Solid-liquid Coexistence Research Articles

Liquid-Solid Equilibrium and Intermediate Phase Formation during Solidification in Mg-1.3 at%Zn-1.7 at%Y Alloy

Mg-Y-Zn alloys have attracted much attention in recent years due to their excellent mechanical properties, which are said to be the effect of the intermediate phase (X phase) in the Mg solid solution (α phase). The formation behavior of the X phase has been reported to be a eutectic type reaction in the Mg-Y-Zn pseudo-binary system. However, the phase diagram of this system still requires clarification. In the present study, the 97.0Mg-1.3Zn-1.7Y (at%) alloy was prepared to clarify the liquidus and solidus in the liquid-solid coexistence region, and the solubility limit in the α phase. The chemical compositions of the alloy were analyzed by electron probe X-ray microanalysis after isothermal heat treatment. The results indicate that the Mg-Y-Zn ternary system can be represented as a pseudo-binary system between the α and X phases. In addition, the solidus line for the α phase has been clarified. The solubility limit for the α phase is not much different from that of the previous report, whereas the formation behavior of the X phase is not in the manner of a eutectic reaction, but rather that of a peritectic one.

The solid-to-liquid transition region of an Ag–Pd bimetallic nanocluster

The solid–liquid transition region of an Ag–Pd bimetallic cluster composed of 201 Ag and201 Pd atoms is investigated by means of molecular dynamics simulation. An effective andconvenient method is reported for finding that region. For external energy, Agand Pd clusters were caused to collide with various initial velocities under thecondition of a microcanonical ensemble. The Ag201–Pd201 bimetallic cluster wasobserved to completely melt at 900 K with a solid–liquid coexistence range of 50 K.

Phase diagram for a model of urate oxidase

Urate oxidase from Asperigillus flavus has been shown to be a model protein for studying the effects of polyethylene glycol (PEG) on the crystallization of large proteins. Extensive experimental studies based on small angle x-ray scattering [Vivares and Bonnete, J. Phys. Chem. B 108, 6498 (2004)] have determined the effects of salt, pH, temperature, and most importantly PEG on the crystallization of this protein. Recently, some aspects of the phase diagram have also been determined experimentally. In this paper, we use Monte Carlo techniques to predict the phase diagram for urate oxidase in solution with PEG, including the liquid-liquid and liquid-solid coexistence curves. The model used includes an electrostatic interaction, van der Waals attraction, and a polymer-induced depletion interaction [Vivares et al., Eur. Phys. J. E 9, 15 (2002)]. Results from the simulation are compared with experimental results.

Homogeneous nucleation and growth of melt in copper

Molecular dynamics simulations are conducted to investigate homogeneous nucleation and growth of melt in copper described by an embedded-atom method (EAM) potential. The accuracy of this EAM potential for melting is validated by the equilibrium melting point obtained with the solid-liquid coexistence method and the superheating-supercooling hysteresis method. We characterize the atomistic melting process by following the temperature and time evolution of liquid atoms. The nucleation behavior at the extreme superheating is analyzed with the mean-first-passage-time (MFPT) method, which yields the critical size, steady-state nucleation rate, and the Zeldovich factor. The value of the steady-state nucleation rate obtained from the MFPT method is consistent with the result from direct simulations. The size distribution of subcritical nuclei appears to follow a power law similar to three-dimensional percolation. The diffuse solid-liquid interface has a sigmoidal profile with a 10%-90% width of about 12 A near the critical nucleation. The critical size obtained from our simulations is in reasonable agreement with the prediction of classical nucleation theory if the finite interface width is considered. The growth of melt is coupled with nucleation and can be described qualitatively with the Johnson-Meh-Avrami law. System sizes of 10(3)-10(6) atoms are explored, and negligible size dependence is found for bulk properties and for the critical nucleation.

