provide an inversion result that is at least as good as the most accurate OS method.
5.3
Effect of density on inversions
In the previous subsection, we performed inversions at one specific density for each of the potentials studied. In order to study how the relative performance of the different inversion schemes depends on density, in this subsection we compare the accuracy of the HMSA and the HDPC scheme with the HNC and PY schemes for a few representative potentials as we increase the density of the monolayer. We first consider the exponential decay potential without an oscillatory tail which has a soft-core. In figures5.5(a),5.5(b)and5.5(c)we present inversion results for this potential at the reduced densities of ρ∗ = ρσ2 = 0.008, 0.01, 0.015 respectively. We see that while HNC and PY yield reasonably accurate results at the lowest density, the accuracy of these schemes becomes progressively worse as we go to higher densities. On the other hand, both HMSA and HDPC yield very accurate inversion results for ρ∗ = 0.008, 0.01 which are comparable to each other. However as noted in the previous subsection, the HDPC scheme breaks-down at ρ∗ = 0.015 because at this density, the area fraction of the reference HD fluid (η = 0.637 as estimated from the compressibility equation eqn.(4.27)) is too close to the close packed area fraction of η0 = 0.9069. In contrast, the HMSA scheme remains
numerically stable at this density and still provides very accurate inversion results. Next we consider the LJ potential which has a strong repulsion and an attractive well. In figures5.6(a),5.6(b)and5.6(c)we present inversion results for the LJ potential at the reduced densities of ρ∗ = ρσ2 = 0.4, 0.6, 0.7 respectively.
5.3 Effect of density on inversions
(a) ρ∗ = 0.008
(b) ρ∗= 0.01
(c) ρ∗= 0.015
5.3 Effect of density on inversions
Just as for the case of the exponential decay potential, both HNC and PY yield accurate inversion results at the lowest density but become progressively worse as we go to higher densities. In contrast, the HDPC scheme yields accurate results for all the densities studied (though even this scheme becomes slightly less accurate at the highest density of ρ∗ = 0.7). Interestingly, while the HMSA scheme yields accurate results for ρ∗ = 0.4, 0.6 (comparable to HDPC), it performs much worse than HDPC for ρ∗ = 0.7, though it is still marginally better than either HNC or PY.
The poor performance of HMSA at ρ∗ = 0.7 is probably due to the fact that at this density, the LJ fluid is close to an ordering transition. This is evidenced by the fact that the height of the first maximum in g(r) is about 2.7 which is close to the value of 3.3 − 3.4 where the LJ fluid undergoes an ordering transition to either a hexatic or crystalline phase [72]. This is supported further by the fact that the LJ system in 2D appears to enter a metastable phase of solid-liquid coexistence at ρ∗ ≈ 0.7 when T∗ ≈ 1/1.25 [73]. Since HMSA is fundamentally
an integral equation theory for the isotropic fluid state, it is not surprising that the theory breaks down close to an ordering transition. In contrast, for the exponential decay potential, the height of the first maximum in g(r) is about 1.8 for the highest density studied. We therefore expect this system to be deep in the isotropic fluid state where the HMSA scheme is valid. Interestingly, the HDPC scheme yields accurate inversion results for all the densities studied for the LJ potential even though it is also based on integral equation theory. This suggests that the key approximation behind this scheme, i.e. that the actual bridge function is equal to the HD bridge function, remains accurate even close to an ordering transition. Note that unlike the exponential decay potential case,
5.3 Effect of density on inversions
(a) ρ∗= 0.4
(b) ρ∗= 0.6
(c) ρ∗= 0.7
Figure 5.6: Results illustrating the effect of density on the inversion for LJ potential, i.e. eqn.(5.4).
5.3 Effect of density on inversions
the HDPC scheme remains numerically stable at the highest density studied for the LJ potential. This is because of the harsh repulsion the LJ potential possess at low r that drives long-range statistical ordering within the fluid. This means that even close to an ordering transition, the area fraction of the reference HD fluid can still be quite far from the close packing density. For example for ρ∗ = 0.7 where the height of the first maximum in g(r) is 2.7, the area fraction of the reference HD fluid is 0.605. In contrast for the exponential decay potential, for ρ∗ = 0.01 where the height of the first maximum in g(r) is 1.52, the area fraction of the reference HD fluid is 0.552, while for ρ∗ = 0.015 where the height of the first maximum in g(r) is 1.71, the area fraction of the reference HD fluid is 0.637. Comparing our results for the exponential decay potentials and the LJ potential, we conclude that the accuracy of the HMSA and HDPC schemes are superior to HNC and PY, especially as we go to higher densities. For densities away from an ordering transition and the close packed density of the reference HD fluid, both HMSA and HDPC schemes yield very accurate inversion results which are comparable to each other. For densities close to an ordering transition (as evidenced by the height of the first maximum in g(r) approaching the value of 3), the HDPC scheme is much more accurate compared to the HMSA scheme. The HDPC scheme is therefore better suited to studying hard-core monolayers (e.g., monolayers interacting via the LJ potential) where there can be significant statistical ordering in the monolayer for relatively small area fractions of the corresponding reference HD fluid. However for densities close to the close packing density of the reference HD fluid, the HDPC scheme becomes unstable while the HMSA scheme remains numerically stable. The HMSA scheme is therefore better suited to studying soft-core monolayers (e.g., monolayers interacting via