Ross David S, Thurston George M, Lutzer Carl V
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, New York 14623, USA.
J Chem Phys. 2008 Aug 14;129(6):064106. doi: 10.1063/1.2937902.
In this paper we present a method for determining the free energies of ternary mixtures from light scattering data. We use an approximation that is appropriate for liquid mixtures, which we formulate as a second-order nonlinear partial differential equation. This partial differential equation (PDE) relates the Hessian of the intensive free energy to the efficiency of light scattering in the forward direction. This basic equation applies in regions of the phase diagram in which the mixtures are thermodynamically stable. In regions in which the mixtures are unstable or metastable, the appropriate PDE is the nonlinear equation for the convex hull. We formulate this equation along with continuity conditions for the transition between the two equations at cloud point loci. We show how to discretize this problem to obtain a finite-difference approximation to it, and we present an iterative method for solving the discretized problem. We present the results of calculations that were done with a computer program that implements our method. These calculations show that our method is capable of reconstructing test free energy functions from simulated light scattering data. If the cloud point loci are known, the method also finds the tie lines and tie triangles that describe thermodynamic equilibrium between two or among three liquid phases. A robust method for solving this PDE problem, such as the one presented here, can be a basis for optical, noninvasive means of characterizing the thermodynamics of multicomponent mixtures.
在本文中，我们提出了一种根据光散射数据确定三元混合物自由能的方法。我们使用了一种适用于液体混合物的近似方法，将其表述为一个二阶非线性偏微分方程。这个偏微分方程（PDE）将强度自由能的海森矩阵与前向光散射效率联系起来。这个基本方程适用于相图中混合物热力学稳定的区域。在混合物不稳定或亚稳的区域，合适的偏微分方程是凸包的非线性方程。我们给出了这个方程以及在浊点轨迹处两个方程之间过渡的连续性条件。我们展示了如何离散化这个问题以获得其有限差分近似，并提出了一种求解离散化问题的迭代方法。我们给出了使用实现我们方法的计算机程序进行计算的结果。这些计算表明，我们的方法能够从模拟光散射数据中重建测试自由能函数。如果浊点轨迹已知，该方法还能找到描述两个或三个液相之间热力学平衡的系线和系三角。一种求解此类偏微分方程问题的稳健方法，比如本文所提出的方法，可成为表征多组分混合物热力学性质的光学、非侵入性手段的基础。