Glass Transition Line in C60: A Mode-Coupling/Molecular-Dynamics Study

We report a study of the mode-coupling theory (MCT) glass transition line for the Girifalco model of C60 fullerene. The equilibrium static structure factor of the model, the only required input for the MCT calculations, is provided by molecular dynamics simulations. The glass transition line develops inside the metastable liquid-solid coexistence region and extends down in temperature, terminating on the liquid side of the metastable portion of the liquid-vapor binodal. The vitrification locus does not show re-entrant behavior. A comparison with previous computer simulation estimates of the location of the glass line suggests that the theory accurately reproduces the shape of the arrest line in the density-temperature plane. The theoretical HNC and MHNC structure factors (and consequently the corresponding MCT glass line) compare well with the numerical counterpart. Our results confirm the conclusion drawn in previous works about the existence of a glassy phase for the fullerene model at issue.

Freezing of Lennard-Jones fluid in cylindrical nanopores under tensile conditions

We have shown the Lennard-Jones (LJ) phase diagram for a slit-shaped nanopore by molecular simulations and thermodynamically predicted the results with no adjustable parameter. With this success, LJ phase diagrams are predictable. In this study, the freezing of an LJ CH4 capillary condensate under a tensile condition in a nonstructural carbon nanopore with a cylindrical geometry was examined using molecular dynamics (MD) simulation. We employ a unit cell in contact with a bulk vapor phase, which is useful for the determination of the bulk vapor pressure in equilibrium with the molecules in a pore. The MD simulation results show liquid-solid (amorphous) phase transitions with a variation in the bulk vapor pressure. The frozen particles are arranged in concentric circular regions along the wall similar to those reported by Maddox and Gubbins. The freezing points are determined from the variations in density, enthalpy, arrangement, and structural functions. The obtained liquid-solid coexistence points are found to exhibit a significant dependence of the freezing point on the equilibrium bulk vapor pressure, forming an extraordinarily skewed curve on the p-T diagram, in contrast to the bulk phase coexistence that is represented by an almost vertical line. The origin of the significant dependence is considered to be the Laplace effect on the capillary condensate similar to the case with a slit-shaped pore. A simple model, which the authors earlier presented for slit-shaped nanopores, successfully predicted the p-T relation of the freezing point for cylindrical nanopores as well.

Solid–liquid phase equilibria for mixtures containing diatomic Lennard–Jones molecules

Solid—liquid phase diagrams are calculated for systems containing pure Lennard–Jones dumbbell molecules and their mixtures. We begin by calculating the equations of state for systems containing C 2H 6, CO 2 and F 2, all of which are modeled by a two-center Lennard–Jones potential for linear diatomic molecules. We then use the Frenkel–Ladd thermodynamic integration method to calculate the free energies. The equations of state and the free energies are used to obtain solid–liquid coexistence points which are needed to start the Gibbs–Duhem integration. The solid–liquid phase equilibria for pure C 2H 6, CO 2 and F 2 and binary mixtures of Lennard–Jones dumbbells are predicted using the Gibbs–Duhem integration method. We investigate the effects of molecular size and intermolecular attractions on the solid–liquid phase diagrams of binary Lennard–Jones mixtures.

New parametrization method for dissipative particle dynamics

We introduce an improved method of parametrizing the Groot-Warren version of dissipative particle dynamics (DPD) by exploiting a correspondence between DPD and Scatchard-Hildebrand regular solution theory. The new parametrization scheme widens the realm of applicability of DPD by first removing the restriction of equal repulsive interactions between like beads, and second, by relating all conservative interactions between beads directly to cohesive energy densities. We establish the correspondence by deriving an expression for the Helmoltz free energy of mixing, obtaining a heat of mixing which is exactly the same form as that for a regular mixture (quadratic in the volume fraction) and an entropy of mixing which reduces to the ideal entropy of mixing for equal molar volumes. We equate the conservative interaction parameters in the DPD force law to the cohesive energy densities of the pure fluids, providing an alternative method of calculating the self-interaction parameters as well as a route to the cross interaction parameter. We validate the new parametrization by modeling the binary system SnI(4)SiCl(4), which displays liquid-liquid coexistence below an upper critical solution temperature around 140 degrees C. A series of DPD simulations were conducted at a set of temperatures ranging from 0 degrees C to above the experimental upper critical solution temperature using conservative parameters based on extrapolated experimental data. These simulations can be regarded as being equivalent to a quench from a high temperature to a lower one at constant volume. Our simulations recover the expected phase behavior ranging from solid-liquid coexistence to liquid-liquid coexistence and eventually leading to a homogeneous single phase system. The results yield a binodal curve in close agreement with the one predicted using regular solution theory, but, significantly, in closer agreement with actual solubility measurements.

