A global investigation of phase equilibria using the perturbed-chain statistical-associating-fluid-theory approach

The recently developed perturbed-chain statistical-associating-fluid theory (PC-SAFT) is investigated for a wide range of model parameters including the parameter m representing the chain length and the thermodynamic temperature T and pressure p. This approach is based upon the first-order thermodynamic perturbation theory for chain molecules developed by Wertheim [M. S. Wertheim, J. Stat. Phys. 35, 19 (1984); ibid. 42, 459 (1986)] and Chapman et al. [G. Jackson, W. G. Chapman, and K. E. Gubbins, Mol. Phys. 65, 1 (1988); W. G. Chapman, G. Jackson, and K. E. Gubbins, ibid. 65, 1057 (1988)] and includes dispersion interactions via the second-order perturbation theory of Barker and Henderson [J. A. Barker and D. Henderson, J. Chem. Phys. 47, 4714 (1967)]. We systematically study a hierarchy of models which are based on the PC-SAFT approach using analytical model calculations and Monte Carlo simulations. For one-component systems we find that the analytical model in contrast with the simulation results exhibits two phase-separation regions in addition to the common gas-liquid coexistence region: One phase separation occurs at high density and low temperature. The second demixing takes place at low density and high temperature where usually the ideal-gas phase is expected in the phase diagram. These phenomena, which are referred to as "liquid-liquid" and "gas-gas" equilibria, give rise to multiple critical points in one-component systems, as well as to critical end points and equilibria of three fluid phases, which can usually be found in multicomponent mixtures only. Furthermore, it is shown that the liquid-liquid demixing in this model is not a consequence of a "softened" repulsive interaction as assumed in the theoretical derivation of the model. Experimental data for the melt density of polybutadiene with molecular mass Mw=45,000 gmol are correlated here using the PC-SAFT equation. It is shown that the discrepancies in modeling the polymer density at ambient temperature and high pressure can be traced back to the liquid-liquid phase separation predicted by the equation of state at low temperatures. This investigation provides a basis for understanding possible inaccuracies or even unexpected phase behavior which can occur in engineering applications of the PC-SAFT model aiming at predicting properties of macromolecular substances.     

二氧化碳-水混合体系的相平衡和汽-液-液三相点研究

在微扰链统计缔合流体理论(PC-SAFT)的基础上建立了适用于二氧化碳-水体系相行为研究的状态方程, 以汽-液平衡和液-液平衡实验数据关联体系的交叉作用参数, 关联结果与实验数据吻合良好. 预测了二氧化碳-水体系存在汽-液-液三相平衡的温度和压力区间, 确定了三相点的汽-液-液三相密度及其与温度和压力的关系.     

Systematic investigation of global phase behavior of polymer mixtures in the pressure-temperature plane

We investigate the critical lines of polymer mixtures in the presence of their vapor phase at the mathematical double point, where two critical lines meet and exchange branches, and its environment. The model used combines the lattice gas model of Schouten, ten Seldam and Trappeniers with the Flory-Huggins theory. The critical line structure is displayed for various combinations of the chain length and system parameters in the pressure (P)-temperature (T) plane, as is usually done with experimental results. This type of work sheds light on the essential transition mechanism involved in the phase diagram's change of character, such as multi-critical points and mathematical double points, which are of great practical importance in supercritical fluid extraction processes. The P, T diagrams are discussed in accordance with the Scott and van Konynenburg binary phase diagram classification. We found that our P, T plots were in agreement with type II, type III, or type IV phase diagram behaviors. We also found that some of our phase diagrams represent the liquid-liquid equilibria in polymer solutions and mixtures.     

Identification of the phase of a fluid using partial derivatives of pressure, volume, and temperature without reference to saturation properties: Applications in phase equilibria calculations

The phase of a fluid (liquid, vapor) at a given state is normally identified by comparing the properties of the fluid at the given state with saturation properties. In this paper we present a thermodynamic method for determining the phase of a fluid from the partial derivatives of pressure, volume and temperature without reference to saturated properties. The method has immediate application in a variety of phase equilibria calculations, particularly for liquid-liquid or vapor-liquid-liquid equilibria calculations in process simulators.     

