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

On Casimir and Helmholtz Fluctuation-Induced Forces in Micro- and Nano-Systems: Survey of Some Basic Results

Fluctuations are omnipresent; they exist in any matter, due either to its quantum nature or to its nonzero temperature. In the current review, we briefly cover the quantum electrodynamic Casimir (QED) force as well as the critical Casimir (CC) and Helmholtz (HF) forces. In the QED case, the medium is usually a vacuum and the massless excitations are photons, while in the CC and HF cases the medium is usually a critical or correlated fluid and the fluctuations of the order parameter are the cause of the force between the macroscopic or mesoscopic bodies immersed in it. We discuss the importance of the presented results for nanotechnology, especially for devising and assembling micro- or nano-scale systems. Several important problems for nanotechnology following from the currently available experimental findings are spelled out, and possible strategies for overcoming them are sketched. Regarding the example of HF, we explicitly demonstrate that when a given integral quantity characterizing the fluid is conserved, it has an essential influence on the behavior of the corresponding fluctuation-induced force.

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.

Manipulation and amplification of the Casimir force through surface fields using helicity

We present both exact and numerical results for the behavior of the Casimir force in O(n) systems with a finite extension L in one direction when the system is subjected to surface fields that induce helicity in the order parameter. We show that for such systems, the Casimir force in certain temperature ranges is of the order of L^{-2}, both above and below the critical temperature, T_{c}, of the bulk system. An example of such a system would be one with chemically modulated bounding surfaces, in which the modulation couples directly to the system's order parameter. We demonstrate that, depending on the parameters of the system, the Casimir force can be either attractive or repulsive. The exact calculations presented are for the one-dimensional XY and Heisenberg models under twisted boundary conditions resulting from finite surface fields that differ in direction by a specified angle, and the three-dimensional Gaussian model with surface fields in the form of plane waves that are shifted in phase with respect to each other. Additionally, we present exact and numerical results for the mean-field version of the three-dimensional O(2) model with finite surface fields on the bounding surfaces. We find that all significant results are consistent with the expectations of finite-size scaling.

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.

Line contribution to the critical Casimir force between a homogeneous and a chemically stepped surface

Recent experimental realizations of the critical Casimir effect have been implemented by monitoring colloidal particles immersed in a binary liquid mixture near demixing and exposed to a chemically structured substrate. In particular, critical Casimir forces have been measured for surfaces consisting of stripes with periodically alternating adsorption preferences, forming chemical steps between them. Motivated by these experiments, we analyze the contribution of such chemical steps to the critical Casimir force for the film geometry and within the Ising universality class. By means of Monte Carlo simulations, mean-field theory and finite-size scaling analysis we determine the universal scaling function associated with the contribution to the critical Casimir force due to individual, isolated chemical steps facing a surface with homogeneous adsorption preference or with Dirichlet boundary condition. In line with previous findings, these results allow one to compute the critical Casimir force for the film geometry and in the presence of arbitrarily shaped, but wide stripes. In this latter limit the force decomposes into a sum of the contributions due to the two homogeneous parts of the surface and due to the chemical steps between the stripes. We assess this decomposition by comparing the resulting sum with actual simulation data for the critical Casimir force in the presence of a chemically striped substrate.

Critical Casimir force in the presence of random local adsorption preference

We study the critical Casimir force for a film geometry in the Ising universality class. We employ a homogeneous adsorption preference on one of the confining surfaces, while the opposing surface exhibits quenched random disorder, leading to a random local adsorption preference. Disorder is characterized by a parameter p, which measures, on average, the portion of the surface that prefers one component, so that p=0,1 correspond to homogeneous adsorption preference. By means of Monte Carlo simulations of an improved Hamiltonian and finite-size scaling analysis, we determine the critical Casimir force. We show that by tuning the disorder parameter p, the system exhibits a crossover between an attractive and a repulsive force. At p=1/2, disorder allows to effectively realize Dirichlet boundary conditions, which are generically not accessible in classical fluids. Our results are relevant for the experimental realizations of the critical Casimir force in binary liquid mixtures.

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.

