Scalar Order Parameter Research Articles

Nonsymmetrical thermal conductivity along the director field in ferroelectric nematic liquid crystals.

The synthesis of ferroelectric nematic liquid crystals (FNLCs) concludes the long wait for their existence and potential usage in multiple liquid crystal based applications. In FNLCs, electric polarization in the nematic phase significantly decreases the switching time of in-on display pixels. In this article, we report the occurrence of translation symmetry breaking for heat propagation along the director field n[over ̂] in the ferroelectric nematic phase. Due to the C_{∞V} symmetry of such a phase and close similarity to the bent-core polar liquid crystal phase, a rank-3 tensor describes its scalar order parameter and algebraic deductions. The finite element simulations show the occurrence of the nonsymmetrical thermal conductivity along n[over ̂]. The preferential heat transport in FNLCs can allow them to work as an all-thermal monophase non-nanostructured single-material thermal rectifier. We expect that this study will contribute towards the FNLCs application as functional layers and inks.

An introduction to phase ordering in scalar active matter

AbstractThese notes provide an introduction to phase ordering in dry, scalar active matter. We first briefly review Model A and Model B, the long-standing continuum descriptions of ordering in systems with a non-conserved and conserved scalar order parameter. We then contrast different ways in which the field theories can be extended so that the phase ordering persists, but in systems that are active and do not reach thermodynamic equilibrium. The active models allow a wide range of dynamical steady states not seen in their passive counterparts. These include microphase separation, active foams and travelling density bands.

Ordering kinetics in the active Ising model.

We undertake a numerical study of the ordering kinetics in the two-dimensional (2D) active Ising model (AIM), a discrete flocking model with a conserved density field coupled to a nonconserved magnetization field. We find that for a quench into the liquid-gas coexistence region and in the ordered liquid region, the characteristic length scale of both the density and magnetization domains follows the Lifshitz-Cahn-Allen growth law, R(t)∼t^{1/2}, consistent with the growth law of passive systems with scalar order parameter and nonconserved dynamics. The system morphology is analyzed with the two-point correlation function and its Fourier transform, the structure factor, which conforms to the well-known Porod's law, a manifestation of the coarsening of compact domains with smooth boundaries. We also find the domain growth exponent unaffected by different noise strengths and self-propulsion velocities of the active particles. However, transverse diffusion is found to play the most significant role in the growth kinetics of the AIM. We extract the same growth exponent by solving the hydrodynamic equationsof the AIM.

Learning fast, accurate, and stable closures of a kinetic theory of an active fluid

Important classes of active matter systems can be modeled using kinetic theories. However, kinetic theories can be high dimensional and challenging to simulate. Reduced-order representations based on tracking only low-order moments of the kinetic model serve as an efficient alternative, but typically require closure assumptions to model unrepresented higher-order moments. In this study, we present a learning framework based on neural networks that exploits rotational symmetries in the closure terms to learn accurate closure models directly from kinetic simulations. The data-driven closures demonstrate excellent a-priori predictions comparable to the state-of-the-art Bingham closure. We provide a systematic comparison between different neural network architectures and demonstrate that nonlocal effects can be safely ignored to model the closure terms. We develop an active learning strategy that enables accurate prediction of the closure terms across the entire parameter space using a single neural network without the need for retraining. We also propose a data-efficient training procedure based on time-stepping constraints and a differentiable pseudo-spectral solver, which enables the learning of stable closures suitable for a-posteriori inference. The coarse-grained simulations equipped with data-driven closure models faithfully reproduce the mean velocity statistics, scalar order parameters, and velocity power spectra observed in simulations of the kinetic theory. Our differentiable framework also facilitates the estimation of parameters in coarse-grained descriptions conditioned on data.

Extended free-energy functionals for achiral and chiral ferroelectric nematic liquid crystals: theory and simulation.

