Alexei Krekhov

Flexoelectricity and pattern formation in nematic liquid crystals.

We present in this paper a detailed analysis of the flexoelectric instability of a planar nematic layer in the presence of an alternating electric field (frequency ω), which leads to stripe patterns (flexodomains) in the plane of the layer. This equilibrium transition is governed by the free energy of the nematic, which describes the elasticity with respect to the orientational degrees of freedom supplemented by an electric part. Surprisingly the limit ω→0 is highly singular. In distinct contrast to the dc case, where the patterns are stationary and time independent, they appear at finite, small ω periodically in time as sudden bursts. Flexodomains are in competition with the intensively studied electrohydrodynamic instability in nematics, which presents a nonequilibrium dissipative transition. It will be demonstrated that ω is a very convenient control parameter to tune between flexodomains and convection patterns, which are clearly distinguished by the orientation of their stripes.

Ferrofluid pipe flow in an oscillating magnetic field

Ferrofluid pipe flow in an oscillating magnetic field along the pipe axis is studied theoretically in a wide range of the flow rate. The field-dependent part of viscosity (it can be positive or negative) reveals significant dependence on the flow vorticity, i.e., ferrofluids exhibit non-Newtonian behavior. This is manifested in an alteration of the velocity profile—it ceases to be parabolic—and deviation of the flow rate from the value prescribed by Poiseuille’s formula. The presented model based on the conventional ferrohydrodynamic equations and an assumption of the ferrofluid structure fits well experimental data recently obtained by Schumacher, Sellien, Konke, Cader, and Finlayson [“Experiment and simulation of laminar and turbulent ferrofluid pipe flow in an oscillating magnetic field,” Phys. Rev. E 67, 026308 (2003)].

Thermal Diffusion in Polymer Blends: Criticality and Pattern Formation

We discuss the dynamic critical properties of a binary blend of the two polymers poly(dimethyl siloxane) (PDMS) and poly(ethyl-methyl siloxane) (PEMS), and we have investigated experimentally and theoretically patterning and structure formation processes above and below the spinodal in the case of a spatially varying temperature. Asymptotic critical scaling is found close to T c in the range 6 ×10− 4 ≤ e ≤ 0. 2 of the reduced temperature e and a mean field behavior for large values of e. The thermal diffusion coefficient D T is thermally activated but does not show the critical slowing down of the Fickian diffusion coefficient D, which can be described by crossover functions for D. The Soret coefficient \({S}_{\mathrm{T}} = {D}_{\mathrm{T}}/D\) diverges at the critical point with a critical exponent − 0. 67 and shows a crossover to the exponent − 1 of the structure factor in the classical regime. Thermal activation processes cancel out and do not contribute to S T. The divergence of S T also leads to a very strong coupling of the order parameter to small temperature gradients, which can be utilized for laser patterning of thin polymer films. For a quantitative numerical model all three coefficients D, D T, and S T have been determined within the entire homogeneous phase and are parameterized by a pseudospinodal model. It is shown that equilibrium phase diagrams are no longer globally valid in the presence of a temperature gradient, and systems with an upper critical solution temperature (UCST) can be quenched into phase separation by local heating. Below the spinodal there is competition between the spontaneous spinodal demixing patterns and structures imposed by means of a focused laser beam utilizing the Soret effect. Elongated structures degrade to spherical objects due to surface tension effects leading to pearling instabilities. Grids of parallel lines can be stabilized by enforcing certain boundary conditions. Phase separation phenomena in polymer blends belong to the universality class of pattern forming systems with a conserved order parameter. In such systems, the effects of spatial forcing are rather unexplored and, as described in this work, spatial temperature modulations may cause via the Soret effect (thermal diffusion) a variety of interesting concentration modulations. In the framework of a generalized Cahn–Hilliard model it is shown that coarsening in the two-phase range of phase separating systems can be interrupted by a spatially periodic temperature modulation with a modulation amplitude beyond a critical one, where in addition the concentration modulations are locked to the periodicity of the external forcing. Accordingly, temperature modulations may be a useful future tool for controlled structuring of polymer blends. In the case of a traveling spatially periodic forcing, but with a modulation amplitude below the critical one, the coarsening dynamics can be enhanced. With a model of phase separation, taking into account thermal diffusion, essential features of the spatio-temporal dynamics of phase separation and thermal patterning observed in experiments can be reproduced. With a directional quenching an effective approach is studied to create regular structures during the phase separation process. In addition, it is shown that the wavelength of periodic stripe patterns is uniquely selected by the velocity of a quench interface. With a spatially periodic modulation of the quench interface itself, cellular patterns can also be generated.

Periodic structures in binary mixtures enforced by Janus particles

Phase separation in binary mixtures in the presence of Janus particles has been studied in terms of a Cahn-Hilliard model coupled to the Langevin equations describing the particle dynamics. We demonstrate that the phase separation process is arrested leading to unexpected regular stripe patterns in the concentration field. The underlying pattern forming mechanism has been elucidated: The twofold absorption properties on the surface of Janus particles with respect to the two components of a binary mixture trigger in their neighborhood spatial concentration variations. They result in an effective interaction between the particles mediated by the binary mixture. Our findings open a route to design composite materials with nanoscale lamellar morphologies where the pattern wavelength can be tuned by changing the wetting properties of the Janus particles.