Determining landscape-based criteria for freezing of liquids

The correlation between statistical properties of the energy landscape and the number of accessible configurational states, as measured by the exponential of the excess entropy (e(Se)), are studied in the case of a simple Lennard-Jones-type liquid in the neighborhood of the thermodynamic freezing transition. The excess entropy Se is defined as the difference between the entropy of the liquid and that of the ideal gas under identical temperature and pressure conditions and is estimated using the pair correlation contribution, S2. Landscape properties associated with three categories of configurations are considered: instantaneous configurations, inherent saddles, and inherent minima. Landscape properties studied include the energy and the key parameters of the Hessian eigenvalue distribution as well as the mean distances between instantaneous configurations and the corresponding inherent saddles and minima. The signatures of the thermodynamic freezing transition are clearest in the case of inherent structure properties which show, as a function of e(S2), a pronounced change in slope in the vicinity of the solid-liquid coexistence. The mean distance between instantaneous and saddle configurations also shows a similar change in slope when the system crosses from the stable to the supercooled regime. In the case of inherent saddles, the minimum eigenvalue acts as a similar indicator of the thermodynamic freezing transition but the average and maximum eigenvalues do not carry similar signatures. In the case of instantaneous configurations, a weak indicator of the thermodynamic freezing transition is seen in the behavior of the fraction of negative curvature directions as a function of the exponential of the excess entropy.

Hydrodynamic multi-phase model for simulation of laser-induced non-equilibrium phase transformations

A hydrodynamic model based on transport equations and semi-empirical multiphase equations-of-state for metals is developed and applied to simulation of short pulse laser melting and resolidification. New computational algorithms are developed for modeling of two-phase zones of solid-liquid and liquid-gas coexistence, as well as for explicit tracking of interfaces between the phases. The model accounts for both heterogeneous and homogeneous melting mechanisms. A series of simulations are performed for bulk aluminum irradiated by a picosecond laser pulse at a wide range of laser fluence. The effect of non-equilibrium conditions and homogeneous melting on the melting/resolidification times and the maximum depth of melting is investigated. Three distinct stages are identified in the melting/resolidification process, namely, the fast homogeneous melting of the overheated surface region, a slower increase of the melting depth due to the advancement of the sharp melting front formed at the end of the homogeneous melting, and the reverse propagation of the liquid-crystal interface in recrystallization. Computational results are in a good qualitative agreement with the results of recent molecular dynamics simulations of laser melting.

Solid–liquid transitions of sodium chloride at high pressures

We investigate solid-liquid transitions in NaCl at high pressures using molecular dynamics simulations, including the melting curve and superheating/supercooling as well as solid-solid and liquid-liquid transitions. The first-order B1-B2 (NaCl-CsCl type) transition in solid is observed at high pressures besides continuous liquid structure transitions, which are largely analogous to the B1-B2 transition in solid. The equilibrium melting temperatures (T(m)) up to megabar pressure are obtained from the solid-liquid coexistence technique and the superheating-supercooling hysteresis method. Lindemann's vibrational and Born's mechanical instabilities are found upon melting. The Lindemann frequency is calculated from the vibrational density of states. The Lindemann parameter (fractional root-mean-squared displacement) increases with pressure and approaches a constant asymptotically, similar to the Lennard-Jones system. However, the Lindemann melting relation holds for both B1 and B2 phases to high accuracy as for the Lennard-Jonesium. The B1 and B2 NaCl solids can be superheated by 0.18T(m) and 0.24T(m), and the NaCl liquid, supercooled by 0.22T(m) and 0.32T(m), respectively, at heating or cooling rates of 1 K/s and 1 K/ps. The amount of maximum superheating or supercooling and its weak pressure dependence observed for NaCl are in accord with experiments on alkali halides and with simulations on the Lennard-Jones system and Al.

