Perspectives on a Few Puzzles in Phase Transformations: When Should the Farthest Reach the Earliest?

We briefly review the facts concerning two important aspects of phase transitions, namely, critical and coarsening phenomena. A discussion of the universal features, highlighting the current challenges, is provided. Following this, we elaborate on a topic of much recent interest, viz., the Mpemba effect, a puzzle that found mention even in the works of Aristotle. After a description of the debated case of faster freezing of a hotter sample of liquid water, into ice, than a colder one, when quenched to the same subzero temperature, we discuss more modern interest. There one asks, should a hotter body of a material equilibrate faster than a colder one when quenched to a common lower temperature? Within this broad scenario, we focus on magnetic systems. A surprising observation of the effect during the para- to ferromagnetic transition, in a simple model system, viz., the nearest-neighbor Ising model, without any built-in frustration, is described. Some associated future directions are pointed out. A discussion is provided by considering the effect as a kinetic outcome in the background of critical phenomena. A picture is drawn by putting emphasis on the role of spatial correlations in the initial configurations alongside discussing the importance of frustration and metastability in evolution from one state to another. In connection with dynamical freezing, concerning metastability, we have introduced the complex Ginzburg-Landau equation that has relevance in phase transitions, chemical oscillations, and elsewhere. For this model and a few other cases also, we have described how a lack of order or correlation in certain parameters can lead to quicker evolution.

Phase behavior and dynamics in a colloid-polymer mixture under spherical confinement.

From studies via molecular dynamics simulations, we report results on structure and dynamics in mixtures of active colloids and passive polymers that are confined inside a spherical container with a repulsive boundary. All interactions in the fully passive limit are chosen in such a way that in equilibrium coexistence between colloid-rich and polymer-rich phases occurs. For most part of the studies the chosen compositions give rise to Janus-like structure: nearly one side of the sphere is occupied by the colloids and the rest by the polymers. This partially wet situation mimics approximately a neutral wall in the fully passive scenario. Following the introduction of a velocity-aligning activity to the colloids, the shape of the polymer-rich domain changes to that of an ellipsoid, around the long axis of which the colloid-rich domain attains a macroscopic angular momentum. In the steady state, the orientation of this axis evolves via diffusion, magnitude of which depends upon the strength of activity, but only weakly.

Effects of alignment activity on the collapse kinetics of a flexible polymer.

The dynamics of various biological filaments can be understood within the framework of active polymer models. Here we consider a bead-spring model for a flexible polymer chain in which the active interaction among the beads is introduced via an alignment rule adapted from the Vicsek model. Following quenching from the high-temperature coil phase to a low-temperature state point, we study the coarsening kinetics via molecular dynamics (MD) simulations using the Langevin thermostat. For the passive polymer case the low-temperature equilibrium state is a compact globule. The results from our MD simulations reveal that though the globular state is also the typical final state in the active case, the nonequilibrium pathways to arrive at such a state differ from the picture for the passive case due to the alignment interaction among the beads. We notice that deviations from the intermediate "pearl-necklace"-like arrangement, which is observed in the passive case, and the formation of more elongated dumbbell-like structures increase with increasing activity. Furthermore, it appears that while a small active force on the beads certainly makes the coarsening process much faster, there exists a nonmonotonic dependence of the collapse time on the strength of active interaction. We quantify these observations by comparing the scaling laws for the collapse time and growth of pearls with the passive case.

How do clusters in phase-separating active matter systems grow? A study for Vicsek activity in systems undergoing vapor-solid transition.

Via molecular dynamics simulations, we have studied the kinetics of vapor-"solid" phase transition in an active matter model in which self-propulsion is introduced via the well-known Vicsek rule. The overall density of the particles is chosen in such a way that the evolution morphology consists of disconnected clusters that are defined as regions of high density of particles. Our focus has been on understanding the influence of the above-mentioned self-propulsion on structure and growth of these clusters by comparing the results with those for the passive limit of the model that also exhibits vapor-"solid" transition. While in the passive case growth occurs due to a standard diffusive mechanism, the Vicsek activity leads to very rapid growth, via a process that is practically equivalent to the ballistic aggregation mechanism. The emerging growth law in the latter case has been accurately estimated and explained by invoking information on velocity and structural aspects of the clusters into a relevant theory. Some of these results are also discussed with reference to a model for active Brownian particles.

Should a hotter paramagnet transform quicker to a ferromagnet? Monte Carlo simulation results for Ising model.

