Liquid-liquid phase transitions for soft-core attractive potentials

Using event-driven molecular dynamics simulations, we study a three-dimensional one-component system of spherical particles interacting via a discontinuous potential combining a repulsive square soft core and an attractive square well. In the case of a narrow attractive well, it has been shown that this potential has two metastable gas-liquid critical points. Here we systematically investigate how the changes of the parameters of this potential affect the phase diagram of the system. We find a broad range of potential parameters for which the system has both a gas-liquid critical point $C_1$ and a liquid-liquid critical point $C_2$. For the liquid-gas critical point we find that the derivatives of the critical temperature and pressure, with respect to the parameters of the potential, have the same signs: they are positive for increasing width of the attractive well and negative for increasing width and repulsive energy of the soft core. This result resembles the behavior of the liquid-gas critical point for standard liquids. In contrast, for the liquid-liquid critical point the critical pressure decreases as the critical temperature increases. As a consequence, the liquid-liquid critical point exists at positive pressures only in a finite range of parameters. We present a modified van der Waals equation which qualitatively reproduces the behavior of both critical points within some range of parameters, and gives us insight on the mechanisms ruling the dependence of the two critical points on the potential’s parameters. The soft-core potential studied here resembles model potentials used for colloids, proteins, and potentials that have been related to liquid metals, raising an interesting possibility that a liquid-liquid phase transition may be present in some systems where it has not yet been observed.

Figure 1

The generic soft-core potential with attractive well with parameters $w_A∕a$, $w_R∕a$, and $U_R∕U_A$. We use the parameters listed in Table .

Figure 2

The MD $P\text{−}\rho$ phase diagram for the potential in the inset, with the parameter set (ii) in Table . The long-dashed lines are the fits of the calculated state points (circles) at constant $T$. The isotherms (from top to bottom) are for $k_{B}T∕U_A=1.30$, 1.29, 1.28, 1.27, 1.26, 1.25, and 1.24 at low $\rho$ and $k_{B}T∕U_A=0.62$, 0.60, 0.58, 0.57, 0.55, 0.53, and 0.50 at high $\rho$. The fits are calculated by considering $P$ a polynomial function of both $T$ and $\rho$. The isotherms show two regions with negative slope, i.e., mechanically unstable, delimited by the spinodal lines (solid bold lines). Each spinodal line is associated with a first-order phase transition. By using the Maxwell construction, we estimate the coexisting regions associated to each spinodal line, delimited by the phase transition line (bold dashed line). The coexisting regions are clearly separated at the considered temperatures. The phase transition line at low $\rho$ is indistinguishable from the spinodal line at this scale. The points where the coexisting lines merge with the spinodal lines are, by definition, the critical points $C_1$ (at low $\rho$) and $C_2$ (at high $\rho$). No spontaneous crystal nucleation is observed in the explored region of the phase diagram.

Figure 3

As in Fig. 2, for parameter set (iii) in Table . The isotherms in the low-$\rho$ region (from top to bottom) are for $k_{B}T∕U_A=1.25$, 1.24, 1.23, 1.22, 1.20, 1.18, 1.16, and in the high-$\rho$ region are for $k_{B}T∕U_A=0.72$, 0.70, 0.68, 0.65, 0.62, 0.60. Spontaneous crystal nucleation is observed for $T<T_{C_2}$ and $\rho >{\rho }_{C_2}$.

Figure 4

As in Fig. 2, for parameter set (iv) in Table . The isotherms (from top to bottom) in the low-$\rho$ region are for $k_{B}T∕U_A=1.20$, 1.15, 1.10, 1.05, 1.00, and in the high-$\rho$ region are for $k_{B}T∕U_A=0.77$, 0.75, 0.73, 0.72, 0.70, 0.69. No spontaneous crystal nucleation is observed in the explored region of the phase diagram.

Figure 5

As in Fig. 2, for parameter set (xi) in Table . The isotherms (from top to bottom) in the low-$\rho$ region are for $k_{B}T∕U_A=1.53$, 1.52, 1.515, 1.51, 1.50, 1.48, 1.46, and in the high-$\rho$ region are for $k_{B}T∕U_A=0.70$, 0.68, 0.66, 0.64, 0.63, 0.62, 0.61. $C_2$ is at negative pressure.

Figure 6

As in Fig. 2, for parameter set (xii) in Table . The isotherms (from top to bottom) in the low-$\rho$ region are for $k_{B}T∕U_A=1.83$, 1.82, 1.815, 1.81, 1.80, 1.79, 1.75, 1.70, and in the high-$\rho$ region are for $k_{B}T∕U_A=0.98$, 0.96, 0.64, 0.92, 0.90. $C_2$ is at a negative pressure.

Figure 7

As in Fig. 2, for parameter set (xiv) in Table . The isotherms (from top to bottom) in the low-$\rho$ region are for $k_{B}T∕U_A=1.65$, 1.62, 1.60, 1.58, 1.55, 1.50, 1.45, and in the high-$\rho$ region are for $k_{B}T∕U_A=0.60$, 0.59, 0.58, 0.57, 0.56, 0.54. $C_2$ is at negative pressures.

