Noura Dawass

4 66 S URFACE E FFECTS 4.1. I NTRODUCTION When studying small systems, of the order of few molecular diameters, thermo- dynamics of small systems is applied [78, 79] . Using Hill’s formulation of small- system thermodynamics [79] , it is shown that properties of small systems can be written in terms of volume and surface contributions [122] . In Ref. [122] , Hill’s thermodynamics was applied to several properties, such as pressure. From the volume contribution of pressure, the homogeneous pressure is obtained, while the Gibbs surface relation was obtained from the surface contribution [122] . This last contribution is proportional to the surface tension. In the case of KBIs, the surface term, or contribution, F ∞ , can also be defined from the Gibbs surface equation [122] . From a microscopic point of view, it originates from interactions between molecules inside the subvolume and molecules across the boundary of the subvolume [74, 80] (see also section 1.3.2) . These surface effects vanish in the thermodynamic limit, but for systems used in MD simulations these ef- fects cannot be neglected [22] . As a result, the quantitative and qualitative study of surface contributions is important for estimating G ∞ αβ from integrals of finite subvolumes G V αβ . The scaling of finite integrals G V (Eq. (1.25) ) with the size of the subvolumes L is used to compute KBIs in the thermodynamic limit G ∞ (for convenience, indi- cies α and β indicating the different components will be dropped from this point onwards). Specifically, G ∞ is computed from extrapolating the linear part of the scaling of G V with 1/ L to the limit 1/ L → 0 [74, 80, 122] . A disadvantage of this approach is that a linear regime is not always easily identified [80] . To avoid extrapolating G V , Krüger and Vlugt [81] proposed a direct estima- tion of KBIs in the thermodynamic limit: G ∞ ≈ G k ( L ) = Z L 0 £ g ( r ) − 1 ¤ u k ( r )d r (4.1) The accuracy of the estimation depends on the function u k ( r ) [129] , where the index k indicates the level of estimation. Krüger and Vlugt [81] considered three different estimations ( k = 0,1 and 2) and found that integrals computed using the function u 2 ( r ) provided the best estimation of G ∞ , u 2 ( r ) = 4 π r 2 µ 1 − 23 8 x 3 + 3 4 x 4 + 9 8 x 5 ¶ (4.2) where x is the dimensionless distance x = r / L . KBIs computed using Eq. (4.1) and Eq. (4.2) will be denoted by G 2 . To derive the expression for G 2 , the start- ing point was the scaling of KBIs with 1/ L . First, an explicit estimation of F ∞ in Eq. (1.28) was derived. In the work of Krüger and Vlugt [81] , F ∞ has the following form