Noura Dawass

5 98 P ROPERTIES OF U REA -C HOLINE C HLORIDE M IXTURES 5.2.2. M AXWELL –S TEFAN DIFFUSION COEFFICIENTS OF PSEUDO – BINARY MIXTURES The MS diffusion coefficient in a three-dimensional system can be computed from the Onsager coefficients ( Λ i j ). Onsager coefficients are defined as the cross-correlation of the displacement of the molecules of species i and j in a multicomponent mixture: [66, 152] Λ i j = lim t →∞ 1 2 t 1 3 N ¿ Ã N i X k = 1 [ r k , i ( t ) − r k , i (0)] ! . Ã N j X l = 1 [ r k , j ( t ) − r k , j (0)] ! À (5.25) where N , N i and N i are the total number of molecules, number of molecules of species i and j , respectively. The matrix of Onsager coefficients has a symmetric nature. Thus, we can correlate the Onsager coefficients of a binary mixture in terms of the molar masses of the two components α and β ( M α and M β ): [153] Λ αβ = − · M α M β ¸ Λ αα = − · M β M α ¸ Λ ββ (5.26) For binary mixtures, there is a single MS and Fick difussion coefficient defined. The MS diffusion coefficient ¯ D is related to the Onsager coefficients by: [69] ¯ D = " ( M β + x α ( M α − M β )) 2 x α x β M 2 β # Λ αα (5.27) In a conventional ternary system consisting of molecules of α , θ and γ , there are six Onsager coefficients as Λ αα , Λ αθ , Λ αγ , Λ θθ , Λ θγ , and Λ γγ . In a pseudo- binary system, there are only three Onsager coefficients that are interdependent according to Eq. (5.26) . A pseudo-binary system consist of an independent and an indistinguishable species, which could be described as molecules of α and β =( θ + γ ), respectively. For this pseudo-binary system, the three Onsager coeffi- cients are Λ αα , Λ αβ , and Λ ββ . To compute Onsager coefficients of the pseudo-binary system Λ αα , Λ αβ , and Λ ββ , we can use the Onsager coefficients of the ternary system following the steps below: 1. PerformMD simulations of the ternary systems consisting of molecules of types α , θ and γ . 2. Compute Onsager coefficients for this ternary system. ( Λ αα , Λ αθ , Λ αγ , Λ θθ , Λ θγ , and Λ γγ ) using MSDs obtained from the OCTP tool in LAMMPS [154] . Eq. (5.25) requires information on the total number of molecules, which is N = N α + N θ + N γ .