Pseudo-Casimir effect in nematic liquid crystals in frustrating geometries

The Casimir-like effect induced by active nematics

We consider an active nematic phase and use hydrodynamical equations of it to model the activity as an internal field. The interaction of this field with the nematic director in a confined geometry is included in the Hamiltonian of the system. Based on this model Hamiltonian and the standard field theoretical approach, the Casimir-like force induced between the boundaries of such a confined film is discussed. The force depends on the geometrical shape and the dynamical character of the constituents of our active phase, as well as the anchoring conditions. For homeotropically aligned rod-like particles which in principle tend to align along a planar flow field, extensile activity enhances the attraction present in a thin nematic film. As the film thickness increases the force reduces. Beyond a critical thickness, a planar flow field instantaneous to a bend distortion sets in. Near but below the threshold of this activity-induced instability, the force crosses zero and repulsively diverges right at the critical threshold of this so-called flow instability. For contractile rods, in the same geometry as above, the structure is stable and the Casimir-like force diminishes by an exponential factor as a function of the film thickness. On the other side for a planar director alignment, rod-like contractile particles can induce opposite shear flows at the boundaries creating a splay distortion for the director between the plates. In this configuration, we obtain the same universal pretransitional behavior for the force as above. Vice versa, for extensile rod-like particles in this geometry, the director fluctuations become massive and the Casimir-like force diminishes again by an exponential factor as the film thickness increases. The effect of the active field on thermal fluctuations of the director and the fluctuation-induced Casimir force per area is derived through a "semi"-dynamical approach as well. However, the results of the calculation due to a mathematical sum over the fluctuating modes do not lead to an approved closed form.

Effect of Materials Parameters on the Shape of Face-On Lamellae in Semi-Conducting Polymers: Insights From Qualitative Theory

Polymer semiconductors frequently form crystals or mesophases with lamellae, that comprise alternating layers of stacked backbones and side chains. Controlling lamellar orientation in films is essential for obtaining efficient charge carrier transport. Herein, lamellar orientation is investigated in an application-relevant setup: lamellae assembled on a substrate that strongly favors face-on orientation, but exposed to a film surface that promotes orientation along an "easy" direction, other than face on. It is assumed that the face-on order propagates from the substrate, but the lamellae bend to reduce their surface energy. A qualitative free-energy model is developed. The deformation is investigated as a function of film thickness, effective Young modulus, anchoring coefficient, and easy direction at the free surface. The calculations highlight the importance of elastic constants - lamellae can substantially deform already when Young moduli are only an order of magnitude smaller than the values that are reported for crystals. Softer Young moduli are expected when lamellar assembly occurs in a non-solidified mesophase that can be an equilibrium or (more speculatively) a transient state prior to crystallization. The alternative scenario of a two-layered film is also evaluated, where edge-on and face-on grains form, respectively, at the free surface and substrate.

Director anchoring on a simple edge dislocation at the surface of induced smectic-C_{A} films

