Article pubs.acs.org/Langmuir
Colloidal Particles in Thin Nematic Wetting Films Haifa Jeridi,† Mykola Tasinkevych,‡,§ Tahar Othman,† and Christophe Blanc*,∥ †
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Université de Tunis El Manar, Faculté des Sciences de Tunis, LR99ES16 Laboratoire Physique de la Matière Molle et de la Modélisation Electromagnétique, 2092, Tunis, Tunisia ‡ Max-Planck-Institut für Intelligente Systeme, Heisenbergstraße 3, D-70569 Stuttgart, Germany § Institut für Theoretische Physik IV, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany ∥ Laboratoire Charles Coulomb, UMR 5221, CNRS-Université de Montpellier, 34095 Montpellier, France
ABSTRACT: We experimentally and theoretically study the variety of elastic deformations that appear when colloidal inclusions are embedded in thin wetting films of a nematic liquid crystal with hybrid anchoring conditions. In the thickest films, the elastic dipoles formed by particles and their accompanying defects share features with the patterns commonly observed in liquid crystal cells. When the film gets thinner than the particles size, however, the capillary effects strongly modify the appearance of the elastic dipoles and the birefringence patterns. The influence of the film thickness and particles sizes on the patterns has been explored. The main experimental features and the transitions observed at large scalewith respect to the inclusions’ sizeare explained with a simple two-dimensional Ansatz, combining capillarity and nematic elasticity. In a second step, we discuss the origin of the variety of observed textures. Developing a three-dimensional Landau-de Gennes model at the scale of the particles, we show that the presence of free interfaces and the beads confinement yield metastable configurations that are quenched during the film spreading or the beads trapping at interfaces.
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INTRODUCTION Similar to surfactant molecules, colloidal particles can spontaneously accumulate at the interface between two immiscible fluids.1−3 The interest in such systems has grown due to their self-assembly properties and the great variety of resulting organizations for microscopic and macroscopic particles.4−9 These two-dimensional (2D) colloidal assemblies at interfaces are not only of broad importance in the design of soft colloidal materials for technological and industrial applications,10 but they are also relevant in the field of biophysical science11−16 where proteins, viruses, or bacteria deal with membranes and thin biological films. The attachment of colloidal particles to a fluid interface or their confinement in thin liquid films is typically accompanied by interfacial deformations that minimize the interfacial area.17 The overlap of the menisci around each particles gives rise to long-range capillary forces that often dominate the other interactions. In thin films, the capillary interactions are attractive in most of the cases16,18−21 and are modulated by the wetting properties of the particles, the surface tension and surface elasticity. In simple fluids, the capillary forces compete mostly with electrostatic interactions.10 When inclusions are attached at complex fluid interfaces such as a bilayer membrane11,13,15,22 or a liquid crystal interface,23−27 other © 2016 American Chemical Society
long-range interactions related to bulk distortions of the surrounding fluid appear. For instance, a nematic liquid crystal phase is characterized by a preferred orientationthe director n(r)of its long rigid molecules. Inclusions trapped even at a single interface of a nematic film may orient the director either parallel,23 or perpendicular24 to their surface, yielding far-field distortions of the director and the appearance of elastic dipoles. These elastic effects strongly modify the colloidal behavior with respect to the one observed in simple fluids, mainly under capillary forces. We have, for instance, recently shown that the elastic distortions of a thin nematic film caused by the presence of solid particles trapped at both interfaces were sufficient to prevent their expected capillary aggregation.28 In a thin nematic film, the orientational elasticity changes the usual picture of a simple fluid film. It not only modifies the shape of the menisci through the presence of a large additional disjoining pressure,28 but also yields in-plane far-field distortions of the director field around the particles. This results in lateral long-range elastic interactions between particles that can be strong enough to overcome the capillary interactions. In this previous work, we Received: July 20, 2016 Revised: August 17, 2016 Published: August 18, 2016 9097
DOI: 10.1021/acs.langmuir.6b02701 Langmuir 2016, 32, 9097−9107
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modulation of the director field n.33,34 In the experiments, we have focused only on thicker films (h > 0.4 μm) for which the director field nH is homogeneously distorted as shown in Figure 1. The field is characterized by a rapid distortion in the z
have elucidated the mechanisms at work by analyzing in depth the behavior of a specific nematic texture found around a spherical inclusion, namely the “giant dipole” texture, and its relationships with the capillary distortions caused by the microsphere. This texture was characterized by a preferred radial director field around the particle (promoted by a perpendicular anchoring of the nematic director at the bead surface) and the presence of a topological defect at far distance (typically a hundred times the particle size), mandatory to restore a uniform director field far away. We also showed that simple 2D approximations of the free interfacial and elastic energies were sufficient to capture the evolutions of both this “giant dipole” texture and the distortions with the film thickness.28 Here, we go much further in our experimental and theoretical investigations and explain all nematic textures induced by microparticles confined in thin nematic films. This work confirms that the in-plane elasticity is influenced by the capillary effects but also reveals the important role of the quenching of topological defects when particles get trapped at interfaces. We first report all observed elasto-capillary patterns formed by an isolated inclusion and their evolution with the films thickness. We demonstrate that these large scale distortions can be all reproduced with a single 2D model combining capillarity and nematic elasticity but with various boundary conditions. Second, using a full three-dimensional Landau-de-Gennes model of the nematic order parameter, we have explored the evolution of the topological defects at the scale of the solid inclusions. This theoretical approach explains the origin of the variety of the observed textures, but also allows discussion of their relative stability and the transitions observed between them.
