Fluid-solid Equilibrium Research Articles

Supercritical Fluid Extraction of Flavonoids and Terpenoids from Herbal compounds: Experiments and Mathematical modeling

Supercritical extraction of flavonoid compounds from Dorema aucheri Boiss., a native Iranian plant using carbon dioxide was studied. A combined theory of broken and intact cell and fluid-solid discontinuous phase equilibrium approach was utilized to explain the role of operating variables on the extraction yields. Effects of temperature, pressure, solvent flow rate and solid particle size on the extraction yield were evaluated using the mathematical modeling. The model accuracy was evaluated in two cases, with and without mass transfer coefficient as a fitting parameter. The theoretical work was also repeated for supercritical extraction of terpenoid compounds from Baizhu to further evaluation of model applicability in supercritical systems. The average absolute deviations of model predictions from experimental data for Baizhu and Dorema aucheri, using no fitting parameters, were between 3.61-8.04% and 5.38-12.62 % respectively. The same deviations using one fitting parameter showed 3.37-6.33% and 1.58-7.40 % deviation, respectively.

Sintering at Particle Scale: An Eulerian Computing Framework to Deal with Strong Topological and Material Discontinuities

This work presents a numerical modelling approach of particle packing consolidation, at the particle scale, based on specific numerical methods implemented in a high-performance computing framework. Typically, the sintering process triggers several mass transport paths, thermally activated, that are driven by geometrical as well as physical gradients and laplacians. Computing precisely such major characteristics is of paramount importance but represents a real scientific challenge, which have not been fully solved yet but which must however be tackled to gain precious insights into sintering mechanisms which are seldom accessible at this scale. An Eulerian-based formulation is then proposed here to deal with the strong topological changes related to particle sintering. Also, a specific attention is paid to the precise and robust computation of high-order derivatives which are known to control the physics of surface solid diffusion, namely the surface laplacian of the curvature. Besides, the hydrostatic pressure gradient is known to control the volume diffusion path, it results from the coupled fluid-solid mechanical equilibrium, including surface tension, which must be solved precisely. Furthermore, a mesh adaptation technique allows the particles surface description to be improved, while the number of mesh elements is kept reasonable. Once verified on test-cases, this numerical approach is applied to several 3D granular packings undergoing micro-structural changes under combined surface and volume diffusion.

Fluid-solid equilibrium of carbon dioxide as obtained from computer simulations of several popular potential models: The role of the quadrupole

In this work the solid-fluid equilibrium for carbon dioxide (CO2) has been evaluated using Monte Carlo simulations. In particular the melting curve of the solid phase denoted as I, or dry ice, was computed for pressures up to 1000 MPa. Four different models, widely used in computer simulations of CO2 were considered in the calculations. All of them are rigid non-polarizable models consisting of three Lennard-Jones interaction sites located on the positions of the atoms of the molecule, plus three partial charges. It will be shown that although these models predict similar vapor-liquid equilibria their predictions for the fluid-solid equilibria are quite different. Thus the prediction of the entire phase diagram is a severe test for any potential model. It has been found that the Transferable Potentials for Phase Equilibria (TraPPE) model yields the best description of the triple point properties and melting curve of carbon dioxide. It is shown that the ability of a certain model to predict the melting curve of carbon dioxide is related to the value of the quadrupole moment of the model. Models with low quadrupole moment tend to yield melting temperatures too low, whereas the model with the highest quadrupole moment yields the best predictions. That reinforces the idea that not only is the quadrupole needed to provide a reasonable description of the properties in the fluid phase, but also it is absolutely necessary to describe the properties of the solid phase.

High pressure–high temperature phase diagram of ammonia

The high pressure (P)-high temperature (T) phase diagram of solid ammonia has been investigated using diamond anvil cell and resistive heating techniques. The III-IV transition line has been determined up to 20 GPa and 500 K both on compression and decompression paths. No discontinuity is observed at the expected location for the III-IV-V triple point. The melting line has been determined by visual observations of the fluid-solid equilibrium up to 9 GPa and 900 K. The experimental data are well fitted by a Simon-Glatzel equation in the covered P-T range. These transition lines and their extrapolations are compared to the reported ab initio calculations.

