On inferring liquid-liquid phase boundaries and tie lines from ternary mixture light scattering

We investigate the possibility of using light scattering data in the single-phase regions of a ternary liquid mixture phase diagram to infer ternary mixture coexistence curves, and to infer tie lines joining the compositions of isotropic liquid phases in thermodynamic equilibrium. Previous analyses of a nonlinear light scattering partial differential equation (LSPDE) show that it provides for reconstruction of ternary [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)] and quaternary [C. Wahle, D. Ross, and G. Thurston, J. Chem. Phys. 137, 034202 (2012)] mixing free energies from light scattering data, and that if the coexistence curves are already known, it can also yield ternary tie lines and triangles [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008)]. Here, we show that the LSPDE can be used more generally, to infer phase boundaries and tie lines from light scattering data in the single-phase region, without prior knowledge of the coexistence curve, if the single-phase region is connected. The method extends the fact that the reciprocal light scattering intensity approaches zero at the thermodynamic spinodal. Expressing the free energy as the sum of ideal and excess parts leads to a natural family of Padé approximants for the reciprocal Rayleigh ratio. To test the method, we evaluate the single-phase reciprocal Rayleigh ratio resulting from the mean-field, regular solution model on a fine grid. We then use a low-order approximant to extrapolate the reciprocal Rayleigh ratio into metastable and unstable regions. In the metastable zone, the extrapolation estimates light scattering prior to nucleation and growth of a new phase. In the unstable zone, the extrapolation produces a negative function that in the present context is a computational convenience. The original and extrapolated reciprocal light scattering are jointly used as input to solving the LSPDE to deduce the mixing free energy and its convex hull. When projected onto the composition triangle, the boundary of the convexified part of the free energy is the phase boundary, and lines on the convexified region along which the second directional derivative is zero are the tie lines. We find that the tie lines and phase boundaries so deduced agree well with their exact values. This work is a step toward developing methods for inferring phase boundaries from real light scattering intensities measured with noise, from mixtures having compositions on a coarser grid.