Figure 1. (a) Nematic layer in hybrid anchoring conditions. The c director is defined as the projection of n onto the xy plane and is characterized by its angular orientation φ. (b) Side view sketch of a microsphere embedded in a thin 5CB layer of thickness h0 at far distance.
direction and a much slower one across the xy plane, thus defining a two-dimensional polar director field c and the corresponding scalar angular orientation field, φ(x,y), such as c = (cos φ, sin φ). In the wetting films, solid beads orient the director field perpendicularly to their spherical surface due to the surface treatments and are thus equivalent to a topological defect of strength s = +1 for the c-director field. When the thickness h of the NLC layer is thicker than the diameter 2R of the beads, the latter are either confined in the film or trapped at a single interface. In both cases, individual beads induce a localized distortion of the nematic field similar to the one observed in nematic cells,24,35 with the apparent formation of a s = −1 topological defect of the c-director field which permits to recover a uniform texture at large distance. When the film is thinner than the beads diameter 2R (h < 2R), the particles are necessarily trapped at both interfaces. Due to the Young relations at the triple lines, they induce a thickness distortion h(x,y) (see Figure 1b) at the origin of distorted cdirector fields at large scale and distant defects.28 Optical Observations and Measurements. The thin films (in the cuvette or in the Langmuir trough) were observed under an upright microscope (LEICA DM 2500P) equipped with SONY digital cameras (either grayscale of 1024 × 768 pixel resolution or color of 1600 × 1200 pixel resolution) and a Instec heating stage regulated at 0.1 °C. The films are studied either in polarized transmission mode or in reflection mode at 546 nm. Quantitative birefringence measurements of the film were additionally performed with an Abrio system (CRI Inc.) adapted to the microscope. In this imaging system,36 the sample is illuminated with a filtered light polarized circularly (546 nm) and the traditional Berek compensator is replaced by a universal liquid crystal compensator. In full frame acquisition, the optical retardation and the azimuthal orientation of the optical axis can be determined in 1024 × 1392 pixels regions, with a resolution better than 1 nm for the retardation and 1° for the azimuthal orientation φ of the slow axis (i.e the c-director). The maximum value of the measured retardance δM given by the Abrio System is 273 nm, but the retardance δ can be easily obtained in case of a monotonous evolution by unwinding δM: we indeed have δ = δM in the [0−273 nm] range modulo 546 nm and δ = 546−δM in the [273 nm-546 nm] range. The Abrio retardation maps can be translated in film thickness maps. Assisted by the thin-film interference patterns obtained in reflection mode, the local thickness is obtained from the retardation, using a calibration method exposed in ref 28, which
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EXPERIMENTAL SECTION Films Preparation and Their Structure. The thin films are obtained by spreading droplets of 4-n-pentyl-4′cyanobiphenyl (5CB from Synthon Chemicals), containing 0.1 wt % of silica beads (Bangslabs), on the surface of 1 mm-thick water layer containing 5 wt % of poly(vinyl alcohol) (PVA from Sigma-Aldrich, Mr = 20.000 g/mol) in a glass cuvette of diameter 20 mm following the procedure described in ref 28. The solid microspheres were previously treated with a monolayer of N,N-dimethyl-N-octadecyl-3-aminopropyl trimethoxysilyl chloride (DMOAP, from Sigma-Aldrich), to ensure a strong perpendicular anchoring of the liquid crystal molecules at their surfaces.29 After 5CB deposition, the film either directly spreads at room temperature or is briefly heated to 40 °C in the isotropic state of 5CB and then cooled back to room temperature. Following a resting period of at least 30 min, the film and its textures are then characterized. In some experiments, the thin films were prepared at the surface of a homemade Langmuir microtrough of total dimensions (4 × 2.7x 0.6 cm3) with a transparent glass bottom. The position of the Polytetrafluoroethylene barrier is controlled mechanically and allows a typical film thickness variation of 400%. The presence of PVA ensures a complete wetting of 5CB on water. It also defines a strong degenerate parallel anchoring of the nematic director on the water30 without disturbing the strong homeotropic anchoring of the silica particles.31 The texture of NLC films is then distorted vertically, due to hybrid anchoring conditions.32 Such films, when their thickness h is lower than ∼0.4 μm, show an elastic instability in the plane with the formation of periodic domain patterns and a xy 9098
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Langmuir leads to h = δ/Δn̅ with Δn̅ ≈ 0.08 at room temperature (23 °C).