Application of the perturbed chain SAFT equation of state to complex polymer systems using simplified mixing rules

A simplified perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state is applied to polymer systems that include a variety of non-associating (esters, cyclic hydrocarbons), polar (ketones) as well as associating (amines, alcohols) solvents. The solvent pure-component parameters that are not available in the literature are estimated by correlating vapor-pressure and liquid-density data. The performance of the simplified PC-SAFT is compared to the original PC-SAFT equation of state for polymer systems of varying complexity. It is shown that the applied simplification is not at the expense of the accuracy of equation of state, while the computational time and complexity are significantly reduced, especially for associating systems. With no binary interaction parameter, simplified PC-SAFT is successfully able to predict vapor–liquid equilibria of polymers with non-associating solvents. In the case of associating solvents, a small binary interaction parameter k_ij is usually needed for the satisfactory correlation of the experimental data.     

A perturbed hard-sphere-chain equation of state for nematic liquid crystals and their mixtures with polymers

A perturbed hard-sphere-chain (PHSC) equation of state is presented to compute nematicisotropic equilibria for thermotropic liquid crystals, including mixtures. The equation of state consists of an isotropic term and an anisotropic term given by the Maier-Saupe theory whose contribution disappears in the isotropic phase. The isotropic contribution is the recently presented PHSC equation of state for normal fluids and polymers which uses a reference equation of state for athermal hard-sphere chains and a perturbation theory for the squarewell fluid of variable well width. The PHSC equation of state gives excellent correlations of pure-component pressure-volume-temperature data in the isotropic region and, combined with the Maier-Saupe theory, correlates the dependence of nematic-isotropic transition temperature on the pressure. Theory also predicts a nematic-isotropic biphasic region and liquid-liquid phase separation in a temperature-composition diagram of binary mixtures containing a nematic liquid crystal and a normal fluid or polymer. Theory and experiment show good agreement for pure fluids as well as for mixtures.     

Density of methanol-carbon dioxide mixtures at three temperatures: Comparison with vapor-liquid equilibria measurements and results obtained from chromatography

The density of methanol-carbon dioxide mixtures has been measured at 40, 50, and 60°C and the data presented as constant density curves in plots of pressure against composition. Mixtures containing from 1 to 23.5% methanol were covered. The density information was compared with liquid-vapor equilibria data from the literature and with results from chromatographic separations using both sub- and supercritical conditions. The results indicate that phase transitions do not usually occur when the concentration of the modifier is increased while holding temperature and pressure constant. Similarly, no phase transition occurs when the temperature is increased from 28 to 80°C, at a constant (high) pressure and modifier concentration. Binary fluids with many desirable compositions cannot be prepared at low densities: densities below 0.4–0.5 g/cm³ cannot be obtained from methanol-carbon dioxide mixtures because the fluids separate into two phases. This means that density programming with many binary fluid combinations will be of secondary importance compared with composition programming.     

Modeling of three-phase vapor–liquid–liquid equilibria for a natural-gas system rich in nitrogen with the SRK and PC-SAFT EoS

In this work, we present the modeling of three-phase vapor–liquid–liquid equilibria for a mixture of natural gas (Hogback gas) containing high concentrations in nitrogen (51.8 mol%) with the SRK and PC-SAFT equations of state. The interest of studying this mixture is due to the experimental evidence of the occurrence of multiple equilibrium liquid phases for this mixture over certain ranges of temperature and pressure. The calculation of the multiphase equilibria was carried out by using an efficient numerical procedure based on the minimization of the system Gibbs energy and thermodynamic stability tests to find the most stable state of the system. The results of the calculated vapor–liquid–liquid equilibria (VLLE) show that the PC-SAFT equation of state predicts satisfactorily the phase behavior that experimentally exhibits this mixture, whereas the SRK equation of state predicts a three-phase region wider than the experimentally observed. The two-phase boundary for this mixture was also calculated through flash calculations, and the results showed that this mixture does not present any gas-liquid critical point.     

球腔中氢键流体的吸附-解吸附转变:表面调控研究

集中讨论了球形微腔表面对腔中氢键流体相态结构的调控机制. 为了揭示微腔表面对腔中氢键流体相平衡的影响, 首先根据吸附-解吸附原理并利用经典流体的密度泛函理论计算了微腔中氢键流体的平衡密度分布, 进而通过吸附-解吸附等温线及巨势等温线绘制出体系的相图. 在此基础上, 重点考察了球腔尺寸、 表面作用强度和作用力程对氢键流体毛细凝聚及层化转变的影响. 结果表明, 这些因素可以有效地调控体系毛细凝聚和层化转变的临界约化温度、 临界密度和相区大小等特征, 从而阐明了表面调控的主要机制. 研究结果为设计相关吸附材料提供了理论参考.     

