Statics and dynamics of point boojums, line and modified Saturn ring topological defects in nematic confined geometry.

In this paper we study the static structure and the dynamics of topological defects associated with isotropic droplets in nematic environment. Investigations were made in confined geometry of optical cells when the droplet size was of the order of or larger than the gap of the cell. We observed the coexistence of point boojums and Saturn ring or modified Saturn ring defects. We found transformation of the Saturn ring defect to two localized broad defects at increasing the droplet size. At droplet coalescence antipodes of point and localized broad defects were born and the dynamics of their annihilation with existing defects was investigated. We found strong difference in the process of annihilation of point and localized broad defects. Microscope images of isotropic droplets in nematic environment in a planar cell. The director orientation far from the droplets is in horizontal direction. The photographs were taken with crossed vertical and horizontal polarizers (a) and with a single horizontal polarizer (b). The cell thickness is 100 µm. Droplet diameter is less than the cell thickness. 1 and 2 are point boojums, L is the Saturn ring defect.

Quasi-two-dimensional coalescence of nematic and isotropic droplets and Rayleigh-Plateau instability in flat optical cells.

We investigated the coalescence of nematic droplets in an isotropic environment and that of isotropic droplets in a nematic environment in quasi-two-dimensional geometry of a flat optical cell. Two different regimes of coalescence were found. In the circular meniscus between the nematic and isotropic regions both nematic and isotropic phases exist. As a result, two bridges form at coalescence: a nematic and an isotropic bridge. In this work, we focus on the situation when nematic wets the cell surface. The coalescence of nematic droplets starts near the cell surfaces where the droplet bridge from the nematic phase is formed. An outer bridge connecting the isotropic environment is localized in the middle of the cell. When the outer bridge gets thinner it becomes unstable and breaks up. A series of pinch-offs leads to the formation of satellite droplets. On the contrary, when isotropic droplets coalesce, the coalescence starts in the middle of the cell and breaking of the bridges occurs without instability and without the formation of satellite droplets. Breakup of the outer bridge is a new example of Rayleigh-Plateau instability in addition to actively studied transformation and breaking of filaments and stretched droplets.

Dynamics of island-meniscus coalescence in free-standing smectic films.

In free-standing smectic films islands (regions of larger thickness than the film) can be considered as two-dimensional analogues of liquid droplets in a three-dimensional medium. The dynamics of droplet coalescence is an important but up to now incompletely solved problem in non-equilibrium mechanics. Here, we report on our investigations of island coalescence with the film meniscus. This phenomenon is analogous to the coalescence of a 3D droplet with a flat liquid surface. We found that the time evolution of island dimension is described by universal power-law dependencies for different stages of coalescence. Limited agreement with existing theory was found. In particular, in the final stage of coalescence the domain dynamics differs from theoretical predictions.

Structures in the meniscus of smectic membranes: the role of dislocations?

We report an experimental investigation of the structure of periodic patterns observed in the meniscus of free-standing smectic films. Combination of polarizing optical microscopy and phase shifting interferometry enabled us to obtain new information on the structure of the meniscus, and in particular, on the topography of the smectic-air interface. We investigate the profile of the undulations in the striped structure in the thin part of the meniscus, change of the stripe period with the meniscus thickness and subsequent transition into a two-dimensional structure. It is shown that the two-dimensional structure has an unusual complex profile of "egg-box" type. The striped texture occurs upon cooling from the nontilted smectic-A to the smectic-C* phase, whereas the two-dimensional pattern is present in both phases. We discuss the possible origin of the modulated structures, the role of the dislocations in the meniscus, the elasticity of smectic layers, and the mechanical stress induced by dislocations.

Ordered structures formed by nematic topological defects and their transformation with changing the Euler characteristics.

Ordered chain structures from topological defects of opposite charges ("necklaces" of defects) were prepared and their dynamics and cooperative rearrangement were investigated. We studied topological defects in nematic films with change of the Euler characteristic induced by temperature. Topological defects emerged due to competing surface anchoring on the nematic-isotropic and nematic-solid interfaces. Transformation of the structure with a circular chain from topological defects to the structure with a single defect and then to a structure without defects takes place as the nematic geometry changes. The temporal evolution of the number of topological defects at their annihilation in the chains differs from coarsening in two-dimensional (2D) and 3D geometry.

Dynamics of viscous droplet coalescence in the confined geometry of optical cells.