conditions (see Figure 2). An elastic dipole is here defined by the distance d between the point defect and the center of the
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NUMERICAL METHODS Numerical Modeling in 2D. The free energy of the thin film consists in capillary and elastic energies, both dependent on n and h fields. The decoupling of the liquid crystal distortions between a homogeneous vertical distortion and the in-plane c-director field, however, greatly simplifies the full variational problem. First, the thickness variation is found to be almost independent of the slow-varying c-director field. For a given thickness h0 at far distance, the capillary distortion around the microsphere is then axisymmetric. The experimental profiles have been shown to be well fitted with strong anchoring hypotheses at interfaces, which yields the analytical radial dependence28 h(r) = h0 + AthK0(qr) where q(h0) ≈ βthh0−3/2, with βth = 0.049 μm1/2 and Ath(h0) is numerically computed from the Young relations at the triple lines. K0 is the modified Bessel function of the second kind of order 0 and the quantity q−1 is a characteristic capillary length related to the presence of a disjoining pressure,16 here due to the part of nematic elastic energy stored in the vertical distortion. We have
Figure 2. Finite-element domain Ω used in our numerical computations corresponding to a half disk centered on the midpoint between the microparticle (magnified region) and a −1 defect. We have computed the director field orientation φ(x,y) with the two possible boundary conditions corresponding either to a parallel or an antiparallel (values between brackets) dipole with strong anchoring conditions.
particle. The free energy F2D is then minimized, for a given thickness h0 and a distance d, using the nonlinear conjugate gradient algorithm of the Freefem++ package, which is iterated until the relative variation of the norm of the gradient of the functional F2D with respect to all degrees of freedom is less than 10−5 (to improve the numerical accuracy, we have used an adaptive grid). The absolute minima of F2D are then found by varying d. Numerical Modeling in 3D. In order to get insights on the full distortions of the n-director in the vicinity of the beads and to focus on the role of the confinement, we also performed numerical minimization of Landau−de Gennes free energy functional. For this more accurate but also computationally demanding method, we have only considered localized distortions due to a particle confined either at a single or at both interfaces of a thin film, assuming that nematic−air and nematic−water interfaces, and consequently the whole nematic film, are flat. Landau−de Gennes Free Energy. Within the framework of Landau-de Gennes theory, liquid crystals are described by a traceless symmetric tensor order parameter (OP) Qij,i,j = 1,..3, and the corresponding Landau−de Gennes free energy functional FLdG may be written as
π
q(h0) = 2 · K (γ1−1 + γ2−1) ·h0−3/2 where K is the splay-bend modulus in the one-constant approximation, γ1 and γ2 being the respective interfacial tensions of air-5CB and PVA/water-5CB interfaces. In a second step, the nematic texture at large scale (with respect to the beads size) can be computed numerically with the above fixed thickness variation, using an approximation of the bulk elastic energy37 for a nonflat film with hybrid anchoring conditions. The 2D nematic bulk free energy density is then given by28
f2D
2 1 1 ⎛ Θ + c ·∇xy h ⎞ 2 = 2(∇xy φ) + K ⎜ ⎟ 2 ⎝ 2 h ⎠
(1)
where φ(x,y) is the azimuthal angle of the c director in the x−y plane (see Figure 1a) and h(x,y) the varying thickness. The first term accounts for the in-plane splay-bend elasticity of c with an elastic modulus 2 ≈ K/2 derived from the 3D splay-bend elastic modulus (in 3D, the in-plane distortion is maximal at the lower surface but vanishes at the upper interface due to the homeotropic anchoring). The second term accounts for the vertical splay-bend distortion of the film. Its main part is due to the angular distortion Θ = π/2 of n in case of a uniform thickness (see Figure 1b) and accounts for the disjoining pressure discussed above. In a nonuniformly thick film, the angular distortion Θ is corrected by c·▽xyh at first order. The part of the elastic free energy to minimize with φ(x,y) is then F2D =
⎛
∫ ∫ ⎜⎝f2D −
K Θ2 ⎞ ⎟h(x , y) dx dy 2h2 ⎠
FLdG =
⎛
∫V ⎝aQ ij2 − bQ ijQ jkQ ki + c(Q ij2)2 + ⎜
+
⎞ L2 ∂jQ ij∂kQ ik ⎟dV + W ⎠ 2
L1 ∂kQ ij∂kQ ij 2
∫∂V fs ds
(3)
where summation over repeated indices is assumed. The nematic-to-isotropic phase transition is controlled by the temperature T dependent a parameter, which is assumed in the form a(T) = a0(T − T*) with a0 being a material dependent constant, and T* the supercooling temperature of the isotropic phase. The coupling constants b, c are temperature independent, and L1, L2 are phenomenological parameters that can be related to the Frank-Oseen bend, splay, and twist elastic
