Noura Dawass

3 52 F INITE -S IZE E FFECTS 3.4.2. F INITE - SIZE EFFECTS OF RADIAL DISTRIBUTION FUNCTIONS Here, we study the RDF-related effects discussed in section 3.3. RDFs obtained from simulations of closed, and finite systems have to be corrected. Figure 3.4 shows the enhanced scaling of G V αβ with 1/ R when applying an RDF correction. The correctionmethods for g ( r ) are applied to KBIs computed from closed, finite simulations of the WCA mixture described in section 3.4.2. In the case of obtain- ing the RDFs from MD simulations, the extrapolation to G ∞ αβ is not straightfor- ward. In the following section we show how to identify the linear regime. I DENTIFYING THE LINEAR REGIME We consider the RDF computed for the binary WCA mixture, while applying the van der Vegt correction, since we find that it provides the most accurate KBIs out of the three methods studied in this work (details are provided in section 3.4.2) . The MD simulation details and system conditions are provided in section 3.3.4. The study is performed to identify the extrapolation range from the scaling of σ / R with G V αβ from simulation boxes with L box = 10, L box = 20, and L box = 40. To find the best linear range, the start of the fitting range, 1/ a , and the end, 1/ b , are varied (see Figure 3.4 (b)). For each extrapolation range, the square of the correlation coefficient (denoted as c 2 in this work) is computed to assess the lin- earity of the selected range. Also, the difference between the extrapolated KBIs, G ∞ αβ , and the integrals computed from a large system ( L box = 80) are computed, using Eq. (3.12) . To relate the tested ranges to the dimensions of the system, a and b are related to the molecular diameter σ . The variable a is set based on how many molecular diameters should be discarded at the beginning of the distances at which the RDF is computed. The starting point for the 1/ R extrapolation could be varied as follows, 1 a = 1 x σ (3.13) Once x , and subsequently a , is set the end point of the extrapolation range could be again related to σ , 1 b = 1 a + y σ (3.14) In this section, we demonstrate how to find the best extrapolation range for the case of the values of G V 22 . To determine the accuracy of the computed KBIs, dif- ferences between values of G V 22 of finite simulation boxes and G V 22 of a very large simulation box ( L box = 80) are considered (difference%). The results for the ef- fect of varying x and y on the quality of the linear fit ( c 2 ) and the accuracy in KBIs computations (difference%) are shown in Figure 3.5, 3.6, and 3.7 for the box sizes L box = 40, L box = 20 and L box = 10, respectively. In these figures, the values of − ln(1 − c 2 ) and − ln(difference) are plotted as functions of x and y . Also, in