Polar nematic liquid crystals are new classes of condensed-matter states, where the inversion symmetry common to the traditional apolar nematics is broken. Establishing theoretical descriptions for the novel phase states is an urgent task. Here, we develop a Landau-type mean-field theory for both the achiral and chiral ferroelectric nematics. In the polar nematic states, the inversion symmetry breaking adds three new contributions: an additional odd elastic term (corresponding to the flexoelectricity in symmetry) to the standard Oseen-Frank free energy, electrostatic effect and an additional Landau term relating to the gradient of local polarization. The coupling between the scalar order parameter and polarization order should be considered. In the chiral and polar nematic state, we reveal that the competition between the twist elasticity and polarity dictates effective compressive energy arising from the quasi-layer structure. The polarization gradient is an essential term for describing the ferroelectric nature. Besides, we successfully simulate an experimentally reported structural transition in ferroelectric nematic droplets from a concentric-vortex-like to a line-disclination-mediated topology based on the developed theory. The approaches provide theoretical foundations for testing and predicting polar structures in emerging polar liquid crystals.

A numerical study of interface dynamics in fluid materials

This paper deals with the approximation of the dynamics of two fluids having non-matching densities and viscosities. The modeling involves the coupling of the Allen-Cahn equation with the time-dependent Navier-Stokes equations. The Allen-Cahn equation describes the evolution of a scalar order parameter that assumes two distinct values in different spatial regions. Conversely, the Navier-Stokes equations govern the movement of a fluid subjected to various forces like pressure, gravity, and viscosity. When the Allen-Cahn equation is coupled with the Navier-Stokes equations, it is typically done through a surface tension term. The surface tension term accounts for the energy required to create an interface between the two phases, and it is proportional to the curvature of the interface. The Navier-Stokes equations are modified to include this term, which leads to the formation of a dynamic interface between the two phases. The resulting system of equations is known as the two-phase Navier-Stokes/Allen-Cahn equations. In this paper, the authors propose a mathematical model that combines the Allen-Cahn model and the Navier-Stokes equations to simulate multiple fluid flows. The Allen-Cahn model is utilized to represent the diffuse interface between different fluids, while the Navier-Stokes equations are employed to describe the fluid dynamics. The Allen-Cahn-Navier-Stokes model has been employed to simulate the generation of bubbles in a liquid subjected to an acoustic field. The model successfully predicted the size of the bubbles and the frequency at which they formed. The numerical outcomes were validated against experimental data, and a favorable agreement was observed.

Novel turbulence and coarsening arrest in active-scalar fluids.

We uncover a new type of turbulence - activity-induced homogeneous and isotropic turbulence - in a model that has been employed to investigate motility-induced phase separation (MIPS) in a system of microswimmers. The active Cahn-Hilliard-Navier-Stokes (CHNS) equations, also called active model H, provide a natural theoretical framework for our study. In this CHNS model, a single scalar order parameter ϕ, positive (negative) in regions of high (low) microswimmer density, is coupled with the velocity field u. The activity of the microswimmers is governed by an activity parameter ζ that is positive for extensile swimmers and negative for contractile swimmers. With extensile swimmers, this system undergoes complete phase separation, which is similar to that in binary-fluid mixtures. By carrying out pseudospectral direct numerical simulations (DNSs), we show, for the first time, that (a) this model develops an emergent nonequilibrium, but statistically steady, state (NESS) of active turbulence, for the case of contractile swimmers, if ζ is sufficiently large and negative, and (b) this turbulence arrests the phase separation. We quantify this suppression by showing how the coarsening-arrest length scale does not grow indefinitely, with time t, but saturates at a finite value at large times. We characterise the statistical properties of this active-scalar turbulence by employing energy spectra and fluxes and the spectrum of ϕ. For sufficiently high Reynolds numbers, the energy spectrum (k) displays an inertial range, with a power-law dependence on the wavenumber k. We demonstrate that, in this range, the flux Π(k) assumes a nearly constant, negative value, which indicates that the system shows an inverse cascade of energy, even though energy injection occurs over a wide range of wavenumbers in our active-CHNS model.

Statistical field theory of mechanical stresses in Coulomb fluids: general covariant approach vs Noether’s theorem

In this paper, we introduce a statistical field theory that describes the macroscopic mechanical forces in inhomogeneous Coulomb fluids. Our approach employs the generalization of Noether’s first theorem for the case of a fluctuating order parameter to calculate the stress tensor for Coulomb fluids. This tensor encompasses the mean-field stress tensor and fluctuation corrections derived through the one-loop approximation. The correction for fluctuations includes a term that accounts for the thermal fluctuations of the local electrostatic potential and field in the vicinity of the mean-field configuration. This correlation stress tensor determines how electrostatic correlation affects local stresses in a nonuniform Coulomb fluid. We also use a previously formulated general covariant methodology (Brandyshev and Budkov 2023 J. Chem. Phys. 158 174114) in conjunction with a functional Legendre transformation method and derive within it the same total stress tensor. We would like to emphasize that our general approaches are applicable not only to Coulomb fluids but also to nonionic simple or complex fluids, for which the field-theoretic Hamiltonian is known as a function of the relevant scalar order parameters.