(2)
The equilibrium states of the c-director are found by minimizing F2D with finite-element methods, using the partial differential equation solver Freefem++.38 In the current work, we have focused on elastic dipoles formed by the bead and its accompanying s = −1 defect, that can be either parallel or antiparallel to the far-distance director c0. Using the mirror symmetry along the x axis, we have worked on half-disk finiteelement domains Ω of radius L = 400R with adequate boundary
(
constants by using an uniaxial Ansatz Q ij = ninj −
δij 3
)3Q /2, b
where Qb is the bulk value of the scalar orientational order parameter, and ni are the Cartesian components of the director field. This results in K11 = K33 = 9Q2b(L1 + L2/2)/2, and K22 = 9099
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Langmuir 9Q2bL1/2. In general, K11 and K33 are different, but in most cases the difference is small and the LdG free energy is deemed adequate. The first integral in eq 3 is taken over the threedimensional domain V occupied by the liquid crystal, whereas the second integral is taken over the domain boundary ∂V (e.g., the surface of colloidal particles) and accounts for nonrigid anchoring surface boundary conditions with an anchoring strength coefficient W. For a planar degenerate anchoring we use
2
2 ⊥ 3 f s = (Q̃ ij − Q̃ ij )2 + Q̃ ij − 2 Q b2 , 3 9
(
)
where
δ
ij Q̃ ij = Q ij + Q b 2 , and Q̃ ⊥ij = (δik − NiNk)Q̃ kl(δlj − NlNj), with N being the normalized outward normal vector to the confining surface. For weak homeotropic (perpendicular) anchoring we use f⊥s = (Qij − Qsij)2, where the surface-preferred value of the tensor order parameter is Qsij = 3Qb(NiNj − δij/3)/ 2. The uniaxial nematic with Q b = b/8c(a + 1 − 8τ /9 )is thermodynamically stable at τ ≡ 24ac/b2 < 1. We use a0 = 0.044 × 106 J/m3, b = 0.816 × 106 J/m3, c = 0.45 × 106 J/m3, L1 = 6 × 10−12 J/m, and L2 = 12 × 10−12 J/m, which are typical values for 5CB40 and T* = 307 K. For these values of the model parameters, the bulk correlation length is ξ = 2√2c(3L1 + 2L2)/b ≅ 15 nm at the isotropic−nematic coexistence at τ = 1.41 Numerical Approach. In the following, the Landau-de Gennes free energy of eq 3 is minimized numerically using finite elements method with the adaptive mesh refinement. The surface of the colloidal particle is represented by a union of triangles using open source GNU Triangulated Surface Library,42 and then the nematic-containing domain V is discretized by using the Quality Tetrahedral Mesh Generator.43 Linear triangular and tetrahedral elements are used and the integration over the elements is performed numerically by using fully symmetric Gaussian quadrature rules.44−46 Consequently, the discretized FLdG is minimized exploiting INRIA’s M1QN3 optimization routine. More detailed description of the numerical procedures is given in ref 47. Geometry and Initial Conditions. We consider a volume V = L × L × Lz where a particle is embedded. We assume that the nematic−air interface is at z = 0, and the nematic−water interface is at z = −Lz. Nonrigid planar/perpendicular anchoring is assumed at the nematic−water/−air interface respectively with the anchoring strength W = 10−2 J/m2. At the side walls, x = ± L/2 and y = ± L/2, we assume rigid boundary conditions, which are determined by the “hybrid initial” conditions (see below). The colloidal particle of radius R has its center rc = (0,0,zc). As the initial conditions we use a combination of an uniaxial hybrid nematic configuration and the dipole ansatz of Lubensky et al.48 Thus, at a point r which satisfies ∥r−rc∥ > 1.5R, we set the initial nematic director
Figure 3. A schematic illustration of the initial conditions used for the 3D minimization of the Landau-de Gennes free energy in eq 3. (a) A colloidal particle with radius R is trapped in the nematic film sandwiched between air (z = 0, perpendicular anchoring) and water (z = −Lz, planar degenerate anchoring). The center rc of the colloidal particle is at (0,0,zc). Lubensky’s dipole ansatz is applied for ∥r − rc∥ > 1.5R, while for larger ∥r−rc∥ a hybrid uniaxial nematic ordering is adopted, which interpolates the planar director orientation at the nematic−water interface with the perpendicular one at the nematic− air interface. (b,c) In the bulk region of the nematic film a hybrid (without Lubensky’s hedgehog), director nH is used, and at the nematic−water interface a pair of 2D defects with the strength s = +1 (marked by blue circles) s = −1 (marked by red circles), separated by a distance d, is initially placed. In panels b and c, the vector joining the −1 defect with the +1 one is oriented antiparallel (parallel) with c0.
water interface with tangentially degenerate boundary conditions contains a pair of self-compensating 2D defects with the strength s = +1 (blue circles in Figure 3b,c) or s = −1 (red circles). The defects are separated by the distance d, the varying of which causes different ni(r) to be generated.