Phase behavior of a hard sphere Maier-Saupe nematogenic system in three dimensions

We present a detailed computer simulation and integral equation study of the phase behavior of a nematogenic system composed of hard spheres with embedded three-dimensional Maier-Saupe spins. For this well-known system, we map the gas-liquid equilibrium, which is coupled to a first-order isotropic-nematic transition. The anisotropic integral equation theory is found to yield excellent agreement with the simulation data within the fluid regime. Additionally, we determine the fluid-solid equilibrium transition by means of computer simulation.

Thermodynamic modelling of supercritical fluid–solid phase equilibrium data

The design and development of processes involving supercritical fluids depend on how easy the phase equilibrium can be accurately modelled and predicted. In the work described herein, the supercritical fluid–solid equilibrium has been considered. Modelling the fluid–solid equilibrium is associated with a number of drawbacks, even when it is possible to obtain the experimental solubility data for the solute in the supercritical fluid. In most cases it is necessary to introduce additional adjustment parameters into the model. The developed program, realized in Visual Basic ® language, is based on the fitting of two parameters – the binary interaction parameter ( k 12) and the solid sublimation pressure ( P 2 sat ). This program can be used for any fluid–solid equilibrium even when both parameters are known or supposed. The model has been applied to several systems and, as example, in this work, the Penicillin G-CO 2 phase equilibrium data are shown. The results obtained allow affirm that the thermodynamic model applied to fluid–solid equilibrium calculations is useful to predict the behaviour of this system.

Simulating Fluid–Solid Equilibrium with the Gibbs Ensemble

The Gibbs ensemble is employed to simulate fluid–solid equilibrium for a shifted-force Lennard-Jones system. This is achieved by generating an accurate canonical Helmholtz free-energy model of the (defect-free) solid phase. This free-energy model is easily generated, with accuracy limited only by finite-size effects, by a single isothermal–isobaric simulation at a pressure not too far from coexistence for which the chemical potential is known. We choose to illustrate this method at the known triple-point because the chemical potential is easily calculated from the coexisting gas. Alternatively, our methods can be used to locate fluid–solid coexistence and the triple-point of pure systems if the chemical potential of the solid phase can be efficiently calculated at a pressure not too far from the actual coexistence pressure. Efficient calculation of the chemical potential of solids would also enable the Gibbs ensemble simulation of bulk solid–solid equilibrium and the grand-canonical ensemble simulation of bulk solids.

Fluid–solid equilibria of flexible and linear rigid tangent chains from Wertheim’s thermodynamic perturbation theory

An extension of Wertheim’s first-order thermodynamic perturbation theory is proposed to describe the global phase behavior of linear rigid tangent hard sphere chains. The extension is based on a scaling proposed recently by Vega and McBride [Phys. Rev. E 65, 052501 (2002)] for the equation of state of linear chains in the solid phase. We have used the Einstein-crystal methodology, the Rahman–Parrinello technique, and the thermodynamic integration method for calculating the free energy and equation of state of linear rigid hard sphere chains with different chain lengths, including the solid–fluid phase equilibria. Agreement between the simulation data and theoretical predictions is excellent in all cases. Once it is confirmed that the proposed theory can be used to describe correctly the equation of state, free energy, and solid–fluid phase transitions of linear rigid molecules, a simple mean-field approximation at the level of van der Waals is included to account for segment–segment attractive interactions. The approach is used to determine the global phase behavior of fully flexible and linear rigid chains of varying chain lengths. The main effect of increasing the chain length in the case of linear rigid chains is to decrease the fluid densities at freezing, so that the triple-point temperatures increase. As a consequence, the range of temperatures where vapor–liquid equilibria exist decreases considerably with chain length. This behavior is a direct result of the stabilization of the solid phase with respect to the liquid phase as the chain length is increased. The vapor–liquid equilibria are seen to disappear for linear rigid chains formed by more than 11 hard sphere segments that interact through an attractive van der Waals mean-field contribution; in other words, long linear rigid chains exhibit solid–vapor phase behavior only. In the case of flexible chains, the fluid–solid equilibrium is hardly affected by the chain length, so that the triple-point temperature reaches quickly an asymptotic value. In contrast to linear rigid chains, flexible chains present quite a broad range of temperatures where vapor–liquid equilibria exist. Although the vapor–liquid equilibria of flexible and linear rigid chain molecules are similar, the differences in the type of stable solid they form and, more importantly, the differences in the scaling of thermodynamic properties with chain length bring dramatic differences to the appearance of their phase diagrams.

