Solid Coexistence Research Articles

Multiple-Phase Hydrate Equilibria of the Ternary Carbon Dioxide, Methane, and Water Mixtures

Three-phase equilibria consisting of vapor, water-rich liquid, and solid hydrate were measured for the aqueous solutions containing two guest molecules of carbon dioxide and methane in the temperature range of 272−284 K and at pressures of 15, 20, 26, 35, and 50 bar. At the specified isobaric condition the three-phase equilibrium temperatures become higher as the relative concentrations of carbon dioxide to methane increase. The upper quadruple points at which the four phases of vapor, water-rich liquid, CO2-rich liquid, and solid hydrate coexist were also measured to examine the phase boundary of the hydrate stability region. At five different temperatures the concentrations of vapor, water-rich liquid, and carbon dioxide-rich liquid were measured and examined through five triangular diagrams. For the prediction of hydrate phase equilibria, the vapor and liquid phases were treated with the Soave−Redlich−Kwong equation of state (SRK-EOS) incorporated with the second-order modified Huron−Vidal (MHV2) mixin...

Monte Carlo simulations of complete phase diagrams for binary Lennard–Jones mixtures

Vapor–liquid, vapor–solid, liquid–liquid, and liquid–solid coexistence lines are calculated for binary mixtures of Lennard–Jones spheres with diameter ratio σ 11/ σ 22=0.85, well-depth ratio ϵ 11/ ϵ 22=0.45, and binary interaction parameters δ 12=1.0, 0.9, and 0.75, using Monte Carlo simulation and the Gibbs–Duhem integration technique. These calculations allow us to construct complete phase diagrams, i.e. showing equilibrium between vapor, liquid, and solid phases. For the mixture with δ 12=1, we find a completely miscible vapor–liquid coexistence region with a eutectic solid–liquid coexistence region. These two regions are separated by a completely miscible liquid phase. For the mixtures with δ 12<1, we find that the vapor–liquid and solid–liquid coexistence regions interfere. This interference results in a vapor–solid coexistence region bounded above and below by solid–liquid–vapor coexistence lines. We also find that the mixtures with δ 12<1 have a region of liquid–liquid immiscibility that is metastable with respect to the solid–fluid phase equilibria.

Ab Initio Calculation of Phase Diagrams of Oxides

We show how Monte Carlo simulations with the explicit interchange of cations, the semigrand-canonical ensemble and configurational bias techniques, can be used to calculate phase diagrams for oxides, including both solid and liquid phases. We illustrate our approach with the system CaO/MgO where our techniques take full account of local structural distortion and clustering due to the large mismatch between the sizes of the two cations. All the characteristic features of the MgO/CaO phase diagram, including the eutectic point and the regions of liquid−solid coexistence, are reproduced.

Phase behavior of C60 by computer simulation using ab-initio interaction potential

A first-principles intermolecular potential recently proposed by Pacheco and Ramalho [Phys Rev Lett 1997, 79, 3873–3876] has been used with the Gibbs ensemble and Gibbs–Duhem integration Monte Carlo methods to simulate the vapor–liquid and fluid–solid coexistence properties of C60. The critical properties were calculated by fitting the results to the laws of rectilinear diameters and order parameter scaling. The triple-point properties were determined from the limiting behavior of the Gibbs ensemble vapor–liquid simulations at the lowest temperature range. A stable liquid phase is predicted for temperatures between 1570±20 and 2006±27 K and densities between 0.444±0.003 and 1.05±0.01 nm−3. The estimated critical and triple-point pressures are, respectively, 35±6 and 5±16 bars. We show for the first time, to our knowledge, that it is possible, strictly by computer simulation, to estimate a triple point for C60 in accordance with the predictions of theoretical methods and the basic concepts of thermodynamics. The liquid and fluid radial distribution functions indicate the presence of solid or glasslike features. This may support the suggestion of a more cooperative interaction of clusters in C60. A comparison of our results with the data obtained by other authors is presented and discussed. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem 84: 375–387, 2001

Phase diagram, morphology development and vulcanization induced phase separation in blends of syndiotactic polypropylene and ethylene–propylene diene terpolymer