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ni = n H ≡ ( −z , 0, z + Lz)/ z 2 + (Lz + z)2 , with the initial degree of the nematic orientational order Q0 = Qb; and for ∥r− rc∥ ≤ 1.5R, we set ni as given by the first three terms on the right hand side of eq 31 in ref 48. Lubensky’s Ansatz describes a dipolar nematic configuration composed of the colloidal particle and a companion hedgehog hyperbolic point defect. This so-called elastic dipole is oriented along the z-axis. In our calculations, we place this elastic dipole at various orientations, specified by an angle, with respect to the z-axis (see Figure 3a). Additionally, we use initial conditions where ni = nH in the bulk region, but the nematic director ns(r) at the nematic−
RESULTS AND DISCUSSION Nematic colloids are mostly characterized by the presence of topological defects induced by the particles. As a general rule, the defects closely accompany the inclusions, forming with the latter elastic dipoles or quadrupoles with a spatial extension similar to the size of the particle.35 Here, in wetting nematic films, elastic distortions are much more extended. We have already reported the existence of giant dipoles in ref 28: an isolated microsphere, equivalent to a + 1 defect for the twodimensional c director field, induces a point defect of charge 9100
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At first sight the particle also produces an accompanying defect, but the latter remains in its close neighborhood while a distortion is nevertheless observed at large distances. Apart from the presence of nearby s = −1 defects, such “butterfly textures” are mainly characterized by a π-wall distortion on the opposite side of the defect. Some of these textures are stable over time, but others (slightly different in aspect: see below) may transform into giant dipoles, especially when the thickness is varied. When heating to the isotropic phase and cooling back, however, the “butterfly” textures are always replaced by a giant dipole in the thinnest films. We determine and further analyze below the main characteristics of the two different textures and then discuss their formation mechanisms. Giant Elastic Dipole Textures. The film thickness variation and the nematic director patterns caused by an inclusion (see Figure 5a,b) have already been examined in depth for 4 μm diameter beads.28 We recall here their main features. In this previous study, we established that the distortion of the thickness was well reproduced by the expression h(r) = h0+AK0(qr) (see Numerical Methods section). The nematic field was then explained by the distortion of the interfaces caused by the particle and by the spontaneous orientation of the polar director c toward negative gradient of the thickness in order to decrease the elastic free energy (see eq 1). For beads of 4 μm diameter, the giant dipoles were observed only for the thinnest films. Experimentally, the equilibrium bead-defect distance de indeed first increases with the thickness to a maximum, then it gradually decreases over a small thickness range until a sudden drop occurs, revealing a first-order transition between giant dipoles and classic short ones. When varying the film thickness slowly, an intermediary region is observed, where both types of dipoles (short and giant dipoles) coexist. This first-order transition has been reproduced with the 2D modeling, using the two-dimensional c-director
−1, located at far distance (up to 100 μm for 4 μm diameter particles: see Figure 4). For the thinnest films, only giant
Figure 4. Two different coexisting textures observed when 4 μm diameter microspheres are trapped in a thin hybrid nematic film (h0 ≈ 0.9 μm). Here, a giant dipole (on the left of each image) coexists with a “butterfly” texture (right). (a) Polarizing optical micrographs and (b) corresponding retardation maps obtained with Abrio birefringence measurement system.
dipoles are observed provided that the nematic film is heated briefly in the isotropic state (≈ 40 °C) and then cooled back to room temperature (23 °C). However, when the nematic film is not heated to the isotropic state after spreading, other uncommon textures are observed. Without heating, approximately 70% of the dispersed particles spontaneously form giant dipoles, and the rest (30% of the particles) give rise to very different textures such as the one shown in Figure 4.
Figure 5. (a) Giant elastic dipole observed between crossed polarizers. The counter defect is located at a far distance from the 4 μm silica bead. (b) Corresponding retardation and director field of the nematic film obtained from Abrio measurement system. Insets show the c-director field and the retardation around the bead. (c) Evolution of the distance de between a 7 μm particle and its defect while decreasing (blue arrow) or increasing (red arrow) the film thickness h0. A hysteresis is present in the two-phase region. (d) Experimental and computed phase diagram of the texture with the particles size. 9101
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Langmuir free energy density given in eq 1. This model also explained the existence of the two-phase region, where the total free elastic energy F2D shows two local minima [corresponding to the cases of the classic short and of the giant dipole28] when the beaddefect distance d varies. We have examined how these features change with the beads size. The same experimental scenario is observed for beads of varying size in the 1−20 μm range, except that larger particles induce a stronger capillary distortion for a film of same thickness resulting in larger dipole moments. Additionally, the two-phase region is shifted to higher thicknesses. For example, we have monitored the configurations around isolated particles of 7 μm diameter while slowly changing the film thickness with the Langmuir trough. In the two-phase region, an expected hysteresis (shown on Figure 5c) is indeed observed. Starting with a short dipole in a thick region, a first transition to a giant dipole is observed at h0 ≈ 4 μm when the defect escapes from the particle. The equilibrium distance between them increases. When increasing the film thickness, the transition from giant to short dipole is now detected at a higher thickness h0 ≈ 4.6 μm. However, the two-phase region, determined with many beads, is experimentally broader than this hysteresis, presumably due to the dispersion in bead size (and/or contact angles). For the 7 μm microspheres, the coexistence of short and giant dipoles is observed in the 3.2−4.6 μm thickness range, whereas it was in the 1.6−2.6 μm range for 4 μm diameter beads.28 This change of behavior is also well captured by the 2D modeling, which predicts a phase diagram close to the experimental observations (see Figure 5d where we have accurately established the boundaries for monodisperse 4 and 7 μm beads but also observed that simulated boundaries correctly described the phase diagram for typically hundreds of beads). Finally, we noted that for giant dipoles, the elastic dipole moment p, defined from the defect toward the particle, is always observed to be parallel to c0 (Figure 5b), the far-distance c-director. “Butterfly” Textures. In the case of the “butterfly” patterns, the thickness of the film around the particle varies also axisymmetrically. The capillary distortion is nearly the same as the one of the giant dipole (e.g., see Figure 6c) and is also satisfactorily given by h(r) = h0+AK0(qr). Concerning the texture, the c-director field is mainly characterized by a s = −1 topological defect on one side of the particle (see insets of Figure 6b) and a π-wall on the other side. In one type of “butterfly” texture, the defect always remains located at a short distance from the particle, independently from the film thickness. The elastic dipole moment p is then antiparallel to c0 (Figure 6b). Equivalently, the particle is a “sink” for the cdirector field (inset of Figure 6b), whereas it is a “source” (inset of Figure 5b) for the parallel dipole. The “butterfly” texture is, however, not specific to antiparallel dipoles only. Starting from thick films showing the two types of short dipoles (parallel and antiparallel, see Figure 7a) formed by 4 μm particles, the surrounding textures change when slowly decreasing the film thickness. In films thinner than 3.5 μm, birefringence patterns appear on large scales, indicating that the c-director tends to follow the thickness gradient around particles. Both parallel and antiparallel dipoles then show “butterfly” textures but of slightly different aspects (see Figure 7b). The antiparallel one never transforms spontaneously into a giant dipole. Its experimental evolution is shown in Figure 7c when varying the thickness of the film. On the contrary, the parallel “butterfly” texture can be observed only in a limited range:between 3 and 1.6 μm. When entering deeply into the
Figure 6. (a) “Butterfly” texture observed between crossed polarizers. (b) Corresponding retardation and director fields obtained from Abrio system. The associated defect (see insets) is located near the 4 μm silica bead. (c) Evolution of the thickness of the nematic film h as a function of the distance r from the bead for the “giant dipole” (black points) and the “butterfly” (red triangles) textures shown in Figure 4b.
