Thin Nematic Films.

Our experimental systems consist of thin films of nematic liquid crystal [4-n-pentyl-4′-cyanobiphenyl (5CB)] spread at the surface of a thick water layer (Fig. 1A). The degree of wetting of high-purity 5CB on pure water is insufficient to give satisfactory film-forming properties (11). We have found that adding 5 wt% of polyvinylalcohol (PVA) ensures complete wetting. After briefly heating to the isotropic state (above T=35 °C) and cooling back to room temperature, the films show a uniform thickness and are highly stable with time. This system has several advantages over the well-studied films spread on glycerin (12, 13): Due to its slight solubility in 5CB, glycerin indeed tends to form droplets in the NLC after some time (14⇓–16), disturbing the system and hindering detailed observations (17). Films containing solid microspheres with perpendicular anchoring are similarly obtained by spreading 5CB containing 0.1 wt% of silica particles of nominal radius R≈ 2 ± 0.2 μm over the PVA solution.

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Fig. 1.

(A) In a hybrid nematic film, the director n rotates between θp and θh at the interfaces. (B) This defines a polar c director field characterized by its angular orientation φ. (C) In a 10-μm-thick film, a 4-μm microsphere is accompanied by a defect located at a short distance (a possible configuration is sketched in Inset). (D) In a thinner film, a giant dipole is observed between crossed polarizers. The microsphere induces a radial director in its neighborhood (Inset) and the counter defect is found much farther away.

The 5CB NLC shows a strong perpendicular anchoring on air (13) and a degenerate parallel (planar) anchoring at the water/PVA solution (18). The difference in anchoring conditions Θ=θp−θh (Fig. 1A) leads to a continuous 3D distortion of the director n. Such hybrid anchoring nematic films (mainly studied on glycerin) have already been thoroughly examined both experimentally and theoretically (12, 13, 16, 19⇓–21). Thick films are expected to show a simple and homogeneous structure in the layer plane (defining a 2D c director sketched in Fig. 1B) but when the thickness h decreases to a fraction of microns, a spatial xy modulation of the director field and periodic domain patterns, due to an elastic instability, are observed (22, 23). We checked that the particle-free films deposited on PVA/water behave similarly. The simple uniform texture is observed down to a thickness of ∼0.4–0.5 μm and thinner films display also periodical instabilities (22). In the following, we mostly focus on films whose thickness is between 0.4 μm and 3 μm. In this range, the comparison of the retardation maps and the interference patterns in reflection mode (details in SI Text and Fig. S1) indicates that the local film thickness h is proportional to the retardation: h=δ/Δn¯ with Δn¯=0.080±0.004. This defines an effective birefringence Δn¯ slightly smaller (∼15%) than the value expected in the hypothesis of strong anchorings at both interfaces; i.e., θh=0 and θp=π/2 (SI Text). Throughout the text, we have used this experimental relation to derive the thickness profiles from the retardation maps.

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Fig. S1.

Dependence of the optical retardation δ on the thickness h of a hybrid aligned nematic film of 5CB. The same 5CB lens floating on the water just after deposition is observed both in polarized reflection (A) and with the Abrio system (B). The location of destructive wave interferences observed in reflection mode is given by the condition n¯.h=jλ/2 (j∈ℕ). When the director c is perpendicular to the polarizer P, n¯=no (horizontal window in A). The measurement (B) of the corresponding optical retardation δ shows that a linear approximation δ=Δn¯h correctly describes the dependence in the range of interest (C, black triangles), with Δn¯=0.08±0.005. This value is also confirmed by the measurements when n¯=ne¯ (vertical window in A and blue circles in C). Red lines are the corresponding linear fits whereas the green curve that fits the triangles is obtained assuming weak anchoring conditions with Wh=6⋅10−5 J⋅m⁻² and Wp=4⋅10−5 J⋅m⁻².

Giant Elastic Dipoles.