For quicker formation of ice, before inserting inside a refrigerator, heating up of a body of water can be beneficial. We report first observation of a counterpart of this intriguing fact, referred to as the Mpemba effect (ME), during ordering in ferromagnets. By performing Monte Carlo simulations of a generic model, we have obtained results on relaxation of systems that are quenched to sub-critical state points from various temperatures above the critical point. For a fixed final temperature, a system with higher starting temperature equilibrates faster than the one prepared at a lower temperature, implying the presence of ME. The observation is extremely counter-intuitive, particularly because of the fact that the model has no in-built frustration or metastability that typically is thought to provide ME. Via the calculations of nonequilibrium properties concerning structure and energy, we quantify the role of critical fluctuations behind this fundamental as well as technologically relevant observation.

Relaxation in a phase-separating two-dimensional active matter system with alignment interaction.

Via computer simulations, we study kinetics of pattern formation in a two-dimensional active matter system. Self-propulsion in our model is incorporated via the Vicsek-like activity, i.e., particles have the tendency of aligning their velocities with the average directions of motion of their neighbors. In addition to this dynamic or active interaction, there exists passive inter-particle interaction in the model for which we have chosen the standard Lennard-Jones form. Following quenches of homogeneous configurations to a point deep inside the region of coexistence between high and low density phases, as the systems exhibit formation and evolution of particle-rich clusters, we investigate properties related to the morphology, growth, and aging. A focus of our study is on the understanding of the effects of structure on growth and aging. To quantify the latter, we use the two-time order-parameter autocorrelation function. This correlation, as well as the growth, is observed to follow power-law time dependence, qualitatively similar to the scaling behavior reported for passive systems. The values of the exponents have been estimated and discussed by comparing with the previously obtained numbers for other dimensions as well as with the new results for the passive limit of the considered model. We have also presented results on the effects of temperature on the activity mediated phase separation.

Aging phenomena during phase separation in fluids: decay of autocorrelation for vapor-liquid transitions.

We performed molecular dynamics simulations to study relaxation phenomena during vapor-liquid transitions in a single component Lennard-Jones system. Results from two different overall densities are presented: one in the neighborhood of the vapor branch of the coexistence curve and the other being close to the critical density. The nonequilibrium morphologies, growth mechanisms and growth laws in the two cases are vastly different. In the low density case growth occurs via diffusive coalescence of droplets in a disconnected morphology. On the other hand, the elongated structure in the higher density case grows via advective transport of particles inside the tube-like liquid domains. The objective in this work has been to identify how the decay of the order-parameter autocorrelation, an important quantity to understand aging dynamics, differs in the two cases. In the case of the disconnected morphology, we observe a very robust power-law decay, as a function of the ratio of the characteristic lengths at the observation time and at the age of the system, whereas the results for the percolating structure appear rather complex. To quantify the decay in the latter case, unlike the standard method followed in a previous study, here we have performed a finite-size scaling analysis. The outcome of this analysis shows the presence of a strong preasymptotic correction, while revealing that in this case also, albeit in the asymptotic limit, the decay follows a power-law. Even though the corresponding exponents in the two cases differ drastically, this study, combined with a few recent ones, suggests that power-law behavior of this correlation function is rather universal in coarsening dynamics.

Finite-size scaling study of aging during coarsening in non-conserved Ising model: The case of zero temperature quench.

Following quenches from random initial configurations to zero temperature, we study aging during evolution of the ferromagnetic (nonconserved) Ising model towards equilibrium, via Monte Carlo simulations of very large systems, in space dimensions d = 2 and 3. Results for the two-time autocorrelations exhibit scaling with respect to ℓ/ℓw, where ℓ and ℓw are the average domain sizes at t and tw(⩽t), the observation and waiting times, respectively. The scaling functions are shown to be of power-law type for ℓ/ℓw → ∞. The exponents of these power-laws have been estimated via a novel application of the finite-size scaling method and discussed with reference to the available results from non-zero temperatures. While in d = 2 we do not observe any temperature dependence, in the case of d = 3 the outcome for quench to zero temperature appears different from the available results for high temperatures, which we explain via structural consideration. We also present results on the freezing phenomena that this model exhibits at zero temperature. Furthermore, from simulations of a very large system, thereby avoiding the freezing effect, it has been confirmed that the growth of average domain size in d = 3, that remained a puzzle in the literature, follows the Lifshitz-Allen-Cahn law in the asymptotic limit. We presented results for different acceptance probabilities for the spin flip trial moves. We observe slower growth for lower probability, even though the asymptotic exponent remains the same.

Kinetics of Vapor-Solid Phase Transitions: Structure, Growth, and Mechanism.