The anomalous melting behavior of water in aqueous PVME solutions

The Wertheim lattice thermodynamic perturbation theory is used to predict the liquid–liquid and solid–liquid coexistence data for a model polymer solution. The theory predicts bimodal LCST phase behavior and an unusual step with composition in the solid–liquid equilibrium of the solvent. The theoretical solid–liquid equilibrium calculations are used to interpret experimental data obtained for aqueous solutions of poly(vinyl methyl ether) (PVME), which is known to show bimodal LCST phase behavior. An experimental method is proposed, employing Fourier transform infrared (FTIR) spectroscopy to determine the equilibrium melting line of water in the presence of PVME. In addition, the complete melting line of water is obtained by partial integration of the melting endotherm observed using modulated temperature differential scanning calorimetry (MTDSC). Both, the FTIR and MTDSC methods are in good agreement, experimentally confirming the predicted step with composition in the solid–liquid equilibrium. This peculiar concentration dependence of the melting curve of ice provides a new explanation for the inhibited crystallization of water in aqueous PVME solutions, since the actual supercooling (at high polymer concentration) is smaller than it could be anticipated for a conventional course of the melting curve. Hence, the vicinity of the glass transition region in these highly concentrated polymer mixtures leads to a dramatic slowing down of the nucleation rate and thus the subsequent crystallization. Moreover, the atypical shape of the equilibrium melting line also provides a new explanation for the double melting endotherm observed in (MT)DSC experiments, which is conventionally attributed to the melting at different temperatures of bound and free water.

Melting, freezing and nucleation in nanoclusters of potassium chloride

In a recent article we have reported extensive molecular dynamics simulations for the melting freezing and nucleation in unconstrained nanoclusters of KCl. Based on that study we propose, in the present article, a theoretical model for the solid-liquid coexistence in finite systems, at virtually zero external pressure and no vaporisation. The main trends of the phase coexistence behaviour, namely the starting and the end points, are explained by the model as a function of system size. Other specific properties of clusters, eventually accessible by experiment, are defined and their values predicted. On the absence of available experimental data. the model is tested against simulation results with fairly good accordance.

Phase Diagrams of Binary Crystalline−Crystalline Polymer Blends

A thermodynamically self-consistent theory has been developed to establish binary phase diagrams for two-crystalline polymer blends by taking into consideration all interactions including amorphous-amorphous, crystal-amorphous, amorphous-crystal, and crystal-crystal interactions. The present theory basically involves combination of the Flory-Huggins free energy for amorphous-amorphous isotropic mixing and the Landau free energy of polymer solidification (e.g., crystallization) of the crystalline constituents. The self-consistent solution via minimization of the free energy of the mixture affords determination of eutectic, peritectic, and azeotrope phase diagrams involving various coexistence regions such as liquid-liquid, liquid-solid, and solid-solid coexistence regions bound by liquidus and solidus lines. To validate the present theory, the predicted eutectic phase diagrams have been compared with the reported experimental binary phase diagrams of blends such as polyethylene fractions as well as polycaprolactone/trioxane mixtures.

Bridge functions near the liquid-vapor coexistence curve in binary Lennard-Jones mixtures

We have carried out extensive molecular-dynamics simulation studies of binary Lennard-Jones mixtures to calculate directly the bridge function at state points lying in a very narrow single fluid phase region between the vapor-liquid and solid-liquid coexistence lines [Lamm and Hall, Fluid Phase Equilib. 182, 37 (2001); 194-197, 197 (2002)]. By varying the density close to the liquid-vapor coexistence line, significant deviations are observed at intermediate distances between the simulated bridge function and two widely used approximate closures in the integral equation theory of liquids, viz. the hybrid mean spherical approximation and the Duh-Henderson closures. The overall qualitative agreement remains the same with small variation in temperature that brings the system closer to either the liquid-vapor or liquid-solid coexistence curve. We also report a comparison of the direct and indirect correlation functions obtained from our simulation studies as well as from the integral equation theory of liquids. Our results emphasize the need for developing new closures applicable to binary fluid mixtures over a wide range of thermodynamic parameters.