Miscibility phase diagram in blends of syndiotactic polypropylene (sPP) and ethylene–propylene diene terpolymer has been established by means of differential scanning calorimetry, light scattering and optical microscopy. In descending order of temperature, the olefinic blends reveal a complex phase diagram involving a lower critical solution temperature (LCST), a single phase region followed by a depression of melting transition and a crystal transition (crystallization) of sPP. Below the liquid–solid crystal coexistence line, phase separation occurs in competition with crystallization of sPP in the blends. Temporal evolution of structure factors and the emergence of phase separated domains in these blends have been investigated by time-resolved light scattering and optical microscopy following several temperature quenches from a single phase into the LCST immiscibility gap. The temporal evolution of structure factors has been analyzed in the context of nonlinear dynamical scaling laws. Vulcanization induced phase separation has been undertaken at various temperatures in the single-phase region. Of particular interest is that the higher the reaction temperature (i.e. the faster the chemical reaction), the smaller the domain size.

Monte Carlo Study of Dense Monolayer and Bilayer Films on the (100) Plane of Face-Centered Cubic Crystals

A Monte Carlo simulation method in the canonical and in the grand canonical ensembles is used to study the behavior and properties of dense monolayer and bilayer films formed on the (100) plane of model face-centered cubic crystals. Systems with different effects due to the periodicity of the gas−solid potential are considered, and the mechanism of melting in the first and the second adsorbed layer is discussed. It is demonstrated that the film structure is very sensitive to the gas−solid potential corrugation, as well as to the temperature and the surface coverage. In particular, it is shown that monolayer films formed on weakly corrugated surfaces exhibit the incommensurate (dense) phase at sufficiently low temperatures, and the two-dimensional liquid phase of lower density, when the temperature is higher than a certain transition temperature, which depends on the surface corrugation. In the case of intermediate corrugation, the incommensurate solid coexists with the registered (1 × 1) phase, while for highly corrugated surfaces only the registered monolayers are stable. It is also shown that the melting temperature for the first layer is very sensitive to the surface corrugation, while in the second layer we do not observe such strong effects.

Solid–liquid equilibria for solutions of binary globular‐protein mixtures

AbstractA theory of the van der Waals form is used to establish solid–fluid phase diagrams for aqueous solutions containing two kinds of globular proteins. Theory is based on a one‐fluid model that uses a composition‐dependent hard‐sphere diameter and a composition‐dependent solvent‐mediated effective temperature to represent protein solutions. The hard‐sphere reference equation of state is based on the model of Young that correlates computer simulations of fluid–solid coexistence curves for binary hard‐sphere mixtures. The attractive perturbation term uses the inverse‐power potential with variable exponent n. Based on the correlation of George and Wilson and Rosenbaum et al., the optimum range of effective temperature is determined for protein crystallization. For binary protein systems, as the size difference increases, the mutual solubility of unlike proteins declines sharply in the precipitated solid phase. The theoretical results obtained here may be useful for design of protein‐separation processes by crystallization.

Structural evolution of phase-separating model colloidal liquids by Brownian dynamics computer simulation

Brownian dynamics, (BD), simulation has been used to follow the phase separation of Lennard-Jones-type particles quenched from a supercritical state point into the vapor–liquid or vapor–solid co-existence parts of their phase diagrams. Calculations were performed with spherical particles interacting via 12:6, 24:12, and 36:18 interaction laws at subcritical temperatures and low-volume fractions (φ⩽0.2). Structural properties were followed as the systems evolved using pictures of the configurations, radial distribution function, and the low-angle scattering peak of the structure factor. The time dependence of the interaction energy was also followed. The scaling behavior of these quantities as a function of time was found to be similar to that observed in light scattering experiments during the phase separation of real colloidal systems. The aggregate structure that developed with time was sensitive to the range of the attractive part of the potential and its underlying phase diagram (the 36:18 system does not have a liquid phase). The 12:6 systems soon formed compact structures, whereas the systems generated using the shorter-ranged potentials persisted in a more diffuse, tenuous network for the duration of the simulations. Apart from at very short times for all potential laws, the only convincing evidence for a long-lived fractal structure was for the 36:18 systems at the lowest quench temperatures (kT/ε=0.3, where ε is the depth of the potential). The local structure in the dense regions of the network was sensitive to the range of the potential, exhibiting in the vapor–solid co-existence part of the phase diagram glassylike features for the 12:6 systems and crystalline local order for the 24:12 and 38:18 states. The 12:6 systems close to the metastable region of the vapor–liquid two-phase part of the phase diagram exhibited latency in the appearance and growth of the small angle scattering peak. The 24:12 and 36:18 systems also displayed latency at the higher temperatures both in the growth of the peak height and its movement to lower scattering vectors.