The kinetics of the separation between low and high density phases in a single component Lennard-Jones model is studied via molecular dynamics simulations, at very low temperatures, in the space dimension d=2. For densities close to the vapor branch of the coexistence curve, disconnected nanoscale clusters of the high density phase exhibit essentially ballistic motion. Starting from nearly circular shapes, at the time of nucleation, these clusters grow via sticky collisions, gaining filamentlike nonequilibrium structure at a later time, with a very low fractal dimensionality. The origin of the latter is shown to lie in the low mobility of the constituent particles, in the corresponding cluster reference frame, due to the (quasi-long-range) crystalline order. Standard self-similarity in the domain pattern, typically observed in the kinetics of phase transitions, is found to be absent. This invalidates the common method, that provides a growth law comparable to that in solid mixtures, of quantifying growth. An appropriate alternative approach, involving the fractality, quantifies the growth of the characteristic "length" to be a power law with time, the exponent being strongly temperature dependent. The observed growth law is in agreement with the outcome of a nonequilibrium kinetic theory.

Pattern, growth, and aging in aggregation kinetics of a Vicsek-like active matter model.

Via molecular dynamics simulations, we study kinetics in a Vicsek-like phase-separating active matter model. Quantitative results, for isotropic bicontinuous pattern, are presented on the structure, growth, and aging. These are obtained via the two-point equal-time density-density correlation function, the average domain length, and the two-time density autocorrelation function. Both the correlation functions exhibit basic scaling properties, implying self-similarity in the pattern dynamics, for which the average domain size exhibits a power-law growth in time. The equal-time correlation has a short distance behavior that provides reasonable agreement between the corresponding structure factor tail and the Porod law. The autocorrelation decay is a power-law in the average domain size. Apart from these basic similarities, the overall quantitative behavior of the above-mentioned observables is found to be vastly different from those of the corresponding passive limit of the model which also undergoes phase separation. The functional forms of these have been quantified. An exceptionally rapid growth in the active system occurs due to fast coherent motion of the particles, mean-squared-displacements of which exhibit multiple scaling regimes, including a long time ballistic one.

Finite-size scaling study of dynamic critical phenomena in a vapor-liquid transition.

Via a combination of molecular dynamics (MD) simulations and finite-size scaling (FSS) analysis, we study dynamic critical phenomena for the vapor-liquid transition in a three dimensional Lennard-Jones system. The phase behavior of the model has been obtained via the Monte Carlo simulations. The transport properties, viz., the bulk viscosity and the thermal conductivity, are calculated via the Green-Kubo relations, by taking inputs from the MD simulations in the microcanonical ensemble. The critical singularities of these quantities are estimated via the FSS method. The results thus obtained are in nice agreement with the predictions of the dynamic renormalization group and mode-coupling theories.

Droplet growth during vapor-liquid transition in a 2D Lennard-Jones fluid.

Results for the kinetics of vapor-liquid phase transition have been presented from the molecular dynamics simulations of a single component two-dimensional Lennard-Jones fluid. The phase diagram for the model, primary prerequisite for this purpose, has been obtained via the Monte Carlo simulations. Our focus is on the region very close to the vapor branch of the coexistence curve. Quenches to such region provide morphology that consists of disconnected circular clusters in the vapor background. We identified that these clusters exhibit diffusive motion and grow via sticky collisions among them. The growth follows power-law behavior with time, exponent of which is found to be in nice agreement with a theoretical prediction.

Activity mediated phase separation: Can we understand phase behavior of the nonequilibrium problem from an equilibrium approach?

We present results for structure and dynamics of mixtures of active and passive particles, from molecular dynamics (MD) simulations and integral equation theory (IET) calculations, for a physically motivated model. The perfectly passive limit of the model corresponds to the phase-separating Asakura-Oosawa model for colloid-polymer mixtures in which, for the present study, the colloids are made self-propelling by introducing activity in accordance with the well known Vicsek model. Such activity facilitates phase separation further, as confirmed by our MD simulations and IET calculations. Depending upon the composition of active and passive particles, the diffusive motion of the active species can only be realized at large time scales. Despite this, we have been able to construct an equilibrium approach to obtain the structural properties of such inherently out-of-equilibrium systems. In this method, effective inter-particle potentials were constructed via IET by taking structural inputs from the MD simulations of the active system. These potentials in turn were used in passive MD simulations, results from which are observed to be in fair agreement with the original ones.

Study of critical dynamics in fluids via molecular dynamics in canonical ensemble.

With the objective of understanding the usefulness of thermostats in the study of dynamic critical phenomena in fluids, we present results for transport properties in a binary Lennard-Jones fluid that exhibits liquid-liquid phase transition. Various collective transport properties, calculated from the molecular dynamics (MD) simulations in canonical ensemble, with different thermostats, are compared with those obtained from MD simulations in microcanonical ensemble. It is observed that the Nosé-Hoover and dissipative particle dynamics thermostats are useful for the calculations of mutual diffusivity and shear viscosity. The Nosé-Hoover thermostat, however, as opposed to the latter, appears inadequate for the study of bulk viscosity.