We present a detailed analysis of c-director anchoring measurements on simple edge dislocations at the surface of smectic-C_{A} films (steps). Indications show that the c-director anchoring on the dislocations originates from a local and partial melting of the dislocation core that depends on the anchoring angle. The SmC_{A} films are induced on isotropic puddles of 1-(methyl)-heptyl-terephthalylidene-bis-amino cinnamate molecules by the surface field, while the dislocations are located at the isotropic-smectic interface. The experimental setup is based on the connection of a three-dimensional smectic film sandwiched between a one-dimensional edge dislocation on its lower surface, and a two-dimensional surface polarization spread over the upper surface. Applying an electric field produces a torque that balances the anchoring torque of the dislocation. The film distortion that results is measured under a polarizing microscope. Exact calculations on these data, anchoring torque versus director angle, yield the anchoring properties of the dislocation. A specificity of our sandwich configuration is to improve the measurement quality by a factor of N^{3/2}∼600, where N=72 is the number of smectic layers in the film. We fit a second-order Fourier series on the torque-anchoring angle data, which has the advantage of converging uniformly over the entire anchoring angle range, i.e., over more than 70^{∘}. The two corresponding Fourier coefficients, k_{a1}^{F2} and k_{a2}^{F2}, are anchoring parameters that generalize the usual anchoring coefficient. When changing the electric field E, the anchoring state evolves along paths in a torque-anchoring angle diagram. Two cases occur depending on the angle α_{∞} of E relative to the unit vector S, perpendicular to the dislocation and parallel to the film. When α_{∞}<130^{∘}, the operating point Q that represents the anchoring state in the diagram follows reversible and "at-equilibrium" paths. Its free displacement velocity is infinitely slow, so that we have to push it with electric torque steps smaller than the experimental error bar δΓ∼10^{-14}N. On the other hand, for α_{∞}>130^{∘}, Q describes a hysteresis loop similar to the usually encountered ones in solids. This loop connects two states that exhibit broken and nonbroken anchorings, respectively. The paths that join them in an out-of-equilibrium process are irreversible and dissipative. When coming back to a nonbroken anchoring state, both the dislocation and smectic film spontaneously heal back in the very same state they were before the anchoring broke. The process does not produce any erosion thanks to their liquid nature, including at the microscopic scale. The energy that is dissipated on these paths is roughly estimated in terms of the c-director rotational viscosity. Similarly, we can evaluate the maximum time of flight along the dissipative paths to be of the order of a few seconds, which is consistent with qualitative observations. In contrast, the paths located inside each domain of these anchoring states are reversible and can be followed in an "at equilibrium" manner all along. This analysis should provide a basis for understanding the structure of multiple edge dislocations in terms of parallel simple edge dislocations interacting with each other through pseudo-Casimir forces arising from c-director thermodynamic fluctuations between them.

Casimir-like interactions and surface anchoring duality in bookshelf geometry of smectic-A liquid crystals

We analyze the transverse intersubstrate Casimir-like force, arising as a result of thermal fluctuations of the liquid crystalline layers of a smectic-A film confined between two planar substrates in a bookshelf geometry, in which the equidistant smectic layers are placed perpendicular to the bounding surfaces. We discuss the variation of the interaction force as a function of the intersubstrate separation in the presence of surface anchoring to the substrates, showing that the force induced by confined fluctuations is attractive and depends on the penetration length as well as the layer spacing. The strongest effect occurs for tightly confined fluctuations, in which the surface anchoring energy is set to infinity, where the force per area scales linearly with the thermal energy and inversely with the fourth power of the intersubstrate separation. By reducing the strength of the surface anchoring energy, the force first becomes weaker in magnitude but then increases in magnitude as the surface anchoring strength is further reduced down to zero, in which case the force tends to that obtained in the limit of strong anchoring.

Fluctuation-induced forces in nematics with a foreign anisotropy in the bulk

Within a linear coupling between orientational order of nematic liquid crystal and anisotropic mesoscopic particles immeresed in the nematic, the pseudo-Casimir effect is investigated. A quenched disorder in the alignment of the particles, which is on average in the direction of the nematic director, induces an inter-substrate force as this composite is confined by two flat parallel surfaces a distance d apart. The disorder-induced force decays as -d ^-1 in the weak coupling regime. The force magnitude increases with the variance of the disorder and decreases on increasing the correlation length of the disorder. If the disorder is considered to be annealed, the disorder effects are not decoupled from the thermal effects and thus the form of the nematic fluctuation-induced force does not alter. The force is affected by the disorder only through a re-normalization of the mean particles' pinning strength. The trend for this modified thermal-induced force with respect to the variance and the correlation length of the disorder remains as in the quenched case, where the pseudo-Casimir force was decomposed into two distinct thermal- and disorder-induced components.

Pseudo-Casimir forces in nematics with disorders in the bulk

A nematic liquid-crystalline slab is considered in which some rod-like particles are randomly distributed. The particles are locally elongated either homeotropic or planar with respect to the confining substrates of the cell. We consider thermal fluctuations of a nematic director which is aligned perpendicular to the confining substrates due to strong homeotropic anchoring at the substrates. The resulting fluctuation-induced force across the cell is analyzed for an annealed disorder in the anchoring of the nematic director at the dispersed mesoscopic particles. Within the saddle-point approximation to free energy of the system, the effect of the disorder is renormalization of the strength of the mean anchoring which is assumed to be homeotropic. By increasing the variance of the disorder, the modes become less massive and deviations from the mean behavior become larger, so that the disorder-free universal long-range attraction, due to the soft modes, is approached.

