. b) For the same interparticle distance the inner line defects can attach to the surface of distinct colloidal particles creating a bond. A zoom of the inner regions is shown in c) and d). The color code represents the component perpendicular to the plane of observation. The isosurfaces correspond to scalar order parameter Q = 0.25 and are shown 18 to visualize the defects. ACS Paragon Plus Environment

Page 19 of 26

(2)

F /kBT

600

L2 = 2L1

580 560 540 520 -30

0

30

(2)

F /kBT

-40

L2 = 0

-60 -80 -100 -120 -30

0

30

F /kBT

-420

(2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

L2 = -L1

-440 -460 -480 -500 -30

0

30

θ (°) Figure 2: Pair interaction energy as a function of the particles orientation relative to the far field director, θ , for different elastic constants L2 /L1 = −1, 0, 2, at a distance d = 2.05R (near contact). 19 ACS Paragon Plus Environment

Langmuir

900 L2 = 2L1 L2 = 0 L2 = -L1

F /kBT

600

(2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 26

300 0 -300 -600

2

2.2

2.4

2.6

2.8

3

d/R Figure 3: Pair interaction energy F (2) as a function of the distance between particles d at relative orientation parallel to the far field director (θ = 0), for different elastic constants L2 /L1 = −1, 0, 2. Open symbols correspond to the unbonded configuration, while full symbols correspond to the BBC configuration.

Figure 4: Schematic representation of two boojum-colloids interacting close to a flat wall (in grey). The global orientation of the director field is parallel to z-axis. At the wall the director field is planar and parallel to the z-axis. 20 ACS Paragon Plus Environment

Page 21 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Figure 5: Interaction energy of boojum-colloids near a flat wall Fwall , given by the asymptotic theory, as a function of relative orientation θ and the distance between particles d, for the reference 21 D = 3R (top) from the wall. K is the elastic particle located at distances D =ACS 2R (bottom) and Paragon Plus Environment constant in the one-constant approximation and Q is the quadrupolar moment of the boojumcolloid. The color and isolines are represented to guide the eye.

Langmuir

(2)

F /kBT

100

d = 2.05 R d = 2.50 R d = 3.00 R d = 3.50 R d = 4.00 R

50

D = 3R

0 -50

-100 -90

-60

100

(2)

F /kBT

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 26

-30

0

d = 2.05 R d = 2.50 R d = 3.00 R d = 3.50 R d = 4.00 R

50

30

60

D = 2R

0 -50

-100 -90

-60

-30

0

30

60

θ (°) Figure 6: Pair interaction energy as a function of the relative orientation θ for several inter-particle separations d/R = 2.05, 2.5, 3.0, 3.5, 4.0. The distance between the reference particle and the wall is D = 3R (top) and D = 2R (bottom). Each curve is limited by the angle corresponding to the other particle touching the wall.


Page 23 of 26

200

F /kBT

100

(2)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

0 D = 1.5 R D = 1.5 R (confined) D = 4.0 R (confined) D = 4.0 R

-100 -200

2

2.2

2.4

2.6

2.8

3

d/R Figure 7: Pair interaction energy F (2) as a function of the inter-particle distance for relative orientation θ = 0 and D/R = 1.5, 4. A nematic cell (under confinement) is also shown, with the particles located in the center of the cell. Full symbols correspond to the BBC branch, while the open symbols correspond to the unbonded configuration, as indicated by the insets. The metastability of these states is not represented for clarity.


Langmuir

0 D = 1.5 R D = 1.5 R (confined) D = 4.0 R D = 4.0 R (confined) in bulk

-30

(2)

F /kBT

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 26

-60 -90

-120

-30

-20

-10

0

10

θ (°) Figure 8: Pair interaction energy as a function of the relative orientation for two cells. The reference particle is place at the center of the cell at D/R = 1.5, 4.0 from the wall, and the interparticle separation is d = 2.05R. The curves for particles near a single wall were added for comparison. At D = 4.0R the profile is symmetric in θ , but for clarity we do not represent the data for θ > 20◦ . The data points are omitted.


Page 25 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

(b)

(a)

Figure 9: Schematic representation of templated assembly through walls decorated with spherical pillars. The pillars have planar anchoring, the far field director is parallel to the wall, and the free particle has (a) planar or (b) homeotropic anchoring.


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Graphic for Table of Contents


Page 26 of 26

Bonded Boojum-Colloids in Nematic Liquid Crystals - Langmuir (ACS

Recommend Documents