Noura Dawass

4.3. R ESULTS 4 87 Table 4.5: Surface term in the thermodynamic limit F ∞ for a WCA system at T = 2 and ρ = 0.6 (dimensionless units). Values of F ∞ are computed from systems with various number of particles N and using the different methods listed in Table 4.2. N Direct estimation of F ∞ 2 Scaling of LG V with L Scaling of LF V with L 500 n/a 0.8168 ± 0.0008 n/a 1000 n/a 0.8082 ± 0.0002 0.804 ± 0.004 5000 0.801 ± 0.002 0.8036 ± 0.0002 0.8004 ± 0.0002 10000 0.8013 ± 0.0004 0.8034 ± 0.0001 0.8013 ± 0.0003 30000 0.795 ± 0.005 0.7979 ± 0.0006 0.79 ± 0.01 50000 0.79 ± 0.01 0.793 ± 0.001 0.78 ± 0.02 Table 4.6: Surface term in the thermodynamic limit F ∞ for a LJ system at T = 2 and ρ = 0.4 (di- mensionless units). Values of F ∞ are computed from systems with various number of particles N and using the different methods listed in Table 4.2. N Direct estimation of F ∞ 2 Scaling of LG V with L Scaling of LF V with L 500 n/a − 0.2483 ± 0.0001 n/a 1000 n/a − 0.3320 ± 0.0008 n/a 5000 − 0.53 ± 0.04 − 0.460 ± 0.004 − 0.5718 ± 0.0001 10000 − 0.52 ± 0.03 − 0.464 ± 0.004 − 0.5433 ± 0.0004 30000 − 0.60 ± 0.06 − 0.543 ± 0.008 − 0.6315 ± 0.0009 E FFECT OF SYSTEM SIZE AND DENSITY Figure 4.13(a) shows the dependence of F ∞ on the inverse of the system size. Specifically, F ∞ decreases as (1/ N ) 1/3 . This is observed for surface terms com- puted using the scaling of LF V with L as well as values computed using the scal- ing of LG V with L . In Figur e 4.13(a), the error bars of the values of F ∞ vary with N in a similar manner to the values of G ∞ . Statistical uncertainties of the systems studied in this work are provided in Tables 4.5 and 4.6. As for KBIs, the surface terms F ∞ can be determined accurately as a func- tion of the density. Figure 4.14 (b) shows the values of F ∞ with density for LJ and WCA fluids. The surface term is estimated for the range ρ = 0.1 − 0.8. At ρ → 0, F ∞ is computed analytically using Eq. (4.5) and g ( r ) = exp[ − β u ( r )] [26] . In the low density limit, the surface terms computed in this work approach the theoret- ical value. In the high density limit, differences between surface term of LJ and WCA disappear due to dominating repulsive interactions, which are the same for the LJ and WCA potentials. From Figure 4.9 (b) and Figure 4.14 (b), a compari- son between the values of G ∞ and F ∞ can be made. Both values change in the