Modeling multiple time scales during glass formation with phase-field crystals

Deformable active nematic particles and emerging edge currents in circular confinements

We consider a microscopic field theoretical approach for interacting active nematic particles. With only steric interactions the self-propulsion strength in such systems can lead to different collective behaviour, e.g. synchronized self-spinning and collective translation. The different behaviour results from the delicate interplay between internal nematic structure, particle shape deformation and particle-particle interaction. For intermediate active strength an asymmetric particle shape emerges and leads to chirality and self-spinning crystals. For larger active strength the shape is symmetric and translational collective motion emerges. Within circular confinements, depending on the packing fraction, the self-spinning regime either stabilizes positional and orientational order or can lead to edge currents and global rotation which destroys the synchronized self-spinning crystalline structure.

Two-dimensional localized states in an active phase-field-crystal model

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations, and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.

Predictive local field theory for interacting active Brownian spheres in two spatial dimensions

We present a predictive local field theory for the nonequilibrium dynamics of interacting active Brownian particles with a spherical shape in two spatial dimensions. The theory is derived by a rigorous coarse-graining starting from the Langevin equations that describe the trajectories of the individual particles. For high accuracy and generality of the theory, it includes configurational order parameters and derivatives up to infinite order. In addition, we discuss possible approximations of the theory and present reduced models that are easier to apply. We show that our theory contains popular models such as Active Model B+  as special cases and that it provides explicit expressions for the coefficients occurring in these and other, often phenomenological, models. As a further outcome, the theory yields an analytical expression for the density-dependent mean swimming speed of the particles. To demonstrate an application of the new theory, we analyze a simple reduced model of the lowest nontrivial order in derivatives, which is able to predict the onset of motility-induced phase separation of the particles. By a linear stability analysis, an analytical expression for the spinodal corresponding to motility-induced phase separation is obtained. This expression is evaluated for the case of particles interacting repulsively by a Weeks-Chandler-Andersen potential. The analytical predictions for the spinodal associated with these particles are found to be in very good agreement with the results of Brownian dynamics simulations that are based on the same Langevin equations as our theory. Furthermore, the critical point predicted by our analytical results agrees excellently with recent computational results from the literature.

Description of order-disorder transitions based on the phase-field-crystal model

Order-disorder transition is an attractive topic in the research field of phase transformation. However, how to describe order-disorder transitions on atomic length scales and diffusional time scales is still challenging. Inspired from high-resolution transmission electron microscopy, we proposed an approach to describe ordered structures by introducing an order parameter into the original phase-field-crystal model to reflect the atomic potential distribution. This new order parameter contains information about kinds of atoms, showing that different kinds of sublattices in ordered structures can be distinguished by the amplitude of the order parameter. Two case studies, growth of ordered precipitations and evolution of antiphase domains, are also presented to demonstrate the capabilities of this approach.

Hydrodynamic theory of freezing: Nucleation and polycrystalline growth

Structural aspects of crystal nucleation in undercooled liquids are explored using a nonlinear hydrodynamic theory of crystallization proposed recently [G. I. Tóth et al., J. Phys.: Condens. Matter 26, 055001 (2014)JCOMEL0953-898410.1088/0953-8984/26/5/055001], which is based on combining fluctuating hydrodynamics with the phase-field crystal theory. We show that in this hydrodynamic approach not only homogeneous and heterogeneous nucleation processes are accessible, but also growth front nucleation, which leads to the formation of new (differently oriented) grains at the solid-liquid front in highly undercooled systems. Formation of dislocations at the solid-liquid interface and interference of density waves ahead of the crystallization front are responsible for the appearance of the new orientations at the growth front that lead to spherulite-like nanostructures.

Incorporating Noise Quantitatively in the Phase Field Crystal Model via Capillary Fluctuation Theory

A tacit assumption underlying most phase field models of nonequilibrium phase transformations is that of scale separation. Stochastic order parameter field theories utilize noise to separate atomic-scale fluctuations from the slowly varying fields that describe microstructure patterns. The mesoscale distribution of such stochastic variables is generally assumed to follow Gaussian statistics, with their magnitude following fluctuation-dissipation relations. However, there is still much debate about how atomic-scale fluctuations map onto the mesoscale upon coarse graining of microscopic theories. This Letter studies interface fluctuations in the phase field crystal (PFC) model and proposes a self-consistent method for relating how the effective noise strength and spectral filtering of the noise in the PFC model, and similar types of microscopic models, should be defined so as to attain the spectrum of mesoscale capillary fluctuations quantitatively.