After the deposition of particle-free films of homogeneous thickness, defects heal spontaneously in a few tens of minutes. The films show uniform thickness at equilibrium and homogeneous textures with minimal azimuthal distortion in the viewfield. On the contrary, films containing microspheres show persistent distortions that strongly depend on the film thickness. When the NLC layer is thicker than the bead diameter (typically between 4 μm and 10 μm), the distortions are localized: Individual beads are classically accompanied by a nearby hyperbolic hedgehog defect (Fig. 1C). The beads are then either confined within the film or trapped at a single interface, similarly to what has been analyzed and discussed in refs. 24 and 25, where microspheres were trapped at the air–5CB interface of films planar anchored on glass substrates. More surprisingly, when the thickness of the film decreases typically to the micrometer, giant stable dipoles are observed (Fig. 1D). The bead is equivalent to a +1 defect for the c director, which confirms (26) that perpendicular anchoring on the particle has remained strong despite the direct contact with the PVA solution. The induced defect of charge −1 is located at a distance up to ∼100 μm from the microsphere. To understand the origin of such unusual textures we have extensively studied the films and the director fields in the 0.4- to 4-μm thickness range. The bead diameter being larger than the film thickness, a capillary distortion is expected. Retardation maps reveal that the thickness of a film indeed increases in the vicinity of the particle. The deformation of the interface is roughly axisymmetric around the microsphere (Fig. 2A) and almost independent of the azimuthal director field (−1 defects have only minor influence on the film thickness). This reveals that the capillary effects largely dominate the 2D elasticity related to the distortions of the c director. However, the influence of the nematic elasticity on the interfaces should not be entirely discarded because a much larger elastic energy is stored in 3D via the hybrid anchoring conditions. At first order, the Frank elastic energy density of a hybrid NLC film indeed reduces to (27)fd=12K(∂θ∂z)2,[1]where K is the nematic splay-bend modulus in the one-constant approximation. In the limit of strong anchoring conditions with Θ=π/2, this yields a thickness-dependent surface free elastic energy Fd=Kπ2/8h (27), at the origin of a thin film disjoining pressure ΠD=−∂Fd/∂h=Kπ2/8h2. When weak anchoring conditions and an evolution of Θ with the thickness are considered (SI Text), a similar form is obtained with ΠD=−∂Fd/∂h=KΘ2/2h2.

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Fig. 2.

(A) Retardation and director field of a nematic film around a silica microsphere in a thin film. The director field reveals a strong perpendicular anchoring on the particle, equivalent to a +1 defect for the c director and the presence of a −1 defect far from the bead. The retardation has an axial symmetry, which reveals that capillary effects dominate the azimuthal elasticity. (B) Side view sketch of a microsphere embedded in a thin NLC film of thickness h0 at far distance. Capillary effects fix the contact angles ϑ1 and ϑ2, which yields a distortion h(x,y) of the interfaces. Pf(h), Pa, and Pw are the pressures in the film, in air, and in the PVA solution.

Capillary Effects.