The fluid–solid equilibrium for a charged hard sphere model revisited

The global phase diagram of a system of charged hard spheres, composed of positive and negative ions of the same size, is obtained by means of computer simulations. Thermodynamic integration and Einstein crystal calculations are used to determine the free energies of the different possible solid structures. In this way, the fluid–solid and solid–solid phase transitions are located. Gibbs–Duhem integration is used to trace the full coexistence curves between the different phases involved. Three different solid structures are found to be stable for the model considered; namely, a cesium chloride structure (CsCl), a substitutionally disordered close packed structure which is faced centered cubic (fcc), and a tetragonal ordered structure with a fcc arrangement of atoms if the charge of the ions is not considered. At high temperatures, freezing leads to the substitutionally disordered close packed structure. This solid structure undergoes an order–disorder transition at low temperatures transforming into the tetragonal solid. At low temperatures freezing leads to the cesium chloride structure (CsCl) which undergoes a phase transition to the tetragonal structure at high pressures. The tetragonal solid is the stable solid phase at low temperatures and high densities. In a narrow range of temperatures direct coexistence between the fluid and the tetragonal solid is observed. Three triple points are found for the model considered. The usual vapor–liquid–CsCl solid triple point occurs at T*=0.0225. In addition, a fluid-fcc disordered-tetragonal triple point is located at T*=0.245 and, finally, a fluid-CsCl-tetragonal triple point appears at T*=0.234. The results presented here can be used to test the performance of the different theoretical treatments of freezing available in the literature.

The phase diagram of the two center Lennard-Jones model as obtained from computer simulation and Wertheim’s thermodynamic perturbation theory

The global phase diagram (i.e., vapor–liquid and fluid–solid equilibrium) of two-center Lennard-Jones (2CLJ) model molecules of bond length L=σ has been determined by computer simulation. The vapor–liquid equilibrium conditions are obtained using the Gibbs ensemble Monte Carlo method and by performing isobaric-isothermal NPT calculations at zero pressure. In the case of the solid phase, two close-packed solid structures are considered: In the first structure, the molecules are located in layers and all molecular axes point in the same direction; and in the second structure, the atoms form a close-packed arrangement but the molecular axes of the diatomic molecules have random orientations. It is shown that at the vapor–liquid–solid triple-point temperature, the orientationally disordered solid is the stable structure for the solid phase of this model. The vapor–liquid-disordered solid triple-point temperature of the 2CLJ model, with bond length L=σ, is located at T*=0.650(4). This is very close to the triple-point temperature of the Lennard-Jones monomer system (T*=0.687). At very low temperatures, the ordered solid is the stable phase. The vapor-ordered solid-disordered solid triple point is situated at T*=0.282. The simulation data are compared to Wertheim’s first-order thermodynamic perturbation theory (TPT1) for the fluid and solid phases. It is found that Wertheim’s TPT1 not only provides an accurate description of the equation of state in both the fluid and solid phases, but also provides accurate values of the free energies. The prediction of Wertheim’s TPT1 for the global phase diagram of the 2CLJ model shows excellent agreement with the simulation results, illustrating the possibility of using Wertheim’s perturbation theory to determine not only the vapor–liquid equilibria but also the global phase diagram of simple chain model molecules.

Salt-induced protein precipitation in aqueous solution: Single and binary protein systems

A molecular-thermodynamic model is developed for the salt-induced protein precipitation. The protein molecules interact through four intermolecular potentials. An equation of state is derived based on the statistical mechanical perturbation theory with the modifiedChiew’s equation for the fluid phase, Young’s equation for the solid phase as the reference system and a perturbation based on the protein-protein effective two body potential. The equation of state provides an expression for the chemical potential of the protein. In a single protein system, the phase separation is represented by fluid-fluid equilibria. The precipitation behaviors are simulated with the partition coefficient at various salt concentrations and degree of pre-aggregation effect for the protein particles. In a binary protein system, we regard the system as a fluid-solid phase equilibrium. At equilibrium, we compute the reduced osmotic pressure-composition diagram in the diverse protein size difference and salt concentrations.