two-phase region (thickness about 2 μm) it becomes metastable and may suddenly transform into a giant dipole texture, the defect escaping far from the particle. Textures and 2D Numerical Modeling. The evolution of all these patterns could be satisfactorily reproduced using the 2D modeling approach (see Figure 7c−e). An antiparallel dipole is formed when the boundary conditions of the parallel dipole are shifted by π (see Figure 2). The evolution of the free energy F2D (eqs 1 and 2), with d (the bead-defect distance) is, however, different for parallel and antiparallel dipoles (see Figure 8a). For parallel dipoles, a single minimum is found in the thickest films. In the two-phase region, the free energy shows two local minima corresponding to the giant dipole and to the parallel “butterfly” textures. In thin films, only the “giant dipole” minimum remains. For the antiparallel dipole, a single local minimum located near the bead is always observed. This can be understood qualitatively since an escape of the defect from the particle in a parallel (respectively antiparallel) dipole mainly consists in favoring the alignment of c with −∇h (respectively with ∇h) in a large region around the particle. The formation of giant dipoles is then favored for parallel dipoles but disfavored for antiparallel ones. Note finally that the antiparallel dipole energy is systematically higher than the one of the parallel dipole (see Figure 8a). This could explain why a transition toward the parallel dipole is systematically observed24 when the sample is briefly heated in the isotropic state (T about 40 °C) and then cooled back to room temperature. The persistence of an antiparallel dipole is nevertheless compatible with the 2D model, since such a dipole cannot spontaneously transform into a parallel one. Such a transformation would require at least a surface anchoring breaking (a possible 9102
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Figure 8. (a) Evolution of the free energy F2D (eq 2) of a parallel dipole (left) and an antiparallel one (right) with the bead-defect distance d (bead diameter 4 μm). The equilibrium textures are marked by green disks and are accompanied by the corresponding birefringence patterns [the half upper part of the simulated director field correspond to the boundary conditions of Figure 2]. For all thicknesses, the parallel dipole has the lowest free energy. In the frame of the 2D model, however, an antiparallel dipole cannot spontaneously transform into a parallel one unless anchoring breaking occurs at the bead interface, such as in the boojums-assisted transformation shown in panel b.
Figure 7. (a) A parallel dipole and an antiparallel one observed between crossed polarizers in a thick nematic film (h0 ≈ 4 μm). (b) “Butterfly” textures are observed for both dipoles when slowly decreasing the film thickness (h0 ≈ 2 μm). Experimental (c) and computed (d) evolution of the “butterfly” texture of an antiparallel dipole with the film thickness. (e) When decreasing the thickness, a similar evolution is obtained for the parallel dipole before switching to the giant dipole texture. In computed textures, the far-distance director c0 points upward.
