Monte Carlo calculations of the equation of state of the square‐well fluid as a function of well width

A Thermodynamic Approach for the Prediction of Oiling Out Boundaries from Solubility Data

Many pharmaceutical molecules, fine chemicals, and proteins exhibit liquid–liquid phase separation (LLPS, also known as oiling out) during solution crystallization. LLPS is of significant concern in crystallization process development, as oiling out can compromise the effectiveness of a crystallization and can lead to operational problems. A comprehensive methodology that allows a process scientist/engineer to characterize the various phase boundaries relevant to oiling out is currently lacking. In this work, we present a modeling framework useful in predicting the binodal, spinodal, and gelation boundaries starting from the solubility data of a solute that is prone to oiling out. We collate the necessary theoretical concepts from the literature and describe a unified approach to model the phase equilibria of solute–solvent systems from first principles. The modeling effort is validated using experimental data reported in the literature for various solute–solvent systems. The predictive methods presented in this work can be easily implemented and help a process engineer establish the design space for a crystallization process that is affected by liquid–liquid phase separation.

Self-diffusion coefficient of the square-well fluid from molecular dynamics simulations within the constant force approach

We present a systematic study of the self-diffusion coefficient for a fluid of particles interacting via the square-well pair potential by means of molecular dynamics simulations in the canonical (N, V, T) ensemble. The discrete nature of the interaction potential is modeled by the constant force approximation, and the self-diffusion coefficient is determined for several fluid densities at supercritical thermodynamic states. The dependence of the self-diffusion coefficient on the potential range λ is analyzed in the range of 1.1 ≤ λ ≤ 1.5. The obtained simulation results are in agreement with the self-diffusion coefficient predicted by the Enskog method. Additionally, we show that the diffusion coefficient is very sensitive to the potential range λ. Our results for the self-diffusion coefficient times density extrapolate well to the values in the zero-density limit obtained from the Chapman-Enskog theory for dilute gases. The constant force approximation used in this work to model the discrete pair potentials has shown to be an excellent scheme to compute the transport properties of square-well fluids using molecular dynamics simulations. Finally, the simulation results presented here are useful for improving theoretical approaches, such as the Enskog method.

Third-order thermodynamic perturbation theory for effective potentials that model complex fluids

We have performed Monte Carlo simulations to obtain the thermodynamic properties of fluids with two kinds of hard-core plus attractive-tail or oscillatory potentials. One of them is the square-well potential with small well width. The other is a model potential with oscillatory and decaying tail. Both model potentials are suitable for modeling the effective potential arising in complex fluids and fluid mixtures with extremely-large-size asymmetry, as is the case of the solvent-induced depletion interactions in colloidal dispersions. For the former potential, the compressibility factor, the excess energy, the constant-volume excess heat capacity, and the chemical potential have been obtained. For the second model potential only the first two of these quantities have been obtained. The simulations cover the whole density range for the fluid phase and several temperatures. These simulation data have been used to test the performance of a third-order thermodynamic perturbation theory (TPT) recently developed by one of us [S. Zhou, Phys. Rev. E 74, 031119 (2006)] as compared with the well-known second-order TPT based on the macroscopic compressibility approximation due to Barker and Henderson. It is found that the first of these theories provides much better accuracy than the second one for all thermodynamic properties analyzed for the two effective potential models.

Thermodynamic properties of model solids with short-ranged potentials from Monte Carlo simulations and perturbation theory

Monte Carlo simulations have been performed to determine the excess energy and the equation of state of fcc solids with Sutherland potentials for wide ranges of temperatures, densities, and effective potential ranges. The same quantities have been determined within a perturbative scheme by means of two procedures: (i) Monte Carlo simulations performed on the reference hard-sphere system and (ii) second-order Barker-Henderson perturbation theory. The aim was twofold: on the one hand, to test the capability of the "exact" MC-perturbation theory of reproducing the direct MC simulations and, on the other hand, the reliability of the Barker-Henderson perturbation theory, as compared with direct MC simulations and MC-perturbation theory, to determine the thermodynamic properties of these solids depending on temperature, density, and potential range. We have found that the simulation data for the excess energy obtained from the two procedures are in close agreement with each other. For the equation of state, the results from the MC-perturbation procedure also agree well with the direct MC simulations except for very low temperatures and extremely short-ranged potentials. Regarding the Barker-Henderson perturbation theory, we have found that in general the second-order approximation does not provide significant improvement over the first-order one.

