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Hard spheres confined to narrow quasi-one-dimensional cylindrical channels form perfect helical structures at close packing. Here, we use molecular dynamics simulation to show that the thermodynamics, structure and dynamics of the fluid below close packing are dominated by the presence of topological defects that reverse the local twist direction of the helix. When compressed from a random, low density state, or decompressed from high density ordered states with zero excess helical twist, the system equilibrates to an achiral fluid that exhibits two heat capacity maxima along the equation of state. The low density heat capacity maximum corresponds to the onset of helix formation and the high density maximum occurs when the system rapidly loses defects in a Schottky-like anomaly. The local twist auto-correlation function in the achiral fluid exhibits a stretched exponential decay and the structural relaxation times undergo a fragile-to-strong crossover located at the high density heat capacity maximum. We also study the effect of excess helical twist by using initial starting configurations consisting of two helical domains with opposite twist directions of different lengths. This leads to the formation of topologically protected states that are characterized by the presence of loosely bound defect pairs which become more tightly bound with increasing excess helical twist. The local twist auto-correlation function in the chiral fluid decays as a power law at long times. The possible kinetic or thermodynamic origin of this topological protection is discussed., Comment: New results, extended discussion. 13 figures, 16 pages

It has been established empirically that the rate of addition of molecules to the crystal during crystal growth from the melt is proportional to exp(-|{\Delta}S_fus|/R) where {\Delta}S_fus is the entropy of fusion. Here we show that this entropic slowdown arises directly from the separation of the entropy loss and energy loss processes associated with the freezing of the liquid. We present a theoretical treatment of the kinetics based on a model flat energy landscape and derive an explicit expression for the coupling magnitude in terms of the crystal-melt interfacial free energy. The implications of our work for nucleation kinetics are also discussed.

Granular materials, such as sand or grain, exhibit many structural and dynamic characteristics similar to those observed in molecular systems, despite temperature playing no role in their properties. This has led to an effort to develop a statistical mechanics for granular materials that has focused on establishing an equivalent to the microcanonical ensemble and a temperature-like thermodynamic variable. Here, we expand on these ideas by introducing a granular potential into the Edwards ensemble, as an analogue to the chemical potential, and explore its properties using a simple model of a granular system. A simple kinetic Monte Carlo simulation of the model shows the effect of mass transport leading to equilibrium and how this is connected to the redistribution of volume in the system. An exact analytical treatment of the model shows that the compactivity and the ratio of the granular potential to the compactivity determine the equilibrium between two open systems that are able to exchange volume and particles, and that mass moves from high to low values of this ratio. Analysis of the granular potential shows that adding a particle to the system increases the entropy at high compactivity, but decreases the entropy at low compactivity. Finally, we demonstrate the use of a small system thermodynamics method for the measurement of granular potential differences., Comment: 11 Pages, 7 Figures. Extended Introduction and Discussion. New section on Measuring granular potential

Studies of nucleation generally focus on the properties of the critical cluster, but the presence of defects within the crystal lattice means that the population of nuclei necessarily evolve through a distribution of pre-critical clusters with varying degrees of structural disorder on their way to forming a growing stable crystal. To investigate the role pre-critical clusters play in nucleation, we develop a simple thermodynamic model for crystal nucleation in terms of cluster size and the degree of cluster order that allows us to alter the work of forming the pre-critical clusters without effecting the properties of the critical cluster. The steady state and transient nucleation behaviour of the system are then studied numerically, for different microscopic ordering kinetics. We find that the models exhibits a generic order-disorder transition in the pre-critical clusters. Independent of the type of ordering kinetics, increasing the accessibility of disordered pre-critical clusters decreases both the steady state nucleation rate and the nucleation lag time. Furthermore, the interplay between the free energy surface and the microscopic ordering kinetics leads to three distinct nucleation pathways., Comment: 9 pages. 5 Figures in main text. 3 Figures in Supplemental Information

