On the design of experiments for determining ternary mixture free energies from static light scattering data using a nonlinear partial differential equation

Abstract

We mathematically design sets of static light scattering experiments to provide for modelindependent measurements of ternary liquid mixing free energies to a desired level of accuracy. A parabolic partial differential equation (PDE), linearized from the full nonlinear PDE [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008)], describes how data noise affects the free energies to be inferred. The linearized PDE creates a net of spacelike characteristic curves and orthogonal, timelike curves in the composition triangle, and this net governs diffusion of information coming from light scattering measurements to the free energy. Free energy perturbations induced by a light scattering perturbation diffuse along the characteristic curves and towards their concave sides, with a diffusivity that is proportional to the local characteristic curvature radius. Consequently, static light scattering can determine mixing free energies in regions with convex characteristic curve boundaries, given suitable boundary data. The dielectric coefficient is a Lyapunov function for the dynamical system whose trajectories are PDE characteristics. Information diffusion is heterogeneous and system-dependent in the composition triangle, since the characteristics depend on molecular interactions and are tangent to liquid-liquid phase separation coexistence loci at critical points. We find scaling relations that link free energy accuracy, total measurement time, the number of samples, and the interpolation method, and identify the key quantitative tradeoffs between devoting time to measuring more samples, or fewer samples more accurately. For each total measurement time there are optimal sample numbers beyond which more will not improve free energy accuracy. We estimate the degree to which many-point interpolation and optimized measurement concentrations can improve accuracy and save time. For a modest light scattering setup, a sample calculation shows that less than two minutes of measurement time is, in principle, sufficient to determine the dimensionless mixing free energy of a non-associating ternary mixture to within an integrated error norm of 0.003. These findings establish a quantitative framework for designing light scattering experiments to determine the Gibbs free energy of ternary 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.…”

“…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. The formulation given in Eqs.…”

“…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.…”