Flashing flexodomains and electroconvection rolls in a nematic liquid crystal

Pattern forming instabilities induced by ultralow frequency sinusoidal voltages were studied in a rod-like nematic liquid crystal by microscopic observations and simultaneous electric current measurements. Two pattern morphologies, electroconvection (EC) and flexodomains (FD), were distinguished; both appearing as time separated flashes within each half period of driving. A correlation was found between the time instants of the EC flashes and that of the nonlinear current response. The voltage dependence of the pattern contrast C(U) for EC has a different character than that for the FD. The flattening of C(U) at reducing the frequency was described in terms of an imperfect bifurcation model. Analysing the threshold characteristics of FD the temperature dependence of the difference |e_1-e_3| of the flexoelectric coefficients were also determined by considering elastic anisotropy.

Anomalous diffusion in viscosity landscapes

Anomalous diffusion of Brownian particles in inhomogeneous viscosity landscapes is predicted by means of scaling arguments, which are substantiated through numerical simulations. Analytical solutions of the related Fokker–Planck equation in limiting cases confirm our results. In the case of an ensemble of particles starting at a spatial minimum (maximum) of the viscous damping, we find subdiffusive (superdiffusive) motion. Superdiffusion also occurs in the case of a monotonically varying viscosity profile. We suggest different systems for related experimental investigations.

Flexoelectricity and competition of time scales in electroconvection

An unexpected type of behavior in electroconvection (EC) has been detected in nematic liquid crystals (NLCs) under the condition of comparable time scales of the director relaxation and the period of the driving ac voltage. The studied NLCs exhibit standard EC (s-EC) at the onset of the instability, except one compound in which nonstandard EC (ns-EC) has been detected. In the relevant frequency region, the threshold voltage for conductive s-EC bends down considerably, while for dielectric s-EC it bends up strongly with the decrease of the driving frequency. We show that inclusion of the flexoelectric effect into the theoretical description of conductive s-EC leads to quantitative agreement, while for dielectric s-EC a qualitative agreement is achieved. The frequency dependence of the threshold voltage for ns-EC strongly resembles that of the dielectric s-EC.

Fast propagation regions cause self-sustained reentry in excitable media

Significance Rotating spiral waves have been observed in a wide variety of nonlinear spatially distributed systems in physics, chemistry, and biology, called excitable media. In medicine, they are associated with cardiac arrhythmias. Nevertheless, how exactly these waves are initiated in excitable tissue remains largely unknown. Several candidate mechanisms have been proposed over the last century, largely based on the assumption of wave breaking due to unexcitable obstacles or the medium’s refractoriness. Here, we have identified a generic mechanism for self-sustained spiral wave generation in inhomogeneous excitable media by localized regions with the fast propagation velocity. We believe our theory creates constructive options to fight these deadly phenomena. Self-sustained waves of electrophysiological activity can cause arrhythmia in the heart. These reentrant excitations have been associated with spiral waves circulating around either an anatomically defined weakly conducting region or a functionally determined core. Recently, an ablation procedure has been clinically introduced that stops atrial fibrillation of the heart by destroying the electrical activity at the spiral core. This is puzzling because the tissue at the anatomically defined spiral core would already be weakly conducting, and a further decrease should not improve the situation. In the case of a functionally determined core, an ablation procedure should even further stabilize the rotating wave. The efficacy of the procedure thus needs explanation. Here, we show theoretically that fundamentally in any excitable medium a region with a propagation velocity faster than its surrounding can act as a nucleation center for reentry and can anchor an induced spiral wave. Our findings demonstrate a mechanistic underpinning for the recently developed ablation procedure. Our theoretical results are based on a very general and widely used two-component model of an excitable medium. Moreover, the important control parameters used to realize conditions for the discovered phenomena are applicable to quite different multicomponent models.

Director Precession and Nonlinear Waves in Nematic Liquid Crystals under Elliptic Shear

Elliptic shear applied to a homeotropically oriented nematic above the electric bend Freedericks transition (FT) generates slow precession of the director. The character of the accompanying nonlinear waves changes from diffusive phase waves to dispersive ones exhibiting spirals and spatiotemporal chaos as the FT is approached from above. An exact solution of the flow alignment equations captures the observed precession and predicts its reversal for non-flow-aligning materials. The FT transforms into a Hopf bifurcation opening the way to understand the wave phenomena.

Stability and Orientation of Lamellae in Diblock Copolymer Films

The dynamics of microphase separation and the orientation of lamellae in diblock copolymers are investigated in terms of a mean-field model. The formation of lamellar structures and their stable states are explored and it is shown that lamellae are stable not only for the period of the structure corresponding to the minimum of the free energy. The range of wavelengths of stable lamellae is determined by an efficient functional approach introduced with this work. The effects of the interaction of diblock copolymers with two confining substrates on the lamellae orientation are studied by an extensive analysis of the total free energy. By changing the wetting property at one boundary, a transition from a preferentially perpendicular to a parallel lamellar orientation with respect to the confining plates is found, which is rather independent of the distance between the boundaries. Simulations of the dynamics of microphase separation reveal that the time scale of the lamellar orientational order dynamics, which is quantitatively characterized in terms of an orientational order parameter and the structure factor, depends significantly on the properties of the confining boundaries as well as on the quench depth.