The global phase diagram of the Gay–Berne model

The phase diagram of the Gay–Berne model with anisotropy parameters κ=3, κ′=5 has been evaluated by means of computer simulations. For a number of temperatures, NPT simulations were performed for the solid phase leading to the determination of the free energy of the solid at a reference density. Using the equation of state and free energies of the isotropic and nematic phases available in the existing literature the fluid–solid equilibrium was calculated for the temperatures selected. Taking these fluid–solid equilibrium results as the starting points, the fluid–solid equilibrium curve was determined for a wide range of temperatures using Gibbs–Duhem integration. At high temperatures the sequence of phases encountered on compression is isotropic to nematic, and then nematic to solid. For reduced temperatures below T=0.85 the sequence is from the isotropic phase directly to the solid state. In view of this we locate the isotropic–nematic–solid triple point at TINS=0.85. The present results suggest that the high-density phase designated smectic B in previous simulations of the model is in fact a molecular solid and not a smectic liquid crystal. It seems that no thermodynamically stable smectic phase appears for the Gay–Berne model with the choice of parameters used in this work. We locate the vapor–isotropic liquid–solid triple point at a temperature TVIS=0.445. Considering that the critical temperatures is Tc=0.473, the Gay–Berne model used in this work presents vapor–liquid separation over a rather narrow range of temperatures. It is suggested that the strong lateral attractive interactions present in the Gay–Berne model stabilizes the layers found in the solid phase. The large stability of the solid phase, particularly at low temperatures, would explain the unexpectedly small liquid range observed in the vapor–liquid region.

Extending Wertheim’s perturbation theory to the solid phase of Lennard-Jones chains: Determination of the global phase diagram

Wertheim’s first order thermodynamic perturbation theory (TPT1) [M. S. Wertheim, J. Chem. Phys. 87, 7323 (1987)] is extended to model the solid phase of chains whose monomers interact via a Lennard-Jones potential. Such an extension requires the free energy and contact values of the radial distribution function for the Lennard-Jones reference system in the solid phase. Computer simulations have been performed to determine the structural properties of the monomer Lennard-Jones system in the solid phase for a broad range of temperatures and densities. Computer simulations of dimer Lennard-Jones molecules in the solid phase have also been carried out. The theoretical results for the equation of state, the internal energy, and the sublimation curve of the dimer model in the solid phase are in excellent agreement with the simulation data. The extended theory is used to determine the global (solid–liquid–vapor) phase diagram of the LJ dimer model; the theoretical estimate of the triple point temperature for the LJ dimer is T*=0.653. Similarly, Wertheim’s TPT1 is used to determine the global phase diagram of chains formed by up to 8 monomer units. It is found that the calculated triple point temperature is hardly affected by the chain length, and that for large chain lengths the fluid–solid equilibrium coexistence densities are virtually independent of the number of monomers in the chain when the densities are expressed in monomer units. This is in agreement with experimental indications observed in polyethylene, where both the critical and the triple point temperatures tend to finite values for large molecular weights.

Extending Wertheim’s perturbation theory to the solid phase: The freezing of the pearl-necklace model

An extension of Wertheim’s first order perturbation theory [M. S. Wertheim, J. Chem. Phys. 87, 7323 (1987)] for chain molecules in the solid phase is proposed. In particular the solid phase of a model of freely jointed chains of tangent hard spheres has been considered. The equation of state and free energies from theory are compared with simulation results obtained previously by Malanoski and Monson [A. P. Malanoski and P. A. Monson, J. Chem. Phys. 107, 6899 (1997)] for chains with up to eight monomer units. The agreement between theory and simulation is good for the densities close to the melting, and deteriorates for densities close to close-packing. We also evaluated theoretically the fluid-solid equilibrium of the pearl-necklace model and compare with simulation results. The theory reproduces reasonably well the trends observed in the simulation results.

Liquid-solid transition of the ferromagnetic Heisenberg fluid: Simulation, density functional, and perturbation theories

The fluid-solid equilibrium of a system of hard spheres with embedded classical Heisenberg spins is studied by means of computer simulation and various combinations of density functional theories with the spin-spin contributions treated at the level of first-order perturbation theory. The phase boundaries are determined from the simulation data using free energy calculations in the ferromagnetic and isotropic phases. Estimates of the location of the Curie line in the solid are extracted from inspection of the evolution of magnetization with temperature. The agreement between theory and simulation is relatively good for the ferromagnetic transition and merely qualitative as far as the fluid-solid transition is concerned. The theoretical approach investigated here tends to underestimate the stability of the liquid phase.

