Noura Dawass

3.2. F INITE - SIZE EFFECTS OF SUBVOLUMES 3 41 3.2. F INITE - SIZE EFFECTS OF SUBVOLUMES Computing KBIs using molecular simulations of finite simulation boxes results in finite-size effects. These effects impact the accuracy of the computed KBIs from finite subvolumes, and hence the KBIs at the thermodynamic limit. In this section, we describe the system and simulation method used to study the finite- size effects. Also, we relate the surface area of the subvolume embedded in the simulation box to the finite-size effects of the subvolume. To deal solely with effects originating from the finite size of the system, and not RDF related effects, we study a fluid that is described by the analytic RDF of Eq. (2.6) . As we will consider a pure component fluid, for simplicity the indices α and β are dropped in this section. Also, the parameter σ is set to unity through- out this work, so we use L box for the size of the box instead of L box / σ . We show how finite KBIs (Eq. (1.25) ) scale with the inverse of the size of the spherical sub- volume, V (the subvolume can have any other shape but in this study, we choose to consider spherical subvolumes). To quantify the inaccuracies resulting from finite size effects of these subvol- ume, the values of G V αβ are computed from simulations of different sizes of the subvolume, R (the radius of the spherical subvolume), for a specific simulation box size L box (side length of the cubic simulation box). KBIs in the thermody- namic limit G ∞ αβ are obtained from extrapolating KBIs of finite subvolumes to infinite subvolume size ( R →∞ ). These KBIs are then compared to integrals ob- tained from the direct numerical integration of Eq. (1.25) , with the RDF at each distance computed analytically from Eq. (2.6) . To quantify the inaccuracies in the KBIs due to finite-size effects, the differences between KBIs from the numer- ical integration of Eq. (1.25) and G ∞ from simulation of finite simulation boxes are compared. Furthermore, we examine the distances up to which the compu- tations of the KB integral are performed (i.e the appropriate subvolume sizes). In molecular simulation, RDFs are typically computed up to half the length of the simulation box ( L box /2) and as a result the computed KBIs are limited to this range. However, in simulations one can in principle extend r up to p 3 2 of the box length [124] . In this work, the RDF is extended up to p 2 2 of the box length (the range p 2 2 < L box < p 3 2 involves complex calculations that will not be considered further). In the results section we show how this extension affects the computa- tions of KBIs. When the radius of the subvolume, R , is larger than half the box size, the surface area and volume of the subvolume are computed by disregard- ing the sphere caps falling outside the simulation box (see Figure 3.1 (b)). Using R − L box /2 for the height of the cap falling outside the simulation box, we obtain the following equations for the truncated surface area A trunc. , and truncated vol- ume V trunc. [82] ,