Phase behavior of active swimmers in depletants: molecular dynamics and integral equation theory.

We study the structure and phase behavior of a binary mixture where one of the components is self-propelling in nature. The interparticle interactions in the system are taken from the Asakura-Oosawa model for colloid-polymer mixtures for which the phase diagram is known. In the current model version, the colloid particles are made active using the Vicsek model for self-propelling particles. The resultant active system is studied by molecular dynamics methods and integral equation theory. Both methods produce results consistent with each other and demonstrate that the Vicsek model-based activity facilitates phase separation, thus, broadening the coexistence region.

Finite-size scaling study of shear viscosity anomaly at liquid-liquid criticality.

We study the equilibrium dynamics of a symmetrical binary Lennard-Jones fluid mixture near its consolute criticality. Molecular dynamics simulation results for the shear viscosity, η, from a microcanonical ensemble are compared with those from a canonical ensemble with various thermostats. It is observed that the Nosé-Hoover thermostat is a good candidate for this purpose, and is therefore adopted for the quantification of the critical singularity of η, to avoid the temperature fluctuations (or even drifts) that are often encountered in microcanonical simulations. Via a finite-size scaling analysis of our simulation data we have been able to confirm that the shear viscosity exhibits a weak critical singularity in agreement with the theoretical predictions.

Nonuniversal aging during phase separation with long-range interaction.

Issues concerning the kinetics of phase transitions are not well established for the cases where the order parameter remains conserved with time, particularly when the interatomic interactions are long-range in nature. Here we present results on structure, growth, and aging from Monte Carlo simulations of the two-dimensional long-range Ising model. In our computer simulations, random initial configurations, for 50:50 compositions of up and down spins, mimicking high-temperature equilibrium states, have been quenched to temperatures inside the coexistence curve. Our analyses of the simulation data, for such a protocol, show interesting dependence of the aging exponent, λ, on σ, the parameter, within the Hamiltonian, that controls the range of interaction. These nonuniversal values of λ are compared with a theoretical result for lower bounds. For this purpose, we extracted information on relevant aspects of structural properties during the evolution. To estimate λ, as is necessary, we also calculated the average domain size and analyzed its time dependence to obtain the growth exponent α which also is nonuniversal. The trends in the values of λ and α, as well as an anomaly in structure, suggest that a crossover from the long-range to the short-range variety occurs at σ≃1. The location of this boundary and the nonuniversality provide a picture that is surprisingly different from that of the corresponding static critical phenomena. Furthermore, our results suggest an important scaling law combining α and λ.

Effects of density conservation and hydrodynamics on aging in nonequilibrium processes.

Aging in kinetics of three different phase transitions, viz., magnetic, a binary solid, and a single component fluid, are studied via Monte Carlo and molecular dynamics simulations in three space dimensions with the objective of identifying the effects of order-parameter conservation and hydrodynamics. We observe that the relevant autocorrelations exhibit power-law decay in a ferromagnet and binary solid but with different exponents. At early time the fluid autocorrelation function nicely follows that of the binary solid, the order parameter being conserved for both of them, as opposed to a ferromagnet. At a late time the fluid data crosses over to an exponential decay which we identify as a hydrodynamic effect and we provide analytical justification for this behavior.

A comparative study of nonequilibrium dynamics in complex and real Ginzburg-Landau equations.

Complex and real Ginzburg-Landau equations have been numerically studied by implementing Euler discretization technique. In addition to characterizing the differences and similarities of patterns involving these two continuum dynamical equations, in a wide range of appropriate parameter space, we have also made quantitative comparisons of growth dynamics in the two cases. In most part of the above-mentioned parameter space the complex Ginzburg-Landau equation exhibits frozen spiral dynamics. Results on the unlocking of this freezing are also presented.

Temperature and composition dependence of kinetics of phase separation in solid binary mixtures.

We present results on the effects of temperature variation and composition in the kinetics of phase separation in solid binary mixtures (A1 + A2) from Monte Carlo simulations of the Ising model in two dimensions. The simulation results are understood via an appropriate application of the finite-size scaling theory. At moderately high temperatures, for symmetric (50 : 50) compositions of A1 and A2 particles the average size of the domains exhibits power-law growth with the exponent having a Lifshitz-Slyozov value of 1/3 from very early time. However, our analysis shows that for low enough temperatures, the growth exponent at an early time is smaller than the Lifshitz-Slyozov value. For composition dependence, we find that at moderate temperature, even for extreme off-critical composition, the curvature dependent correction to the growth law is weak which is counter-intuitive in case of droplet morphology. This is, however, consistent with the recent understanding on the curvature dependence of surface tension. Results from rather general studies on the finite-size effects with the variations of temperature and composition have also been presented.