Critical Casimir forces between homogeneous and chemically striped surfaces

Recent experiments have measured the critical Casimir force acting on a colloid immersed in a binary liquid mixture near its continuous demixing phase transition and exposed to a chemically structured substrate. Motivated by these experiments, we study the critical behavior of a system, which belongs to the Ising universality class, for the film geometry with one planar wall chemically striped, such that there is a laterally alternating adsorption preference for the two species of the binary liquid mixture, which is implemented by surface fields. For the opposite wall we employ alternatively a homogeneous adsorption preference or homogeneous Dirichlet boundary conditions, which within a lattice model are realized by open boundary conditions. By means of mean-field theory, Monte Carlo simulations, and finite-size scaling analysis we determine the critical Casimir force acting on the two parallel walls and its corresponding universal scaling function. We show that in the limit of stripe widths small compared with the film thickness, on the striped surface the system effectively realizes Dirichlet boundary conditions, which generically do not hold for actual fluids. Moreover, the critical Casimir force is found to be attractive or repulsive, depending on the width of the stripes of the chemically patterned surface and on the boundary condition applied to the opposing surface.

Nematic liquid crystal in the wedge and edge geometry in the case of homeotropic alignment

Nematic liquid crystal confined to a wedge or edge is studied on the assumption that the confining surfaces provide strong and weak homeotropic anchorings, respectively. Both infinite and finite systems are considered. The model based on the Frank-Oseen and Rapini-Papoular formalisms predicts two textures of opposite rotations of the director as in the case of strong anchoring on both surfaces. However, the presence of weak anchoring results in a length scale lambda which characterizes the crossover between the regions close to the apex and far from it. The ratio lambda/b , where b is the extrapolation length, is a function of the opening angle alpha. Both stable and metastable textures are considered and the mechanism by which a texture loses its stability is found. It is related to the formation of a defect-like structure at the surface of weak anchoring whose distance from the apex is lambda(alpha) and the loss of stability is signalled by the divergence of lambda. Only in the limit alpha --> 2tau, the defect-like structure transforms into a defect of strength -1/2 located at a finite distance from the apex.

Interplay of critical Casimir and dispersion forces

Using general scaling arguments combined with mean-field theory we investigate the critical (T approximately Tc) and off-critical (T not equal Tc) behavior of the Casimir forces in fluid films of thickness L governed by dispersion forces and exposed to long-ranged substrate potentials which are taken to be equal on both sides of the film. We study the resulting effective force acting on the confining substrates as a function of T and of the chemical potential mu. We find that the total force is attractive both below and above Tc. If, however, the direct substrate-substrate contribution is subtracted, the force is repulsive everywhere except near the bulk critical point (Tc, mu(c)), where critical density fluctuations arise, or except at low temperatures and (L/a)(beta(Delta)(mu))=O(1), with Delta(mu)=mu-mu(c)<0 and a the characteristic distance between the molecules of the fluid, i.e., in the capillary condensation regime. While near the critical point the maximal amplitude of the attractive force if of order of L(-d) in the capillary condensation regime the force is much stronger with maximal amplitude decaying as L(-1). In the latter regime we observe that the long-ranged tails of the fluid-fluid and the substrate-fluid interactions further increase that amplitude in comparison with systems with short-range interactions only. Although in the critical region the system under consideration asymptotically belongs to the Ising universality class with short-ranged forces, we find deviations from the standard finite-size scaling for xi(ln)(xi/xi0(+/-)) >>L even for xi, L>>xi0(+/-), where xi[t=(T-Tc)/Tc-->+/-0,Delta(mu)=0]=xi0(+/-)/t/-nu, is the bulk correlation length. In this regime the dominant finite-size contributions to the free energy and to the force stem from the long-ranged algebraically decaying tails of the interactions; they are not exponentially small in L, as it is the case there in systems governed by purely short-ranged interactions, but exhibit a power law decay in L. Essential deviations from the standard finite-size scaling behavior are observed also within the finite-size critical region L/xi=O(1) for films with thicknesses L less than or approximately equal Lcrit, where Lcrit=xi0(+/-)(16/s/)nu/beta, with nu and beta as the standard bulk critical exponents and with s=O(1) as the dimensionless parameter that characterizes the relative strength of the long-ranged tail of the substrate fluid over the fluid-fluid interaction. We present the modified finite-size scaling pertinent for such a case and analyze in detail the finite-size behavior in this region. The standard finite-size scaling behavior is recovered only for L>>Lcrit.

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.

Forces between chemically structured substrates mediated by critical fluids

We consider binary liquid mixtures close to their critical points confined by two parallel, geometrically flat, and chemically structured substrates. Universal order parameter profiles are calculated within mean field theory for periodic patterns of stripes with alternating preferences for the two species of the mixture and with different relative positions of the two substrates. From the order parameter profiles the effective forces between the two plates are derived. The tuning of Casimir amplitudes is discussed.

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.