Noura Dawass

1.3. K IRKWOOD –B UFF I NTEGRALS FROM MOLECULAR SIMULATIONS 1 11 In 2013, Krüger et al. [74] derived an expression for KBIs of finite and open subvolumes embedded in a reservoir. Similar to the original KB theory [12] for infinitely large and open systems, Krüger et al. [74] derived an expression that relates local density fluctuations inside the subvolume with the integral of the RDF of the system. This was achieved by first considering the average densities and the fluctuations in density of the open subvolume, V , embedded in a large reservoir. The open subvolume, V is grand-canonical. The system is character- ized by the following variables: temperature ( T ), volume of the subvolume ( V ), and chemical potentials ( µ α and µ β for a binary system). In this ensemble, we consider the average number of molecules, 〈 N α 〉 , and the average number of α and β pairs, 〈 N α N β 〉 , expressed as integrals of the one molecule density ( ρ (1) α ( r 1 )) and the two molecule density ( ρ (2) αβ ( r 1 , r 2 )), Z V ρ (1) α ( r 1 )d r 1 = 〈 N α 〉 (1.15) Z V Z V ρ (2) αβ ( r 1 , r 2 )d r 1 d r 2 = 〈 N α N β 〉− δ αβ 〈 N α 〉 (1.16) Integration of the local densities over the subvolume V yields the average num- ber of molecules in the grand-canonical ensemble [13] . Subsequently, the den- sity fluctuations in the subvolume V are expressed as: Z V d r 1 Z V d r 2 [ ρ (2) αβ ( r 1 , r 2 ) − ρ (1) α ( r 1 ) ρ (1) β ( r 2 )] = 〈 N α N β 〉−〈 N α 〉〈 N β 〉− δ αβ 〈 N α 〉 (1.17) For fluid systems, ρ (1) α ( r 1 ) and ρ (2) αβ ( r 1 , r 2 ), can be replaced by c α , and c α c β g αβ ( r 12 ) due to translational and rotational invariance, respectively. Here, c α is the macroscopic number density given by c α = 〈 N α 〉 / V . The function g αβ ( r 12 ) is the RDF and r 12 = | r 1 − r 2 | . For a finite multicomponent fluid, the integral, G V αβ , is defined by simply dividing Eq. (1.17) by c α c β V : G V αβ ≡ 1 V Z V Z V £ g αβ ( r 12 ) − 1 ¤ d r 1 d r 2 ≡ V 〈 N α N β 〉−〈 N α 〉〈 N β 〉 〈 N α 〉〈 N β 〉 − V δ αβ 〈 N β 〉 (1.18) In the limit V → ∞ and for homogeneous conditions, the double integrals of Eq. (1.18) can be reduced to a single integral by applying the transformation: r 2 → r = r 1 − r 2 , which yields the original expression for the KB integral for in- finitely large systems (Eqs. (1.3) and (1.4) ). However, for a finite subvolume, V , applying this transformation is not possible since the domain of integration over