To study the solution growth of crystals composed of chiral organic molecules, a spin-one Ising lattice gas model is proposed. The model turns out to be equivalent to the Blume-Emery-Griffiths model, which shows an equilibrium chiral symmetry breaking at low temperatures. The kinetic Monte Carlo simulation of crystal growth, however, demonstrates that Ostwald ripening is a very slow process with a characteristic time proportional to the system size: The dynamics is nonergodic. It is then argued that by incorporating grinding dynamics, homochirality is achieved in a short time, independent of the system size. Grinding limits cluster sizes to a certain range independent of system size and at the same time keeps the supersaturation so high that population numbers of average-sized clusters grow. If numbers of clusters for two types of enantiomers differ by chance, the difference is amplified exponentially and the system rapidly approaches the homochiral state. Relaxation time to the final homochiral state is determined by the average cluster size. We conclude that the system should be driven and kept in a nonequilibrium state to achieve homochirality.
We investigate a nonequilibrium coarsening dynamics of a one-dimensional Ising spin system with chirality. Only spins at domain boundaries are updated so that the model undergoes a coarsening to either of equivalent absorbing states with all spins + or -. Chirality is imposed by assigning different transition rates to events at down (+-) kinks and up (-+) kinks. The coarsening is characterized by power-law scalings of the kink density $ρ\sim t^{-δ}$ and the characteristic length scale $ξ\sim t^{1/z}$ with time $t$. Surprisingly the scaling exponents vary continuously with model parameters, whi…
A cluster growth model is proposed to study chiral symmetry breaking in stirred crystallization from an achiral element. Achiral monomers are assumed to coagulate to form chiral clusters from dimers to hexamers. Due to the stirring, the hexamers break into dimers. The coagulation of two dimers into a tetramer also occurs. The fixed point analysis of coupled rate equations of cluster densities and their numerical integrations show that the chiral symmetry becomes broken if dimers are critical and able to dissociate back to achiral monomers. The chirality selection is understood by showing that,…
We study the pattern formation of chiral charges in the presence of reactions. We show that, in contrast to the original Turing's mechanism of pattern formation in diffusion-reaction systems, the interplay between chiral effects and reactions can lead to a new kind of instability against spatially inhomogeneous perturbations, and furthermore, to a helical pattern formation, even without diffusion. This provides a new physical mechanism that can generate a macroscopic helical structure from microscopic chirality, including chirality of elementary particles, via nonequilibrium processes.
We investigate the statistical mechanics of chirality and biaxiality in liquid crystals through a variety of theoretical approaches, including Monte Carlo simulations, lattice mean-field theory, and Landau theory. All of these calculations show that there is an important interaction between cholesteric twist and biaxial order: The twist acts as a field on the biaxial order, and conversely, the biaxial order increases the twist, i.e. reduces the pitch. We model the behavior of chiral biaxial liquid crystals as a function of temperature, and discuss how the predictions can be tested in experimen…
To study the establishment of homochirality observed in the crystal growth experiment of chiral molecules from a solution under grinding, we extend the lattice gas model of crystal growth as follows. A lattice site can be occupied by a chiral molecule in R or S form, or can be empty. Molecules form homoclusters by nearest neighbor bonds. They change their chirality if they are isolated monomers in the solution. Grinding is incorporated by cutting and shafling the system randomly. It is shown that Ostwald ripening without grinding is extremely slow to select chirality, if possible. Grinding alo…
The local equilibrium approach previously developed by the Authors [J. Mabillard and P. Gaspard, J. Stat. Mech. (2020) 103203] for matter with broken symmetries is applied to crystalline solids. The macroscopic hydrodynamics of crystals and their local thermodynamic and transport properties are deduced from the microscopic Hamiltonian dynamics. In particular, the Green-Kubo formulas are obtained for all the transport coefficients. The eight hydrodynamic modes and their dispersion relation are studied for general and cubic crystals. In the same twenty crystallographic classes as those compatibl…
We propose a model for charge density waves in ring shaped crystals, which depicts frustration between intra- and inter-chain couplings coming from cylindrical bending. It is then mapped to a three dimensional uniformly frustrated XY model with one dimensional anisotropy in connectivity. The nonequilibrium relaxation dynamics is investigated by Monte Carlo simulations to find a phase transition which is quite different from that of usual whisker crystal. We also find that the low temperature state is a three dimensional phase vortex lattice with a two dimensional phase coherence in a cylindric…