Spinodal-Assisted Crystallization in Polymer Melts

Recent experiments in some polymer melts quenched below the melting temperature have reported spinodal kinetics in small-angle X ray scattering before the emergence of crystalline structure. To explain these observations we propose that the coupling between density and chain conformation induces a liquid-liquid binodal within the equilibrium liquid--crystalline solid coexistence region. A simple phenomenological theory is developed to illustrate this idea, and several experimentally testable consequences are discussed. Shear is shown to enhance the kinetic role of the hidden binodal.

Size is everything

Bulk matter melts and freezes in a predictable manner. Not so clusters of atoms in their tens, hundreds or thousands, which have very different properties. They have no fixed melting temperature at a given pressure, but instead have melting ranges of temperature in which solid and liquid coexist. New work provides the first quantitative demonstration of how the melting temperature of such clusters depends on their size. But the pattern that emerges is highly irregular, and the challenge of finding its physical basis remains.

On thermodynamics of supercooled liquids and glasses

Thermodynamics of glass-forming liquids and glasses is considered taking into account the contribution of heterophase fluctuations to the free energy. The role of heterophase fluctuations in liquids and glasses is played by nuclei of noncrystalline solids and nuclei of the liquid phase respectively. Crystalline solid state is not considered. Free energy is described on the bases of the droplet model and the model of mixed state. It is shown that three regions of stability of different structural states of the system exist on the (P,T) plane. Two of them are stability regions for droplet structural states, while the mixed state is stable in the third region in which heterophase fluctuations have the form of isolated drops. In this state, two infinite percolating clusters (liquid and noncrystalline solid) coexist. The width of the temperature interval [Tt1,Tt2] in which the mixed state is stable increases with configuration entropy (complexity) of the noncrystalline solid fraction. In the approach used, weak first-order phase transitions are observed at T=Tt1 and T=Tt2. It is shown that first- and second-order phase transitions are possible in the mixed state. The criteria for the existence of these phase transitions are obtained. Equations of state and expressions for thermodynamic parameters (entropy, heat capacity, compressibility, and thermal expansion coefficient) are derived for the droplet and mixed states.

Precipitation in a CrCr 2Nb alloy

The microstructure of a CrCr 2Nb two-phase alloy containing 12 at. % Nb was examined in the as-cast and annealed conditions using optical and electron microscopy techniques. In the as-cast condition, a Cr solid-solution matrix and a eutectic mixture of Cr solid solution and C15 Cr 2Nb coexist. In the Cr matrix, adjacent to the CrCr 2Nb interface, dislocations are observed and are likely a consequence of the thermal mismatch between the two phases. The matrix solid solution, away from the eutectic Cr 2Nb, also exhibits a high dislocation density and intense deformation bands; no solid-state precipitation of Cr 2Nb in the matrix had occurred. Upon annealing for as little as 3 h at 1200°C, however, precipitation occurs; these precipitates, with the C15 crystal structure, have a rod morphology with a hexagonal cross-section and faceted end-caps. These precipitates are twinned, the twinning system being the same as in the eutectic Cr 2Nb. Mismatch dislocations are observed in the precipitate/matrix interface and dislocations arising from thermal contraction mismatch are also present in the Cr matrix adjacent to the precipitate/matrix interface. The precipitate-matrix orientation relationship was (101) m//(1 1 ) p and [ 1 01] m// 1 01] p. Exposure at 1200°C for extended times results in some precipitate coarsening. A second type of precipitate with a cube-on-cube orientation with the matrix and a lattice parameter of a ≅ 11.5 A ̊ was occasionally observed, persisted upon extended annealing, and was identified as an M 6C-type carbide (M = Cr, Nb) that may be stabilized by other interstitial contaminants such as oxygen.