The dynamics of quasi-two-dimensional coalescence of isotropic droplets in nematic liquid crystal environment was studied. Investigations were made in confined geometry of a Hele-Shaw optical cell with different transverse droplet sizes. The existence of three distinct dynamic regimes was found for coalescence, namely, short-, middle-, and long-time regimes. The fast dynamics of bridge transformation was visualized. At short time the dynamics of droplet transformation is similar to the transformation of free (three-dimensional) droplets. At later stages, two regimes of the coalescence at different timescales are determined by Poiseuille flow. Experimental data are discussed on the basis of existing theories.

Peculiarities of focal conic structure formed near the cholesteric-isotropic phase transition.

Focal conic structure formed near the transition from the cholesteric liquid crystal phase into isotropic liquid is studied in planar cells with a combination of optical methods. Employing a procedure with slow variation of temperature, we manage to obtain regions of periodic focal conic structure with two-dimensional square ordering in the plane of the cell. Using ordered samples and materials with selective reflection in the visible spectral range, we obtain spectral and microscopic optical data which deepen our knowledge of the focal conic structure.

Undulation instabilities in the meniscus of smectic membranes.

Using optical microscopy, phase shifting interferometry, and atomic force microscopy, we characterize the undulated structures which appear in the meniscus of freestanding ferroelectric smectic-C* films. We demonstrate that these periodic structures correspond to undulations of the smectic-air interface. The resulting striped pattern disappears in the untilted smectic-A phase. The modulation amplitude and wavelength of the instability both depend on meniscus thickness. We study the temperature evolution and propose a model that qualitatively accounts for the observations.

Dynamics of capillary coalescence and breakup: Quasi-two-dimensional nematic and isotropic droplets.

We observed the formation of small satellite droplets from the bridge at droplet coalescence. Investigations were made using a Hele-Shaw cell in the two-phase region at the nematic-isotropic phase transition. In previous works on coalescence it was considered that before the start of coalescence there exists a bridge between the outer fluid connecting regions on the two sides of the droplets (outer bridge). After the start of coalescence, a bridge connecting the two droplets appears (droplet bridge) and the outer bridge is broken. We have shown that there are coalescence processes where after the start of coalescence both the droplet bridge and the outer bridge can exist. This cardinally changes the coalescence process. During the first coalescence stage the size of the outer bridge decreases and the size of the droplet bridge increases. During the second stage the outer bridge becomes unstable which leads to pinch-off, formation of pointed end domains, secondary instability, splitting of pointed end domains, and formation of satellite droplets. We found the linear dependence of the minimum bridge radius on time near bridge breakup. This behavior confirms the capillary viscous regime of bridge breakup. Our work connects two areas of fluid dynamics: coalescence and breakup with formation of satellite droplets.

Transient hexagonal structures in sheared emulsions of isotropic inclusions on smectic bubbles in microgravity conditions.

We describe the collective behavior of isotropic droplets dispersed over a spherical smectic bubble, observed under microgravity conditions on the International Space Station (ISS). We find that droplets can form two-dimensional hexagonal structures changing with time. Our analysis indicates the possibility of spatial and temporal periodicity of such structures of droplets. Quantitative analysis of the hexagonal structure including the first three coordination circles was performed. A peculiar periodic-in-time ordering of the droplets, related to one-dimensional motion of droplets with non-uniform velocity, was found.

Coalescence of holes in two-dimensional free-standing smectic films.

We investigate in free-standing smectic films coalescence of holes (circular regions with thickness smaller than the surrounding film). This process can be considered as a two-dimensional analog of coalescence of bubbles in a three-dimensional fluid. A high speed video camera was used to study the evolution of domains at different stages of coalescence. Special attention was given to investigations of the dependence of the size of the bridge between two holes at the initial stage of coalescence, which was considered in numerous theoretical works and bears information on the coalescence mechanism. It is established that the scaling law is applicable for the description of the transformation of bridges for holes of different radius R. We found that in the regime corresponding to the experimental situation the length of the bridge H increases with the scaling law H/R=(t/τ_{R})^{1/2}. The characteristic time τ_{R} determined from the scaling law is larger than the theoretical time, which can be connected with dissipation of energy both in the film and inside the holes.

Coalescence of viscous two-dimensional smectic islands.

Freestanding smectic films give a unique possibility to study two-dimensional coalescence. We report experimental investigations in freestanding films and detailed analysis of coalescence of islands, circular regions of larger thickness than the surrounding film. The driving force of island coalescence is the dislocation tension on the boundary between the island and the film. The obtained experimental results enable one to perform complex analysis of two-dimensional coalescence in Stokes regime and compare it to theoretical predictions. The applicability of scaling arguments for the description of the peculiarities of domain dynamics is demonstrated. The whole process of coalescence is well described by the analytical solution adapted to our case of islands in freestanding smectic films.

Erratum: Coalescence of viscous two-dimensional smectic islands [Phys. Rev. E 99, 062702 (2019)].

