Noura Dawass

3 58 F INITE -S IZE E FFECTS C OMPARING CORRECTION METHODS Using RDFs computed from finite and closed simulation boxes leads to a sys- tematic error in KBIs computations. These RDF finite-size effects should be cor- rected. In Figure 3.4 we already showed the scaling of G V 22 with σ / R (a) when using RDFs that are not corrected and (b) when applying the van der Vegt correc- tion [113] . For all the system sizes used, implementing the correction enhances the linear scaling of the finite-volume KBIs. In addition to the van der Vegt cor- rection, we considered the 1/ N correlation method [74] , and the method pro- posed by Cortes-Huerto et al. [83] We compare between the correction methods by considering the differences between the corrected RDF obtained from small systems and the RDF computed using a large system ( L box = 80). Figure 3.8 shows the quantity ln | g ( r ) − g L box = 80 ( r ) | for all distances, where the RDFs are computed for the WCA binary mixture using L box = 10, 20, and 40. Generally, the differences decrease with larger box sizes, and the differences are larger for small r . Also, for all system sizes, the van der Vegt correction and the correction of Cortes-Huerto et al. result in lower deviations from the finite RDF than in the case of the 1/ N correlation. The KBIs G ∞ 11 , G ∞ 12 , and G ∞ 22 are computed for the binary WCA mixture using three different simulation boxes, with L box = 10, L box = 20, and L box = 40. In the case of the 1/ N correlation, the values of G ∞ αβ are obtained by using two simula- tion box sizes that are not very different. For instance, to compute KBIs from a simulation box with L box = 10, RDFs from simulations of boxes with L box = 10 and L box = 11, at the same density and temperature, are required. The ob- tained KBIs from each correction method are then compared to integrals com- puted using a system with L box = 80. The differences (%) between KBIs com- puted using L box = 80 and KBIs G ∞ 11 , G ∞ 12 , and G ∞ 22 from finite simulation boxes ( L box = 10, 20, 40) are presented in Table 3.2. The differences are computed when the RDF is not corrected and when the correctionmethods discussed in this work are applied. Overall, the differences decrease with larger simulation boxes, with the exception of the 1/ N correlation. Additionally, for box sizes up to L box = 20, the van der Vegt method results in the lowest differences. For large boxes, Ta- ble 3.2 shows that a RDF correction is not needed and in the case of L box = 40 the differences resulting from not using a correction and the differences result- ing from the correction methods are not comparable. Other than the individ- ual KBIs, we also consider the quantity, G f = G 11 + G 22 − 2 G 12 , which is useful when computing the thermodynamic factor Γ from KBIs [64] . For L box = 80, we find that G f / σ 3 = − 3.38 +− 0.005. Deviations from this value are presented in Ta- ble 3.3 for the case of not using a RDF correction as well as when using the three