The inherent structure landscape for a system of hard spheres confined to a hard cylindrical channel, such that spheres can only contact their first and second neighbours, is studied using an analytical model that extends previous results [Phys. Rev. Lett. 115, 025702 (2015)] to provide a comprehensive picture of jammed packings over a range of packing densities. In the model, a packing is described as an arrangement of $k$ helical sections, separated by defects, that have alternating helical twist directions and where all spheres satisfy local jamming constraints. The structure of each helical section is determined by a single helical twist angle, and a jammed packing is obtained by minimizing the length of the channel per particle with respect to the $k$ helical section angles. An analysis of a small system of $N=20$ spheres shows that the basins on the inherent structure landscape associated with these helical arrangements split into a number of distinct jammed states separated by low barriers giving rise to a degree of hierarchical organization. The model accurately predicts the geometric properties of packings generated using the Lubachevsky and Stillinger compression scheme ($N=10^4$) and provides insight into the nature of the probability distribution of helical section lengths., Comment: 11 Figures

We test the theoretical free energy surface (FES) for two-step nucleation (TSN) proposed by Iwamatsu [J. Chem. Phys. 134, 164508 (2011)] by comparing the predictions of the theory to numerical results for the FES recently reported from Monte Carlo simulations of TSN in a simple lattice system [James, et al., J. Chem. Phys. 150, 074501 (2019)]. No adjustable parameters are used to make this comparison. That is, all the parameters of the theory are evaluated directly for the model system, yielding a predicted FES which we then compare to the FES obtained from simulations. We find that the theoretical FES successfully predicts the numerically-evaluated FES over a range of thermodynamic conditions that spans distinct regimes of behavior associated with TSN. All the qualitative features of the FES are captured by the theory and the quantitative comparison is also very good. Our results demonstrate that Iwamatsu's extension of classical nucleation theory provides an excellent framework for understanding the thermodynamics of TSN., Comment: 15 pages, 15 figures

The diffusion of particles trapped in long narrow channels occurs predominantly in one dimension. Here, molecular dynamics simulation is used to study the inertial dynamics of two-dimensional hard disks, confined to long, narrow, structureless channels with hard walls in the no-passing regime. We show that the diffusion coefficient obtained from the mean squared displacement can be mapped onto the exact results for the diffusion of the strictly one-dimensional hard rod system through an effective occupied volume fraction obtained from either the equation of state or a geometric projection of the particle interaction diameters., Comment: 7 pages, 10 figures

Fluids confined to quasi-one-dimensional channels exhibit a dynamic crossover from single file diffusion to normal diffusion as the channel becomes wide enough for particles to hop past each other. In the crossover regime, where hopping events are rare, the diffusion coefficient in the long time limit can be related to a hopping time that measures the average time it takes a particle to escape the local cage formed by its neighbours. In this work, we show that a transition state theory that calculates the free energy barrier for two particles attempting to pass each other in the small system isobaric ensemble is able to quantitatively predict the hopping time in a system of two-dimensional soft repulsive discs [$U(r_{ij})=(\sigma/r_{ij})^{\alpha}$] confined to a hard walled channel over a range of channel radii and degrees of particle softness measured in terms of $1/\alpha$. The free energy barrier exhibits a maximum at intermediate values of $\alpha$ that moves to smaller values of $1/\alpha$ (harder particles) as the channel becomes narrower. However, the presence of the maximum is only observed in the hopping times for wide channels because the interaction potential dependence of the kinetic prefactor plays an increasingly important role for narrower channels. We also begin to explore how our transition state theory approach can be used to optimize and control dynamics in confined quasi-one-dimensional fluids., Comment: 9 pages, 10 figures, 1 table

We consider the thermodynamic behavior of local fluctuations occurring in a stable or metastable bulk phase. For a system with three or more phases, a simple analysis based on classical nucleation theory predicts that small fluctuations always resemble the phase having the lowest surface tension with the surrounding bulk phase, regardless of the relative chemical potentials of the phases. We also identify the conditions at which a fluctuation may convert to a different phase as its size increases, referred to here as a "fluctuation phase transition" (FPT). We demonstrate these phenonena in simulations of a two dimensional lattice model by evaluating the free energy surface that describes the thermodynamic properties of a fluctuation as a function of its size and phase composition. We show that a FPT can occur in the fluctuations of either a stable or metastable bulk phase and that the transition is first-order. We also find that the FPT is bracketed by well-defined spinodals, which place limits on the size of fluctuations of distinct phases. Furthermore, when the FPT occurs in a metastable bulk phase, we show that the superposition of the FPT on the nucleation process results in two-step nucleation (TSN). We identify distinct regimes of TSN based on the nucleation pathway in the free energy surface, and correlate these regimes to the phase diagram of the bulk system. Our results clarify the origin of TSN, and elucidate a wide variety of phenomena associated with TSN, including the Ostwald step rule., Comment: 35 pages, 12 figures