observed in the final thin films. However, if the same sample is briefly heated in the isotropic phase and cooled back (either in the initial thick films before trapping at the second interface or in the final film), only parallel dipoles are formed. These observations show that the trapping at a single interface is sufficient to explain the existence and metastability of the two types of short dipoles. The lower interface, however, also influences the resulting textures obtained by dry deposition. In films thinner than 7 μm, a noticeable amount of parallel dipoles are obtained after the deposition. 3D Modeling of the Elastic Dipoles. Motivated by these observations, we have focused on the “exact” 3D structure of parallel and antiparallel short dipoles and their formation when the beads are trapped at a single interface. Performing numerical minimization of the 3D Landau−de Gennes free energy functional in eq 3, we were able to reproduce the parallel (e.g., Figure 9a−c, Figure 10b−d, Figure 12a,b and Figure 13b−d) and antiparallel dipoles (Figure 9e,f, Figure 10f, Figure 12c,d and Figure 13a,e) textures. First we consider the case when the particle is trapped at the upper nematic−air interface. Our findings demonstrate that the surface-to-surface distance S between the colloidal particle and the lower nematic-water interface has the strongest impact on the particle-induced nematic configuration. For large thicknesses S , shown in Figure 9, the lower nematic−water interface is essentially uniform, with the surface director ns(r) being parallel to the far-field director. The system reveals a host of metastable n(r)-configurations and the most typical ones are shown in Figure 9. The director field n(r) of the parallel dipole structure (Figure 9a−c) has either hyperbolic hedgehog defects as it is presented in Figure 9b, or off c-equatorial −1/2 disclination loops (Figure 9a,c). The core of the hedgehog
transition involving 2D boojum defects is sketched in Figure 8b), but this is unlikely, due to the strong anchoring provided by the silanization of the bead surface.29 The Role of the Confinement and 3D Modeling. The 2D approach is particularly relevant for explaining the fardistance textures of giant dipoles and “butterfly” patterns. However, it is expected to become inaccurate in the regime where the short dipoles exist. Additionally, it cannot explain the metastability of the antiparallel dipole since the c-director model fails to accurately describe the immediate surroundings of the microspheres in 3D. We have therefore refined our analysis and looked for the physical origin and the stability of the two types of dipoles (parallel/antiparallel), both experimentally and theoretically. Nature of the Defects under Progressive Film Thinning. From an experimental point of view, in order to explain the noticeable presence of both dipole types in nonheated thin films, we have followed the evolution of the textures before trapping of the beads at both interfaces. Using the Langmuir trough, we have first prepared a thick film (typically h0 > 7 μm) and then deposited dry particles of 4 μm diameter at the upper surface, using the method described in ref 24. After having sufficiently decreased the thickness, the beads get trapped at the lower surface, as shown by the sudden appearing of a thickness distortion. With this method, only antiparallel dipoles are 9103
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Figure 9. Typical configurations of a nematic film of thickness Lz = 2 μm, with a colloidal particle of radius R = 1 μm trapped at the homeotropic nematic−air interface. The center of the colloidal particle is placed at rc = (0,0,zc) with zc = −R/4. All the configurations are “equilibrated” starting from the initial conditions as shown in Figure 3a and varying the hedgehog angle θ: (a) θ = 63°, (b) θ = 46°, (c) θ = 34°, (d) θ = 0°, (e) θ = −23°; and (f) θ = −46°. Blue rods depict cross-sectional view of the nematic director field n(r) in the plane x = 0; black rods show n(r) at the planar nematic−water interface at z = −Lz, defect lines in a, c, d−f are shown as red tubes. The thinner red tubes laying at the colloid−nematic−air triple line do not represent defect lines, but only correspond to the nematic regions with a reduced value of the scalar order parameter Q < Qb. The core of the hedgehog “point” defect in panel b has an open structure of a small, on the order of several nematic correlation length ξ, disclination ring.
Figure 10. Nematic film thickness Lz = 2 μm, R = 1 μm. The colloid center is placed on the z-axis, at zc = −3R/4. (a) Isotropic initial conditions; (b) θ = 57°; (c) θ = 40°; (d) θ = 34°; (e) only hybrid nematic, without Lubensky’s dipole inclusion; and (f) θ = −40°. The nematic configurations around colloidal particle, with n(r) shown by rods and defect lines as red tubes.
12b,c,d. For small enough S , the parallel dipole can be found in three different configurations. The first two are similar to the earlier case of large S and have the structure with either a hyperbolic hedgehog such as in Figure 10b and Figure 12a or off-equatorial declination loops of various extent (Figure 10c,d). The third class of configurations features the presence at the lower nematic−water interface of self-compensating surface topological defects of opposite strengths, see Figure 12b right. There are two defects of strength s = +1/2 (see inset in Figure 12b, right panel) and two defects of strength s = −1/2 where s is defined as the number of times ns(r) rotates by 2π as one circumnavigates the defect core once. The surface defects with the same strength are connected by disclination lines passing through the bulk nematic. Each pair may be thought of
defect is opened into a small ring, which is tilted away from the z-axis toward negative x (see Figure 3a) in order to match with the hybrid n(r) structure. The antiparallel dipole (Figure 9e,f) has a more complex structure, where some portion of a −1/2 disclination loop is trapped at the air−nematic−colloid triple line, while the rest of the loop lays within the nematic domain. Sometime the “bulk” part of the loop may encircle the colloid at the side with positive x realizing an antiparallel dipole. Upon decreasing S at fixed Lz, the surface director ns(r) at the lower nematic−water interface with tangential degenerate boundary conditions acquires a nontrivial structure. The last can be nonsingular, i.e., such that contains no defects (see Figure 10 and Figure 12a), as well as singular with 2D topological defects as presented in Figure 11, and Figure 9104
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Figure 11. Lz = 2 μm, R = , zc = −3R/4. The nematic configurations around colloidal particle, with n(r) shown by rods and defect lines as red tubes for (a) θ = −29° and (b) θ = −34°. The right column shows the structure of the surface boojums generated by the proximity of the colloidal particle at the planar nematic−water interface.
