Equilibrium properties of penetrable soft spheres

Figures & data

Figure 1. Model potential given by Equations (1(1) $\varphi (r)=\epsilon [W_{c}w(r/R_c)-W_{d}w(r/R_d)],$(1) ) and (2(2) $w(x)=\{\begin{array}{ll}1-6x^2+6x^3& (x\le 1/2)\\ 2(1-x{)}^3& (1/2<x\le 1)\\ 0& (x>1).\end{array}$(2) ). $ϵ=18.75$, $W_c=2$, $R_c=0.8$, and $R_d=1$.

Figure 2. Density dependence of the difference in the radial distribution functions between HNC and MC for the system interacting with the reference (${\varphi }_{\mathrm{r}}$) and the full (φ) potentials. $W_d=1$.

Figure 3. Pressure versus density isotherms. $W_d=1$.

Figure 4. The radial distribution function from HNC and an MC simulation. To improve visibility, graphs for $\rho =14$ and 18 are shifted upwardly by 1 and 2, respectively. $k_{\mathrm{B}}T=1$ and $W_d=1$.

Figure 5. The relative importance in the thermodynamic perturbation theory of the second order term ${\psi }_{\mathrm{a}}^{(2)}$ in comparison to the first order term ${\psi }_{\mathrm{a}}$ given by Equations (27(27) ${\psi }_{\mathrm{a}}^{(2)}(T,\rho )\approx -\frac{1}{4}\beta \rho {\left(\frac{\mathrm{\partial }{p}_{\mathrm{ref}}}{\mathrm{\partial }\rho }\right)}_T^{-1}\int {[{\varphi }_{\mathrm{a}}(r)]}^2{g}_{\mathrm{r}}(r)\mathrm{d}\mathbf{r},$(27) ) and (12(12) ${\psi }_{\mathrm{a}}(T,\rho )=\frac{1}{2}\beta \rho \int {\varphi }_{\mathrm{a}}(r)g_{\mathrm{r}}(r)\mathrm{d}\mathbf{r}.$(12) ), respectively. $W_d=1$.

Figure 6. Phase diagram showing vapour-liquid phase coexistence.

Figure 7. Vapor phase density at saturation.

Figure 8. Temperature dependence of the the liquid phase density at p = 0.01, p = 1, and p = 10. $W_d=0.95$. For each method and at each temperature, the density is larger for a higher pressure.

Figure 9. Surface tension γ of the vapour-liquid interface at saturation.

Figure 10. The density ${\rho }_{\star }$, beyond which a high density fluid phase is mechanically unstable, plotted versus temperature. An open symbol indicates a fluid phase with a negative pressure.

Figure 11. Liquid phase density at saturation (${\rho }_{\mathrm{l}}^{\mathrm{eq}}$) shown with the onset of mechanical instability (${\rho }_{\star }$) and the maximum density (${\rho }_{\max }$) above which iterative solution of Equations (7(7) $h_{\mathrm{r}}(r)=c_{\mathrm{r}}(r)+\rho \int c_{\mathrm{r}}(|\mathbf{r}-{\mathbf{r}}^{\mathrm{\prime }}|)h_{\mathrm{r}}(r^{\mathrm{\prime }})\mathrm{d}{\mathbf{r}}^{\mathrm{\prime }}$(7) ) and (8(8) $c_{\mathrm{r}}(r)=h_{\mathrm{r}}(r)-\ln [h_{\mathrm{r}}(r)+1]-\beta {\varphi }_{\mathrm{r}}(r),$(8) ) fails. $W_d=1.024$. An open symbol for ${\rho }_{\star }$ indicates a negative pressure.

Figure 12. Density dependence of the radial distribution function at r = 0.01 using the HNC closure. $k_{\mathrm{B}}T=1$ and $W_d=1.024$. The filled circules indicate the values at the onset of mechanical instability ($\rho ={\rho }_{\star }$).