To evaluate analytically the distortion of a film, we have considered weakly distorted interfaces and use an expansion (28) of the disjoining pressure in a thin film around the far distance thickness h0: ΠD(h)≈ΠD(h0)+∂ΠD/∂h|h0(h−h0). Using the geometric notations of Fig. 2B, the interface positions z1 and z2 check the linearized Young–Laplace equations (the gravity-induced effects are negligible at the micrometer scale and the slopes are small),Δz1=q12(z1−z2−h0)Δz2=−q22(z1−z2−h0),[2]where qi=−∂ΠD/∂h|h0/γi (i=1,2) compares the effects of disjoining pressure and of the interface tensions γ1 and γ2 (respectively for the air–5CB and the PVA/water–5CB interfaces). The axisymmetric solution gives a thickness profile (details in SI Text)h(r)=z1(r)−z2(r)=h0+AK0(qr),[3]where r is the radial distance from the bead center, q=q12+q22, and K0 is the modified Bessel function of the second kind of order 0. The constant of integration A can be determined using Young’s relation at the triple lines as a boundary condition (full analytical details in SI Text and Fig. S2). We have determined the profile of several tens of beads and found that Eq. 3 correctly describes the change of thickness around them (Fig. 3A). Moreover, it allows us to test the elastic origin of the disjoining pressure: The expression ΠD=Kπ2/8h2 yields qth=βelh0−3/2, where βel=K(γ1−1+γ2−1)π/2. This dependence is compatible with our observations as shown in Fig. 3B. The best-fitting value βfit=0.038±0.001 μm^1/2 is smaller but of the same magnitude as the computed value at T=23 °C, βth=0.049 μm^1/2 obtained from the average constant K≈7 pN (29), γ1=34.5±0.5 mN⋅m⁻¹ (30), and γ2=9±0.5 mN⋅m⁻¹, where the two latter values were respectively checked or measured from pendant and rising drops experiments. If we consider weak anchoring conditions within our simplified model, the dependence of q on h can be obtained numerically. Reduced values qw are obtained, especially at low thicknesses and low anchoring energies, but the magnitude remains the same. However, the comparison of these curves with the experimental data (taking also into account the birefringence measurements) shows that the associated anchoring energies of the liquid crystals at interfaces cannot be much lower than W≈5⋅10−5 J⋅m⁻² (Fig. 3 and SI Text).

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Fig. 3.

(A) Thickness of a nematic film vs. its distance r from an embedded bead. Experimental points are obtained by angular averaging of the thickness obtained from the retardation map (Inset). The solid curve is a fit using Eq. 3. (B) Evolution of q with the thickness. The fitting lines (main text) confirm that the disjoining pressure mainly results from nematic elasticity. Taking into account the finite anchoring energies reduces the values of q especially at low thicknesses. The green dashed-dotted line shows a fitting curve qw obtained using weak anchoring conditions for possible anchoring energies Wh=6⋅10−5 J⋅m⁻² and Wp=4⋅10−5 J⋅m⁻².

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Fig. S2.

(Circles) Fitting parameter of the experimental profiles Afit with the far thickness h0. The solid lines are the theoretical values Ath computed for the estimated bounding values of the particles radius R=2±0.3 μm.

Elasto-Capillarity in Nematic Films.

Now that the profile of the film thickness is explained, let us consider the nematic field around the particle. As said above, giant dipoles have been seen only in deformed films. However, the largest dipoles are not necessarily observed in the most distorted films (which are the thinnest ones). The equilibrium distance de between the bead center and the defect has been measured in samples of different thickness. For a few samples, we have also followed the evolution of this distance while increasing (or decreasing) the thickness by adding (removing) a small amount of 5CB. For each measurement, we have waited tens of minutes to reach an equilibrium distance. Fig. 4A shows the evolution of de from 100 microspheres. Up to a thickness about 1.6 μm, the data are only slightly dispersed (presumably due to size dispersion of beads) and each mark corresponds to an average of about 10 individual measurements. In this region, we observe only giant dipoles with an increase of the distance with the thickness. Above 2.6 μm, on the contrary only the short common dipoles are observed with a typical distance de<4 μm. In the thickness range 1.6–2.6 μm, a coexistence of giant and short dipoles is observed in samples at equilibrium. When adding 5CB, the distance of a given giant dipole increases (purple triangle trajectory in Fig. 4A), reaches a maximum, and then rapidly decreases toward the close contact.

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Fig. 4.

(A) Evolution of the distance particle center defect de with the film thickness h0. Below 1.6 μm, the experimental data are grouped and each mark (brown circles) corresponds to the average of about 10 independent measurements in a 0.1-μm thickness range. Above 2.6 μm, the defects are always located in the vicinity of the particle (individual measurements). In the 1.6- to 2.6- μm range, the data are highly scattered with defects either close to the particle or at a large distance from it (open squares). Individual dipoles (colored markers with arrows) have also been followed while continuously increasing or decreasing the film thickness (main text). (B) Evolution of de with h0 computed using the ansatz of Eq. 4 with Θ=π/2 and K=K/2. Solid black diamonds indicate local minima of the elastic free energy, whereas open red diamonds show the absolute minima. The letters A, B, and C indicate three typical situations detailed in Fig. 5.

