Noura Dawass

3.4. R ESULTS AND DISCUSSION 3 47 used. The KBIs for finite subvolumes, G V scales with the inverse of the sphere size, 1/ R . For each box size, the linear part of the scaling of G V is extrapolated up to 1/ R → 0, to find G ∞ . Figure 3.2 (a) shows the scaling behavior for the case of simulating the sys- temwhen χ = 2. The regime where the scaling is linear depends on the size of the simulated box. Larger simulation boxes provide larger linear regimes. The accu- racy of the computations of the KBIs in the thermodynamic limit G ∞ depends on the size of the simulation box. The computed KBIs from the MC simulations are compared to KBIs in the thermodynamic limit, G ∞ ,num , obtained by numer- ically integrating Eq. (1.25) up to very large distances. The absolute differences between the numerically integrated KBIs G ∞ ,num , and KBIs from simulations are computed using Difference% = | G ∞ ,num − G ∞ | | G ∞ ,num | ∗ 100% (3.12) In Table 3.1, the differences (%) are listed when using the system sizes shown in Figure 3.2 and for three χ parameters, χ = 1, 2, and 4. For these parameters, the values of G ∞ ,num / σ 3 are − 1.785, − 2.041, and − 2.172 respectively. The values of G ∞ / σ 3 were obtained by extrapolating the linear part of the lines in Figure 3.2, which extend until R = L box /2 (indicated by a dot for each line). In general, for all fluctuation length parameters, χ , the difference decreases with the system size. For simulation boxes with L box = 7.5 or 10 the difference is equal to or larger than 1%, but the deviation decreases by approximately 75% and 90% when increasing L box to 15 σ and 20 σ , respectively. Finally, obtaining the KBIs from simulation boxes with L box = 40 σ and L box = 50 σ results in marginal differences. The finite-size effect of the subvolumes is shown more clearly when plotting A / V instead of 1/ R as shown in Figure 3.2 (b). This is due to the fact that the linear scaling of G V with 1/ R is correct up to R = L box /2 ( A / V ∼ 1/ R ). When R is larger than L box /2, parts of the sphere fall outside the simulation box and for these distances the ratio A trunc. / V trunc. (Eq. (3.1) and (3.2) ) is used instead of A / V . Another important observation made from Figure 3.2 (b) is related to the size of subvolume used to compute KBIs. For each simulation box size, the dot in the function G V indicates the point where the radius of the spherical subvol- ume equals L box /2. As shown in Figure 3.2 (b), increasing R beyond this value will not extend the linear regime. This finding is manifested when looking at the correlation between G V and the ratio between the area and volume of the sphere ( A / V ) which is presented in Figure 3.2 (b). As shown by Eq. (1.27) , G V scales with A / V , but this scaling does not continue when R is larger than half the simulation box size. When the size of the subvolume extends beyond the simulation box the number of molecules surrounding the embedded subvolume decreases greatly,