Investigation of the surface tension of methane and n-alkane mixtures by perturbed-chain statistical associating fluid theory combined with density-gradient theory

The perturbed-chain statistical associating fluid theory (PC-SAFT) and density-gradient theory are used to construct an equation of state to describe the phase behavior of binary methane–n-alkane mixtures. With the molecular parameters and influence parameters regressed from bulk properties and surface tensions of pure fluids, respectively as input, both the bulk and interfacial properties are investigated. The surface tension of the binary systems methane–propane, methane–pentane, methane–heptane and methane–decane are predicted, and the results are satisfactory compared with the experimental data. Our results show that PC-SAFT combined with density-gradient theory is able to describe the interfacial properties of binary methane–n-alkane mixtures in wide temperature and pressure ranges, and illustrate the influence of the equilibrium bulk properties and chain length of n-alkane molecule on the interfacial properties.     

Modeling of thermodynamic behavior of PVT properties and cloud point temperatures of polymer blends and polymer blend + carbon dioxide systems using non-cubic equations of state

In recent years, many factors influencing phase behavior of polymer blends have been studied because of their widely technological importance, as a simple method of formulating new materials with tailored properties which make them suitable for a variety of applications. This work has three main goals which were reached by using the Perturbed Chain Statistical Associating Fluid Theory (PC-SAFT) and the Sanchez–Lacombe (SL) non-cubic equations of state (EoS), which in previous works have shown their ability to handle long chain and associating interactions. First, both equations of state were tested with the correlation of the specific volumes of pure blends (PBD/PS, PPO/PS, PVME/PS, PEO/PES) and the prediction of the specific volumes for blends; second, the modeling of blend miscibilities in the liquid–liquid equilibria (LLE) of PBD/PS, PPG/PEGE, PVME/PS, PEO/PES, and PnPMA/PS blends; third, the modeling of the phase behavior of PS/PVME blends at various compositions in the presence of CO₂. PC-SAFT and SL pure-component parameters were regressed by fitting pure-component data of real substances (liquid pressure–volume–temperature, PVT, data for polymers and vapor pressure and saturated liquid molar volume for CO₂) and the fluid phase behavior of blend systems were simulated fitting one binary interaction parameter (k_ij) by regression of experimental data using the modified likelihood maximum method. Results were compared with experimental data obtained from literature and an excellent agreement was obtained with both EoS, which were also capable of predicting the fluid phase behavior corresponding to the critical solution temperatures (LCST: lower critical solution temperature, UCST: upper critical solution temperature) of blends.     

Polymorphism in simple liquids: a Gibbs ensemble Monte Carlo study

We perform Gibbs ensemble Monte Carlo (GEMC) simulations of a one-component system of hard spheres with a repulsive shoulder and an attractive well. We show the existence of two distinct liquid-gas and liquid-liquid phase equilibria. The GEMC estimate of the critical parameters, as following from an interpolation of the binodal points, is only slightly influenced by finite size effects. The liquid-gas critical temperature and pressure are lower than those of the liquid-liquid phase separation. A discussion of our findings in comparison with those of previous numerical studies is also presented.     

Liquid-liquid phase equilibria in polymer solutions and polymer mixtures

The pressure dependence of liquid-liquid equilibria in weakly interacting binary macromolecular systems (homopolymer solutions and blends) will be discussed. The common origin of the separate high-temperature/low-temperature and high-pressure/low-pressure branches of demixing curves will be demonstrated by extending the study into the region of metastable liquid states including the undercooled, overheated and stretched states (i.e. states at negative pressures). The seemingly different response of the UCST-branch of solutions and blends when pressurized (pressure induced mixing for most polymer solutions, pressure induced demixing for most blends) will be explained in terms of the location of a hypercritical point found either at positive (most solutions) or negative pressure (most blends). Further, it is shown that the pressure dependence of demixing of homopolymer solutions and blends may be described using a ‘master-curve’ which, however, is sometimes partly masked by degradation or by vapour-liquid and/or solid-liquid phase transitions. Experimental results demonstrating the extension of liquid-liquid phase boundary curves into the metastable regions will be presented, and the existence of solubility islands in the vicinity of the hypercritical points discussed.     