Figure 8

Estimation of the critical point $C_2$ by the isochoric method for the set of potential parameters $w_A∕a=0.5$, $w_R∕a=0.5$, $U_R∕U_A=2$. Inset: $P$ at constant $a^3\rho =0.492$ for the MD calculation during the slow cooling described in the text. For $k_{B}T∕U_A>0.5$ the errors on the estimate of the state points are of the order of the nonmonotonic jumps. The interpolating line is a quadratic fit of the calculated points, and gives an estimate of the isochore at $a^3\rho =0.492$ for $k_{B}T∕U_A>0.5$. Main panel: quadratic fits of isochores for $a^3\rho =0.492$, 0.435, 0.405, 0.387, and 0.361 (from top to bottom). The critical point $C_2$ is located at the highest-$T$ intersection of two isochores (region inside the circle). The indeterminacy of this intersection gives an estimate of the error on the values $T_{C_2}=0.53\pm 0.03$, $P_{C_2}=1.05\pm 0.03$, and ${\rho }_{C_2}=0.39\pm 0.05$.

Figure 9

The behavior of the density, temperature, and pressure of the low-density critical point $C_1$ (open circles) and high-density critical point $C_2$ (filled squares) for variations of the potential parameters (a)–(c) $w_A$, (d)–(f) $w_R$, and (g)–(i) $U_R$. The other two parameters are constant: in (a)–(c) $w_R∕a=0.5$ and $U_R∕U_A=2$, in (d)–(f) $w_A∕a=0.7$ and $U_R∕U_A=2$, and in (g)–(i) $w_A∕a=0.9$ and $w_R∕a=0.5$. Where not shown, errors are smaller than the symbol size. Lines are guides for the eye.

Figure 10

The gas-LDL critical point ($C_1$) and LDL-HDL critical point ($C_2$) in the $P-T$ plane for varying attractive width $w_A$ and constant $w_R∕a=0.5$, $U_R∕U_A=2.0$. Symbols denote $w_A∕a=0.3$ (circles), 0.4 (left triangles), 0.5 (diamonds), 0.6 (up triangles), 0.7 (right triangles), 0.8 (down triangles), and 0.9 (squares). Open symbols are for $C_1$ and filled symbols are for $C_2$. The arrows denote the direction of increasing $w_A$.

Figure 11

The gas-LDL critical point $\left(C_1\right)$ and LDL-HDL critical point $\left(C_2\right)$ in the $P\text{−}T$ plane, for varying shoulder width $w_R∕a$ and constant $w_A∕a=0.7$, $U_R∕U_A=2.0$. Symbols denote $w_R∕a=0.4$ (circles), 0.5 (up triangles), 0.6 (diamonds), 0.7 (left triangles), 0.8 (down triangles), and 0.9 (squares). Open symbols are for $C_1$ and filled symbols are for $C_2$. The arrows denote the direction of increasing $w_R$.

Figure 12

The gas-LDL critical point $\left(C_1\right)$ and LDL-HDL critical point $\left(C_2\right)$ in the $P\text{−}T$ plane for varying repulsive energy $U_R∕U_A$ and constant $w_A∕a=0.9$, $w_R∕a=0.5$. Symbols denote $U_R∕U_A=2.0$ (circles), 2.5 (up triangles), 3.0 (diamonds), 3.5 (down triangles), and 4.0 (squares). Open symbols are for $C_1$ and filled symbols are for $C_2$. The arrows denote the direction of increasing $U_R$.

Figure 13

(a) The behavior of the effective excluded volume $B(\rho ,T)$ for a one-dimensional system with $B_2∕B_1=2$ for densities $B_1\rho =0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9$ from top to bottom. The thick curve indicates the behavior for $B_1\rho =0.5=B_1∕B_2$. (b) The isochores of MVDWE for $B_2∕B_1=1.4$, $A∕\left(B_1{U}_R\right)=2.2$ on the $P\text{−}T$ plane. An open circle indicates the low-density critical point. A filled circle indicates the high-density critical point. A dashed line indicates the low-density critical isochore $B_1{\rho }_{C_1}\approx 0.25$. A dot-dashed line indicates the high-density critical isochore $B_1{\rho }_{C_2}\approx 0.70$. The thick line indicates $B_1\rho \approx 0.71=B_1∕B_2$. Note that isochores start to develop density anomaly below the high-density critical point.

Figure 14

The behavior of the density, temperature, and pressure of the high-density critical point $C_2$ for variations of the MVDWE parameters (a)–(c) $A$, (d)–(f) $B_2$, and (g)–(i) $U_R$. The other two parameters are constant: in (a)–(c) $B_2∕B_1=1.1(엯),1.5(◻),2.0(◇),3.0(▵)$; in (d)–(f) $A∕\left(B_1{U}_R\right)=0.2(엯),1.0(◻),2.0(◇),3.0(▵)$, in (g)–(i) $B_2∕B_1=1.1(엯),1.3(*),1.5(◻),2.0(◇),3.0(▵)$ Lines are guides for the eye.

Figure 15

Inset: the single square-well potential defined by the well width $w_A$. Main panel: symbols represent the values of the normalized temperature $T_c$ (circles), the density ${\rho }_c$ (diamonds), and the pressure $P_c$ (squares) of the gas-liquid critical point for different values of the parameter $w_A$. Where not shown, errors are smaller than the symbol size. Lines are the linear fits $T_c$, $P_c$, and ${\rho }_c$ as functions of $w_A$, with the parameters in Table .