Mathematical and computational aspects of quaternary liquid mixing free energy measurement using light scattering

We provide a mathematical and computational analysis of light scattering measurement of mixing free energies of quaternary isotropic liquids. In previous work, we analyzed mathematical and experimental design considerations for the ternary mixture case [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008); C. Wahle, D. Ross, and G. Thurston, J. Chem. Phys. 137, 034201 (2012)]. Here, we review and introduce dimension-free general formulations of the fully nonlinear partial differential equation (PDE) and its linearization, a basis for applying the method to composition spaces of any dimension, in principle. With numerical analysis of the PDE as applied to the light scattering implied by a test free energy and dielectric gradient combination, we show that values of the Rayleigh ratio within the quaternary composition tetrahedron can be used to correctly reconstruct the composition dependence of the free energy. We then extend the analysis to the case of a finite number of data points, measured with noise. In this context the linearized PDE describes the relevant diffusion of information from light scattering noise to the free energy. The fully nonlinear PDE creates a special set of curves in the composition tetrahedron, collections of which form characteristics of the nonlinear and linear PDEs, and we show that the information diffusion has a time-like direction along the positive normals to these curves. With use of Monte Carlo simulations of light scattering experiments, we find that for a modest laboratory light scattering setup, about 100-200 samples and 100 s of measurement time are enough to be able to measure the mixing free energy over the entire quaternary composition tetrahedron, to within an L(2) error norm of 10(-3). The present method can help quantify thermodynamics of quaternary isotropic liquid mixtures.