Fluctuation-induced interactions in nematics with disordered anchoring energy

We examine fluctuation-induced (pseudo-Casimir) interactions in nematic liquid-crystalline films confined between two surfaces, where one of the surfaces imposes a strong homeotropic anchoring (ensuring a uniform mean director profile), while the other one is assumed to be a chemically disordered substrate exhibiting an annealed distribution of anchoring energies. We employ a saddle-point approximation to evaluate the free energy of interaction mediated between the two surfaces and investigate how the interaction force is influenced by the presence of disordered surface anchoring energy. It is shown that the disorder results in a renormalization of the effective surface anchoring parameter in a way that it leads to quantitative and qualitative changes (including a change of sign at intermediate inter-surface separations) in the pseudo-Casimir interaction force when compared with the interaction force in the absence of disorder.

Pseudo-Casimir interactions across nematic films with disordered anchoring axis

We study the effective pseudo-Casimir interaction forces mediated by a nematic liquid-crystalline film bounded by two planar surfaces, one of which imposes a random (disordered) distribution of the preferred anchoring axis in the so-called easy direction. We consider both the case of a quenched as well as an annealed disorder for the easy direction on the disordered surface and analyze the resultant fluctuation-induced interaction between the surfaces. In the case of quenched disorder, we show that the disorder effects appear additively in the total interaction and are dominant at intermediate inter-surface separations. Disorder effects are shown to be unimportant at both very small and very large separations. In the case of annealed disorder its effects are non-additive in the total inter-surface interaction and can be rationalized in terms of a renormalized extrapolation length.

Surface-induced weak orientational order and role of isotropic-nematic interface fluctuations in the appearance of an induced nematic film

Recently the nontrivial spatial and temperature dependence of the surface-induced weak planar orientational order parameter Q(z, T) was determined just above the isotropic-nematic (IN) phase transition point (Ji-H. Lee et al., Phys. Rev. Lett. 102, 167801 (2009)). In this paper we present a theoretical explanation of the observed behaviour. We obtain expressions for the short-range and long-range contributions to the interface potential of the induced nematic film and specify the repulsive character of the interaction between the soft IN interface and the external bounding substrate. It is shown that the small value of the IN interfacial tension results in the renormalization of the repulsive interaction potential due to the thermal fluctuations of the soft IN interface. This leads to an increase of the equilibrium thickness of the induced nematic film and the appearance of a step-like orientational order parameter profile. We find that only renormalized short-range and thermal pseudo-Casimir interactions are essential for the appearance of the induced nematic film, which provide the observed thickness, h ~ 30 nm, of this film. The long-range van der Waals interaction is shown to be negligibly small and the dominant role is played by the renormalized short-range repulsion. Fitting of the experimental order parameter profiles (Ji-H. Lee et al. (2009)) with the expressions based on these interactions makes it possible to determine the material parameters of the system, including the amplitudes of the surface interaction, the IN interfacial tension and the interfacial coherence length. The agreement between theory and experiment confirms the importance of the interface fluctuation renormalization of the interface potentials for soft interfaces.

Competition between capillarity, layering and biaxiality in a confined liquid crystal

