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.     

Polydispersity treated by continuous thermodynamics

Continuous thermodynamics is a suitable concept for performing phase equilibrium calculations of polydisperse systems such as polymer solutions or polymer mixtures. In contrast to the traditional pseudo-component method the continuous molar mass distribution density function obtained by fractionation or gel permeation chromatography is used directly for thermodynamic calculations. This paper describes the application of continuous thermodynamics to calculate cloud-point curves, shadow curves and coexistence curves for solutions and mixtures of homopolymers and random copolymers. Particular attention is devoted to special problems such as fractionation or the transition from polymers to oligomers requiring the application of sophisticated Gibbs energy functions. In all cases the use of continuous thermodynamics leads to a drastic reduction of the required computer time. For Schulz-Flory distributions the occurring integrals often may be calculated analytically leading to simple closed end formulas.     

假二元微乳液体系标度律

微乳液是在表面活性物质存在下形成的油 -水混合热力学稳定体系 ,在一定条件下 ,表面活性剂双 -2 -乙基己基硫代琥珀酸钠 (AOT) -水 -正构烷烃可形成油包水型微乳液体系 ,其中分散颗粒的大小仅由水与 AOT的物质的量比 x确定[1] .这类微乳液体系有一个低临界温度 Tc,当温度高于 Tc时体系分成两相 ,两相中分散颗粒的大小相同 ,但颗粒浓度不同 ,因此 ,可看成一个假二元体系 .近年来 ,有关这类微乳液体系在临界区域与温度相关的标度律是否符合 3 D-Ising模型方面存在争论 [2～ 4 ] .本文通过精确测量 AOT-水 -正构烷烃假二元微乳液体系在…     

Effect of polydispersity on the cloud-point curves of polymer mixtures

A method is presented for the calculation of cloud-point curves of polymer–polymer mixtures when the polymers involved are polydisperse. The method is based on the Flory–Huggins free energy of mixing with a concentration-independent χ parameter. Numerical results are given for cases in which the molecular weight distributions are represented by the Schulz–Flory type. When the two polymers have similar average molecular weights and polydispersities, the cloud-point curves become flatter as the polydispersity increases. When the two polymers have similar average molecular weights but differ in polydispersity, the cloud-point curves become more skewed as the difference in the polydispersity increases. The results point out that, if the polydispersity effect is not properly accounted for, the value of χ deduced from experimental cloud points is liable to be in error, especially with regard to its temperature coefficient and its concentration dependence.     

Phase behavior of elemental aluminum using monte carlo simulations

Monte Carlo simulations are presented for two models of aluminum: an embedded-atom model and an explicit many-body model. Vapor/liquid coexistence curves are determined using Gibbs ensemble Monte Carlo simulations. The normal boiling points predicted by both models are somewhat higher (by about 10%) than the experimental value. Isothermal constant-stress simulations are used to simulate solid Al from 300 K to the triple point. The solid structures are at least metastable in the face-centered cubic configuration, and the specific heat is determined to be lower than the experimental value. The melting point for the embedded-atom model determined via thermodynamic integration along a pseudo-supercritical path is approximately 20% higher than the experimental value.     

Calculating poly(ethylene-co-acrylic acid)-solvent phase behavior with the SAFT equation of state

Statistical Associating Fluid Theory (SAFT) is used to model the cloud-point behavior of poly(ethylene-co-acrylic acid), with up to 7 mol % acid content, in propane, butane, propylene, butene, and dimethyl ether at temperatures to 250°C and pressure to 2600 bar. The values for the pure component temperature-independent segment volumes, nonspecific interaction energies, and the numbers of segments per molecule are equal to those used for polyethylene, because these copolymers contain modest amounts of acrylic acid repeat units. Two different approaches are used to determine values of the pure component energy of hydrogen bonding, ?/k, and the binary interaction parameter, k_ij. In one approach, ?/k for acid dimerization is obtained from literature spectroscopic data and a constant value of k_ij is fit to each copolymer-solvent cloud-point curve. Increasing the value of k_ij shifts the predicted cloud-point curves to higher temperatures and pressures. For the five solvents used in this study, k_ij decreased steadily in the range of 0.040 to ?0.025 as the acid content in the copolymer increased. The predicted cloud-point curves are in good agreement with experimental data, and the impact of hydrogen bonding on the phase behavior is well represented, even if k_ij is set equal to zero. For the second approach, ?/k is set to ～ 90% of the value obtained from spectroscopic data as determined from a fit of a single poly(ethylene-co-acrylic acid)-butane cloud-point curve, while k_ij is fit to the corresponding polyethylene-solvent system. This approach requires less mixture data than the previous approach, and the calculated cloud-point curves are also in good agreement with experimental data, except for the EAA-DME systems. © 1995 John Wiley & Sons, Inc.     