This corrects the article DOI: 10.1103/PhysRevE.99.062702.

Field-induced structures and transitions in chiral antiferroelectric liquid crystals.

The transformation of a chiral antiferroelectric liquid crystal in an electric field was calculated. Minimization of the free energy was performed with respect to both the phase and the modulus of the two-component order parameter. Competition between the electric field, which favors a planar structure, and chirality and the anticlinic ordering leads to frustration and sequential formation of a distorted helix, a soliton state, transition to an unwound distorted antiferroelectric, and finally the planar synclinic state. In the soliton and distorted antiferroelectric states an anomalous electroclinic effect (the decrease of the tilt angle and layer polarization) was found. The antiferroelectric soliton can be removed by a large electric field only via the transition to the synclinic state.

Influence of chirality on director configuration and droplet interaction in ferroelectric free-standing films.

Droplets in smectic free-standing films interact due to elastic distortion of the c-director field and formation of topological defects. Our experiments show that chirality of the liquid-crystal medium plays a key role in the interaction between droplets. We find that the configuration of the c-director field near the droplets and the position of the topological defects on the droplet boundary depend on droplet size and film polarity. An intermediate c-director configuration between dipolar and quadrupolar is formed near droplets in smectic films due to the competition between elasticity and chirality. We observed that the distance between droplets in self-organized structures depends on the position of defects on the droplet boundary and significantly changes with respect to that for dipolar droplets. Our results open the way to modify the droplet interaction and the structures formed by droplets.

Rearrangement of topological defects and anchoring on the inclusion boundary in ferroelectric smectic membranes.

We report experiments on a ferroelectric membrane and droplets with tunable surface properties. In smectic membranes the configuration of the c -director field near inclusions may be rearranged drastically with temperature. The transformation of the c -director field results from the competition between the elastic and polar properties of the membranes. We demonstrate that anchoring conditions on the inclusion boundary are not fixed but depend on the temperature. A dipolar c -director configuration near droplets can evolve to a mixed configuration and to a quadrupolar one. These modifications of the c -director field near the inclusions lead to a change of the interaction between the inclusions, their self-organization, and even to the destruction of structures already formed by the inclusions. Our observations open new possibilities for manipulating inclusions and controlling their self-organization.

Orientational action of edge dislocations on the director field in antiferroelectric smectic-C_{A}

We report imaging of the director field near edge dislocations in thermotropic antiferroelecric smectic-C_{A}^{*} (SmC_{A}^{*}) liquid crystal. Measurements were made in freestanding films with thickness from two to ten smectic layers. We find two different orientations of the molecular tilt plane with respect to the edge dislocation line. The orientation is determined by the value of the Burgers vector of the dislocation. Elementary edge dislocation and dislocations with a Burgers vector equal to an odd number of layers orient the tilt plane perpendicular on the two sides of the dislocation. Dislocations with a Burgers vector equal to an even number of layers orient the molecular tilt plane parallel to the dislocation line. Difference in the orientation for an odd Burgers vector can be attributed to breaking of antiferroelectric symmetry by the edge dislocation.

Director configuration and self-organization of inclusions in two-dimensional smectic membranes.

The behavior of isolated inclusions (nematic droplets, smectic islands) and formation of chains and clusters from inclusions in oriented smectic membranes have been studied. Investigations of inclusions were performed in membranes in which the molecular ordering was oriented by an external magnetic field. At planar boundary conditions on the interface between the membrane and inclusions different configurations of the c-director field were observed: Coulombic, dipolar, quadrupolar, and mixed. The observed orientation of inclusions and their interactions and self-organization correlate with the predictions of the theory based on the electromagnetic analogy. Chaining and formation of superstructures differ in oriented and nonoriented membranes.

Two-dimensional hexagonal smectic structure formed by topological defects.

A two-dimensional hexagonal smectic structure formed by point topological defects and intersecting defect walls was discovered. This unique structure was predicted theoretically about 30 years ago but not observed. For a long time the hexagonal structure was a challenge for experimentalists. A different type of self-organization in smectic films was found and used to form the hexagonal structure. Methods applied for building the hexagonal phase can be used for the formation of complicated liquid-crystal structures.

Interaction of surfaces in smectic membranes and their instability near thinning transitions.

We report measurements of the interaction between surfaces of the presmectic membrane above the temperature of transition to the phase without layer ordering. Investigations were performed employing cholesteric droplets embedded in the membrane in the temperature range of thinning transitions. Upon heating, the difference between the membrane tension and surface tension of the bulk sample decreases sufficiently, which leads to membrane instability. After the thinning transition, the membrane returns to a stable state with a larger value of surface interaction.