It is common practice in molecular dynamics and Monte Carlo computer simulations to run multiple, separately-initialized simulations in order to improve the sampling of independent microstates. Here we examine the utility of an extreme case of this strategy, in which we run a large ensemble of $M$ independent simulations (a "swarm"), each of which is relaxed to equilibrium. We show that if $M$ is of order $10^3$, we can monitor the swarm's relaxation to equilibrium, and confirm its attainment, within $\sim 10\bar\tau$, where $\bar\tau$ is the equilibrium relaxation time. As soon as a swarm of this size attains equilibrium, the ensemble of $M$ final microstates from each run is sufficient for the evaluation of most equilibrium properties without further sampling. This approach dramatically reduces the wall-clock time required, compared to a single long simulation, by a factor of several hundred, at the cost of an increase in the total computational effort by a small factor. It is also well-suited to modern computing systems having thousands of processors, and is a viable strategy for simulation studies that need to produce high-precision results in a minimum of wall-clock time. We present results obtained by applying this approach to several test cases., Comment: 12 pages. To appear in Eur. Phy. J. E, 2017

We use molecular simulation to study the structural and dynamic properties of glassy nanoclusters formed both through the direct condensation of the vapor below the glass transition temperature, without the presence of a substrate, and \textit{via} the slow supercooling of unsupported liquid nanodroplets. An analysis of local structure using Voronoi polyhedra shows that the energetic stability of the clusters is characterized by a large, increasing fraction of bicapped square antiprism motifs. We also show that nanoclusters with similar inherent structure energies are structurally similar, independent of their history, which suggests the supercooled clusters access the same low energy regions of the potential energy landscape as the vapor condensed clusters despite their different methods of formation. By measuring the intermediate scattering function at different radii from the cluster center, we find that the relaxation dynamics of the clusters are inhomogeneous, with the core becoming glassy above the glass transition temperature while the surface remains mobile at low temperatures. This helps the clusters sample the highly stable, low energy structures on the potential energy surface. Our work suggests the nanocluster systems are structurally more stable than the ultra-stable glassy thin films, formed through vapor deposition onto a cold substrate, but the nanoclusters do not exhibit the superheating effects characteristic of the ultra-stable glass states., Comment: 4 Figures

Particles confined to a single file, in a narrow quasi-one dimensional channel, exhibit a dynamic crossover from single file diffusion to Fickian diffusion as the channel radius increases and the particles can begin to pass each other. The long time diffusion coefficient for a system in the crossover regime can be described in terms of a hopping time, which measures the time it takes for a particle to escape the cage formed by its neighbours. In this paper, we develop a transition state theory approach to the calculation of the hopping time, using the small system isobaric--isothermal ensemble to rigorously account for the volume fluctuations associated with the size of the cage. We also describe a Monte Carlo simulation scheme that can be used to calculate the free energy barrier for particle hopping. The theory and simulation method correctly predict the hopping times for a two dimensional confined ideal gas system and a system of confined hard discs over a range of channel radii, but the method breaks down for wide channels in the hard discs case, underestimating the height of the hopping barrier due to the neglect of interactions between the small system and its surroundings., Comment: 13 Pages and & 7 Figures: Added Discussion corrected typos

Many metastable systems can nucleate to multiple competing stable or intermediate metastable states. In this work, a Potts model, subject to external fields, is used to study the competitive nucleation of two phases attempting to grow on a microscopic impurity. Monte Carlo simulations are used to calculate the free energy surfaces for the system under different conditions, where the relative stability of the phases is adjusted by changing the interaction parameters, and the nucleation rates obtained using multicomponent transition state theory are compared with the rates measured using the survival probability method. We find that the two methods predict similar nucleation rates when the free energy barrier used in the transition state theory is defined as the work required to form a critical embryo from the metastable phase. An analysis of the free energy surfaces also reveals that the competition between the nucleating phases leads to an effective drying of the impurity which slows down the nucleation rate compared to the single phase case., Comment: 8 pages, 7 Figures

