Solid-liquid Coexistence Research Articles

Direct calculation of solid-liquid coexistence points of a binary mixture by thermodynamic integration

We present a new thermodynamic integration method that directly connects the liquid and the solid phases of a binary mixture by a reversible path. The states along the path are simulated in the isothermal-isobaric semigrand canonical ensemble, in which temperature, pressure, the total number of particles, and the fugacity fractions of the components are held fixed. The thermodynamic integration yields the chemical-potential difference between the two phases for one of the components and this information is then used to locate the solid-liquid coexistence points. The melting temperatures predicted by our method agree well with those predicted by the Gibbs-Duhem integration for a truncated and shifted Lennard-Jones system with a cutoff radius of 2.5sigma.

Transition from Icosahedral to Decahedral Structure in a Coexisting Solid-Liquid Nickel Cluster

We have used molecular dynamics simulations to construct a microcanonical caloric curve for a 1415 atom Ni icosahedron. Prior to melting, the Ni cluster exhibits static solid-liquid phase coexistence. Initially, a partial icosahedral structure coexists with a partially wetting melt. However, at energies very close to the melting point the icosahedral structure is replaced by a truncated decahedral structure that is almost fully wet by the melt. This structure remains until the cluster fully melts. The transition appears to be driven by a preference for the melt to wet the decahedral structure.

Time autocorrelation function analysis of master equation and its application to atomic clusters

We derive the energy fluctuation Delta(2)E, and the time autocorrelation kappa(tau) and its Fourier transformation--the fluctuation spectra S(omega)--of the master-equation transition matrix. The contribution from each eigenmode of the transition matrix to these fluctuation quantities reveals the relevant importance of the individual mode in the relaxation processes. The time scales associated with these relaxation processes are determined by the corresponding eigenvalues. Unlike traditional time evolution analysis, the autocorrelation function and fluctuation spectra analysis does not involve an arbitrary initial population. It is also more suitable for analyzing the underlying dynamic, kinetic behavior near the equilibrium and the behavior of the long-time-scale rare events. We utilize our technique to analyze the solid-liquid phase coexistence of the 13-atom Morse cluster and the fcc-to-icosahedral structure transition of the 38-atom Lennard-Jones cluster. For the processes studied, the fluctuation spectra from the master equation simplify the analysis of the transition matrix, and the important relaxation modes are easily extracted.

Nature and extent of melting in superheated solids: Liquid-solid coexistence model

The nature and extent of superheating in solids have remained an open problem for nearly a century. Using a liquid-solid coexistence model, we critically examined melting in the superheating regime. It is found that the solid/liquid density difference and the interface disordering at melting are two key factors determining the nature of the transition and the extent of superheating. A nucleation forbidden gap is found due to the density difference. The nucleation barrier rising from the interface is reduced by the interface disordering. The results obtained from the theoretical model are in excellent agreement with those in atomistic simulations.

The Nonequilibrium Phase and Glass Transition Behavior of β-Lactoglobulin

Concentrated solutions of bovine β-lactoglobulin were studied using osmotic stress and rheological techniques. At pH 6.0 and 8.0, the osmotic pressure was largely independent of NaCl concentration and could be described by a hard sphere equation of state. At pH 5.1, close to the isoelectric point, the osmotic pressure was lower at the lower NaCl concentrations (0 mM, 100 mM) and was fitted by an adhesive hard sphere model. Liquid-liquid phase separation was observed at pH 5.1 at ionic strengths of 13 mM and below. Comparison of the liquid-liquid and literature solid-liquid coexistence curves showed these solutions to be supersaturated and the phase separation to be nonequilibrium in nature. In steady shear, the zero shear viscosity of concentrated solutions at pH 5.1 was observed at shear rates above 50 s −1. With increasing concentration, the solution viscosity showed a progressive increase, a behavior interpreted as the approach to a colloidlike glass transition at ∼60% w/w. In oscillatory shear experiments, the storage modulus crossed the loss modulus at concentrations of 54% w/w, an indication of the approaching glass transition. Comparison of the viscous behavior with predictions from the Krieger-Dougherty equation indicates the hydrodynamic size of the protein decreases with increasing concentration, resulting in a slower approach to the glass transition than a hard sphere system.

