Noura Dawass

1.3. K IRKWOOD –B UFF I NTEGRALS FROM MOLECULAR SIMULATIONS 1 9 where γ α is the activity coefficient of component α . In Eq. (1.12) , the symbol Σ indicates that γ α is differentiated with respect to the mole fraction of component β , x β , while keeping the mole fraction of the other components constant, except the n th one. For a binary mixture at a constant temperature and pressure we have, Γ = 1 + x 1 dln γ 1 d x 1 = 1 + x 2 dln γ 2 d x 2 (1.13) where the sum of the mole fractions ( x 1 + x 2 ) equals unity when the differentia- tion is carried out [65, 68] . For a specific solution, the thermodynamic factor pro- vides an indication of the phase stability, since Γ relates to the second derivative of the Gibbs energy with respect to composition [65] . Γ is positive for a thermo- dynamically stable mixture and negative for an unstable one [65] . As discussed earlier, computing properties such as activity coefficients and their derivatives is challenging for fluids with strong interactions. To avoid simulations that require molecular insertions, thermodynamic factors can be computed from KBIs. For a binary system, Γ can be computed using [13, 64, 69] : Γ αβ = 1 − x α ρ β ( G αα + G ββ − 2 G αβ ) 1 + ρ β x α ( G αα + G ββ − 2 G αβ ) (1.14) The term G αα + G ββ − 2 G αβ describes the strength of α − β interactions compared to α − α and β − β interactions. If this term is negative, then α − β attractive in- teractions are stronger than α − α and β − β interactions and as a result Γ > 1. For an ideal gas, the term G αα + G ββ − 2 G αβ will be zero and hence Γ = 1. We will use this in chapter 5 to analyse the interactions of DESs. Expressions relating Γ i j to KBIs for ternary [13, 70] and quaternary [71] mixtures are available in literature. The thermodynamic factor also plays an important role for correcting finite–size effects of diffusion coefficients [72, 73] . This is discussed further in section 1.4.3. 1.3.2. M ETHODS FOR COMPUTING K IRKWOOD –B UFF I NTEGRALS KBIs can be computed from fluctuations in the number of particles or RDFs, which are both accessed by molecular simulation. KBIs are defined for infinitely large systems while a finite number of molecules are studied by molecular simu- lation. To estimate KBIs in the thermodynamic limit ( G ∞ αβ ) using microscopic information of finite systems, three main approaches can be adopted. The most common approach is to simply truncate KBIs to the size of the simula- tion box, which results in integrals that converge poorly to the thermodynamic limit [13, 74] . In the second approach, RDFs fromfinite systems were extended to the thermodynamic limit, using mathematically involved methods [75, 76] that