The ST2 interaction potential has been used in a large number of simulation studies to explore the possibility of a liquid-liquid phase transition (LLPT) in supercooled water. Using umbrella sampling Monte Carlo simulations of ST2 water, we evaluate the free energy of formation of small ice nuclei in the supercooled liquid in the vicinity of the Widom line, the region above the critical temperature of the LLPT where a number of thermodynamic anomalies occur. Our results show that in this region there is a substantial free energy cost for the formation of small ice nuclei, demonstrating that the thermodynamic anomalies associated with the Widom line in ST2 water occur in a well-defined metastable liquid phase. On passing through the Widom line, we identify changes in the free energy to form small ice nuclei that illustrate how the thermodynamic anomalies associated with the LLPT may influence the ice nucleation process., Comment: 11 pages, 15 figures

We use a series of molecular dynamics simulations, and analytical theory, to demonstrate that a system of hard spheres confined to a narrow cylindrical channel exhibits a continuous phase transition from an isotropic fluid at low densities, to an orientationally ordered, but translationally disordered, helical fluid at high densities. The ordered fluid phase contains small sections of helix separated by topological defects that change the direction of the twist, altering the local chirailty. The defects break up the translational order, but the fluid develops long range orientational order. An analysis of the particle packings show that the length separation between defects controls the geometrical properties of the helical sections, including the orientation, and that pairs of defects experience a weak, but long range attraction resulting from entropic free volumes effects. These collective long range interactions overcome the restrictions on quasi-one-dimensional transitions, even though the particle-particle interaction is short ranged., Comment: 5 pages, 6 figs

Monte Carlo simulation is used to study the dynamical crossover from single file diffusion to normal diffusion in fluids confined to narrow channels. We show that the long time diffusion coefficients for a series of systems involving hard and soft interaction potentials can be described in terms of a hopping time that measures the time it takes for a particle to escape the cage formed by its neighbors in the pore. Free energy barriers for the particle hopping process are calculated and used to show that transition state theory effectively describes the hopping time for all the systems studied, over a range of pore diameters. Our work suggests that the combination of hopping times and transition state theory offers a useful and general framework to describe the dynamics of these highly confined fluids., Comment: 6 figures

Molecular dynamics simulations of miscible and partially miscible binary Lennard--Jones mixtures are used to study the dynamics and thermodynamics of vapor condensation onto a non-volatile liquid drop in the canonical ensemble. When the system volume is large, the driving force for condensation is low and only a submonolayer of the solvent is adsorbed onto the liquid drop. A small degree of mixing of the solvent phase into the core of the particles occurs for the miscible system. At smaller volumes complete film formation is observed and the dynamics of film growth are dominated by cluster-cluster coalescence. Mixing into the core of the droplet is also observed for partially miscible systems below an onset volume suggesting, the presence of a solubility transition. We also develop a non-volatile liquid drop model, based on the capillarity approximations, that exhibits a solubility transition between small and large drops for partially miscible mixtures and has a hysteresis loop similar to the one observed in the deliquescence of small soluble salt particles. The properties of the model are compared to our simulation results and the model is used to study the formulation of classical nucleation theory for systems with low free energy barriers., Comment: 12 Figures

We carry out umbrella sampling Monte Carlo simulations to evaluate the free energy surface of the ST2 model of water as a function two order parameters, the density and a bond-orientational order parameter. We approximate the long-range electrostatic interactions of the ST2 model using the reaction-field method. We focus on state points in the vicinity of the liquid-liquid critical point proposed for this model in earlier work. At temperatures below the predicted critical temperature we find two basins in the free energy surface, both of which have liquid-like bond orientational order, but differing in density. The pressure and temperature dependence of the shape of the free energy surface is consistent with the assignment of these two basins to the distinct low density and high density liquid phases previously predicted to occur in ST2 water., Comment: 8 pages, 9 figures