Equation of state of the hard-disc solid

We present a study of the equation of state of the hard-disc solid, including the freezing transition. The study is based on a weighted-density approximation scheme for the free-energy density functional of hard discs. Using a standard procedure to construct such a functional in terms of a suitably chosen weighting function, augmented to include a cubic (rather than quadratic) approximation for the density dependence of the latter, we obtain the equation of state of the solid and use it to locate the densities where solid and fluid coexist. The equation of state is compared with constant-pressure Monte Carlo simulation data obtained by us and reasonable agreement is found. Also the location of the freezing transition is compared with available estimates based on extensive simulations performed by other groups. Although our mean-field-type theory does not take proper account of fluctuations and defects (which are believed to be important to determine the nature of this transition) are not treated at all, its prediction for the transition densities are quantitatively correct; however, the density gap at the (first-order) transition is overestimated. In addition, we observe that our results for the location of the transition crucially depend on how accurately the weighting function models the direct correlation function of the uniform fluid: As this description becomes better, the agreement with simulations improves.

Bimodality in the dynamic response of a supercooled liquid

In order to understand the emergence and the influence of the “cage effect” in the dynamic response of a liquid, a detailed theoretical study of the frequency dependent friction of a Lennard-Jones liquid is carried out by using the mode coupling theory. The frequency dependent friction becomes increasingly bimodal in nature as the density of the liquid is increased beyond the liquid–solid coexistence line. The bimodality is most clearly manifested in the frequency dependence of the imaginary part of the friction. The mode coupling theory provides a nice physical interpretation of these results which are also in good agreement with the existing computer simulations.

Phase diagrams of systems of particles interacting via repulsive potentials

We use a recently developed density-functional perturbation theory, which has been applied successfully to predict phase diagrams of systems of attractive particles, to describe the phase diagram of particles interacting via repulsive potentials. We consider potentials composed of a hard-sphere core plus a repulsive term. Specifically, we have investigated square shoulder and repulsive Yukawa terms. We show that, when the range of the interaction is very short, the shoulder potential leads to solid–solid coexistence involving two face-centered cubic structures, in analogy to an attractive square-well potential. Comparison with simulation results shows that the theory is quantitatively correct. If the range of the potentials is sufficiently long, we also find that a body-centered cubic structure can be stabilized. By considering the phase behavior at zero temperature, we argue that several triple points, involving coexistence of fluid and/or solid phases, may occur. A repulsive Yukawa term also shows a region of body-centered cubic stability but, contrary to the square shoulder and attractive Yukawa cases, there is no isostructural solid-solid coexistence. The role of the functional dependence of the interaction potential on particle separation at short distances is discussed and shown to be crucial to generate a solid-solid transition in systems of repulsive particles. Available computer simulation results for this system indicate that the density-functional approximation for the hard-sphere system used in this work, as well as all other currently available approximations, although qualitatively correct, may be overestimating the stability of the body-centered cubic phase.

Novel procedure to determine coexistence lines by computer simulation. Application to hard-core Yukawa model for charge-stabilized colloids

We describe a novel procedure to determine a phase coexistence line by computer simulation. The procedure combines elements of the Clausius–Clapeyron integration technique with free-energy difference calculations. As an application, the phase diagram of a simple hard-core Yukawa model of a charge-stabilized colloid, which exhibits liquid–solid and solid–solid phase coexistence lines, is computed.

Numerical study of freezing in polydisperse colloidal suspensions

We have established the solid - fluid coexistence region for a system of polydisperse hard spheres with near-Gaussian diameter distributions, as a function of polydispersity. Significantly, we observe a `terminal' polydispersity above which there can be no fluid - solid coexistence. At the terminus the polydispersity is 5.7% for the solid and 11.8% for the fluid while the volume fractions are 0.588 and 0.547 for the solid and fluid respectively. Substantial fractionation observed at high values of the polydispersity (> 5%) implies that the `constrained eutectic' assumption made in previous theoretical studies is not generally valid. Our results for the terminal polydispersity are consistent with experiments performed on polydisperse colloidal suspensions.

Fluid-solid equilibrium of a charged hard-sphere model.