The effect of confinement on the phase behaviour and structure of fluids made of biaxial hard particles (cuboids) is examined theoretically by means of Onsager second-order virial theory in the limit where the long particle axes are frozen in a mutually parallel configuration. Confinement is induced by two parallel planar hard walls (slit-pore geometry), with particle long axes perpendicular to the walls (perfect homeotropic anchoring). In bulk, a continuous nematic-to-smectic transition takes place, while shape anisotropy in the (rectangular) particle cross-section induces biaxial ordering. As a consequence, four bulk phases, uniaxial and biaxial nematic and smectic phases, can be stabilised as the cross-sectional aspect ratio is varied. On confining the fluid, the nematic-to-smectic transition is suppressed, and either uniaxial or biaxial phases, separated by a continuous transition, can be present. Smectic ordering develops continuously from the walls for increasing particle concentration (in agreement with the supression of nematic-smectic second-order transition at confinement), but first-order layering transitions, involving structures with n and n + 1 layers, arise in the confined fluid at high concentration. Competition between layering and uniaxial-biaxial ordering leads to three different types of layering transitions, at which the two coexisting structures can be both uniaxial, one uniaxial and another biaxial, or both biaxial. Also, the interplay between molecular biaxiality and wall interactions is very subtle: while the hard wall disfavours the formation of the biaxial phase, biaxiality is against the layering transitions, as we have shown by comparing the confined phase behaviour of cylinders and cuboids. The predictive power of Onsager theory is checked and confirmed by performing some calculations based on fundamental-measure theory.

Optical-ellipsometric study of the nematic-to-smectic transition in 8CB films adsorbed on silicon

The nematic-to-smectic-A (NA) transition in 8CB (4-octyl-4'-cyanobiphenyl) is especially interesting because experimentally, it has been observed to be second order, but theoretically, it has been predicted that it must have a latent heat. The effect on the NA transition due to confinement in an adsorbed film has hitherto not been investigated. Previous study of adsorbed 8CB films on silicon for coverages less than 100 nm showed the existence of a broad coexistence region, identified by the formation of thick and thin islands on the surface that extends between the bulk NA and the isotropic-to-nematic transition temperatures. In this paper, optical and ellipsometric measurements of 8CB films as a function of temperature are used to identify the location of the NA transition in the film in relation to the coexistence region. The NA transition temperature in the film is found to occur at 32.2+/-0.4 degrees C independent of film thickness for films between 62 to 270 nm thick, based on the decrease in the film anisotropy. This decrease in the anisotropy is found to be surprisingly abrupt. For thicknesses below 62 nm, the NA transition line is joined to the thin-thick coexistence region found previously.

Simulation and theory of hybrid aligned liquid crystal films

We present a study of the effects of nanoconfinement on a system of hard Gaussian overlap particles interacting with planar substrates through the hard-needle-wall potential, extending earlier work by two of us [D. J. Cleaver and P. I. C. Teixeira, Chem. Phys. Lett. 338, 1 (2001)]. Here, we consider the case of hybrid films, where one of the substrates induces strongly homeotropic anchoring, while the other favors either weakly homeotropic or planar anchoring. These systems are investigated using both Monte Carlo simulation and density-functional theory, the latter implemented at the level of Onsager's second-virial approximation with Parsons-Lee rescaling. The orientational structure is found to change either continuously or discontinuously depending on substrate separation, in agreement with earlier predictions by others. The theory is seen to perform well in spite of its simplicity, predicting the positional and orientational structure seen in simulations even for small particle elongations.

Thin-thick coexistence behavior of 8CB liquid crystalline films on silicon

The wetting behavior of thin films of 4-n-octyl-4'-cyanobiphenyl (8CB) on Si is investigated via optical and x-ray reflectivity measurement. An experimental phase diagram is obtained showing a broad thick-thin coexistence region spanning the bulk isotropic-to-nematic (T(IN)) and the nematic-to-smectic-A (T(NA)) temperatures. For Si surfaces with coverages between 47 and 72 +/- 3 nm, reentrant wetting behavior is observed twice as we increase the temperature, with separate coexistence behaviors near T(IN) and T(NA). For coverages less than 47 nm, however, the two coexistence behaviors merge into a single coexistence region. The observed thin-thick coexistence near the second-order NA transition is not anticipated by any previous theory or experiment. Nevertheless, the behavior of the thin and thick phases within the coexistence regions is consistent with this being an equilibrium phenomenon.

Lateral and normal forces between patterned substrates induced by nematic fluctuations

We consider a nematic liquid crystal confined by two parallel flat substrates whose anchoring conditions vary periodically in one lateral direction. Within the Gaussian approximation, we study the effective forces between the patterned substrates induced by the thermal fluctuations of the nematic director. The shear force oscillates as a function of the lateral shift between the patterns on the lower and the upper substrates. We compare the strength of this fluctuation-induced lateral force with the lateral van der Waals force arising from chemically structured adsorbed monolayers. The fluctuation-induced force in the normal direction is either repulsive or attractive, depending on the model parameters.

