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Center For Applied and Computational Mathematics

Inferring Gibbs Free Energies from Light Scattering Data

Gibbs Free Energy
Faculty: David Ross
  George Thurston
  Carl Lutzer
  Chris Wahle


In this work we use a well-established relation for light scattering from liquid mixtures as a second-order nonlinear partial differential equation, which relates the inverse Hessian of the intensive free energy to the efficiency of light scattered near the forward direction.  For ternary mixtures, this PDE is

Ternary Mixture Equation


Hessian Limiting formula

The dielectric coefficient, Dialectric Coefficient, and the Rayleigh ratio, Rayleigh Ratio are known experimentally as functions of the mole fractions x and y.  The Widom form of the Gibbs free energy ensures that the second normal derivatives of Gibbs Free Energy are logarithmically singular at the boundaries of the mixture triangle.  This condition acts as a boundary condition.  We have devised an algorithm that allows us to solve the PDE with this boundary condition to determine Gibbs Free Energy.


  1. On the Design of Experiments for Determining Ternary Mixture Free Energies from Static Light Scattering Data using a Nonlinear Partial Differential Equation, C. Wahle, D. S. Ross, G. Thurston, J. Chem. Phys.,137, 034201 (2012).

  2. On inferring liquid-liquid phase boundaries and tie lines from ternary mixture light scattering, C. Wahle, D. S. Ross, G. Thurston, J. Chem. Phys.,137, 034203 (2012).

  3. Mathematical and Computational Aspects of Quaternary Liquid Mixing Free Energy Measurement Using Light Scattering, C. Wahle, D. S. Ross, G. Thurston, J. Chem. Phys.,137, 034202 (2012).

  4. On a partial differential equation method for determining the free energies and coexisting phase compositions of ternary mixtures from light scattering data, D.S. Ross, G.M. Thurston and C.V. Lutzer, J. Chem. Phys. 129, 064106 (2008).