Oligomer Compatibility by Continuous Thermodynamics

To describe the details of the liquid-liquid equilibrium of oligomer mixtures, it is essential to use Gibbs free energy functions generalizing the classical Flory Huggins relation by replacing Huggins' X-term by a function depending on the averages of the molecular weight distributions. In this paper the concept of continuous thermodynamics is applied to establish a simple calculation procedure for the number-average segment number. Instead of the well-known sums with respect to the species, integrals occur which, in the case of Schulz-Flory distributions, may be calculated analytically, leading to simple formulas for the cloud-point curve and the shadow curve. The method is applied to model calculations showing that the chosen Gibbs free energy function may account for the details of the liquid-liquid equilibrium of oligomer mixtures and to a real oligomer system taken from the literature.     

The measurements of coexistence curves and light scattering for {xC6H5CN + (1 − x)CH3(CH2)6CH3} in the critical region

The coexistence curves and light scattering data for a critical solution of (benzonitrile + octane) have been reported. The critical exponents relating to the difference in the density variables between two coexisting phases β, the osmotic compressibility γ, and the correlation length ν have been deduced and the values are consistent with the 3D-Ising value in the range close to the critical point. The experimental results of the coexistence curves have also been analyzed to examine the Wegner correction terms and the behavior of the diameter of the coexistence curves. The light scattering data are well described by the crossover model proposed by Anisimov and Sengers, and show a monotonic crossover of the critical exponents γ and ν from its 3D-Ising value to the mean-field value as the temperature departures from the critical point. Furthermore, the dependences of the critical amplitudes on the mass of n-alkane for the binary solutions of (benzonitrile + n-alkane) have been discussed.     

The coexistence surface of the system neon—xenon

Gas—gas and gas—liquid equilibrium has been observed in the system neon—xenon. The coordinates of the critical double point are 1025 atm, ?9.40°C and 0.3875 xenon mole fraction. The critical point exponent β of a number of coexistence curves has been determined.     

An old model for magnetic nematics

A mean-field treatment is given of the off-lattice Krieger-James model of ordered fluids, which reduces to the more familiar Maier-Saupe liquid crystal (Heisenberg fluid) in the absence of ferromagnetic (nematic) interactions. As in the lattice version, isotropic, nematic and ferromagnetic nematic phases are found, but the nematic-ferronematic transition can either change order at a tricritical point, or terminate at a critical end point on the ferronematic-isotropic coexistence curve. In addition it is argued that the sequence of phase diagram topologies, as a function of the relative weights of ferromagnetic and nematic contributions to the free energy, should be similar to that obtained on varying the elongation of dipolar spheroids.     

On the design of experiments for determining ternary mixture free energies from static light scattering data using a nonlinear partial differential equation

We mathematically design sets of static light scattering experiments to provide for model-independent measurements of ternary liquid mixing free energies to a desired level of accuracy. A parabolic partial differential equation (PDE), linearized from the full nonlinear PDE [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008)], describes how data noise affects the free energies to be inferred. The linearized PDE creates a net of spacelike characteristic curves and orthogonal, timelike curves in the composition triangle, and this net governs diffusion of information coming from light scattering measurements to the free energy. Free energy perturbations induced by a light scattering perturbation diffuse along the characteristic curves and towards their concave sides, with a diffusivity that is proportional to the local characteristic curvature radius. Consequently, static light scattering can determine mixing free energies in regions with convex characteristic curve boundaries, given suitable boundary data. The dielectric coefficient is a Lyapunov function for the dynamical system whose trajectories are PDE characteristics. Information diffusion is heterogeneous and system-dependent in the composition triangle, since the characteristics depend on molecular interactions and are tangent to liquid-liquid phase separation coexistence loci at critical points. We find scaling relations that link free energy accuracy, total measurement time, the number of samples, and the interpolation method, and identify the key quantitative tradeoffs between devoting time to measuring more samples, or fewer samples more accurately. For each total measurement time there are optimal sample numbers beyond which more will not improve free energy accuracy. We estimate the degree to which many-point interpolation and optimized measurement concentrations can improve accuracy and save time. For a modest light scattering setup, a sample calculation shows that less than two minutes of measurement time is, in principle, sufficient to determine the dimensionless mixing free energy of a non-associating ternary mixture to within an integrated error norm of 0.003. These findings establish a quantitative framework for designing light scattering experiments to determine the Gibbs free energy of ternary liquid mixtures.     