Temperature dependence of the Casimir-like force in free-standing smectic films

The thermal Casimir-like force in free-standing liquid crystal films close to the smectic-A-nematic transition temperature is computed using a quadratic functional approach. In the framework of a microscopic mean-field model of free-standing smectic-A films, the temperature dependence of the order parameter profiles is computed and later used to estimate the elastic coupling variability in the vicinity of first- and second-order bulk smectic-A-nematic phase transitions. The strong nonuniformity of the coupling constant profiles promotes a significant increase of the fluctuation-induced force over three orders of magnitude, especially in thin films. This result reinforces the possible predominance of the thermal Casimir force as compared to the standard van der Waals interaction in thin smectic-A liquid crystal films.

Casimir interaction in smectic-A liquid crystals caused by coupled fluctuations of positional and orientational order

A theoretical study of the Casimir interaction in smectic-A systems, considering fluctuations of both types of smectic ordering--positional and orientational--including the coupling between them, is presented. Two model systems with plan-parallel geometry are studied: homeotropic cell and free-standing film. At large thicknesses of the system the behavior of the Casimir force is found to be primarily determined by positional fluctuations, whereas at small thicknesses also the orientational degrees of freedom greatly contribute to the interaction. The influence of different coupling strengths between orientational and positional order is presented. The dependence of the Casimir force on the director anchoring and surface-tension parameters is studied. The possibilities of experimental detection of the interaction are discussed.

Twist of cholesteric liquid crystal cells: stability of helical structures and anchoring energy effects

We consider helical configurations of a cholesteric liquid crystal (CLC) sandwiched between two substrates with homogeneous director orientation favored at both confining plates. We study the CLC twist wave number q characterizing the helical structures in relation to the free twisting number q(0) which determines the equilibrium value of CLC pitch P(0) = 2 pi/ q(0) . We investigate the instability mechanism underlying transitions between helical structures with different spiral half-turn numbers. Stability analysis shows that for equal finite anchoring strengths this mechanism can be dominated by in-plane director fluctuations. In this case the metastable helical configurations are separated by the energy barriers and the transitions can be described as the director slippage through these barriers. We extend our analysis to the case of an asymmetric CLC cell in which the anchoring strengths at the two substrates are different. The asymmetry introduces two qualitatively different effects: (a) the intervals of twist wave numbers representing locally stable configurations with adjacent helix half-turn numbers are now separated by instability gaps; and (b) sufficiently large asymmetry, when the difference between azimuthal anchoring extrapolation lengths exceeds the thickness of the cell, will suppress the jumplike behavior of the twist wave number.

Field-driven crossover from attractive-to-repulsive Casimir-like force in smectic films

External fields have a profound effect on the fluctuations of strongly correlated fluids, such as a liquid crystal. Within a harmonic functional integral approach, we compute the fluctuation-induced force between the surfaces of a smectic liquid-crystal film under the presence of an ordering field. In particular, we show that for asymmetrically anchored films, the thermal Casimir interaction energy can be collapsed into a universal form crossing over from a repulsive to an attractive interaction as the film thickness is increased. We discuss the possible relevance of this field effect in nematic-smectic wetting transitions.

Liquid-crystalline Casimir effect in the presence of a patterned substrate

We consider a nematic liquid crystal confined by two parallel planar interfaces, one being laterally homogeneous and the other provided by a substrate endowed with a periodic chemical stripe pattern. Based on continuum theory we analyze the influence of the lateral pattern on the liquid-crystalline Casimir force acting on the interfaces of the nematic cell at distance d due to the thermal fluctuations of the nematic director. For d much larger than the pattern periodicity, the influence of the patterned substrate can be described by a homogeneous, effective anchoring energy. By tuning this parameter we recover previous results for the liquid-crystalline Casimir force. For the general case, i.e., smaller separations, we present numerical results.