Reinforcement learning strategy for control of microstructure evolution in phase field models

We propose a reinforcement learning-based approach for controlling microstructure evolution in phase field models. A key feature of our approach is the replacement of the high-fidelity but computationally expensive phase field simulation with an inexpensive surrogate model of acceptable fidelity. We apply our approach to a 2-D model problem based on the Allen-Cahn equation that represents a two-phase microstructure with a single, scalar order parameter. The control parameters for this model are spatially and temporally varying temperature (T) and bias fields (h). Our approach generates sets of T and h parameters that yield final microstructures qualitatively matching target microstructures. We discuss further developments required to apply this approach to optimal processing of designer microstructures.

Non-Local Q-Tensor Approach for Description of Elastic Deformations of Nematic Liquid Crystals at Sub-Micron Scale

Within the framework of phenomenological approach, a generalized concept of a non-local order parameter, that have the form of a traceless correlation tensor function or a tensor integral operator, is introduced. The main relationships, which determine the equilibrium orientational states and the phase transitions of a deformed nematic liquid crystal, are described. The resulting expression for the free energy is quadratic with respect to the gradients of the order parameter tensor. The limiting transition of the non-local order parameter to the local one leads to an expression that complements the classical Oseen–Frank theory of elasticity. The considered model includes eight independent elastic constants. Three of them relate to "bulk" constants analogous to Frank constants, another three ones relate to "surface" constants, and the remaining two constants determine the anisotropic effect of the gradient of the scalar order parameter on the director field. The constants are necessary for the consideration of a liquid crystal state near defects, as well as for the description of orientation effects caused by the inhomogeneity of order parameter. It is shown that in the free energy expression, the "surface" terms contribute to the director orientation in the case of a non-constant scalar order parameter, even in the approximation of rigid boundary conditions.

The role of multiplicative noise in critical dynamics

We study the role of multiplicative stochastic processes in the description of the dynamics of an order parameter near a critical point. We study equilibrium as well as out-of-equilibrium properties. By means of a functional formalism, we build the Dynamical Renormalization Group equations for a real scalar order parameter with Z2 symmetry, driven by a class of multiplicative stochastic processes with the same symmetry. We compute the flux diagram using a controlled ϵ-expansion, up to order ϵ2. We find that, for dimensions d=4−ϵ, the additive dynamic fixed point is unstable. The flux runs to a multiplicative fixed point driven by a diffusion function G(ϕ)=1+g∗ϕ2(x)/2, where ϕ is the order parameter and g∗=ϵ2/18 is the fixed point value of the multiplicative noise coupling constant. We show that, even though the position of the fixed point depends on the stochastic prescription, the critical exponents do not. Therefore, different dynamics driven by different stochastic prescriptions (such as Itô, Stratonovich, anti-Itô and so on) are in the same universality class.

A theoretical investigation on the relative stability of the columnar hexagonal and columnar square phases of discotic molecules

We have employed a mean field theory, which is a modified Ginzburg-Landau expansion in terms of the scalar order parameter Φ(r), to stabilize the ordered phases such as square and hexagonal phases of the discotic molecules. The Φ(r) represents the local volume fraction differential between the stiff core and the outside alkyl chain regions of the discotic molecules. The current theoretical analysis stabilizes the ordered phases, particularly the square (S) and 2D hexagonal (H) phases. The present study also identifies the direct hexagonal (HI) and reverse hexagonal (HII) phases, with the phases occurring in the following order: disorder(D) - HI - S - HII - disorder(D) with increase in the average volume fraction difference, Φo. Contrarily, the stability area of the S phase accounts for around 10% of the total area of the ordered phase, whereas the HI and HII phases each occupy about 45% of the total area of the ordered phase.