Liquid gallium in confined droplets under high-temperature and high-pressure conditions

Phase transitions and the local structure of micrometric droplets of liquid gallium under pressure were studied by combining extended x-ray absorption fine structure, single-energy x-ray absorption detection (SEXAD), and energy-scanning x-ray diffraction (ESXD). Measurements were performed in a range of pressures and temperatures of $0--6.7\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ and $298--440\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, respectively. The samples for the high-pressure measurements were obtained by using an emulsion of gallium into epoxy resin, a procedure previously developed by the authors. The distribution of droplets was fully characterized by scanning electron microscopy. We found that the liquid can be kept in a metastable state well beyond the liquid-solid coexistence line ($1.9\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$ at $300\phantom{\rule{0.3em}{0ex}}\mathrm{K}$). Considering both the ESXD patterns and the SEXAD scans, we infer that the quantity of crystallized gallium droplets increases as a function of pressure, while no sign of crystallization is observed up to $2.7\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. The structural and crystallization properties of Ga emulsions, including the determination of the short-range radial distribution function, were measured by XAS in an extended range of pressures and temperatures, putting to a test the possible existence of different Ga-liquid polymorphs.

Raman Scattering Spectroscopic Study of n-Pentane under High Pressure

The Raman spectroscopy of n-pentane was investigated in a Moissanite anvil cell from 0.07 GPa to 4.77 GPa at ambient temperature. The result shows that the CH3 symmetric stretching vibration (2877 cm(-1)) and asymmetric stretching vibration (2964 cm(-1)), the CH2 symmetric stretching vibration (2856 cm(-1)) and asymmetric stretching vibration (2935 cm(-1)), and -(CH2)n- in-phase twist mode (1303 cm(-1)) shifted to higher wavenumbers almost linearly with increasing pressure. Around 2.4 GPa an abrupt visible change took place, indicating a sort of phenomenon of freezing due to over-pressurization. In the pressure range of 2.84 to 4.77 GPa a high-pressure phase transition may occur in the crystallized n-pentane. By determining pressure with the method of solid-liquid coexistence, we concluded that the equilibrium freezing pressure of n-pentane is 1.90 +/- 0.05 GPa at 27 degrees C.

Roughening transition of prism faces of ice crystals grown from melt under pressure

By observing the crystal shapes and kinetics of ice Ih growing at high pressures from the melt, it was found that the prism { 1 0 1 ¯ 0 } faces undergo a roughening transition at approximately - 16 ∘ C and 160 MPa (1600 bar) in H 2 O ice, and - 14 ∘ C and 180 MPa (1800 bar) in D 2 O ice. At temperatures above the roughening transition, the prism facets completely vanished and the curves of growth rate versus growth drive were linear. In contrast, below the roughening temperature, regions of prism facets existed and the growth rate curves were approximately quadratic. The basal { 0 0 0 1 } faces persisted up to 0 ∘ C and gave approximately quadratic growth rate curves. The melting curves of ice Ih for H 2 O and D 2 O were accurately measured by direct observations of the solid–liquid coexistence equilibrium and found to agree with previous results obtained with other methods.

2330 5052アルミニウム合金のセミソリッド連続鋳造(S30 半溶融・半凝固加工の最先端)

In this study, we performed semisolid continuous casting of the 5052 aluminum alloy using an inclined cooling plate to obtain the finest grain size in the cast structure. Although the solid-liquid coexistence zone of the 5052 alloy is narrower than that of the AC4CH alloy, the inclined cooling plate is useful to obtain the semisolid slurry of this alloy. The casting conditions in this experiment were pouring temperatures onto the inclined cooling plate, length and angles of the cooling plate and the casting speed. The billet surface is smooth and the average grain size in the cast structure is 60μm except the outer surface.

