The A in SAFT: developing the contribution of association to the Helmholtz free energy within a Wertheim TPT1 treatment of generic Mie fluids

An accurate representation of molecular association is a vital ingredient of advanced equations of state (EOSs), providing a description of thermodynamic properties of complex fluids where hydrogen bonding plays an important role. The combination of the first-order thermodynamic perturbation theory (TPT1) of Wertheim for associating systems with an accurate description of the structural and thermodynamic properties of the monomer fluid forms the basis of the statistical associating fluid theory (SAFT) family of EOSs. The contribution of association to the free energy in SAFT and related EOSs is very sensitive to the nature of intermolecular potential used to describe the monomers and, crucially, to the accuracy of the representation of the thermodynamic and structural properties. Here we develop an accurate description of the association contribution for use within the recently developed SAFT-VR Mie framework for chain molecules formed from segments interacting through a Mie potential [T. Lafitte, A. Apostolakou, C. Avendaño, A, Galindo, C. S. Adjiman, E. A. Müller, and G. Jackson, J. Chem. Phys. <b>139</b>, 154504 (2013)]. As the Mie interaction represents a soft-core potential model, a method similar to that adopted for the Lennard-Jones potential [E. A. Müller and K. E. Gubbins, Ind. Eng. Chem. Res. <b>34</b>, 3662 (1995)] is employed to describe the association contribution to the Helmholtz free energy. The radial distribution function (RDF) of the Mie fluid (which is required for the evaluation of the integral at the heart of the association term) is determined for a broad range of thermodynamic conditions (temperatures and densities) using the reference hyper-netted chain (RHNC) integral-equation theory. The numerical data for the association kernel of Mie fluids with different association geometries are then correlated for a range of thermodynamic states to obtain a general expression for the association contribution which can be applied for varying values of the Mie repulsive exponent. The resulting SAFT-VR Mie EOS allows for a much improved description of the vapour-liquid equilibria and single-phase properties of associating fluids such as water, methanol, ammonia, hydrogen sulphide, and their mixtures. A comparison is also made between the theoretical predictions of the degree of association for water and the extent of hydrogen bonding obtained from molecular simulations of the SPC/E and TIP4P/2005 atomistic models.

分子缔合的精准表征是高级状态方程（equations of state, EOSs）的核心要素，可为氢键（hydrogen bonding）发挥关键作用的复杂流体的热力学性质提供严谨描述。将Wertheim提出的适用于缔合体系的一级热力学微扰理论（first-order thermodynamic perturbation theory, TPT1），与单体流体的结构及热力学性质的精准表征相结合，构成了统计缔合流体理论（statistical associating fluid theory, SAFT）家族状态方程的理论基础。在SAFT及相关状态方程中，缔合作用对自由能的贡献对描述单体所用的分子间势能的性质极为敏感，且至关重要地取决于热力学与结构性质的表征精度。本文针对近期开发的、针对由米氏势能（Mie potential）相互作用的链段构成的链状分子的SAFT-VR米氏状态方程框架，构建了一种精准的缔合作用描述方法[T. Lafitte, A. Apostolakou, C. Avendaño, A. Galindo, C. S. Adjiman, E. A. Müller, and G. Jackson, *J. Chem. Phys.* **139**, 154504 (2013)]。由于米氏相互作用属于软核势能模型，本文采用了与伦纳德-琼斯势（Lennard-Jones potential）相关研究类似的方法[E. A. Müller and K. E. Gubbins, *Ind. Eng. Chem. Res.* **34**, 3662 (1995)]，以描述缔合作用对亥姆霍兹自由能（Helmholtz free energy）的贡献。缔合项核心积分的计算需要用到米氏流体的径向分布函数（radial distribution function, RDF），本文借助参考超净链（reference hyper-netted chain, RHNC）积分方程理论，在宽泛的热力学条件（温度与密度范围）下对其进行了求解。随后，针对一系列热力学状态下不同缔合几何构型的米氏流体的缔合核数值数据进行关联，得到了可适用于不同米氏排斥指数的缔合作用通用表达式。所构建的SAFT-VR米氏状态方程可显著提升对水、甲醇、氨、硫化氢等缔合流体及其混合物的气液平衡（vapour-liquid equilibria）与单相性质的刻画精度。本文还将水的缔合度理论预测结果，与基于SPC/E和TIP4P/2005原子级模型的分子模拟得到的氢键形成程度进行了对比。