Superheating and Solid-Liquid Phase Coexistence in Nanoparticles with Nonmelting Surfaces

We present a phenomenological model of melting in nanoparticles with facets that are only partially wet by their liquid phase. We show that in this model, as the solid nanoparticle seeks to avoid coexistence with the liquid, the microcanonical melting temperature can exceed the bulk melting point and that the onset of coexistence is a first-order transition. We show that these results are consistent with molecular dynamics simulations of aluminum nanoparticles which remain solid above the bulk melting temperature.

Melting, freezing and nucleation in nanoclusters of potassium chloride

Molecular dynamics simulations of the melting, freezing and nucleation are presented for unconstrained nanoclusters of KCl with a number of ions between 512 and 10648. The maximum extent of the probed liquid supercooling is analysed to the light of theoretical predictions and compared with experimental data. The fraction of the solid-like ions in the supercooled liquid is used as an indicator of heterogeneities within the liquid. Induced nucleation by seeding the supercooled liquid indicates that solid-liquid coexistence is stable, and sustained during the lifetime of the clusters, relatively to the supercooled liquid. A phenomenological analysis on the relaxation times of the crystal growth process is made. Critical nuclei sizes computed from the effectiveness of the seeds in the heterogeneous nucleation of the supercooled liquid, and from the residual crystallites in clusters not totally melted, are presented as a function of the temperature. The behavior of the systems is followed through various properties such as liquid and solid molar fractions, enthalpies of melting, heat capacities, self-diffusion coefficients and relaxation times related to the freezing process. The consistency of the simulation results for the heterogeneous nucleation is assessed by means of a classical nucleation model, from which an estimate of the interfacial surface tension is also worked out and compared with experimental data.

Role of Crystal−Amorphous Interaction in Phase Equilibria of Crystal−Amorphous Polymer Blends

A self-consistent theory has been developed for determination of phase diagrams of a crystalline polymer solution. Although the original Flory diluent theory captures the liquidus line, the theory is incapable of accounting for the solidus line due to the inherent assumption of complete immiscibility of solvent in the solid crystal. The present theory considers all possible interactions involving amorphous-amorphous and crystal-amorphous interactions. The self-consistent solutions predict various phase diagrams involving liquid-liquid, pure solid, and liquid-solid coexistence regions bound by liquidus and solidus lines. In the limit of complete insolubility of solvent in neat solid crystal, the original Flory diluent theory is recovered.

Atomistic simulation of solid-liquid coexistence for molecular systems: Application to triazole and benzene

A method recently developed to rigorously determine solid-liquid equilibrium using a free-energy-based analysis has been extended to analyze multiatom molecular systems. This method is based on using a pseudosupercritical transformation path to reversibly transform between solid and liquid phases. Integration along this path yields the free energy difference at a single state point, which can then be used to determine the free energy difference as a function of temperature and therefore locate the coexistence temperature at a fixed pressure. The primary extension reported here is the introduction of an external potential field capable of inducing center of mass order along with secondary orientational order for molecules. The method is used to calculate the melting point of 1-H-1,2,4-triazole and benzene. Despite the fact that the triazole model gives accurate bulk densities for the liquid and crystal phases, it is found to do a poor job of reproducing the experimental crystal structure and heat of fusion. Consequently, it yields a melting point that is 100 K lower than the experimental value. On the other hand, the benzene model has been parametrized extensively to match a wide range of properties and yields a melting point that is only 20 K lower than the experimental value. Previous work in which a simple "direct heating" method was used actually found that the melting point of the benzene model was 50 K higher than the experimental value. This demonstrates the importance of using proper free energy methods to compute phase behavior. It also shows that the melting point is a very sensitive measure of force field quality that should be considered in parametrization efforts. The method described here provides a relatively simple approach for computing melting points of molecular systems.

Solid-liquid phase coexistence and structural transitions in palladium clusters

We use molecular dynamics with an embedded atom potential to study the behavior of palladium nanoclusters near the melting point in the microcanonical ensemble. We see transitions from both fcc and decahedral ground-state structures to icosahedral structures prior to melting over a range of cluster sizes. In all cases this transition occurs during solid-liquid phase coexistence and the mechanism for the transition appears to be fluctuations in the molten fraction of the cluster and subsequent recrystallization into the icosahedral structure.