A direct method for calculating thermodynamic factors for liquid mixtures using the Permuted Widom test particle insertion method

Understanding mass transport in liquids by mutual diffusion is an important topic for many applications in chemical engineering. The reason for this is that diffusion is often the rate limiting step in chemical reactors and separators. In multicomponent liquid mixtures, transport diffusion can be described by both generalized Fick's law and the MaxwellStefan theory. The MaxwellStefan and Fick approaches in an n-component system are related by the so-called thermodynamic factor [R. Taylor and H.A. Kooijman, Chem. Eng. Commun, 102, 87 (1991)]. As Fick diffusivities can be measured in experiments and MaxwellStefan diffusivities can be obtained from molecular simulations/theory, the thermodynamic factors bridge the gap between experiments and molecular simulations/theory. It is therefore desirable to be able to compute thermodynamic factors from molecular simulations. Unfortunately, presently used simulation techniques for computing thermodynamic factors are inefficient and often require numerical differentiation of simulation results. In this work, we propose a modified version of the Widom test-particle method to compute thermodynamic factors from a single simulation. This method is found to be more efficient than the conventional Widom test particle insertion method combined with numerical differentiation of simulation results. The approach is tested for binary systems consisting of Lennard-Jones particles. The thermodynamic factors computed from the simulation and from numerically differentiating the activity coefficients obtained from the conventional Widom test particle insertion method are in excellent agreement.