The fluid-solid equilibrium of a system made of charged hard spheres with positive and negative ions of the same size is considered. At high temperatures freezing occurs in a substitutionally disordered close packed structure, the face centered cubic solid (fcc). At low temperatures freezing occurs in the ordered cesium chloride structure (CsCl). As the latter solid coexists with the fcc structure at high densities, two triple points exist on the phase diagram. By using computer simulation we determine the precise location of both triple points. In the first of them, vapor, liquid, and solid (CsCl) are in equilibrium at ${\mathit{T}}^{\mathrm{*}}$=T/[${\mathit{q}}^{2}$/(\ensuremath{\sigma}k\ensuremath{\epsilon})]=0.0225, where q is the charge of the ions, \ensuremath{\sigma} their diameters, k the Boltzmann constant, and \ensuremath{\epsilon} the dielectric constant. In the other triple point, occurring at ${\mathit{T}}^{\mathrm{*}}$=0.24, the three coexisting phases are the fluid, a CsCl solid, and the fcc solid. The vapor-liquid-solid triple point temperature is found to be about one-third of the critical temperature, in good agreement with the experimental ratio for a number of molten salts. An implementation of the cell theory for the solid phases of charged hard spheres is presented. It is shown that this simple theory provides a reasonable description of the properties of solid charged hard spheres. When the theory is combined with an accurate theory for the fluid phase a very satisfactory description of the phase diagram of charged hard spheres is obtained. The cell theory predictions are better than those recently reported using a density functional scheme. \textcopyright{} 1996 The American Physical Society.

Nonergodicity and two subphases in the coexistence region in isomerization dynamics of Ar7-like molecules

It is well-known that a single cluster like Ar7 undergoes ‘‘melting’’ from solidlike to liquidlike states as the energy is increased, the transition of which is not as sudden as the ordinary phase transition though and has a somewhat broad energy range in which solid and liquid coexist. We study a very anomalous dynamics of the coexistence region in the structural isomerization. It is explicitly shown that the time-series of the structural changes both in the purely solidlike and liquidlike phases are stationary, while the coexistence region is found to generate a strongly nonstationary dynamics. The calculated distribution of the residing times for the cluster to stay in one of the possible structures exhibits a nonexponential form having a large hole around the zero lifetime in the coexistence region. Motivated by these strange behaviors, we have calculated the phase-space volumes that are assigned to the individual potential basins, and verified directly that while the pure liquid region is of ergodic nature, the dynamics in the coexistence region is indeed strongly nonergodic. The steep rises of the Lindemann index and the maximum Liapunov exponent in the coexistence region, which were reported before by other authors, are found to be ascribed to the statistical nature rather than the dynamical properties as opposed to the picture suggested by the physical meaning of the indices. It also turns out that the energy range for the coexistence region should be taken wider than considered before and thus extends beyond the ‘‘melting point’’ that is defined usually on the basis of the Lindemann index. Therefore it is appropriate to divide the coexistence region into two subphases. A ‘‘temperature’’ in a microcanonical ensemble is defined so as to characterize the distribution of phase-space volume on a given energy plane. Based on this distribution, we describe a statistical reason why the onset energy of the melting is much higher than those of the transition states.

Density-functional theory and atomistic simulation of the hard-sphere melt-solid interface.

The melt-solid interface of hard spheres is studied by atomistic simulations and density-functional theory (DFT). The Monte Carlo method is used to compute density profiles for the three most important orientations of the fcc-solid--melt interface. The interface ranges in thickness from 4d for the loosely packed (110) interface to 6d for the closely packed (111) interface, where d is the hard-sphere diameter. The planar generalized effective liquid approximation (PGELA) free energy functional, which accurately predicts solid and liquid free energies and coexistence conditions, is used in the DFT analysis of the hard-sphere melt-solid interface. Two parametrizations are compared for variations of the density in the interfacial region. The structural predictions of the interface by the Carnahan-Starling PGELA are in good agreement with the simulations. The calculations of the surface free energies show that the (110) interface has a smaller surface free energy, followed by the (100) interface and then closely by the (111) interface. These results are compared to other recent studies of the hard-sphere melt-solid interface.