Bot Arjen, Venema Paul
Laboratory of Physics and Physical Chemistry of Foods, Department of Agrotechnology and Food Sciences, Wageningen University and Research, Bornse Weilanden 9, NL-6708 WG Wageningen, The Netherlands.
Unilever Foods Innovation Centre, Bronland 14, NL-6708 WH Wageningen, The Netherlands.
ACS Omega. 2023 Jul 26;8(31):28387-28408. doi: 10.1021/acsomega.3c02604. eCollection 2023 Aug 8.
The Edmond-Ogston model for phase separation is extended to ternary polymer mixtures in a common solvent (de facto a quaternary mixture). The model assumes a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers, and the second virial coefficients (, , , , , ) are the six parameters of the model. New results from this model are presented in relation to earlier work on binary mixtures: a necessary condition for the virial coefficients for the occurrence of phase separation in two or three phases, an analysis of the different regions of (local) thermodynamic instability using the Descartes sign rule, an expression for the critical curves, a relation between the tangents in points along the critical curve, a relation between the concentration of components in the different phases according to the so-called Lambert-W function, and a consistency check for the composition of coexisting phases in ternary mixtures. The obtained results are evaluated in the maximally symmetric version of the model, where (, , ) are equal and (, , ) are equal, which leads to two remarkable observations: the concentration range over which two phases are formed is relatively narrow; not all phase separation occurs within a Gibbs triangle, but also, "out-of-Gibbs-triangle" binodals are observed. These results lead to a deeper insight into the phase behavior of ternary mixtures and show promise as a stepping stone toward modeling phase separation in mixtures with many components.
埃德蒙 - 奥格斯顿相分离模型被扩展到在一种常见溶剂中的三元聚合物混合物（实际上是一种四元混合物）。该模型假设亥姆霍兹自由能的维里展开式在聚合物浓度的二阶项处截断，第二维里系数（，，，，，）是该模型的六个参数。本文给出了该模型相对于早期二元混合物研究的新结果：关于两相或三相中发生相分离的维里系数的必要条件，使用笛卡尔符号法则对（局部）热力学不稳定的不同区域的分析，临界曲线的表达式，沿临界曲线各点切线之间的关系，根据所谓的兰伯特 - W函数得到的不同相中各组分浓度之间的关系，以及对三元混合物中共存相组成的一致性检验。在模型的最大对称版本中对所得结果进行了评估，其中（，，）相等且（，，）相等，这导致了两个显著的观察结果：形成两相的浓度范围相对较窄；并非所有相分离都发生在吉布斯三角形内，还观察到了“吉布斯三角形外”的双节线。这些结果使我们对三元混合物的相行为有了更深入的了解，并有望成为迈向多组分混合物相分离建模的垫脚石。
