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.

Nonlocal Dielectric Response of Water in Nanoconfinement

Recent experiments reporting a very low dielectric permittivity for nanoconfined water have renewed the interest in the structure and dielectric properties of water in narrow gaps. Here, we describe such systems with a minimal Landau-Ginzburg field theory composed of a nonlocal bulk-determined term and a local water-surface interaction term. We show how the interplay between the boundary conditions and intrinsic bulk correlations encodes the dielectric properties of confined water. Our theoretical analysis is supported by molecular dynamics simulations and comparison with the experimental data.

Surface alignment disorder and thermal Casimir forces in smectic-A liquid crystalline films

Disorder components in surface anchoring orientation of a smectic-A liquid crystalline film (occurring, e.g., due to surface contamination sources) modify the thermal pseudo-Casimir interaction mediated between the bounding surfaces of the film. By considering a plane-parallel slab with bounding surfaces positioned normal to the layering field of the film, we study the anchoring disorder effects by assuming that the disorder source is present on one of the two substrates, producing a Gaussian-weighted distribution for the preferred molecular anchoring orientation (easy axis) on that substrate, with a finite mean and variance or, more generally, with a homogeneous in-plane, two-point correlation function. We show that the presence of disorder, either of quenched or annealed type, leads to a significant reduction in the magnitude of the net pseudo-Casimir force between the confining substrates of the film. This force can be attractive or repulsive depending on the boundary conditions. In the quenched case, the interaction force reduction is a direct consequence of an additive free energy term dependent on the variance of the disorder, while in the annealed case, the suppression of the interaction force can be understood based on a disorder-renormalized, effective anchoring strength. We predict a regime of behavior, exhibiting non-monotonic dependence for the interaction pressure as a function of separation. We also show that, by increasing the disorder variance, the interaction pressure changes sign and becomes a monotonically decreasing function of the separation between the substrates.

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.

Macroscopic Domains within an Oriented TQ1 Film Visualized Using 2D Polarization Imaging

Large-area self-assembly of functional conjugated polymers holds a great potential for practical applications of organic electronic devices. We obtained well-aligned films of poly[2,3-bis(3-octyloxyphenyl)quinoxaline-5,8-diyl-alt-thiophene-2,5-diyl] (TQ1) using the floating film transfer method. Thereby, a droplet of the TQ1 solution was injected on top of the surface of an immiscible liquid substrate, at the meniscus formed at the edge of a Petri dish, from where the polymer solution and the film spread in one direction. Characterization of the TQ1 film using the recently developed two-dimensional polarization imaging (2D POLIM) revealed large, millimeter-sized domains of oriented polymer chains. The irregular shape of the contact line at the droplet source induced the appearance of disordered stripes perpendicular to the spreading direction. A correlation of polarization parameters measured using 2D POLIM revealed the microstructure of such stripes, providing valuable information for further improvement and possible upscaling of this promising method.

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.

Fluorescence correlation spectroscopy in thin films at reflecting substrates as a means to study nanoscale structure and dynamics at soft-matter interfaces

Structure and dynamics at soft-matter interfaces play an important role in nature and technical applications. Optical single-molecule investigations are noninvasive and capable to reveal heterogeneities at the nanoscale. In this work we develop an autocorrelation function (ACF) approach to retrieve tracer diffusion parameters obtained from fluorescence correlation spectroscopy (FCS) experiments in thin liquid films at reflecting substrates. This approach then is used to investigate structure and dynamics in 100-nm-thick 8CB liquid crystal films on silicon wafers with five different oxide thicknesses. We find a different extension of the structural reorientation of 8CB at the solid-liquid interface for thin and for thick oxide. For the thin oxides, the perylenediimide tracer diffusion dynamics in general agrees with the hydrodynamic modeling using no-slip boundary conditions with only a small deviation close to the substrate, while a considerably stronger decrease of the interfacial tracer diffusion is found for the thick oxides.

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.

Dynamical approach to the Casimir effect

Casimir forces can appear between intrusions placed in different media driven by several fluctuation mechanisms, either in equilibrium or out of it. Herein, we develop a general formalism to obtain such forces from the dynamical equations of the fluctuating medium, the statistical properties of the driving noise, and the boundary conditions of the intrusions (which simulate the interaction between the intrusions and the medium). As a result, an explicit formula for the Casimir force over the intrusions is derived. This formalism contains the thermal Casimir effect as a particular limit and generalizes the study of the Casimir effect to such systems through their dynamical equations, with no appeal to their Hamiltonian, if any exists. In particular, we study the Casimir force between two infinite parallel plates with Dirichlet or Neumann boundary conditions, immersed in several media with finite correlation lengths (reaction-diffusion system, liquid crystals, and two coupled fields with non-Hermitian evolution equations). The driving Gaussian noises have vanishing or finite spatial or temporal correlation lengths; in the first case, equilibrium is reobtained and finite correlations produce nonequilibrium dynamics. The results obtained show that, generally, nonequilibrium dynamics leads to Casimir forces, whereas Casimir forces are obtained in equilibrium dynamics if the stress tensor is anisotropic.

Violation of the action-reaction principle and self-forces induced by nonequilibrium fluctuations

We show that the extension of Casimir-like forces to fluctuating fluids driven out of equilibrium can exhibit two interrelated phenomena forbidden at equilibrium: self-forces can be induced on single asymmetric objects and the action-reaction principle between two objects can be violated. These effects originate in asymmetric restrictions imposed by the objects' boundaries on the fluid's fluctuations. They are not ruled out by the second law of thermodynamics since the fluid is in a nonequilibrium state. Considering a simple reaction-diffusion model for the fluid, we explicitly calculate the self-force induced on a deformed circle. We also show that the action-reaction principle does not apply for the internal Casimir forces between a circle and a plate. Their sum, instead of vanishing, provides the self-force on the circle-plate assembly.

Generalized Casimir forces in nonequilibrium systems

In the present work, we propose a method to determine fluctuation-induced forces in nonequilibrium systems. These forces are the analog of the well-known Casimir forces, which were originally introduced in quantum field theory and later extended to the area of critical phenomena. The procedure starts from the observation that many nonequilibrium systems exhibit fluctuations with macroscopic correlation lengths, and the associated structure factors strongly depend on the wave vectors for long wavelengths; in some cases the correlations become long range, and the structure factors show algebraic divergences in the long-wavelength limit. The introduction of external bodies into such systems in general modifies the spectrum of these fluctuations, changing the value of the renormalized pressure, which becomes inhomogeneous. This inhomogeneous pressure leads to the appearance of a net force between the external bodies. It is shown that the force can be obtained from the knowledge of the structure factor of the homogeneous system. The mechanism is illustrated by means of a simple example: a reaction-diffusion equation, where the correlation function has a characteristic length. The role of this length in the Casimir force is elucidated.

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.

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.

Structural Forces near Phase Transitions of Liquid Crystals

The structure of the fragile liquid-crystalline phases has a strong impact on the forces between bodies immersed in a liquid crystal (LC). We have equipped an atomic force microscope with a precise temperature control and measured various liquid-crystalline structural forces at temperatures close to the phase transitions. The observed forces agree well with predictions of Landau--de Gennes phenomenological theory of LCs, even at a nanoscale length. In addition to this, we have observed a molecular layer, adsorbed on the surfactant-covered glass surface, and determined its thickness and elastic properties.