Experimental Evaluation of Chemical Composition and Viscosity of Melts during Iron Ore Sintering

The viscosity of the melts formed in the sintering reaction was measured for evaluating the fluidity of the melts, which play a crucial role in the sintering process. Using the sphere draw-up method carried out the measurements of high temperature viscosity. With low-melting point compositions of the Fe2O3-CaO-SiO2 system, the viscosity of high SiO2 content melts (Fe2O3/CaO/SiO2=38.5/34.5/27.0 weight basis) was 5 times higher than that of low SiO2 content melts (Fe2O3/CaO=80/20). The viscosity of low SiO2 content increased with addition of Al2O3 and SiO2. Similarly, with melt compositions generated from actual ores, viscosity depended on the contents of Al2O3 and SiO2.Based on the results, the viscosity of the melted liquid was estimated from the chemical composition, and the viscosity of the solid-liquid coexistence phase was also evaluated using a viscosity calculating equation for suspensions. The viscosity distribution of the solid-liquid coexistence phase at 1 300°C in the ternary composition system Fe2O3-CaO-SiO2 was obtained.

セミソリッド鋳造した５０５２アルミニウム合金の凝固組織

In this study, we performed semisolid casting of the 5052 aluminum alloy using an inclined cooling plate to find the casting conditions for the finest grain size in the solidified structure. The casting conditions in this experiment were pouring temperatures onto the inclined cooling plate, length and angles of the cooling plate, and thickness of BN coating on the cooling plate. Although the solid-liquid coexistence zone of the 5052 alloy is narrower than that of the AC4CH alloy, the inclined cooling plate is useful, and we obtained an average grain size of 60 μm. The BN coating film must be as thin as possible because of the significant undercooling of the molten alloy. Greater undercooling causes more increased nucleation and more increased number of crystals to grow on the cooling plate. As a result, fine and granular crystals were obtained. The thermal boundary layer between the BN coating layer and the molten alloy, and the undercooling of the molten alloy were influenced by the flow speed of the molten alloy on the plate.

Toward a robust and general molecular simulation method for computing solid-liquid coexistence

A rigorous and generally applicable method for computing solid-liquid coexistence is presented. The method overcomes some of the technical difficulties associated with other solid-liquid simulation procedures and can be implemented within either a molecular dynamics or Monte Carlo framework. The method consists of three steps: First, relative Gibbs free energy curves are created for the solid and liquid phases using histogram reweighting. Next, the free energy difference between the solid and liquid phases is evaluated at a single state point by integrating along a pseudosupercritical transformation path that connects the two phases. Using this result, the solid and liquid free energy curves are referenced to a common point, allowing a single coexistence point to be determined. Finally, Gibbs-Duhem integration is used to determine the full coexistence curve. To evaluate its utility, this method is applied to the Lennard-Jones and NaCl systems. Results for solid-liquid coexistence agree with previous calculations for these systems. In addition, it is shown that the NaCl model does not correctly describe solid-liquid coexistence at high pressures. An analysis of the accuracy of the method indicates that the results are most sensitive to the transformation free energy calculation.

In-situ observation of alloy phase formation in nanometre-sized particles in the Sn–Bi system

Alloy phase formation in nanometre-sized particles has been studied by in-situ transmission electron microscopy using particles in the Sn–Bi system. Observations have been carried out at a temperature (i.e. 350 K) above the eutectic temperature. For tin-rich compositions, with increasing concentration of bismuth, a particle (approximately 8 nm in size) of the terminal tin solid solution changed directly into a particle of the liquid phase, without going through the stage of solid–liquid coexistence. On the other hand, for bismuth-rich compositions, a particle (approximately 8 nm in size) of the terminal bismuth solid solution changed first to a particle with a crystal–liquid two-phase microstructure and eventually to a particle of the liquid phase, with increasing concentration of tin. Thermodynamic model calculations indicate that the contribution of the solid–liquid interfacial energy to the total Gibbs free energy of an alloy particle with a solid–liquid two-phase microstructure becomes large enough to change the phase equilibrium when the size of the particle is reduced to the nanometre range, and that the difference between the alloy phase formation for the tin-rich and bismuth-rich alloys observed here can be consistently explained in terms of the difference between the relative contributions of the solid–liquid interfacial energy for the two types of alloy.

