Simulations and integral-equation theories for dipolar density interacting disks

Integral equation theories (IETs) based on the Ornstein-Zernike (OZ) relation can be used as an analytical tool to predict structural and thermodynamic properties and phase behavior of fluids with low numerical cost. However, there are no studies of the IETs for the dipolar density interaction potential in two-dimensional systems, a relevant interdomain interaction in lipid monolayers with phase coexistence. This repulsive interaction arises due to the excess dipole density of the domains, which are aligned perpendicular to the interface. This work studies the performance of three closures of the OZ equation for this novel system: Rogers-Young (RY), modified hypernetted chain (MHNC), and variational modified hypernetted chain (VMHNC). For the last two closures the bridge function of a reference system is required, with the hard disk being the most convenient reference system. Given that in two dimensions there is no analytical expressions for the hard disk correlation functions, two different approximations are proposed: one based on the Percus-Yevick (PY) approximation, and the other based on an extension of the hard spheres Verlet-Weis-Henderson-Grundke (LB) parametrization. The accuracy of the five approaches is evaluated by comparison of the pair correlation function and the structure factor with Monte Carlo simulation data. The results show that RY closure is satisfactory only for low-structured regimes. MHNC and VMHNC closures perform globally well, and there are no significant differences between them. However, the reference system in some cases affects their performance; when the pair correlation function serves as the measure, the LB-based closures quantitatively outperform the PY ones. From the point of view of its applicability, LB-based closures do not have a solution for all studied interaction strength parameters, and, in general, PY-based closures are numerically preferable.

Figure 1

Two domains of equal radii $R$ and excess dipolar density $\sigma$ with center-to-center distance $r$.

Figure 2

Dipolar density interaction potential $U_d$ (solid violet line) and the asymptotic expressions for short (dotted-dashed red line) and long distance (dashed black line) given in Eq. (4). In the inset, the asymptotic expression for short distances is also shown as a function of the border-to-border separation $r/R-2$.

Figure 3

Radial distribution functions for $\varphi =0.15$ and $\mathrm{\Gamma }=0.5,2.0,4.0$, and 6.0 in panels (a), (b), (c), and (d), respectively. Circles correspond to MC simulations and solid lines to RY (violet), MHNC-PY (red), MHNC-LB (blue), VMHNC-PY (green), and VMHNC-LB (orange) closures. The symbols in the IETs results were plotted at arbitrary data intervals as a guide to the eye.

Figure 4

Structured factors for $\varphi =0.15$ and $\mathrm{\Gamma }=0.5,2.0,4.0$, and 6.0 in panels (a), (b), (c), and (d), respectively. Circles correspond to MC simulations and solid lines to RY (violet), MHNC-PY (red), MHNC-LB (blue), VMHNC-PY (green), and VMHNC-LB (orange) closures. The symbols in the IETs results were plotted at arbitrary data intervals as a guide to the eye.

Figure 5

Radial distribution functions for $\varphi =0.05$ and $\mathrm{\Gamma }=0.5,1.0,2.0,4.0,6.0$, and 8.0 in panels (a), (b), (c), (d), (e), and (f), respectively. Circles correspond to MC simulations and solid lines to RY (violet), MHNC-PY (red), MHNC-LB (blue), VMHNC-PY (green), and VMHNC-LB (orange) closures. The symbols in the IETs results were plotted at arbitrary data intervals as a guide to the eye.

Figure 6

Structured factors for $\varphi =0.05$ and $\mathrm{\Gamma }=0.5,1.0,2.0,4.0,6.0$, and 8.0 in panels (a), (b), (c), (d), (e), and (f), respectively. Circles correspond to MC simulations and solid lines to RY (violet), MHNC-PY (red), MHNC-LB (blue), VMHNC-PY (green), and VMHNC-LB (orange) closures. The symbols in the IETs results were plotted at arbitrary data intervals as a guide to the eye.

Figure 7

Radial distribution functions for $\varphi =0.25$ and $\mathrm{\Gamma }=0.1,0.5,1.0,2.0,3.0$, and 5.0 in panels (a), (b), (c), (d), (e), and (f), respectively. Circles correspond to MC simulations and solid lines to RY (violet), MHNC-PY (red), MHNC-LB (blue), VMHNC-PY (green), and VMHNC-LB (orange) closures. The symbols in the IETs results were plotted at arbitrary data intervals as a guide to the eye. Panel (a) shows that the closures maintain their excellent performance even when the hard core potential is relevant, i.e., the contact value is different from zero.

Figure 8

Structured factors for $\varphi =0.25$ and $\mathrm{\Gamma }=0.1,0.5,1.0,2.0,3.0$, and 5.0 in panels (a), (b), (c), (d), (e), and (f), respectively. Circles correspond to MC simulations and solid lines to RY (violet), MHNC-PY (red), MHNC-LB (blue), VMHNC-PY (green), and VMHNC-LB (orange) closures. The symbols in the IETs results were plotted at arbitrary data intervals as a guide to the eye. From panel (a) it can be observed that the closures continue to perform very good even when the hard core potential is relevant.