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

We extend methods of solution of a light scattering partial differential equation for the free energy of mixing to apply to connected, isotropic ternary liquid composition domains that do not touch all three binary axes. To do so we mathematically analyze the problem of inferring needed Dirichlet boundary data, and solving for the free energy, with use of hypothetical static light scattering measurements that correspond to dielectric composition gradient vectors that have distinct directions. The physical idea behind the technique is that contrasting absorption properties of mixture components can result in such distinctly directed dielectric composition gradient vectors, due to their differing wavelength dependences of dielectric response. At suitably chosen wavelengths, contrasting light scattering efficiency patterns in the ternary composition triangle can then correspond to the same underlying free energy, and enlarge the scope of available information about the free energy, as shown here. We show how to use distinctly directed dielectric gradients to measure the free energy on both straight lines and curves within the ternary composition triangle, so as to provide needed Dirichlet conditions for light scattering partial differential equation solution. With use of Monte Carlo simulations of noisy light scattering data, we provide estimates of the overall system measurement time and sample spacing needed to determine the free energy to a desired degree of accuracy, for various angles between the assumed dielectric gradient vectors, and indicate how the measurement time depends on instrumental throughput parameters. The present analysis methods provide a way to use static light scattering to measure, directly, mixing free energies of many systems that contain such restricted liquid domains, including aqueous solutions of biological macromolecules, micellar mixtures and microemulsions, and many small molecule systems that are important in separation technology.