The role of the gradient of thickness is of prime importance to explain the formation of the giant dipole. The most economical explanation, at first sight, would evoke the fact that a defect will tend to decrease its core energy by escaping in the thinnest part of the film. Such an explanation has been already proposed to explain why +1/2 defects that are naturally present at the surface of nematic shells (with both planar anchorings) migrate to their thinnest part (31). It should be noted, however, that +1/2 defects are tridimensional disclinations spanning the shells whereas the −1 defects considered here are boojums that do not live in the nematic layers but rather at their surface. Such defects are much less sensitive to the thickness as already noted in ref. 31, where they were also observed in the thick parts of the shells. Moreover, in a recent work (26), we have also reported that particles trapped on shells could give rise to short common dipoles and even be accompanied by −1/2 defects in the thickest part. Another explanation of the presence of the textures has to be found.

To understand why giant dipoles form in the distorted films, one has to consider a refined description of the elastic energy stored in the film. For this, we have explored an ansatz already used to analyze and explain distorted textures found in nonflat hybrid nematic films (20, 32). It consists of an approximation for the elastic energy density of a distorted hybrid NLC film, expressed in terms of 2D operators only (20),f2D=12K(∇xyφ)2+12K(Θ+c.∇xyhh)2,[4]where ∇xy is the 2D gradient and φ is the azimuthal angle of the c director in the xy plane [c=(cos⁡φ,sin⁡φ)]. Twist energy terms are fully neglected in this model. The first term accounts for the xy splay bend distortions of the c director with an elastic modulus that is only about half the 3D modulus (K≈K/2) due to the absence of distortions at the perpendicularly anchored interface. The second term is the already encountered zenithal distortion term with a geometric correction due to the thickness gradient. If the angles θp and θh (Fig. 1A) are defined with respect to the normal of the lower and upper interfaces, the term Θ=θp−θh indeed does not account for the whole angular reorientation of the n director when the interfaces are not parallel. When the gradient of thickness remains small, the geometric angular correction is given by c.∇xyh at first order and the director tends to be parallel with the thickness gradient. The part of the free energy that depends on the 2D c texture is then given by F2D=∬f2D′h(x,y)dxdy, where f2D′h(x,y)=f2D−KΘ2/2h2. Such an elastic energy will tend to favor the alignment of the c director on the gradient of the thickness. Because we use perpendicularly anchored microspheres that also give rise to a radial gradient for the thickness, a radial director field is highly favored in the vicinity of the bead. It is only at large distances, where the film gets a uniform thickness, that the gain in energy of the radial texture is not balanced anymore. Following the classic mechanisms, a defect of opposite charge appears to decrease the distortion energy at very large distance.

To confirm this scenario, we have numerically investigated the evolution of the nematic texture of minimal energy caused by a microsphere. We have found a nonmonotonic behavior of F2D with the distance d between the bead and the topological defect of charge −1. Fig. 4B shows the positions de of the local minima of the energy found as a function of the film thickness h0 with three typical dependencies detailed in Fig. 5. Several features of the experimental observations are found. First, at small thicknesses h0, F2D shows a single minimum (Fig. 4B, case A) with respect to d, corresponding to a giant dipole. In this region, the equilibrium distance de increases with the thickness. The free energy then shows two local minima (Fig. 4B, case B) corresponding to a giant and a classic dipole. In thicker samples, the free elastic energy shows only one minimum corresponding to the short dipole (Fig. 4B, case C). With the 2D ansatz, the main experimental trends are thus found with simulated textures (shown in Fig. 5) very similar to the experimental observations. However, one should note an overestimation of both the threshold thicknesses and the typical distances d in Fig. 4. This could be due to the roughness of the ansatz model, which ignores some components of the elastic energy (such as twist terms), but also to the strong anchoring approximation. Both the threshold thicknesses and d indeed are found to decrease with weaker anchoring conditions (Θ<π/2).

