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

Abstract

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.

Mathematical methods for restricted domain ternary liquid mixture free energy determination using light scattering

“…The answer to the question of how much rotation can be achieved in practice will depend on the system and the scattering probe; and the answer to the question of how much rotation is needed will require statistical analysis of the sort we have published for ternary and quaternary mixture LSPDEs. 2,3 We leave both of these questions for future analysis, but we expect that in practice rotations of the dielectric gradient will be considerably smaller than those illustrated in Fig. 2.…”

“…(2), H g [g(x)] is the Hessian of the free energy g(x) with respect to composition x, ∇ε (ω, x)∇ε(ω, x) is a rank-one matrix, ε(ω, x) is the mixture dielectric coefficient, and g, x, ε, and R have been put in dimensionless form. 2,3 It is important to note that Eqs. (1) and (2) apply not only in singlephase regions of composition space, but also in multiphase regions, provided R is taken to be infinite at such compositions.…”

“…In this paper we extend the light scattering partial differential equation (LSPDE) method of determining mixing free energies [1][2][3][4] to apply to limited composition domains of ternary molecular mixtures. To date, we have developed the LSPDE method for mixtures that have isotropic liquid phases within a simply connected set of compositions that touches all edges of the ternary composition triangle, 2 or composition tetrahedron, for quaternary mixtures.…”

“…To date, we have developed the LSPDE method for mixtures that have isotropic liquid phases within a simply connected set of compositions that touches all edges of the ternary composition triangle, 2 or composition tetrahedron, for quaternary mixtures. 3 However, many important mixtures, such as aqueous, multicomponent protein mixtures, 5-8 micellar mixtures and microemulsions, [9][10][11][12][13] polymer mixtures, 14 and mixtures of small molecules 10,15 do not always have liquid domains of this type, and instead may form liquid crystals, solids, or other states at compositions that prevent the liquid phase compositions from reaching one or more domain edges. Model examples of such regions, computed from a regular solution ternary mixture free energy, 16,17 are shown by the clear zones in Fig.…”

“…composition, g (x), and the single-phase excess light scattering efficiency or Rayleigh ratio, R(ω, x) at frequency ω, as a fully nonlinear partial differential equation, 1,3 λ 0 (x) = 0,…”