Fluctuating dynamics of nematic liquid crystals using the stochastic method of lines

Abstract

We construct Langevin equations describing the fluctuations of the tensor order parameter Q(alphabeta) in nematic liquid crystals by adding noise terms to time-dependent variational equations that follow from the Ginzburg-Landau-de Gennes free energy. The noise is required to preserve the symmetry and tracelessness of the tensor order parameter and must satisfy a fluctuation-dissipation relation at thermal equilibrium. We construct a noise with these properties in a basis of symmetric traceless matrices and show that the Langevin equations can be solved numerically in this basis using a stochastic version of the method of lines. The numerical method is validated by comparing equilibrium probability distributions, structure factors, and dynamic correlations obtained from these numerical solutions with analytic predictions. We demonstrate excellent agreement between numerics and theory. This methodology can be applied to the study of phenomena where fluctuations in both the magnitude and direction of nematic order are important, as for instance, in the nematic swarms which produce enhanced opalescence near the isotropic-nematic transition or the problem of nucleation of the nematic from the isotropic phase.

Stochastic kinetics reveal imperative role of anisotropic interfacial tension to determine morphology and evolution of nucleated droplets in nematogenic films

“…To understand how fluctuations influence the dynamics and microstructural evolution, one needs (i) the theoretical formulation of a stochastic GLdG description of the dynamics and (ii) a numerical prescription to integrate the stochastic equation for the orientation tensor [37] paying special attention to the structure of the noise and satisfying the fluctuation-dissipation theorem (FDT). The first question was addressed by Stratonovich [37] by writing an overdamped Langevin equation in model-A relaxational dynamics that excludes coupling to any external hydrodynamic flow as [38,39]…”

Lattice-Boltzmann-Langevin simulations of binary mixtures

“…We use a stochastic method of lines (SMOL) discretisation [10] to solve the fluctuating Cahn-Hilliard equation for the order parameter. Since it does not contain a pressure term which acts as a Lagrange multiplier in the incompressible Navier-Stokes equations, there is no particular benefit in using a kinetic algorithm with its large number of degrees of freedom in solving for a single scalar variable.…”

“…Since it does not contain a pressure term which acts as a Lagrange multiplier in the incompressible Navier-Stokes equations, there is no particular benefit in using a kinetic algorithm with its large number of degrees of freedom in solving for a single scalar variable. Here, we adopt a semi-discretisation strategy [9,10], discretising the spatial variables to obtain a set of coupled stochastic ordinary differential equations. The spatial discretisations we propose ensure that the conservation law is respected to machine precision and that the fluctuation and dissipation are in balance for all wave vectors.…”

“…In this paper, we solve the model H equations by combining the fluctuating lattice Boltzmann equation (FLBE) [7,8] with a stochastic method of lines (SMOL) [9,10] using both finite difference and finite volume discretisations [11]. The formulation ensures conservation of local densities to machine precision, and a correct balance between fluctuation and dissipation for all the degrees of freedom on the lattice.…”

Theoretically informed Monte Carlo simulation of liquid crystals by sampling of alignment-tensor fields

“…Recall that the alignment tensor is symmetric and traceless, and out of its nine cartesian components, only five are actual degrees of freedom. Therefore, it can be expressed in terms of an orthonormal tensor basis, as originally discussed by Hess and co-workers 52 and, more recently, by Bhattacharjee et al, 11,53 …”