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Fig. 5.

(A–C) Evolution of the dependence of the elastic free energy F2D on d with the film thickness (A) h0=1.2 μm, (B) h0=2.5 μm, and (C) h0=3 μm. Each curve is accompanied by the birefringence pattern simulated from the corresponding ground state texture. A magnification of the vicinity of the microsphere is shown in C.

Colloidal Interactions in a Thin Nematic Film.

As discussed above, the capillary distortion around an isolated colloid is strong enough to be almost independent of the nematic c director, which then adapts to the imposed thickness variation. One could wonder whether the capillary interactions expected between two particles may be compatible with the presence of giant dipoles. It is indeed well known that capillary interactions between particles trapped in a thin film are always attractive (28). We have therefore explored the pair interactions between two trapped colloids by increasing the amount of particles in the deposited films (but still with a density low enough to avoid undesirable aggregation during the brief heating of preparation). When the film thickness h0 is large enough, the common short dipoles are observed and two neighboring particles rapidly come together, forming a dipole pair parallel to the director field. This classic behavior is expected because both capillary and parallel dipolar–dipolar interactions are attractive. More interesting is the case of thin films where giants dipoles are observed. As illustrated in Fig. 6, two parallel dipoles interact attractively until an equilibrium distance is found. The distance ℛ between the two microspheres is comparable to twice the equilibrium distance de between a sphere and its defect. During the approach of the particles, de is hardly modified, which indicates that the attractive capillary interaction is not sufficient to overcome the effective interactions due to the nematic elasticity. If the sample is then heated into the isotropic phase, the particles are submitted to capillary attraction only and they come together in a few minutes, as illustrated in Fig. 6B.

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Fig. 6.

Interactions between two microspheres in a thin film. In nematic phase at room temperature, an appropriate thickness h0 gives rise to the formation of giant dipoles that spontaneously align at an equilibrium distance (A, crossed polarizers). The far-distances elastic forces are sufficient to avoid the collapse of the particles under capillarity, which is observed in a few minutes (B, bright field) when the sample is heated into the isotropic phase (T=40 °C).

The stability of the giant dipoles is due to the fact that the capillary force between immersed particles at a distance ℛ has a magnitude given by ℱc≈2πγ1R2qK1(qℛ) (28), where K1 is the modified Bessel function of the second kind of order 1. Above qℛ≥2, this force decays exponentially, ℱc∝exp(−qℛ). When two dipoles come into contact (at a typical distance of 2de), the radial texture around each particle becomes distorted, which gives rise to a similar repulsive force to the one observed between a particle and its defect, which was discussed in the previous section. The typical magnitude K (Fig. 5) of this repulsive force is larger than ℱc at large distances: For example, the experimental data q≈0.05 μm⁻¹ for a typical thickness of 1 μm give ℱc/K≪1 for ℛ>160 μm.

In conclusion, capillary effects on nematic films strongly modify the known textures found for nematic colloids. Microparticles embedded in thin nematic films give rise to giant elastic dipoles that are not observed in thin nematic cells and that can be used to mediate long-range elastic forces between microparticles. Additionally, the geometry of the dipoles can be simply controlled via the film thickness. From a theoretical point of view, a full variational analysis is formally needed but we have shown that the deformation of the interfaces and the 2D nematic textures could be analyzed separately in a first approximation. In our approach, the nematic distortion through the film thickness is taken into account with a scalar disjoining pressure, and the nematic texture could then be analyzed in a second step with a fixed thickness profile. These results are to our knowledge the first evidence of the role of elastocapillary potentials in nematic films. Moreover, we showed that elastic dipoles and the nematic elasticity are sufficient to counterbalance the strongly attractive capillary interactions between colloidal particles when they are trapped in thin films. These findings may help to direct the arrangement of colloidal particles into rich 2D structures and develop new classes of reconfigurable metamaterials. We hope this work will stimulate experimental and theoretical works on the self-assembly mechanisms that could emerge from such a competition.