Pseudo-Casimir force in chiral smectic liquid crystals

We present a theoretical study of the pseudo-Casimir force in two chiral smectic systems: a homeotropic cell and a free-standing film. We consider the interaction induced by the fluctuations of orientational order. We demonstrate how the character of the force depends on the type of fluctuation modes and on boundary conditions. We focus on the temperature dependence of the force, which is marked by the vicinity of the smectic-A*-->smectic-C* phase transition. We find that at this transition the force diverges if the system is frustrated; otherwise it remains finite. We expose the analogy between the force in these smectic systems and in previously studied nematic systems, thus demonstrating the universality of the pseudo-Casimir interaction.

Order reconstruction in frustrated nematic twist cells

Within the Landau-de Gennes theory of liquid crystals, we study the equilibrium configurations of a nematic cell with twist boundary conditions. Under the assumption that the order tensor Q be uniaxial on both bounding plates, we find three separate classes of solutions, one of which contains the absolute energy minimizer, a twistlike solution that exists for all values of the distance d between the plates. The solutions in the remaining two classes exist only if d exceeds a critical value d(c). One class consists of metastable, twistlike solutions, while the other consists of unstable, exchangelike solutions, where the eigenvalues of Q are exchanged across the cell. When d=d(c), the metastable solution relaxes back to the absolute energy minimizer, undergoing an order reconstruction somewhere within the cell. The critical distance d(c) equals, in general, a few biaxial coherence lengths. This scenario applies to all the values of the boundary twist but pi/2, which thus appears as a very special case, though it is the one more studied in the literature. In fact, when the directors prescribed on the two plates are at right angles, two symmetric twistlike solutions merge continuously into an exchangelike solution at the critical value of d where the latter becomes unstable. Our analysis shows how the classical bifurcation associated with this phenomenon is unfolded by perturbing the boundary conditions.

Surface effects on the amplitude of fluctuation-induced interactions in smectic films

Within a quadratic functional integral approach, we investigate the role played by surface terms in the fluctuation-induced surface-surface interaction of free standing smectic liquid crystals. We show that the typical 1/l decay of the Casimir-type contribution to the free energy of a film with thickness l is replaced by a faster 1/l3 decay at a characteristic surface tension. An intermediate 1/l2 decay can also take place for specific surface parameters with unlike boundary conditions. In all the investigated cases, a repulsive long-range force appears only for mixed boundary conditions with strong anchoring at one surface and weak anchoring at the opposite one. Further, the amplitude of the thermal Casimir energy, besides being influenced by the applied surface tension, depicts a nonmonotonic dependence on the coupling between the outermost film layers, reflecting a crossover between strong and weak anchoring regimes.

Pseudo-Casimir effect in untwisted chiral nematic liquid crystals

We investigate theoretically the pseudo-Casimir force between parallel plates immersed in a chiral nematic liquid crystal. We focus on small-separation limit where the director configuration between the plates inducing strong homeotropic anchoring is uniform. We find that the force is attractive at separations smaller than the crossover distance and repulsive otherwise, and that it diverges logarithmically at the critical distance where the uniform structure is replaced by a distorted structure. We also analyze modifications to the force introduced by magnetic field and comment on the possible detection of the effect.

Casimir effect in fluids above the isotropic-lamellar transition

We study fluctuation-induced interaction in confined fluids above the isotropic-lamellar transition. At an ideally continuous transition, the disjoining pressure has the asymptotic form Pi(d-->infinity) approximately -Ck(B)Tq(2)(0)/d, where d is the interwall distance, q(0) is the wave number of the scattering peak, and C = 1/4pi in the strong anchoring limit. The long-rangedness is enhanced due to continuous distribution of soft modes in the q space. An unconventionally strong Casimir force with a range of several lamella thicknesses is realistic above the transition. We also find an oscillatory force profile near a surface-induced transition.

Casimir torques between anisotropic boundaries in nematic liquid crystals

Fluctuation-induced interactions between anisotropic objects immersed in a nematic liquid crystal are shown to depend on the relative orientation of these objects. The resulting long-range "Casimir" torques are explicitly calculated for a simple geometry where elastic effects are absent. Our study generalizes previous discussions restricted to the case of isotropic walls, and leads to new proposals for experimental tests of Casimir forces and torques in nematics.