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.

Atomic scale imaging of a 2D fluid-solid equilibrium for Cs and O coadsorbed on a Ru(0001) surface

A solid-fluid phase equilibrium has been studied by STM and LEED for coadsorbed Cs and O on a Ru(0001) surface. It is shown that a (√7 × √7)R19° Cs/O structure melts through a coexistance region into a 2D disordered phase upon rising the temperature. STM images recorded at 300 K reveal perfectly ordered islands coexisting with fluid areas which display also small ordered metastable clusters.

Isotopic fronts in hydrothermally mineralized carbonate rocks

Advective-dispersive fluid flow through permeable and porous rock causes systematic alteration of the infiltrated rock. O and Sr isotopes can be used as tracers to monitor exchange processes between mineralizing hydrothermal solutions and carbonate host rocks. Fluid infiltration into rock of initially uniform isotopic composition leads to characteristic changes in δ18O and ɛ87Sr in the rock, that depend on infiltration distance, fluid velocity, diffusivity and reaction kinetics. For fast fluid-solid equilibrium during advective flow, the shape of such alteration profiles can be described by the dimensionless Peclet number (P e ), which expresses the fluid flux, infiltration distance and diffusivity. In the case of mineralization from low to moderate temperature solutions, isotope exchange is not instantaneous, but kinetically controlled, and a similar parameter, the Damkohler number (N d ) expressing fluid flux, isotope exchange rate and infiltration distance, becomes more useful. N d - and P e -values have been estimated for various examples of carbonatehosted hydrothermal mineralization in the Eastern Alps (Austria) and the Benue Trough (Nigeria) by modelling O and Sr isotope data from ore (or gangue minerals) and host rock. The values obtained enable a rough differentiation between pervasive and channelled fluid flow as the prevailing transport mechanism to be made. Results indicate that both highly channelled fluid flow through fractures without significant interaction with the wall-rock, and pervasive fluid flow with infiltration distances into the percolated host rock of the order of 100 m, occur near mineral deposits. In the latter case, the extent of O and Sr isotopic alteration of the host rock can be used as an effective exploration tool.

Depletion interactions and fluid-solid equilibrium in emulsions.

Silicon-oil-in-water emulsions stabilized with sodium dodecylsulfate (SDS), have been studied. They exhibit practically no coalescence, whereas for high surfactant concentrations a ``creaming'' of the emulsion is observed. We demonstrate that this behavior is related to a fluid-solid phase transition due to an attractive interaction induced by the depletion of SDS micelles. A simple model for the fluid-solid transition is proposed, in quantitative agreement with experiment.

Advective-diffusive transport of isotopic fronts: An example from Naxos, Greece

Oxygen isotopic compositions on the margins of marble bands from kyanite-staurolite grade Miocene metamorphic rocks on the island of Naxos in the Aegean have been modified by infiltration of a fluid from underlying pelitic rocks. Detailed isotopic profiles across 1 m wide basal boundary layers fit, within error, the curves predicted for combined advective-diffusive modification of an initial step function in composition. This justifies the twin assumptions of constant porosity within the marbles and local (grain-scale) fluid-solid oxygen isotopic equilibrium on which the transport theory is based. The advective displacements give time integrated fluid fluxes of only 0.2 and 1.0 m3/m2 at the two higher grade localities sampled. Such volumes represent only a fraction of the fluid lost by devolatilisation of the underlying pelitic sequence. Flow regimes for two possible porosity structures can be derived from analysis of boundary layer shapes in terms of diffusive broadening and a gravitationally driven fluid flux. These are: (1) flow along a small (≈ 10−6) interconnected porosity along grain edges over a duration of ≈ 0.3Ma to 1Ma, or (2) flow in cracks with a larger porosity (≈ 10−4) but a much shorter time duration of ≈ 103 years. The time scale calculated for crack flow is that during which cracks were open to longitudinal diffusion. This may reflect the cumulative active time for intermittent deformation events.