Noura Dawass

1 18 A PPLICATIONS OF K IRKWOOD –B UFF I NTEGRALS where c t is the total molar concentration and ∇ x j is the gradient of the mole fraction x j , which is the driving force in Fick’s law. In a molar reference frame, we have Σ n i = 1 J i = 0. MS diffusivities can be computed from MD simulations and Fick diffusivities can be measured by experiments [64, 66, 68, 70, 92] . The MS diffusivity ¯ D i j can be considered as an inverse friction term in an equation where the gradient in chemical potential ∇ µ i of component i is related to differences in the average velocities between species: − 1 RT ∇ µ i = n X j = 1( j 6 = i ) x j ( u i − u j ) ¯ D i j (1.34) where ( u i − u j ) is the difference between the average velocities of components i and j . As chemical potentials cannot be measured directly, it is not possible to directly compare MS diffusivities to experiments. It is more convenient to com- pute MS diffusivites using molecular simulation. Details on this are provided in Refs. [70, 73, 92] . Often, Fick diffusivities depend more strongly on concentra- tion than MS diffusivities [64, 68] . Moreover, it is possible to predict diffusion of multicomponent mixtures ( n > 2) from the knowledge of MS diffusivites of the corresponding binary mixtures [64, 69, 70] . For a mixture with more than two components, Fick diffusivities depend on the type of reference frame, unlike MS diffusivities [64, 68, 93] . In a molar reference frame, Fick diffusivities and the thermodynamic factor can be used to compare MS diffusivites with experimen- tal data [66] , [ D ] = [ B ] − 1 [ Γ ] (1.35) where [ D ] is the Fick diffusion coefficients matrix. The elements of the matrix [ B ] can be computed using B ii = x i ¯ D in + n X j = 1( j 6 = i ) x j ¯ D i j with i = 1, 2, ...( n − 1) (1.36) B i j = − x i µ 1 ¯ D i j − 1 ¯ D in ¶ with i = 1, 2, ...( n − 1) and i 6 = j (1.37) and the elements of matrix [ Γ ] can be expressed as a function of KBIs. For bi- nary systems, the relation between Γ i j and KBIs is provided by Eq. (1.14) . In Refs. [70] and [64] , KBIs from simulations of finite systems were computed us- ing the approach of Krüger and co-workers [74, 80, 81, 84] . KBIs of binary and ternary mixtures were used to compute the thermodynamic factors and convert MS diffusivities computed from simulations to Fick diffusivities measured by ex- periments. The proposedmethod was applied to binary and ternary alcohol mix- tures [64, 70] .