Noura Dawass

2 32 S HAPE E FFECTS 2.3.2. K IRKWOOD –B UFF I NTEGRALS FOR VARIOUS SHAPES We compute KBIs for subvolumes with different geometries. The expression for KBIs of finite subvolumes, G V αβ , is provided by Eq. (1.25) . To investigate shape effects, a liquid with the following analytic RDF model [86, 120] is used, g ( r ) − 1 = ( 3/2 r / σ exp h 1 − r / σ χ i cos £ 2 π ¡ r σ − 21 20 ¢¤ r σ ≥ 19 20 , − 1, r σ < 19 20 (2.6) where σ is the diameter of the particles, and χ is the length scale at which the fluctuations of the RDF decay. This RDF mimics density fluctuations around a central particle for a typical isotropic liquid. The RDF parameters are fixed at σ = 1 and χ = 2. Here, we work with a single-component fluid therefore the in- dices α and β are dropped. The use of an analytic g ( r ) eliminates errors due to uncertainties in numerically obtained RDFs [74, 113] . The functions w ( x ) are ob- tained numerically in tabulated form, and the value of w ( x ) at any x is obtained by interpolation. The integral of Eq. (1.25) is obtained by numerical integration using the trapezoidal rule [121] . In Figure 2.3, we show the KBIs for finite subvolumes, G V / σ 3 , plotted as a function of the inverse of the length of the subvolumes, σ / L . Figure 2.3 (a) shows the KBIs computed for spheroids with different aspect ratios ( a = 1, 2, 5, 10), and Figure 2.3 (b) shows the same for cuboids. In Figure 2.3 (b), we use analytic and numerical functions w ( x ) for spherical subvolumes ( a = 1). Integrating using the analytic or numerical functions w ( x ) yields practically identical values of the KBIs, and differences are of the same order as the error introduced by the nu- merical integration of Eq. (1.25) . Changing the aspect ratio affects the slope of the lines of G V / σ 3 versus σ / L . All lines approach the same value of the KB inte- gral in the limit σ / L → 0, which is expected as in the thermodynamic limit the KB integral should be independent of the shape of the subvolume. Figure 2.3 shows that the shape of the subvolume affects the slope of the plots of G V / σ 3 versus σ / L . The dependence of the slope on the shape of the finite subvolume was previously reported in the work of Strøm et al. [122] using arguments based on small-scale thermodynamics. These authors found that plotting KBIs as a function of the surface to volume ratio of the subvolume should eliminate shape effects. In Figure 2.4 (a), we show the KBIs plotted as a function of the surface area to volume ratio ( A σ / V ) of the subvolume for the following shapes: sphere, cube, spheroid with a = 2, and cuboid with a = 2. As expected from the work by Strøm et al. [122] , all KBIs approach the same value of G ∞ / σ 3 with the same slope. In the thermodynamic limit, G V / σ 3 seems to only be a function of A / V . For large subvolumes, values of w ( r ) at small distances are more significant when inte-