Application of Continuous Thermodynamics to the Stability of Polymer Systems. III

Based on the stability theory of continuous thermodynamics for polymer solutions, necessary and sufficient conditions for multiple critical points are derived assuming the segment-molar excess Gibbs free energy to be independent of the distribution function. Equations for calculating double and triple critical points are given. Higher order critical points may be obtained in a successive way. For polymers possessing a Schulz-Flory molecular weight distribution, general conditions for an m-fold critical point are presented.     

Cloud-Point Curve for the System Copoly(Ethylene-Vinyl Acetate) Plus Methyl Acetate. Measurement and Prediction by Continuous Thermodynamics

The cloud-point curve for the system copoly(ethylene-vinyl acetate) plus methyl acetate has been measured by a simple visual method. The critical point was determined by using the phase volume ratio method. The method of continuous thermodynamics was applied for thermodynamic treatment. The composition of the copolymer is described by a divariate distribution function assumed as a generalized Stockmayer distribution. The activity coefficients were obtained with the aid of the Huggins Chi -parameter concept assuming Chi to be a quadratic polynomial with respect to the weight-average chemical composition of the copolymer. The three model parameters were calculated from the critical point and the slope of the cloud-point curve at the critical point. The cloud-point curve and the shadow curve were predicted from these parameters. The cloud-point curve shows qualitative agreement with experimental data.     

Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems

Computer simulations are employed to obtain subcritical isotherms of small finite sized systems inside the coexistence region. For all temperatures considered, ranging from the triple point up to the critical point, the isotherms gradually developed a sequence of sharp discontinuities as the system size increased from approximately 8 to approximately 21 molecular diameters. For the smallest system sizes, and more so close to the critical point, the isotherms appeared smooth, resembling the continuous van der Waals loop obtained from extrapolation of an analytic equation of state outside the coexistence region. As the system size was increased, isotherms in the chemical potential-density plane developed first two, then four, and finally six discontinuities. Visual inspection of selected snapshots revealed that the observed discontinuities are related to structural transitions between droplets (on the vapor side) and bubbles (on the liquid side) of spherical, cylindrical, and tetragonal shapes. A capillary drop model was developed to qualitatively rationalize these observations. Analytic results were obtained and found to be in full agreement with the computer simulation results. The analysis shows that the shape of the subcritical isotherms is dictated by a single characteristic volume (or length scale), which depends on the surface tension, compressibility, and coexistence densities. For small reduced system volumes, the model predicts that a homogeneous fluid is stable across the whole coexistence region, thus explaining the continuous van der Waals isotherms observed in the simulations. When the liquid and vapor free energies are described by means of an accurate mean-field equation of state and surface tensions from simulation are employed, the capillary model is found to describe the simulated isotherms accurately, especially for large systems (i.e., larger than about 15 molecular diameters) at low temperature (lower than about 0.85 times the critical temperature). This implies that the Laplace pressure differences can be predicted for drops as small as five molecular diameters, and as few as about 500 molecules. The theoretical study also shows that the extrema or apparent spinodal points of the finite size loops are more closely related to (finite system size) bubble and dew points than to classical spinodals. Our results are of relevance to phase transitions in nanopores and show that first order corrections to nucleation energies in finite closed systems are power laws of the inverse volume.     

Supercritical phase equilibria involving solids

The solubility of solids in supercritical solvents is reviewed in a phenomenological discussion of binary and ternary systems containing one highly volatile component. Solubility and selectivity are greatly determined by the course of the binary critical curves, the ternary critical end-point curves, and the locations of the triple points of the solids. The mean-field lattice-gas model is used to review some important molecular parameters.     

