School of Mathematical Sciences, Rochester Institute of Technology, Rochester, New York 14623, USA.
J Chem Phys. 2012 Jul 21;137(3):034201. doi: 10.1063/1.4731694.
We mathematically design sets of static light scattering experiments to provide for model-independent 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.
我们通过数学设计了一系列静态光散射实验，以提供模型独立的三元液体混合自由能测量，达到所需的精度水平。一个抛物型偏微分方程（PDE），由全非线性 PDE [D. Ross, G. Thurston, and C. Lutzer, J. Chem. Phys. 129, 064106 (2008)] 线性化而来，描述了数据噪声如何影响要推断的自由能。线性化 PDE 在组成三角形中创建了一张时空特征曲线网和正交的、类时曲线网，该网控制着来自光散射测量的信息向自由能的扩散。光散射扰动引起的自由能微扰沿着特征曲线及其凹侧扩散，扩散率与局部特征曲率半径成正比。因此，只要有合适的边界数据，静态光散射就可以在特征曲线边界具有凸性的区域确定混合自由能。介电系数是轨迹为 PDE 特征的动力系统的 Lyapunov 函数。在组成三角形中，信息扩散是不均匀的和系统相关的，因为特征取决于分子相互作用，并且在临界点处与液-液分相共存轨迹相切。我们发现了自由能精度、总测量时间、样本数量和插值方法之间的关联关系，并确定了在花费时间测量更多样本或更少样本但更准确之间的关键定量权衡。对于每个总测量时间，都有最佳的样本数量，超过这个数量，更多的样本不会提高自由能精度。我们估计多点插值和优化测量浓度可以提高精度和节省时间的程度。对于一个适度的光散射设置，一个样本计算表明，原则上，少于两分钟的测量时间足以确定非缔合三元混合物的无量纲混合自由能，其积分误差范数为 0.003。这些发现为设计光散射实验以确定三元液体混合物的吉布斯自由能建立了一个定量框架。