Phase equilibria and plate-fluid interfacial tensions for associating hard sphere fluids confined in slit pores

The excess Helmholtz free energy functional for associating hard sphere fluid is formulated by using a modified fundamental measure theory [Y. X. Yu and J. Z. Wu, J. Chem. Phys. 117, 10156 (2002)]. Within the framework of density functional theory, the thermodynamic properties including phase equilibria for both molecules and monomers, equilibrium plate-fluid interfacial tensions and isotherms of excess adsorption, average molecule density, average monomer density, and plate-fluid interfacial tension for four-site associating hard sphere fluids confined in slit pores are investigated. The phase equilibria inside the hard slit pores and attractive slit pores are determined according to the requirement that temperature, chemical potential, and grand potential in coexistence phases should be equal and the plate-fluid interfacial tensions at equilibrium states are predicted consequently. The influences of association energy, fluid-solid interaction, and pore width on phase equilibria and equilibrium plate-fluid interfacial tensions are discussed.     

Phase equilibria of carbon dioxide + poly ethylene glycol + water mixtures at high pressure: Measurements and modelling

Phase equilibria of carbon dioxide + poly ethylene glycol (PEG) of average mol weight 6000 g/mol + water mixtures has been measured by the static method at conditions of interest for the development of Particles from Gas Saturated Solutions (PGSS)-drying processes (pressure from 10 MPa to 30 MPa, temperature from 353 K to 393 K). A thermodynamic model based on the Perturbed-Chain Statistical Associating Fluid Theory (PC-SAFT) equation of state has been developed for correlating experimental data. The model is able to predict the composition of the liquid phase with an average deviation of 8.0%. However, the model does not calculate correctly the concentration of PEG in the gas phase. The model is also capable of predicting VLE data reported in the literature of PEG + CO₂ mixtures with PEGs of molecular weights ranging from 1500 g/mol to 18500 g/mol as well as solid–fluid equilibrium of carbon dioxide + PEG mixtures at pressures below 10 MPa.     

Liquid-vapor and liquid-liquid phase equilibria of the Brodholt-Sampoli-Vallauri polarizable water model

Liquid-vapor and liquid-liquid phase equilibria of the polarizable Brodholt-Sampoli-Vallauri water model have been investigated by Gibbs ensemble Monte Carlo computer simulations. The coexisting liquid and vapor densities and energy of vaporization of the model is found to be in a reasonable agreement with experimental data in the entire temperature range of liquid-vapor coexistence. The critical temperature and density of the model are found to be 615 K and 0.278 gcm(3), respectively, close to the experimental values of 647.1 K and 0.322 gcm(3). In the supercooled state two distinct liquid-liquid coexistence regions are observed. The existence of liquid-liquid phase separation of a polarizable water model is demonstrated for the first time.     

End-grafted low-molecular-weight PNIPAM does not collapse above the LCST

The interfacial properties of end-grafted temperature-responsive poly(N-isopropylacryamide) (PNIPAM) were quantified by direct force measurements both above and below the lower critical solution temperature (LCST) of 32 degrees C. The forces were measured between identical, opposing PNIPAM films and between a PNIPAM film and a lipid membrane. At the grafting densities and molecular weights investigated, the polymer extension did not change significantly above the LCST, and the polymers did not adhere. Below the LCST, the force-distance profiles suggest a vertical phase separation, which results in a diluter outer layer and a dense surface proximal layer. At large separations, the force profiles agree qualitatively with simple polymer theory but deviate at small separations. Importantly, at these low grafting densities and molecular weights, the end-grafted PNIPAM does not collapse above the LCST. This finding has direct implications for triggering liposomal drug release with end-grafted PNIPAM, but it increases the temperature range where these short PNIPAM chains function as steric stabilizers.     

High pressure phase equilibrium properties for ethane+1-butanol system at 313.15 K

Phase equilibria and saturated densities for ethane+1-butanol system at high pressures were measured using a static-circulation apparatus at 313.15 K. The experimental apparatus equipped with three Anton Paar DMA 512S vibrating tube density meters was previously developed for measuring vapor–liquid–liquid equilibrium (VLLE) at high pressures. Co-existing phase composition and saturated density of each phase can be measured by means of the apparatus with a maximum temperature and pressure of 400 K and 20 MPa, respectively. The present experimental results include vapor–liquid equilibria (VLE), liquid–liquid equilibria (LLE), and VLLE. The equilibrium composition and density of each phase were determined by gas chromatography and density measurements, respectively. The experimental data were correlated with various equations of state.