Calculations of free energies in liquid and solid phases: fundamental measure density-functional approach.

In this paper, a theoretical description of the free energies and correlation functions of hard-sphere (HS) liquid and solid phases is developed using fundamental measure density-functional theory. Within the framework of Weeks-Chandler-Andersen perturbation theory, free energies of liquid and solid phases with many interaction potentials can be obtained from these characteristics of the HS system within a single theoretical description. An application to the Lennard-Jones system yields liquid-solid coexistence results in good agreement with the ones from simulations.

Nonequilibrium melting and crystallization of a model Lennard-Jones system.

Nonequilibrium melting and crystallization of a model Lennard-Jones system were investigated with molecular dynamics simulations to quantify the maximum superheating/supercooling at fixed pressure, and over-pressurization/over-depressurization at fixed temperature. The temperature and pressure hystereses were found to be equivalent with regard to the Gibbs free energy barrier for nucleation of liquid or solid. These results place upper bounds on hysteretic effects of solidification and melting in high heating- and strain-rate experiments such as shock wave loading and release. The authors also demonstrate that the equilibrium melting temperature at a given pressure can be obtained directly from temperatures at the maximum superheating and supercooling on the temperature hysteresis; this approach, called the hysteresis method, is a conceptually simple and computationally inexpensive alternative to solid-liquid coexistence simulation and thermodynamic integration methods, and should be regarded as a general method. We also found that the extent of maximum superheating/supercooling is weakly pressure dependent, and the solid-liquid interfacial energy increases with pressure. The Lindemann fractional root-mean-squared displacement of solid and liquid at equilibrium and extreme metastable states is quantified, and is predicted to remain constant (0.14) at high pressures for solid at the equilibrium melting temperature.

Solid–liquid coexistence in ionic systems and the properties of the interface

The crystal–melt interfaces of pure KCl and LiCl crystals with (Li,K)Cl melts of various compositions have been investigated by direct simulation of coexisting crystalline and molten regions. This system forms no solid compounds involving both lithium and potassium and exhibits a eutectic close to the equimolar composition. Simulations at various melt compositions allow the liquidus to be determined and compared with experiment. In order to test the precision achievable by a direct simulation of coexisting liquid and solid phases for the thermodynamic properties, we make preliminary calculations on the melting of a model of NaCl, which has recently been studied by Anwar et al. [2003, J. chem. Phys., 118, 728] using thermodynamic integration. Including anionic polarization, effects in the interionic interaction model lower the melting temperature, despite the fact that there is no discernible effect on the liquid or solid structure in these simple monovalent melts. The method of direct simulation of coexistence also allows for the study of the interfacial structure and dynamics. These can be compared with previous studies [Davidchack, R. L., and Laird, B. B., 1998, J. chem. Phys., 108, 9452; Sibug-Aga, R., and Laird, B. B., 2002, J. chem. Phys., 116, 3410] of the solid–liquid interfaces in binary mixtures of hard spheres of sufficiently different size ratio to prevent solid compound formation. Despite the additional Coulombic interactions the ionic interfaces resemble the hard spheres in many regards, but there are differences. For example, when traversing the interface the perpendicular diffusion rises more rapidly than the parallel diffusion, most likely due to charge ordering.

Constrained fluid lambda-integration: constructing a reversible thermodynamic path between the solid and liquid state.