as a single boojum with the split core, and what is surprising is that this splitting may be of the size of the colloidal particle itself (see the long disclination line joining two −1/2 surface defects in Figure 12b). Topology requires that the total strength of the surface disclinations is always zero (uniform far field c-director). We also observe antiparallel dipoles with nonsingular (Figure 10f) and singular (Figure 11a and Figure 12c,d) surface director field ns(r). In the first case, the dipolar structure of n(r) is realized by the encircling disclination line, which now fully lays in the bulk nematic, contrary to the case in Figure 9f. The singular ns(r) antiparallel structure shown in Figure 12c is similar to the one described above for the parallel dipole. The only difference is that two s = −1/2 defects have migrated to the positive x side. The structure shown in Figure 12d is qualitatively different, as in addition to the two selfcompensating split-core boojums, n(r) also contains a bulk “Saturn” ring disclination. A colloidal particle which is trapped at the nematic−water interface is equivalent to a s = 1 surface defect in ns(r). Based upon topological arguments a compensating s = −1 defect therefore always accompanies the colloidal particle. This particle-induced defect can be “point-like” split-core boojums (see Figure 13d) giving rise to parallel dipoles, or pairs of widely separated s = −1/2 defects connected by the bulk disclination lines (see Figure 13a,c,e). In the last case those can produce both parallel (Figure 13c) and antiparallel (Figure 13a,e) dipoles. Qualitatively similar configurations as those shown in Figure 13 are observed when the colloidal particle is trapped at both interfaces (nematic−air and nematic−water). The short dipoles that are experimentally observed in the thickest films may therefore have a large variety of local structures quenched when the bead got trapped at the first interface it encounters. Upon decreasing h0, the distortion of the film thickness around the bead becomes progressively stronger such that the resulting spontaneous alignment of the far-field c-director along the steepest slope of h strongly changes the film texture. Initially,
Figure 12. Lz = 2 μm, R = 1 μm zc = −0.88R. The depicted nematic configurations are obtained from the initial configurations as shown in Figure 3a with (a) θ = 57°; (b) θ = 40°; (c) θ = −40°. (d) corresponds to an isotropic initial configuration. The particle induced n(r)configurations (shown by rods) and defect lines (red tubes). The right column represents the corresponding director filed at the nematic− water interface (black rods).
when the thickness distortions are still small, all the defects remain trapped in the vicinity of the bead giving rise to the “butterfly” textures. Eventually, with larger variations of h due to the thinning, a class of defects corresponding to the parallel dipoles can be pushed forward and giant dipoles appear.
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SUMMARY AND CONCLUSIONS In this work we have systematically investigated the director and thickness distortions induced by the confinement of microspheres in a thin nematic liquid film. Optical observations and quantitative birefringence measurements reveal the presence of large director distortions at far distance (typically a hundred times the particle size) in the thinnest films where the relative thickness variation is the largest. The experimental dependence of the shape and size of the patterns with the thickness of the film and the size of the bead are well reproduced by a two-steps 2D model. Within this approach the slowly varying in-plane nematic distortion is first neglected to compute the capillary distortion as given by the nematic film 9105
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Figure 13. Colloidal sphere with R = 1 μm is trapped at the lower nematic−water interface. The nematic film thickness Lz = 2 μm, and the position of the center of the colloid zc = −7/4R. (a) θ = −86°, (b) θ = 29°, (c) θ = 74°,(d) θ = 86° and the right panel shows enlarged view of the region around the boojums. (e) Lz = 2 μm and the position of the center of the colloid zc = −1.12R. This configuration is obtained from the isotropic initial conditions. Particle-induced n(r)-configurations (shown by rods) and defect lines (red tubes).
thickness h(r). Next, the in-plane c-director field is obtained by minimizing an effective 2D nematic free energy at a given h(r). Experimentally observed large-scale metastable nematic distortions are well explained by the effective 2D model. This reduced description has uncovered subtle differences between the behavior of parallel and antiparallel elastic dipoles as the initial film thickness h0varies. Thus, in the thickest films the parallel and antiparallel dipoles have very similar structures (see Figure 7a). However, upon decreasing h0 below some threshold value, only the parallel short dipoles exhibit the transformation to the giant dipole configuration (see Figure 7e). The antiparallel short dipoles, in contrast, never transform to the giant ones, but always evolve with the decreasing h0 toward the butterfly configuration (see Figure 7c,d). A systematic study of the short distance 3D nematic distortions around a bead has revealed that a multitude of possible defect configurations accounts for each type of dipole. Even in the absence of capillary deformations, topological defects are indeed quenched by the trapping of a bead at a single interface and more generally by the confinement and the proximity of the interfaces. Finally, our results provide new insights into the effects related to the coupling of orientational elasticity and capillarity. Colloidal self-assembly based on each of these independent mechanisms has been studied in depth and is rather well
understood. Exploiting both of the above mechanisms in a single system will provide a range of unique opportunities to better control effective colloidal interactions, which in turn is expected to generate important synergistic effects and new types of colloidal ordering.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Author Contributions
M.T., T.O., and C.B. designed research; H.J. and C.B. performed experimental research and analyzed data; M.T. and C.B. performed numerical studies; H.J., M.T., and C.B. wrote the paper. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS
This work was partially supported by the Tunisian Ministry for Higher Education, Research and Technology and partially by Laboratoire Charles Coulomb of Montpellier University. 9106
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