(102e) Numerical Aspects of the Saft Equation of State

The Statistical Associating Fluid Theory (or SAFT) equation is a modern equation of state that has attracted considerable attention due to its capabilities to model components of different molecular size and association and solvation effects like hydrogen bonding and donor-acceptor bonding. The original SAFT equation of Chapman et al. (1990) and its variants like simplified SAFT, SAFT-VR and PC-SAFT are based on perturbations from an ideal gas state that contain residual Helmholtz free energy contributions for hard sphere attractive forces, dispersion forces, chain formation effects and association.

Association is an important molecular process that can have significant impact on bulk phase behavior and is present in wide variety of systems ranging from the formation of complexes like dimers to the binding and activity of protein molecules to the formation of micelles, vesicles and lamellae in biological systems. The association term in SAFT requires knowledge of the mole fractions of unbonded sites on molecules and, as Huang and Radosz (1990) clearly state, the equations that define the mole fractions of unbonded association sites on molecules are not, in general, explicit. In fact, they are only explicit for the case of a self-associating pure fluid with one association site (i.e., a pure fluid that forms only dimmers). Therefore, Huang and Radosz suggested a number of approximations to reduce the complexity of these equations to a single quadratic equation for self-associating pure fluids with up to four association sites. These approximations then ensure that the mole fraction of unbonded sites can be solved analytically. Key among the assumptions that Huang and Radosz make is the fact that all non-zero association strengths can be set equal. They argue that this is reasonable in order to reduce the number of SAFT model parameters. However, the approximations of Huang and Radosz of equal association strengths at all sites do not apply to the general situation and are actually unnecessary.

In this work we investigate a number of numerical aspects of the SAFT equation of state that include 1) The existence and uniqueness of solutions to the equations that define the mole fractions of unbonded association sites on molecules for the case of unequal association strengths, 2) The reliable and efficient computation of solutions to the equations that define the mole fractions of unbonded association sites on molecules, 3) The reliable and efficient solution of compressibility or density roots for the calculation of phase properties like fugacities and chemical potentials, and 4) The usefulness of SAFT to model self-assembling systems like lamellae and micelles.

A rigorous and constructive analysis is presented that clearly shows that the equations that define the mole fractions of unbonded association sites on molecules for the case of unequal association strengths proposed by Chapman et al. are a two-step contractive map on the domain [0, 1]. This clearly suggests that direct substitution can be used to reliably solve these equations. However, the rate of convergence is generally slow. Therefore, we propose a quasi-Newton accelerated direct substitution iteration that speeds convergence without sacrificing reliability. Numerical results for a wide variety of multi-component mixtures that exhibit hydrogen bonding are used to support these claim. It is also shown that analytical first and second derivatives of the mole fractions of unbonded sites with respect to density can be obtained using the implicit function theorem but must be computed numerically by solving appropriate systems of linear equations. These derivatives can then be used, along with first and second derivatives of the hard sphere, dispersion, and chain terms, to determine all density roots of the SAFT equation using a terrain methodology. Interestingly, it is shown that the SAFT equation exhibits a large number of density roots as well as a number of asymptotes that are occur as a result of the hard sphere term approaching infinity. Finally, we report on the use of the SAFT equation to model the phase equilibrium of a number of mixtures including a binary surfactant-water mixture that exhibits lamellae. Many geometric illustrations are used to elucidate the key ideas of this study.