A novel lambda-integration path is proposed for calculating the Gibbs free energy difference between any arbitrary solid and liquid state needed for the location of melting lines. This technique involves reversibly forcing a liquid state to a solid state across the phase transition along a nonphysical path, thermodynamically coupling the two states directly. The process eliminates the need for coupling to idealized reference states as is presently performed and hence simplifies the location of phase transitions for computer simulation systems. More specifically the path involves a three stage process, whereby, initially a liquid state is transformed to a weakly attractive fluid using linear lambda-integration scaling of the intermolecular potential. In the second stage, the resulting fluid is then constrained to the required solid configurational phase space via the insertion of a periodic lattice of 3D Gaussian wells. The final stage involves reversing to full strength the main intermolecular potential while gradually turning off the constraining 3D Gaussian lattice finally resulting in a stable (or metastable) solid state. Each stage was found to be completely reversible and the resulting change in free energy was thermodynamically integrable. The methodology is demonstrated and validated by calculating solid-liquid coexistence points using the new technique and comparing to those in present literature for the truncated and shifted Lennard-Jones system. The results are found to be in good agreement. The new method is not limited to melting phase transitions and is readily applicable to any simulation methodology, simulation cell size and/or intermolecular potential including ab initio methods.

Measurement of the Densities of Nickel-Based Ternary, Quaternary and Commercial Alloys

The modified sessile drop method (MSDM) and the modified pycnometric method (MPM) have been employed to measure precisely the densities of liquid nickel-based ternary (Ni-Co-Al) and quaternary (Ni-Co-Al-Cr, Ni-Co-Al-Mo, Ni-Cr-Al-Mo) alloys and four commercial superalloys in liquid and mushy states. There was a good agreement between the values measured by the two methods. The measured densities of liquid model alloys and commercial superalloys in liquid and mushy states decreased linearly with increasing temperature for the experimental temperature range. The temperature coefficient of the density of liquid ternary Ni-Co-Al alloys measured in this work can be represented as a quadratic function of the aluminium concentration in the alloys. The density of liquid Ni-Co-Al alloys can be expressed as a function of both temperature and aluminium concentration in the alloys. The recommended equations for the densities of liquid nickel-based ternary and quaternary alloys and four commercial superalloys in liquid and mushy states were obtained by analysing density values measured by both MSDM and MPM. The calculated values from the recommended density equations show good agreement with those measured by both MSDM and MPM. The density (� ) and its temperature dependency (d�= dT) are important input data for simulation of the solidification of Ni-based superalloys, e.g. the prediction of defects such as gas porosity and microsegregation, as well as the evaluation of the convective contribution to heat transport. Accurate experimental values of the densities of nickel and Ni-based model and commercial alloys (including their dependencies on composition and temperature) are needed to develop models to predict the densities of Ni-based superalloys. In order to predict the densities of Ni-based commercial alloys, it is necessary to have an understanding of the effects of the various alloy elements (such as Cr, Co, Ta, W, Al, Mo, Re, Hf, Nb, Ti, etc.) on the densities of liquid nickel alloys. This can be obtained from accurate measurements of the densities of Ni-based binary alloys in relation to the concentration and temperature. Comparisons of the densities of liquid Ni-based ternary alloys with those for the corresponding binary alloys provide useful information on the effects of the second alloy element on the properties of liquid alloys. For some alloy elements, such as Re, their concentrations in the Ni-based superalloys are relatively low but they are very reactive, which may make the accurate density measurement of their Ni-based binary alloys difficult. The effect of these elements on the densities of Ni-based superalloys can be obtained by comparing the densities of liquid Ni-based commercial alloys with those of liquid Ni- based multi-component model alloys with similar chemical compositions to those of the Ni-based commercial alloys. Furthermore, accurate density values of both Ni-based model and commercial alloys can be used to verify the validity of the models developed to predict the densities of liquid Ni- based superalloys. However, there is a large scatter in the reported density values for liquid nickel and its temperature dependence because of the experimental uncertainties associated with the different techniques. Furthermore, there are very few reports on the measurement of the density of Ni-based alloys in relation to the effects of composition and temperature. 1-7) There are no reports on the measurement of the densities of liquid Ni-based multi-component model alloys such as ternary and quaternary alloys. Therefore, in order to develop a model to predict the densities of commercial Ni-based superalloys, we have used two accurate methods, namely, the modified sessile drop method (MSDM) and the modified pycnometric method (MPM) to measure the densities of liquid nickel and Ni-based model alloys and commercial alloys in the liquid state and in the mushy (ie ''liquid-solid coexistence'') state for some alloys. The principles, advantages and experimental proce- dures of the two methods have been described previously. 5-7) The densities of liquid nickel and liquid Ni-Cr, Ni-Co, Ni-Ta, Ni-W and Ni-Al alloys have been measured precisely with both methods, and density values for Ni-Cr and Ni-Al alloys in mushy state have also been determined. 5-7) In this paper, both MSDM and MPM were employed to measure the densities of liquid ternary model alloys (Ni-Co- Al, molar fraction xNi:xCo � 86:14, Al mass% = 0� 11), liquid quaternary model alloys (Ni-Co-Al-Cr, Ni-Co-Al-Mo and Ni-Cr-Al-Mo), and four commercial Ni-based super- alloys in liquid and mushy states. The chemical compositions of these model alloys were similar to those of commercial Ni- based superalloys. The recommended density values for these alloys have been obtained by combining the measured values determined with the two methods.