We show that a system of hard discs confined to a narrow channel exhibits a fragile-strong fluid crossover located at the maximum of the isobaric heat capacity and that the relaxation times for different channel widths fall onto a single master curve when rescaled by the relaxation times and temperatures of the crossover. Calculations of the configurational entropy and the inherent structure equation of state find that the crossover is related to properties of the jamming landscape for the model but that the Adams-Gibbs relation does not predict the relaxation behavior. We also show that a facilitated dynamics description of the system, where kinetically excited regions are identified with local packing arrangements of the discs, successfully describes the fragile-strong crossover., Comment: 5 pages 5 figs, updated text + Figs, Accepted PRL

We provide a comprehensive picture of the jamming phase diagram by connecting the athermal, granular ensemble of jammed states and the equilibrium fluid through the inherent structure paradigm for a system hard discs confined to a narrow channel. The J-line is shown to be divided into packings that are thermodynamically accessible from the equilibrium fluid and inaccessible packings. The J-point is found to occur at the transition between these two sets of packings and is located at the maximum the inherent structure distribution. A general thermodynamic argument suggests that the density of the states at the configurational entropy maximum represents a lower bound on the J-point density in hard sphere systems. Finally, we find that the granular and fluid systems only occupy the same set of inherent structures, under the same thermodynamic conditions, at two points, corresponding to zero and infinite pressures, where they sample the J-point states and the most dense packing respectively., Comment: 5 pages, 3 Figures

In simulations of the 2D Ising model, we examine heterogeneous nucleation induced by a small impurity consisting of a line of $l$ fixed spins. As $l$ increases, we identify a limit of stability beyond which the metastable phase is not defined. We evaluate the free energy barrier for nucleation of the stable phase, and show that contrary to expectation, the barrier does not vanish on approach to the limit of stability. We also demonstrate that our values for the height of the barrier yield predictions for the nucleation time (from transition state theory) and the size of the critical cluster (from the nucleation theorem) that are in excellent agreement with direct measurements, even near the limit of stability., Comment: 7 pages, 6 figures; to appear in Phys. Rev. E

The heterogeneous condensation of a Lennard-Jones vapor onto a nanoscale seed particle is studied using molecular dynamics simulations. Measuring the nucleation rate and the height of the free energy barrier using the mean first passage time method shows that the presence of a weakly interacting seed has little effect on the work of forming very small cluster embryos but accelerates the rate by lowering the barrier for larger clusters. We suggest that this results from a competition between the energetic and entropic features of cluster formation in the bulk and at the heterogeneity. As the interaction is increased, the free energy of formation is reduced for all cluster sizes. We also develop a simple phenomenological model of film formation on a small seed that captures the general features of the nucleation process for small heterogeneities. A comparison of our simulation results with the model shows that heterogeneous classical nucleation theory provides a good estimate of the critical size of the film but significantly over-estimates the size of the barrier., Comment: 9 pages, 10 figures, In Print J. Chem. Phys

Formulating a statistical mechanics for granular matter remains a significant challenge, in part, due to the difficulty associated with a complete characterization of the systems under study. We present a fully characterized model of a granular material consisting of $N$ two-dimensional, frictionless, hard discs, confined between hard walls, including a complete enumeration of all possible jammed structures. We show the properties of the jammed packings are independent of the distribution of defects within the system and that all the packings are isostatic. This suggests the assumption of equal probability for states of equal volume, which provides one possible way of constructing the equivalent of a microcanonical ensemble, is likely to be vaild for our model. An application of the second law of thermodynamics involving two subsystems in contact shows that the expected spontaneous equilibration of defects between the two is accompanied by an increase in entropy and that the equilibirum, obtained by entropy maximization, is characterized by the equality of compactivities. Finally, we explore the properties of the equivalent to the canonical ensemble for this system., Comment: Accepted PRE

In simulations of supercooled, high-density liquid silica we study a range of temperature T in which we find both crystal nucleation, as well as the characteristic dynamics of a glass forming liquid, including a breakdown of the Stokes-Einstein relation. We find that the liquid cannot be observed below a homogeneous nucleation limit (HNL) at which the liquid crystallizes faster than it can equilibrate. We show that the HNL would occur at lower T, and perhaps not at all, if the Stokes-Einstein relation were obeyed, and hence that glassy dynamics plays a central role in setting a crystallization limit on the liquid state in this case. We also explore the relation of the HNL to the Kauzmann temperature, and test for spinodal-like effects near the HNL., Comment: 4 pages, 4 figures