Prediction of thermodynamic properties of polymer solutions by a group-contribution method

The Flory–Huggins lattice-theory expression for solvent activity in a polymer-solution is commonly used to calculate the thermodynamic interaction parameter χ with the aid of experimental data from vapor pressure osmometry. This expression assumes that χ is independent of composition. However, experimental data for a variety of polymer-solvent mixtures indicate that χ exhibits an appreciable concentration dependence. A group contribution method, UNIFAC (UNIQUAC Functional-Group Activity Coefficients) incorporating the free-volume correction of Oishi and Prausnitz is used to predict the dependence of χ on solvent concentration. Agreement with previously reported experimental data is within 15%. Calculated values of χ obtained from the Flory–Huggins expression for solvent activity and from the corresponding Gibbs free energy of mixing (which does not assume that χ is independent of composition) are compared. Calculations based on the Gibbs free energy of mixing predict a somewhat larger value of χ relative to those based on solvent activity. The specific Gibbs free energy of mixing for polystyrene-solvent mixtures is calculated using the UNIFAC model, and is found to represent qualitatively the phase equilibrium behavior. Quantitative discrepancies are observed, however, for the polystyrene-acetone system in light of the actual experimental solubility reported by Suh and Clark (20). Most of the thermodynamic predictions for polymer-solvent systems investigated herein are correlated qualitatively with the relative mismatch between solubility parameters of both components.     

Water in nanopores. I. Coexistence curves from Gibbs ensemble Monte Carlo simulations

Coexistence curves of water in cylindrical and slitlike nanopores of different size and water-substrate interaction strength were simulated in the Gibbs ensemble. The two-phase coexistence regions cover a wide range of pore filling level and temperature, including ambient temperature. Five different kinds of two-phase coexistence are observed. A single liquid-vapor coexistence is observed in hydrophobic and moderately hydrophilic pores. Surface transitions split from the main liquid-vapor coexistence region, when the water-substrate interaction becomes comparable or stronger than the water-water pair interaction. In this case prewetting, one and two layering transitions were observed. The critical temperature of the first layering transition decreases with strengthening water-substrate interaction towards the critical temperature expected for two-dimensional systems and is not sensitive to the variation of pore size and shape. Liquid-vapor phase transition in a pore with a wall which is already covered with two water layers is most typical for hydrophilic pores. The critical temperature of this transition is very sensitive to the pore size, in contrast to the liquid-vapor critical temperature in hydrophobic pores. The observed rich phase behavior of water in pores evidences that the knowledge of coexistence curves is of crucial importance for the analysis of experimental results and a prerequiste of meaningful simulations.     

Phase transitions in solutions of polystyrene with poly(methyl methacrylate) and polybutadiene under deformation

By the cloud point and static sorption methods, phase diagrams are constructed and the concentration and temperature dependences of the Gibbs free energy of mixing and the interaction parameter are determined under static conditions and in a shear field for the poly(methyl methacrylate)-polystyrene-ethyl acetate, polystyrene-polybutadiene-toluene, polystyrene-polybutadiene, poly(methyl methacrylate)-polystyrene, poly(methyl methacrylate)-ethyl acetate, and polystyrene-ethyl acetate systems. Phase separation in the systems both under heating and cooling, as well as coexistence of three phases, is observed in the polystyrene-poly(methyl methacrylate)-ethyl acetate system. Deformation changes the phase separation temperature by 30–40 K.     

Determination of the solid-fluid coexistence of the n - 6 Lennard-Jones system from free energy calculations

The solid-fluid coexistence properties of the n - 6 Lennard-Jones system, n from 7 to 12, are reported. The procedure relies on determining Helmholtz free energy curves as a function of volume for each phase independently, from several NVT simulations, and then connecting it to points of known absolute free energy. For n = 12 this requires connecting the simulated points to states of very low densities on the liquid phase, and to a harmonic crystal for the solid phase, which involves many extra simulations for each temperature. For the reference points of the remaining systems, however, the free energy at a given density and temperature can be calculated relative to the n = 12 system. The method presented here involves a generalization of the multiple histogram method to combine simulations performed with different potentials, provided they visit overlapping regions of the phase space, and allows for a precise calculation of relative free energies. The densities, free energies, average potential energies, pressure, and chemical potential at coexistence are presented for up to T? = 5.0 and new estimations of the triple points are given for the n - 6 Lennard-Jones system.     

Cloud-point curves for polystyrenes of low polydispersity in cyclohexane solution

Cloud-point curves and relative volumes of the coexisting phases just below the cloud-point temperatures were observed for a commercial sample of “monodisperse” polystyrene and one of its fractions. As expected from theoretical considerations, the cloud-point curve for the more monodisperse sample was flatter near the peak over a greater concentration range than that of the parent sample. Furthermore, the critical point for the fraction appeared to be closer to the peak of the cloud-point curve than was that of the unfractionated polymer. In addition, it was learned that one type of impurity could be detected with extraordinary sensitivity by means of the cloud-point curve.