Stabilized density gradient theory algorithm for modeling interfacial properties of pure and mixed systems

Abstract

Density gradient theory (DGT) allows fast and accurate determination of surface tension and density profile through a phase interface. Several algorithms have been developed to apply this theory in practical calculations. While the conventional algorithm requires a reference substance of the system, a modified "stabilized density gradient theory" (SDGT) algorithm is introduced in our work to solve DGT equations for multiphase pure and mixed systems. This algorithm makes it possible to calculate interfacial properties accurately at any domain size larger than the interface thickness without choosing a reference substance or assuming the functional form of the density profile. As part of DGT inputs, the perturbed chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) was employed for the first time with the SDGT algorithm. PC-SAFT has excellent performance in predicting liquid phase properties as well as phase behaviors. The SDGT algorithm with the PC-SAFT EoS was tested and compared with experimental data for several systems. Numerical stability analyses were also included in each calculation to verify the reliability of this approach for future applications. List of symbols Symbol Units DescriptionA 0 J Homogeneous Helmholtz free energy A id 0 J Ideal gas contribution to A 0 A hs 0 J Hard sphere contribution to A 0 A hc 0 J Hard chain contribution to A 0 A disp 0

A stable algorithm for calculating phase equilibria with capillarity at specified moles, volume and temperature using a dynamic model

“…So a semiimplicit scheme is a preferable choice to preserve the entropy stability and admit reasonable large time steps. The effectiveness of convex-concave splitting techniques has been demonstrated in numerical simulation of the realistic fluids [5,15,24,29]. In Subsection 3.3 , we can prove that the convex-concave splitting of Helmholtz free energy density can result in the entropy stability of numerical schemes.…”

“…unless the time step size is restricted to be sufficiently small [15,24,29]. So a semiimplicit scheme is a preferable choice to preserve the entropy stability and admit reasonable large time steps.…”

Tolman lengths and rigidity constants of multicomponent fluids: Fundamental theory and numerical examples

“…It is challenging to achieve a numerically robust implementation of SGT, and developing various approaches for this is an active research area. [55][56][57] To ensure a correct implementation, the code has been subjected to the following consistency checks:…”

General approach for solving the density gradient theory in the interfacial tension calculations

“…Both the finite difference technique and the finite element method are commonly used [1,5,29,30] for solving the coupled second-order differential equations. The discretization requires a finite value for the interface width, and it is common practice to use a sufficiently large value relative to the 'effective' interface width.…”

“…The discretization requires a finite value for the interface width, and it is common practice to use a sufficiently large value relative to the 'effective' interface width. According to previous literature [1,8,29,30], these algorithms are unstable. To improve the stability, Qiao et al [29] and Mu et al [30] proposed to add a timederivative term.…”