Noura Dawass

1 4 I NTRODUCTION d r R Figure 1.1: A schematic representation of a radial shell in a binary mixture (composed of molecules of types α and β ) formed by considering a distance R from a central molecule of type α . The number of molecules of type β inside radial shell elements with the width d r is used to compute the RDF g αβ ( r ). where r is the particle distance and g ∞ αβ ( r ) is the radial distribution function (RDF) of species α and β for an infinitely large system. In Eq. (1.3) , species α and β can be the same. For a shell with thickness d r centred around a molecule of type α in an infinite system (see Figure 1.1) , the number of molecules of type β is 4 π r 2 d r ρ β and 4 π r 2 d r ρ β g ∞ αβ ( r ) for an ideal gas and real fluid, respectively. Here, ρ β = 〈 N β 〉 / V is the average number density of species β . Integrating from zero to infinity over the excess number of molecules of type β , (4 π r 2 d r ρ β [ g ∞ αβ ( r ) − 1]), yields ρ β G ∞ αβ . Hence, KBIs G ∞ αβ provide the average excess (or depletion) per unit density of molecules of type β around a central molecule of type α , and the affin- ity between components α and β is reflected. It is important to note that this interpretation of KBIs only holds for infinite systems, as indicated by the upper bound of the integral in Eq. (1.3) . Truncating the integral of Eq. (1.3) to a distance R yields the average excess of type β within a sphere of radius R . We will demon- strate later in this thesis (section 2.3) that the resulting truncated integral does not represent the KBIs in the thermodynamic limit. Kirkwood and Buff [12] formulated a relation between integrals over RDFs and fluctuations in the number of molecules in the grand-canonical ensemble,