Effect of pressure on the complete phase behavior of binary mixtures

AbstractUsing Gibbs‐Duhem integration and semigrand canonical Monte Carlo simulations, temperature vs. composition phase diagrams for a binary Lennard‐Jones mixture, σ11/σ22 = 0.85 and ϵ11/ϵ22 = 1.6, are calculated at several reduced pressures in order to examine the effects of pressure on complete phase behavior (that is, equilibrium between vapor, liquid, and solid phases). Interference is observed between the vapor–liquid and solid–liquid coexistence regions at the lowest pressure. As the pressure increases, the vapor–liquid coexistence region shifts to higher temperatures, while the solid–liquid coexistence region remains essentially unaffected. Eventually, the vapor–liquid coexistence region lifts off the solid–liquid coexistence region, ending the interference. Pressure vs. temperature projections for binary Lennard‐Jones mixtures at σ11/σ22 = 0.85, 0.9, and 0.95, and ϵ11/ϵ22 = 0.45 and 1.6 are also presented to explore how the three‐phase loci (solid–liquid–vapor and solid–solid–liquid) change with variations in diameter ratio and well‐depth ratio. It is found that as the diameter ratio decreases, the maximum pressure in the solid–liquid–vapor locus decreases and the characteristic shape of the solid–liquid coexistence region changes from peritectic to eutectic. As the well‐depth ratio decreases, the maximum pressure in the solid–liquid–vapor locus increases. For one mixture, σ11/σ22 = 0.9 and ϵ11/ϵ22 = 1.6, a quadruple point of solid–solid–liquid–vapor coexistence is located. © 2004 American Institute of Chemical Engineers AIChE J, 50: 215–225, 2004

Measurement of the Density of Ni-Cr Alloy by a Modified Pycnometric Method

A modified pycnometric method has been developed to obtain the accurate density value of liquid nickel and Ni-Cr alloy in the liquid and solid-liquid coexistence states. The density of liquid nickel decreases linearly with increasing temperature in the range from the melting point to 1873 K. The density at the melting point and the volume thermal expansion coefficient of liquid nickel are 7.91 Mgm � 3 and 1:87 � 10 � 4 K � 1 , respectively. The density of Ni-Cr alloys in the liquid and solid-liquid coexistence states was found to decrease linearly with increasing temperature and chromium concentration in the alloy. The temperature coefficient of density of Ni-Cr alloys was found to change at the liquidus temperature. The absolute value of temperature coefficient of density in the solid-liquid coexistence state is larger than that of the liquid alloy.