Generalized Form for Finite-Size Corrections in Mutual Diffusion Coefficients of Multicomponent Mixtures Obtained from Equilibrium Molecular Dynamics Simulation

Abstract

The system-size dependence of computed mutual diffusion coefficients of multicomponent mixtures is investigated, and a generalized correction term is derived. The generalized finite-size correction term was validated for the ternary molecular mixture chloroform/acetone/methanol as well as 28 ternary LJ systems. It is shown that only the diagonal elements of the Fick matrix show system-size dependency. The finite-size effects of these elements can be corrected by adding the term derived by Yeh and Hummer (J. Phys. Chem. B 2004, 108, 15873-15879). By performing an eigenvalue analysis of the finite-size effects of the matrix of Fick diffusivities we show that the eigenvector matrix of Fick diffusivities does not depend on the size of the simulation box. Only eigenvalues, which describe the speed of diffusion, depend on the size of the system. An analytic relation for finite-size effects of the matrix of Maxwell-Stefan diffusivities was developed. All Maxwell-Stefan diffusivities depend on the system size, and the required correction depends on the matrix of thermodynamic factors.

Figure 1

Computed elements of the Fick diffusivity matrix for the ternary mixture of (1) chloroform, (2) acetone, and (3) methanol (x_chloroform = x_acetone = 0.3) as a function of the inverse simulation box length L: (a) Diagonal element D_Fick1-1, (b) off-diagonal element D_Fick1-2, (c) off-diagonal element D_Fick2-1, and (d) diagonal element D_Fick2-2. Blue circles are the computed diffusion coefficients in MD simulations. Red squares are corrected diffusivities using the YH correction (eq 8). Dashed lines show extrapolation to the thermodynamic limit, and solid lines are the extrapolated values. All data related to Fick diffusion coefficient computations are provided in the Simulation Details section and in the Supporting Information.

Figure 2

Finite-size corrections required for elements of the Fick diffusivity matrix (a) D_Fick,1-1, (b) D_Fick,1-2, (c) D_Fick,2-1, and D_Fick,2-2 as a function of the YH correction (D^YH, eq 8) for 28 LJ systems containing 500 particles (blue circles), 1000 particles (red squares), 2000 particles (green diamonds), and 4000 particles (magenta pentagons). Reduced temperature is 0.65 and reduced pressure 0.05. The diagonal dashed lines indicate perfect agreement between the YH correction and the required finite size corrections for Fick diffusivities. The vertical dashed lines indicate no finite-size effects of Fick diffusivities. Raw data are provided in the Supporting Information.

Figure 3

MS diffusion coefficients for a mixture of (1) chloroform, (2) acetone, and (3) methanol (x_chloroform = x_acetone = 0.3) as a function of the simulation box length (L). Blue circles are the computed diffusion coefficients in MD simulations. Red squares are the corrected diffusivities using the proposed correction in eq 30. Dashed lines show extrapolation to the thermodynamic limit, and solid lines are the extrapolated values.

Figure 4

Comparison between finite-size corrections required for MS diffusivities (Đ_MS^∞ – Đ_MS^MD) and the proposed correction for MS diffusivities (Đ_MS^correction, eq 30, for 28 LJ systems containing 500 particles (blue circles), 1000 particles (red squares), 2000 particles (green diamonds), and 4000 particles (magenta pentagons). Parts 1-2, 1-3, and 2-3 indicate Đ_1-2, Đ_1-3, and Đ_2-3, respectively. Reduced temperature is 0.65 and reduced pressure 0.05. The dashed lines indicate perfect agreement between the proposed correction and the required finite size corrections for MS diffusivities. Raw data are provided in the Supporting Information.