Noether's second theorem and covariant field theory of mechanical stresses in inhomogeneous ionic liquids.

In this paper, we present a covariant approach that utilizes Noether's second theorem to derive a symmetric stress tensor from the grand thermodynamic potential functional. We focus on the practical case where the density of the grand thermodynamic potential is dependent on the first and second coordinate derivatives of the scalar order parameters. Our approach is applied to several models of inhomogeneous ionic liquids that consider electrostatic correlations of ions or short-range correlations related to packing effects. Specifically, we derive analytical expressions for the symmetric stress tensors of the Cahn-Hilliard-like model, Bazant-Storey-Kornyshev model, and Maggs-Podgornik-Blossey model. All of these expressions are found to be consistent with respective self-consistent field equations.

Local and Global Order in Dense Packings of Semi-Flexible Polymers of Hard Spheres.

The local and global order in dense packings of linear, semi-flexible polymers of tangent hard spheres are studied by employing extensive Monte Carlo simulations at increasing volume fractions. The chain stiffness is controlled by a tunable harmonic potential for the bending angle, whose intensity dictates the rigidity of the polymer backbone as a function of the bending constant and equilibrium angle. The studied angles range between acute and obtuse ones, reaching the limit of rod-like polymers. We analyze how the packing density and chain stiffness affect the chains' ability to self-organize at the local and global levels. The former corresponds to crystallinity, as quantified by the Characteristic Crystallographic Element (CCE) norm descriptor, while the latter is computed through the scalar orientational order parameter. In all cases, we identify the critical volume fraction for the phase transition and gauge the established crystal morphologies, developing a complete phase diagram as a function of packing density and equilibrium bending angle. A plethora of structures are obtained, ranging between random hexagonal closed packed morphologies of mixed character and almost perfect face centered cubic (FCC) and hexagonal close-packed (HCP) crystals at the level of monomers, and nematic mesophases, with prolate and oblate mesogens at the level of chains. For rod-like chains, a delay is observed between the establishment of the long-range nematic order and crystallization as a function of the packing density, while for right-angle chains, both transitions are synchronized. A comparison is also provided against the analogous packings of monomeric and fully flexible chains of hard spheres.

A predictive theoretical model for stretch-induced instabilities in liquid crystal elastomers

ABSTRACTFollowing the experimental lead, we construct a general mathematical model which, depending on whether the uniaxial scalar order parameter is constant or not, can predict either the classical shear striping instability or the molecular auxetic response and mechanical Fréedericksz transition observed in different liquid crystal elastomers. Our theoretical model can shed some light on the role of nematic order in these stretch-induced mechanical instabilities not observed in conventional rubber-like solids.

TADA: The Topology-Accommodating Direction Assignment Algorithm for Liquid Crystals.

Despite the fact that topological defects are a hallmark of liquid crystalline materials, current computational techniques for identifying topological defects in particle-based simulations of these materials─which rest upon Q-tensor theory─do not leverage topological features of the system. In this work, we describe the topology-accommodating direction assignment (TADA) algorithm, a novel approach for identifying disclination cores in liquid crystalline materials, which is sensitive to topology: this method assigns to each mesogen a unique vector, thereby extending the concept of the liquid crystal director field down to the scale of mesogens. In systems containing disclination cores, TADA identifies line segments along which this assigned vector field is discontinuous, with cores located at the interior termination points of these line segments. The mere presence of defects can be identified by searching far away from them. We validate this approach by comparing its results to those obtained using the scalar order parameter for a variety of liquid crystalline assemblies sourced from molecular-dynamics simulations. We also discuss several benefits of the TADA algorithm over existing approaches for identifying topological defects in liquid crystalline materials.

Smectic layering: Landau theory for a complex-tensor order parameter

Composed of microscopic layers that stack along one direction while maintaining fluid-like positional disorder within layers, smectics are excellent systems for exploring topology, defects and geometric memory in complex confining geometries. However, the coexistence of crystalline-like characteristics in one direction and fluid-like disorder within layers makes lamellar liquid crystals notoriously difficult to model—especially in the presence of defects and large distortions. Nematic properties of smectics can be comprehensively described by the -tensor; however, lamellar order can exist independent of nematic alignment. To capture the features of the smectic layering alone, we develop a phenomenological Landau theory for a complex-tensor order parameter , which is capable of describing the local degree of lamellar ordering, layer displacement, and orientation of the layers. This theory accounts for both parallel and perpendicular elastic contributions. In addition to resolving the potential ambiguities inherent to complex scalar order parameter models, this model reduces to previously employed models of simple smectics, and opens new possibilities for numerical studies on smectics possessing many defects, within complex geometries and under extreme confinement.

