Structure and thermodynamics of two-dimensional Yukawa liquids

The thermodynamic and structural properties of two-dimensional dense Yukawa liquids are studied with molecular dynamics simulations. The “exact” thermodynamic properties are simultaneously employed in an advanced scheme for the determination of an equation of state that shows an unprecedented level of accuracy for the internal energy, pressure, and isothermal compressibility. The “exact” structural properties are utilized to formulate a novel empirical correction to the hypernetted-chain approach that leads to a very high accuracy level in terms of static correlations and thermodynamics.

Figure 1

The 2D-YOCP state points that are investigated with NVT MD simulations in the $(log\mathrm{\Gamma }-\kappa )$ phase diagram. All the simulated state points (black open circles) belong to the dense fluid region of the phase diagram which roughly extends between the curve $\mathrm{\Gamma }=0.1{\mathrm{\Gamma }}_{\mathrm{m}}\left(\kappa \right)$ (blue) and the melting curve $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{m}}\left(\kappa \right)$ (red). Here ${\mathrm{\Gamma }}_{\mathrm{m}}\left(\kappa \right)$ denotes the coupling parameter resulting from the analytical parametrization of the melting line of Eq. (1) [15]. The exact numerical values for the simulated state points can be retrieved from the information provided in the inset table: for each screening parameter $\kappa$, the coupling parameter was augmented with a constant step $\mathrm{\Delta }\mathrm{\Gamma }$ between ${\mathrm{\Gamma }}_{\min }$ and ${\mathrm{\Gamma }}_{\max }$.

Figure 2

Radial distribution functions extracted from long MD simulations with the histogram method. Results for constant screening parameter [(a) $\kappa =1$, (b) $\kappa =2$, (c) $\kappa =3]$ and varying coupling parameters within $0.1\le \mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}\left(\kappa \right)\le 1$.

Figure 3

Samples collected for the (a,d) excess internal energy, (b,d) excess pressure, and (c,f) excess inverse isothermal compressibility from the short MD simulations at two 2D-YOCP state points that are defined by $\kappa =3.0$ and $\mathrm{\Gamma }=200$ (green), $\mathrm{\Gamma }=1000$ (magenta). In each plot, the sample average has been demarcated with a black line.

Figure 4

Radial distribution functions for the (a) 3D-YOCP and (b) the 2D-YOCP as obtained from MD simulations (discrete points) and the HNC approach (solid lines). Results for two state points characterized by $\kappa =3.0$ and two values of the normalized coupling parameter $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}$, namely $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.6$ (green) and $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.9$ (magenta). For the 3D-YOCP, ${\mathrm{\Gamma }}_{\mathrm{m}}$ is given in Eq. (4) of Ref. [47] and $d={(4\pi n/3)}^{-1/3}$. For the 2D-YOCP, ${\mathrm{\Gamma }}_{\mathrm{m}}$ is given by Eq. (1) and $d={\left(\pi n\right)}^{-1/2}$.

Figure 5

Radial distribution functions of 2D-YOCP liquids acquired from MD simulations (discrete points), the HNC approach (dashed lines), and the SHNC approach (solid lines). Results for two state points at the opposite ends of the dense fluid region, namely (a) $(\kappa =1.0,\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.8)$ and (b) $(\kappa =1.0,\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.2)$. Recall that the analytical approximation for the melting line ${\mathrm{\Gamma }}_{\mathrm{m}}\left(\kappa \right)$ is given by Eq. (1).

Figure 6

Radial distribution functions of 2D-YOCP liquids acquired from MD simulations (discrete points) and the SHNC approach (solid lines). Each panel focuses on a single screening parameter [(a) $\kappa =0.5$, (b) $\kappa =1.5$, (c) $\kappa =2.5]$ and four coupling parameters $\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\mathrm{m}}=0.1,0.3,0.6,0.9$ color-coded by green, cyan, magenta, and blue, respectively.

Figure 7

Graphical representation of the normalized mapping of the SHNC approach, ${\tilde{\mathrm{\Gamma }}}_{\mathrm{SHNC}}(\mathrm{\Gamma },\kappa )={\mathrm{\Gamma }}_{\mathrm{SHNC}}(\mathrm{\Gamma },\kappa )/\mathrm{\Gamma }$, as a function of the coupling parameter $\mathrm{\Gamma }$. Each color-coded curve corresponds to a different value of the screening parameter $\kappa$ within the range $\kappa \in [0,3]$. The gray dashed line demarcates the upper limit of validity of the SHNC approach obtained by evaluating ${\tilde{\mathrm{\Gamma }}}_{\mathrm{SHNC}}(\mathrm{\Gamma },\kappa )$ at $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{m}}\left(\kappa \right)$, where ${\mathrm{\Gamma }}_{\mathrm{m}}\left(\kappa \right)$ is given by Eq. (1). The black horizontal line illustrates the simplified mapping ${\tilde{\mathrm{\Gamma }}}_{\mathrm{SHNC}}(\mathrm{\Gamma },\kappa )=2$ that emerges by applying the T/2-HNC approach to the 2D-YOCP.