New density functional approach for solid-liquid-vapor transitions in pure materials

A new phase field crystal (PFC) type theory is presented, which accounts for the full spectrum of solid-liquid-vapor phase transitions within the framework of a single density order parameter. Its equilibrium properties show the most quantitative features to date in PFC modeling of pure substances, and full consistency with thermodynamics in pressure-volume-temperature space is demonstrated. A method to control either the volume or the pressure of the system is also introduced. Nonequilibrium simulations show that 2- and 3-phase growth of solid, vapor, and liquid can be achieved, while our formalism also allows for a full range of pressure-induced transformations. This model opens up a new window for the study of pressure driven interactions of condensed phases with vapor, an experimentally relevant paradigm previously missing from phase field crystal theories.

Superadiabatic forces in Brownian many-body dynamics

Theoretical approaches to nonequilibrium many-body dynamics generally rest upon an adiabatic assumption, whereby the true dynamics is represented as a sequence of equilibrium states. Going beyond this simple approximation is a notoriously difficult problem. For the case of classical Brownian many-body dynamics, we present a simulation method that allows us to isolate and precisely evaluate superadiabatic correlations and the resulting forces. Application of the method to a system of one-dimensional hard particles reveals the importance for the dynamics, as well as the complexity, of these nontrivial out-of-equilibrium contributions. Our findings help clarify the status of dynamical density functional theory and provide a rational basis for the development of improved theories.

Phase-field-crystal modeling of glass-forming liquids: spanning time scales during vitrification, aging, and deformation

Two essential elements required to generate a glass transition within phase-field-crystal (PFC) models are outlined based on observed freezing behaviors in various models of this class. The central dynamic features of glass formation in simple binary liquids are qualitatively reproduced across 12 orders of magnitude in time by applying a physically motivated time scaling to previous PFC simulation results. New aspects of the equilibrium phase behavior of the same binary model system are also outlined, aging behavior is explored in the moderate and deeply supercooled regimes, and aging exponents are extracted. General features of the elastic and plastic responses of amorphous and crystalline PFC solids under deformation are also compared and contrasted.

Phase-field-crystal models and mechanical equilibrium

Phase-field-crystal (PFC) models constitute a field theoretical approach to solidification, melting, and related phenomena at atomic length and diffusive time scales. One of the advantages of these models is that they naturally contain elastic excitations associated with strain in crystalline bodies. However, instabilities that are diffusively driven towards equilibrium are often orders of magnitude slower than the dynamics of the elastic excitations, and are thus not included in the standard PFC model dynamics. We derive a method to isolate the time evolution of the elastic excitations from the diffusive dynamics in the PFC approach and set up a two-stage process, in which elastic excitations are equilibrated separately. This ensures mechanical equilibrium at all times. We show concrete examples demonstrating the necessity of the separation of the elastic and diffusive time scales. In the small-deformation limit this approach is shown to agree with the theory of linear elasticity.

Nonlinear hydrodynamic theory of crystallization

We present an isothermal fluctuating nonlinear hydrodynamic theory of crystallization in molecular liquids. A dynamic coarse-graining technique is used to derive the velocity field, a phenomenology which allows a direct coupling between the free energy functional of the classical density functional theory and the Navier-Stokes equation. In contrast to the Ginzburg-Landau type amplitude theories, the dynamic response to elastic deformations is described by parameter-free kinetic equations. Employing our approach to the free energy functional of the phase-field crystal model, we recover the classical spectrum for the phonons and the steady-state growth fronts. The capillary wave spectrum of the equilibrium crystal-liquid interface is in good qualitative agreement with the molecular dynamics simulations.