This paper reviews the complex ordered structures induced by chirality in liquid crystals. In general, chirality favors a twist in the orientation of liquid-crystal molecules. In some cases, as in the cholesteric phase, this favored twist can be achieved without any defects. More often, the favored twist competes with applied electric or magnetic fields or with geometric constraints, leading to frustration. In response to this frustration, the system develops ordered structures with periodic arrays of defects. The simplest example of such a structure is the lattice of domains and domain walls …
Lyotropic cholesteric liquid crystal phases are ubiquitously observed in biological and synthetic polymer solutions, characterized by a complex interplay between thermal fluctuations, entropic and enthalpic forces. The elucidation of the link between microscopic features and macroscopic chiral structure, and of the relative roles of these competing contributions on phase organization, remains a topical issue. Here we provide theoretical evidence of a novel mechanism of chirality amplification in lyotropic liquid crystals, whereby phase chirality is governed by fluctuation-stabilized helical de…
Effect of exchange anisotropy on the relaxation time of spin and vector chirality is studied for the antiferromagnetic classical Heisenberg model on the triangular lattice by using the nonequilibrium relaxation Monte Carlo method. We identify the Berezinskii-Kosterlitz-Thouless (BKT) transition and the chiral transition in a wide range of the anisotropy, even for very small anisotropy of $\sim$ 0.01%. As the anisotropy decreases, both the critical temperatures steeply decrease, while the BKT critical region becomes divergently wide. We elucidate a sharp "V shape" of the phase diagram around th…
We extend the phase field crystal method for nonequilibrium patterning to stochastic systems with external source where transient dynamics is essential. It was shown that at short time scales the system manifests pattern selection processes. These processes are studied by means of the structure function dynamics analysis. Nonequilibrium pattern-forming transitions are analyzed by means of numerical simulations.
The relaxation and complex magnetic susceptibility treatments of a spin-3/2 Blume-Capel model with quenched random crystal field on a two dimensional square lattice are investigated by a method combining the statistical equilibrium theory and the thermodynamics of linear irreversible processes. Generalized force and flux are defined in irreversible thermodynamics limit. The kinetic equation for the magnetization is obtained by using linear response theory. Temperature and also crystal field dependencies of the relaxation time are obtained in the vicinity of phase transition points. We found th…
We review the depinning and nonequilibrium phases of collectively interacting particle systems driven over random or periodic substrates. This type of system is relevant to vortices in type-II superconductors, sliding charge density waves, electron crystals, colloids, stripe and pattern forming systems, and skyrmions, and could also have connections to jamming, glassy behaviors, and active matter. These systems are ideal for exploring the broader issues of characterizing transient and steady state nonequilibrium flow phases as well as nonequilibrium phase transitions between distinct dynamical…
A chiral-order parameter of spins in a crystal can not be inferred from measurements of Bragg reflections, although there are several claims in published literature to the contrary. Appropriate techniques use total scattering, of x-rays or neutrons, that yield the instantaneous value of a spin correlation related to an established measure of spin chirality.
We examine the Bloch-Peierls-Berry dynamics under a classical nonequilibrium dynamical formulation. In this formulation all coordinates in phase space formed by the position and crystal momentum space are treated on equal footing. Explicitly demonstrations of the no (naive) Liouville theorem and of the validity of Darboux theorem are given. The explicit equilibrium distribution function is obtained. The similarities and differences to previous approaches are discussed. Our results confirm the richness of the Bloch-Peierls-Berry dynamics.
The chiral structure of liquid crystalline phases arises due to the intrinsic chirality of the constituent mesogens. While it is seemingly straightforward to quantify the macroscopic chirality by using, for instance, the cholesteric pitch or the optical rotatory power, it is not as simple to quantify the chirality of a single molecule. I will discuss a systematic quantification of molecular chirality and show how the resulting chiral parameters may be used to predict macroscopic chiral structure.
We prove the existence of non-equilibrium phases of matter in the prethermal regime of periodically-driven, long-range interacting systems, with power-law exponent $α> d$, where $d$ is the dimensionality of the system. In this context, we predict the existence of a disorder-free, prethermal discrete time crystal in one dimension -- a phase strictly forbidden in the absence of long-range interactions. Finally, using a combination of analytic and numerical methods, we highlight key experimentally observable differences between such a prethermal time crystal and its many-body localized counterpar…