An exact description of the complete jamming landscape is developed for a system of hard discs of diameter $\sigma$, confined between two lines separated by a distance $1+\sqrt{3/4} < H/\sigma < 2$. By considering all possible local packing arrangements, the generalized ensemble partition function of jammed states is obtained using the transfer matrix method, which allows us to calculate the configurational entropy and the equation of state for the packings. Exploring the relationship between structural order and packing density, we find that the geometric frustration between local packing environments plays an important role in determining the density distribution of jammed states and that structural "randomness" is a non-monotonic function of packing density. Molecular dynamics simulations show that the properties of the equilibrium liquid are closely related to those of the landscape., Comment: 5 Pages, 4 figures

The number of distinguishable inherent structures of a liquid is the key component to understanding the thermodynamics of glass formers. In the case of hard potential systems such as hard discs, spheres and ellipsoids, an inherent structure corresponds to a collectively jammed configuration. This work develops a tiling based approach to counting inherent structures that constructs packings by combining sets of elementary locally jammed structures but eliminates those final packings that either, do not tile space, or are not collectively jammed, through the use of tile incompatibility rules. The resulting theory contains a number of geometric quantities, such as the number of growth sites on a tile and the number of tile compatibilities that provide insight into the number of inherent structures in certain limits. We also show that these geometric quantities become quite simple in a system of highly confined hard discs., Comment: 12 pages, 2 figures. Accepted for Publication : Journal of Non-Crystalline Solids

The free energy of a 600-atom Lennard-Jones cluster is calculated as a function of surface and bulk crystallinity in order to study the structural transformations that occur in the core of medium sized clusters. Within the order parameter range studied, we find the existence of two free energy minima at temperatures near freezing. One minimum, at low values of both bulk and surface order, belongs to the liquid phase. The second minimum exhibits a highly ordered core with a disordered surface and is related to structures containing a single FCC-tetrahedral subunit, with an edge length of seven atoms (l=7), located in the particle core. At lower temperatures, a third minimum appears at intermediate values of the bulk order parameter which is shown to be related to the formation of multiple l=6 tetrahedra in the core of the cluster. We also use molecular dynamics simulations to follow a series of nucleation events and find that the clusters freeze to structures containing l=5,6,7 and 8 sized tetrahedra as well as those containing no tetrahedral units. The structural correlations between bulk and surface order with the size of the tetrahedral units in the cluster core are examined. Finally, the relationships between the formation of FCC tetrahedral subunits in the core, the phase behavior of medium sized clusters and the nucleation of noncrystalline global structures such as icosahedra and decahedra are discussed., Comment: 8 pages, 7 figures

We test classical nucleation theory (CNT) in the case of simulations of deeply supercooled, high density liquid silica, as modelled by the BKS potential. We find that at density $\rho=4.38$~g/cm$^3$, spontaneous nucleation of crystalline stishovite occurs in conventional molecular dynamics simulations at temperature T=3000 K, and we evaluate the nucleation rate J directly at this T via "brute force" sampling of nucleation events. We then use parallel, constrained Monte Carlo simulations to evaluate $\Delta G(n)$, the free energy to form a crystalline embryo containing n silicon atoms, at T=3000, 3100, 3200 and 3300 K. We find that the prediction of CNT for the n-dependence of $\Delta G(n)$ fits reasonably well to the data at all T studied, and at 3300 K yields a chemical potential difference between liquid and stishovite that matches independent calculation. We find that $n^*$, the size of the critical nucleus, is approximately 10 silicon atoms at T=3300 K. At 3000 K, $n^*$ decreases to approximately 3, and at such small sizes methodological challenges arise in the evaluation of $\Delta G(n)$ when using standard techniques; indeed even the thermodynamic stability of the supercooled liquid comes into question under these conditions. We therefore present a modified approach that permits an estimation of $\Delta G(n)$ at 3000 K. Finally, we directly evaluate at T=3000 K the kinetic prefactors in the CNT expression for J, and find physically reasonable values; e.g. the diffusion length that Si atoms must travel in order to move from the liquid to the crystal embryo is approximately 0.2 nm. We are thereby able to compare the results for J at 3000 K obtained both directly and based on CNT, and find that they agree within an order of magnitude., Comment: corrected calculation, new figure, accepted in JCP