Micromechanics of emergent patterns in plastic flows

Crystalline solids undergo plastic deformation and subsequently flow when subjected to stresses beyond their elastic limit. In nature most crystalline solids exist in polycrystalline form. Simulating plastic flows in polycrystalline solids has wide ranging applications, from material processing to understanding intermittency of earthquake dynamics. Using phase field crystal (PFC) model we show that in sheared polycrystalline solids the atomic displacement field shows spatio-temporal heterogeneity spanning over several orders of length and time scales, similar to that in amorphous solids. The displacement field also exhibits localized quadrupolar patterns, characteristic of two dislocations of the opposite sign approaching each other. This is a signature of crystallinity at microscopic scale. Polycrystals being halfway between single crystals and amorphous solids, in terms of the degree of structural order, descriptions of solid mechanics at two widely different scales, namely continuum plastic flow and discrete dislocation dynamics turns out to be necessary here.

Colloidal particles in a drying suspension: a phase field crystal approach

Using a phase field crystal model we study the structure and dynamics of a drop of colloidal suspension during evaporation of the solvent. We model an experimental system where contact line pinning of the drop on the substrate is non-existent. Under such carefully controlled conditions, evaporation of the drop produces an ordered or disordered arrangement of the colloidal residue depending only on the initial average density of solute and the drying rate. We obtain a non-equilibrium phase boundary showing amorphous and crystalline phases of single component and binary mixtures of colloidal particles in the density-drying rate plane. While single-component colloids order in the two-dimensional triangular lattice, a symmetric binary mixture of mutually repulsive particles can be ordered into three triangular sub-lattices in two dimensions. Two of them are occupied by the two species of particles with the third sub-lattice vacant.

Scale-coupling and interface-pinning effects in the phase-field-crystal model

Effects of scale coupling between mesoscopic slowly varying envelopes of liquid-solid profile and the underlying microscopic crystalline structure are studied in the phase-field-crystal (PFC) model. Such scale coupling leads to nonadiabatic corrections to the PFC amplitude equations, the effect of which increases strongly with decreasing system temperature below the melting point. This nonadiabatic amplitude representation is further coarse-grained for the derivation of effective sharp-interface equations of motion in the limit of small but finite interface thickness. We identify a generalized form of the Gibbs-Thomson relation with the incorporation of coupling and pinning effects of the crystalline lattice structure. This generalized interface equation can be reduced to the form of a driven sine-Gordon equation with Kardar-Parisi-Zhang (KPZ) nonlinearity, and can be combined with two other dynamic equations in the sharp interface limit obeying the conservation condition of atomic number density in a liquid-solid system. A sample application to the study of crystal layer growth is given, and the corresponding analytic solutions showing lattice pinning and depinning effects and two distinct modes of continuous vs nucleated growth are presented. We also identify the universal scaling behaviors governing the properties of pinning strength, surface tension, interface kinetic coefficient, and activation energy of atomic layer growth, which accommodate all range of liquid-solid interface thicknesses and different material elastic moduli.

Modeling the structure of liquids and crystals using one- and two-component modified phase-field crystal models

A modified phase-field crystal model in which the free energy may be minimized by an order parameter profile having isolated bumps is investigated. The phase diagram is calculated in one and two dimensions and we locate the regions where modulated and uniform phases are formed and also regions where localized states are formed. We investigate the effectiveness of the phase-field crystal model for describing fluids and crystals with defects. We further consider a two-component model and elucidate how the structure transforms from hexagonal crystalline ordering to square ordering as the concentration changes. Our conclusion contains a discussion of possible interpretations of the order parameter field.

Dynamic density functional theory of solid tumor growth: Preliminary models

Cancer is a disease that can be seen as a complex system whose dynamics and growth result from nonlinear processes coupled across wide ranges of spatio-temporal scales. The current mathematical modeling literature addresses issues at various scales but the development of theoretical methodologies capable of bridging gaps across scales needs further study. We present a new theoretical framework based on Dynamic Density Functional Theory (DDFT) extended, for the first time, to the dynamics of living tissues by accounting for cell density correlations, different cell types, phenotypes and cell birth/death processes, in order to provide a biophysically consistent description of processes across the scales. We present an application of this approach to tumor growth.