Improvement on macroscopic compressibility approximation and beyond

A numerical procedure is proposed to extend the thermodynamic perturbation expansion (TPE) to a higher order. It is shown that the present second order term is superior to that due to a macroscopic compressibility approximation (MCA), a local compressibility approximation, and a superposition approximation by Barker and Henderson [Rev. Mod. Phys. 48, 587 (1976)]. Extensive model calculation and comparison with simulation data available in literature and supplied in the present report indicate that the present third order TPE is superior to a previous second order TPE based on the MCA, two previous perturbation theories, which are respectively based on an analytical mean spherical approximation for an Ornstein-Zernike equation, and an assumed explicit functional form for the Laplace transform of radial distribution function multiplied by radial distance, and a recent generalized van der Waals theory. The present critical temperature for a hard core attractive Yukawa fluid of varying range is in very good agreement with that due to a hierarchical reference theory. The present third order TPE is computationally far more modest than the self-consistent integral equation theory, and therefore is a viable alternative to use of the latter.

Theoretical prediction of the coordination number, local composition, and pressure-volume-temperature properties of square-well and square-shoulder fluids

By assuming a Boltzmann distribution for the molecular equilibrium between local and bulk environments, a general model is derived for the prediction of coordination numbers and local compositions of square-well and square-shoulder fluids. The model has no empirical parameter fitted from the data of square-well and square-shoulder fluids, but is valid from the low-density limit to the high-density limit. The applicable width of well or shoulder covers the commonly used range varying from 1.0 to 2.0. The model can accurately predict the coordination numbers of pure square-well and square-shoulder fluids, so the equation of state derived from it is superior to other equations of state based on the existing coordination number models. The model also accurately predicts the local compositions of mixtures in wide ranges of density and size ratio (1.0-8.0), as well as the configuration energy of lattice gases and highly nonideal lattice mixtures. It is remarkable that the model correctly predicts temperature-dependent coordination numbers and local compositions for both equal- and unequal-sized mixtures at close packing, which cannot be predicted by the existing coordination number models. Our derivation demonstrates that the energy parameters in local composition models should represent the potential difference of a molecule between the local and bulk environments, not the pair-interaction potential, and depend on the system conditions and different kinds of pair-interaction parameters. This result is very useful for the development of local composition and activity coefficient models and the mixing rules of equations of state.

Pair correlation function of short-ranged square-well fluids

We have performed extensive Monte Carlo simulations in the canonical (NVT) ensemble of the pair correlation function for square-well fluids with well widths lambda-1 ranging from 0.1 to 1.0, in units of the diameter sigma of the particles. For each one of these widths, several densities rho and temperatures T in the ranges 0.1< or =rhosigma(3)< or =0.8 and T(c)(lambda) less or approximately T less or approximately 3T(c)(lambda), where T(c)(lambda) is the critical temperature, have been considered. The simulation data are used to examine the performance of two analytical theories in predicting the structure of these fluids: the perturbation theory proposed by Tang and Lu [Y. Tang and B. C.-Y. Lu, J. Chem. Phys. 100, 3079 (1994); 100, 6665 (1994)] and the nonperturbative model proposed by two of us [S. B. Yuste and A. Santos, J. Chem. Phys. 101 2355 (1994)]. It is observed that both theories complement each other, as the latter theory works well for short ranges and/or moderate densities, while the former theory works for long ranges and high densities.

Liquid–vapor interface of square-well fluids of variable interaction range

Properties of the liquid-vapor interface of square-well fluids with ranges of interaction lambda=1.5, 2.0, and 3.0 are obtained by Monte Carlo simulations and from square-gradient theories that combine the Carnahan-Starling equation of state for hard spheres with the second and third virial coefficients. The predicted surface tensions show good agreement with the simulation results for lambda=2 and for lambda=3 in a temperature range reasonably close to the critical point, 0.8</=T/T(c)</=0.95. As expected, the surface tension increases with the range of interaction and decreases monotonically with temperature. A comparison between theory and simulation results is also given for the width of the interface and for the coexistence curves for the different interaction ranges.