Noura Dawass

4 84 S URFACE E FFECTS 4.3.2. E STIMATION OF SURFACE EFFECTS An important objective of this work is to investigate surface effects of finite sys- tems used to compute KBIs. As mentioned earlier, there are three possible ap- proaches to compute the surface term in the thermodynamic limit F ∞ . Similar to the estimation of G ∞ , the surface term of the WCA fluid is computed from systems with varying number of particles N at the same thermodynamic state. In Figure 4.10, estimations of the surface term in the thermodynamic limit F ∞ 2 (Eq. (4.6) ) are presented as a function of L . Unlike the values of G ∞ 2 , a plateau where the values can be averaged is not easily identified. Alternatively, it is possi- ble to consider the scaling of the values of the surface term of finite subvolumes F V (Eq. (4.5) ). Figure 4.11 shows the scaling of LF V with L for the same WCA fluid. As in the case of the scaling of LG V , a linear regime to be fitted is easily identified. The slope corresponds to the value of F ∞ . Additionally, the value of F ∞ can be estimated from the intercept with the vertical axis of the line formed from the scaling of LG V with L . The latter two approaches require smaller sys- tem sizes than the direct estimation of F ∞ 2 . For instance, Figure 4.11 shows that systems with as few as N = 1000 provide a clear linear range that can be used to estimate F ∞ . It is of interest to investigate whether the different available meth- ods to find F ∞ result in matching estimations. The values of F ∞ computed using the three methods considered in this work are listed in Table 4.5. Results are shown for systems with varying number of particles. While acceptable statistics are achieved for most methods and system sizes, the values of F ∞ from the three different methods agree less well than the corresponding values of G ∞ . This can be attributed to the larger statistical errors obtained when compared to estimat- ing G ∞ . As in the case of computing G ∞ , using the scaling of LG V provides estima- tions of the surface term using systems smaller than those required by the other methods. This effect is more significant when looking at a system of LJ parti- cles at a relatively low density. Table 4.6 provides the values of F ∞ for a LJ fluid at ρ = 0.4, computed using the methods studied in this work. In Figures 4.12 and 4.13 the scaling of F ∞ 2 with L as well as the scaling of LF V with L are shown. These plots illustrate that linear regions are not easily identified to compute F ∞ , in contrast to the higher density case with ρ = 0.6 (Figures 4.10 and 4.11) . As a re- sult, when computing surface effects, the scaling of LG V with L is recommended.