Biotropic liquid crystal phase transformations in cellulose-producing bacterial communities

Biological functionality is often enabled by a fascinating variety of physical phenomena that emerge from orientational order of building blocks, a defining property of nematic liquid crystals that is also pervasive in nature. Out-of-equilibrium, "living" analogs of these technological materials are found in biological embodiments ranging from myelin sheath of neurons to extracellular matrices of bacterial biofilms and cuticles of beetles. However, physical underpinnings behind manifestations of orientational order in biological systems often remain unexplored. For example, while nematiclike birefringent domains of biofilms are found in many bacterial systems, the physics behind their formation is rarely known. Here, using cellulose-synthesizing Acetobacter xylinum bacteria, we reveal how biological activity leads to orientational ordering in fluid and gel analogs of these soft matter systems, both in water and on solid agar, with a topological defect found between the domains. Furthermore, the nutrient feeding direction plays a role like that of rubbing of confining surfaces in conventional liquid crystals, turning polydomain organization within the biofilms into a birefringent monocrystal-like order of both the extracellular matrix and the rod-like bacteria within it. We probe evolution of scalar orientational order parameters of cellulose nanofibers and bacteria associated with fluid-gel and isotropic-nematic transformations, showing how highly ordered active nematic fluids and gels evolve with time during biological-activity-driven, disorder-order transformation. With fluid and soft-gel nematics observed in a certain range of biological activity, this mesophase-exhibiting system is dubbed "biotropic," analogously to thermotropic nematics that exhibit solely orientational order within a temperature range, promising technological and fundamental-science applications.

Critical behavior of active Brownian particles: Connection to field theories.

We explore the relation between active Brownian particles, a minimal particle-based model for active matter, and scalar field theories. Both show a liquid-gas-like phase transition toward stable coexistence of a dense liquid with a dilute active gas that terminates in a critical point. However, a comprehensive mapping between the particle-based model parameters and the effective coefficients governing the field theories has not been established yet. We discuss conflicting recent numerical results for the critical exponents of active Brownian particles in two dimensions. Starting from the intermediate effective hydrodynamic equations, we then present a construction for a scalar order parameter for active Brownian particles that yields the active model B+. We argue that a crucial ingredient is the coupling between density and polarization in the particle current. The renormalization flow close to two dimensions exhibits a pair of perturbative fixed points that limit the attractive basin of the Wilson-Fisher fixed point, with the perspective that the critical behavior of active Brownian particles in two dimensions is governed by a strong-coupling fixed point different from Wilson-Fisher and not necessarily corresponding to Ising universality.

Pattern Formation and Aggregation in Ensembles of Solitons in Quasi One-Dimensional Electronic Systems

Broken symmetries of quasi one-dimensional electronic systems give rise to microscopic solitons taking roles of carriers of the charge or spin. The double degeneracy gives rise to solitons as kinks of the scalar order parameter A; the continuous degeneracy for the complex order parameter Aexp(iθ) gives rise to phase vortices, amplitudes solitons, and their combinations. These degrees of freedom can be controlled or accessed independently via either the spin polarization or the charge doping. The long-range ordering in dimensions above one imposes super-long-range confinement forces upon the solitons, leading to a sequence of phase transitions in their ensembles. The higher-temperature T transition enforces the confinement of solitons into topologically bound complexes: pairs of kinks or the amplitude solitons dressed by exotic half-integer vortices. At a second lower T transition, the solitons aggregate into rods of bi-kinks or into walls of amplitude solitons terminated by rings of half-integer vortices. With lowering T, the walls multiply, passing sequentially across the sample. Here, we summarize results of a numerical modeling for different symmetries, for charged and neutral soliton, in two and three dimensions. The efficient Monte Carlo algorithm, preserving the number of solitons, was employed which substantially facilitates the calculations, allowing to extend them to the three-dimensional case and to include the long-range Coulomb interactions.