Hidden scale invariance in molecular van der Waals liquids: A simulation study

Results from molecular dynamics simulations of two viscous molecular model liquids—the Lewis-Wahnström model of orthoterphenyl and an asymmetric dumbbell model—are reported. We demonstrate that the liquids have a “hidden” approximate scale invariance: equilibrium potential energy fluctuations are accurately described by inverse power-law (IPL) potentials, the radial distribution functions are accurately reproduced by the IPL’s, and the radial distribution functions obey the IPL predicted scaling properties to a good approximation. IPL scaling of the dynamics also applies—with the scaling exponent predicted by the equilibrium fluctuations. In contrast, the equation of state does not obey the IPL scaling. We argue that our results are general for van der Waals liquids, but do not apply, e.g., for hydrogen-bonded liquids.

Figure 1

(Color online) (a) $\Delta W\left(t\right)/N$ vs $\Delta U\left(t\right)/N$ for the asymmetric dumbbell model [9]. The strong correlations are quantified by the correlation coefficient, $R\equiv ⟨\Delta W\Delta U⟩/\sqrt{⟨{\left(\Delta W\right)}^2⟩⟨{\left(\Delta U\right)}^2⟩}=0.959$. The dashed line has the slope $\gamma \equiv \sqrt{⟨{\left(\Delta W\right)}^2⟩/⟨{\left(\Delta U\right)}^2⟩}=5.90$. (b) For configurations taken from the simulation reported in Fig. 1a we evaluated $U_{\text{IPL},\text{BB}}\left(t\right)\equiv {\sum }_{i>j}{\varphi }_{\text{IPL},\text{BB}}\left(r_{ij}\left(t\right)\right)$ with ${\varphi }_{\text{IPL},\text{BB}}\left(r\right)\equiv C_{\text{BB}}{ϵ}_{\text{BB}}{\left({\sigma }_{\text{BB}}/r\right)}^{18}$. The constant $C_{\text{BB}}=1.489$ $\left(C_{\text{AA}}=1.075,C_{\text{AB}}=1.237\right)$ was chosen to make the variance of $U_{\text{IPL},\text{BB}}\left(t\right)$ equal to the variance of $U_{\text{BB}}\left(t\right)\equiv {\sum }_{i>j}{\varphi }_{\text{LJ},\text{BB}}\left[r_{ij}\left(t\right)\right]$. $U_{\text{IPL},\text{BB}}\left(t\right)$ strongly correlates with $U_{\text{BB}}\left(t\right)$ with a correlation coefficient of 0.977. In comparison $U_{\text{Diff},\text{BB}}\left(t\right)\equiv U_{\text{BB}}\left(t\right)-U_{\text{IPL},\text{BB}}\left(t\right)$ has small variance, 0.045 times the variance of $U_{\text{BB}}\left(t\right)$ and is only weakly correlated with the latter. Redoing the analysis with $n=3\times 5.90$ [the IPL exponent suggested by the fluctuations, see Fig. 1a] gives almost identical results.

Figure 2

(Color online) (a) Radial distribution functions for the asymmetric dumbbell model [9] (full lines), and the corresponding $r^{-18}$ IPL model (dashed lines, see text). (b) Scaled $g_{BB}\left(r\right)$ for three state points; the two state points with the same $\Gamma ={\rho }^{5.9}/T$ (black full line and red long dashed line) have almost identical (scaled) structure. When increasing $\rho$ at constant $\Gamma$ there is a slight shift of the first peak to smaller distances, consistent with the existence of the fixed bond. Similar scaling is found for the AA and AB interactions (data not shown).

Figure 3

(Color online) (a) Radial distribution functions for the LW-OTP model [10] at three different state points. Two of the state points (full black lines and red dashed lines) have approximately the same $\Gamma ={\rho }^{7.9}/T$. The third state point (full green line) has the same temperature as the first state point and same pressure as the same state point. (b) Same as in (a), but in reduced units. The two state points with the same $\Gamma$ (black full line and red long dashed line) have almost identical (scaled) structure. Similar to what is seen for the asymmetric model (Fig. 2), there is a slight indication of the fixed bond in the first peak of the two state points with the same $\Gamma$.

Figure 4

(Color online) The “slopes,” $\gamma \equiv \sqrt{⟨{\left(\Delta W\right)}^2⟩/⟨{\left(\Delta U\right)}^2⟩}$ plotted as a function of temperature for two molecular models. Both models are strongly correlating. For the Lewis-Wahnström OTP model [10] the zero pressure isobar covers densities in the interval 1.008 to $1.089\text{g}/{\text{cm}}^3$. For the asymmetric dumbbell model [9] the 2.0 GPa isobar covers densities in the interval 1.120 to $1.217\text{g}/{\text{cm}}^3$. For the three dumbbell isochores the pressures at the lowest temperatures are $-0.02$, 0.67, and 1.28 GPa, respectively.

Figure 5

(Color online) The reduced diffusion coefficient for Lewis-Wahnström OTP [10]. Open symbols: $D^{\ast }$ plotted without scaling ($\gamma =0$, $D^{\ast }$ divided by 100). Filled symbols: $D^{\ast }$ scaled according to Eq. (1) with three different scaling exponents: $\gamma =9.0$ (Upper set of curves, $D^{\ast }$ multiplied by 10), $\gamma =7.9$ (Second set of curves), $\gamma =7.0$ (Third set of curves, $D^{\ast }$ divided by 10).

Figure 6

(Color online) The reduced diffusion coefficient for the asymmetric dumbbell model [9]. Open symbols: $D^{\ast }$ plotted without scaling ($\gamma =0$, $D^{\ast }$ divided by 100). Filled symbols: $D^{\ast }$ scaled according to Eq. (1) with four different scaling exponents: $\gamma =7.0$ (Upper set of curves, $D^{\ast }$ multiplied by 100), $\gamma =6.1$ (Second set of curves, $D^{\ast }$ multiplied by 10), $\gamma =5.9$ (Third set of curves), and $\gamma =5.0$ (Fourth set of curves, $D^{\ast }$ divided by 10).

Figure 7

(Color online) Dynamics of the LW OTP model [10] in reduced units for the three state point shown in Fig. 3. (a) Mean-square displacement for the noncentral LJ spheres. (b) Rotational correlation function for the vector perpendicular to the plane of the three LJ spheres, using the first Legendre polynomial.

Figure 8

(Color online) Results from equilibrium molecular dynamics simulations of 512 asymmetric dumbbell molecules [9] with, respectively, a strong dipole moment ($q=\pm 0.5\text{e}$, open symbols, three isochores), and zero dipole moment ($q=0\text{e}$, filled symbols, three isochores and an isobar). (a) Correlation coefficients, $R\equiv ⟨\Delta W\Delta U⟩/\sqrt{⟨\Delta W^2⟩⟨\Delta U^2⟩}$. (b) The “slopes,” $\gamma \equiv \sqrt{⟨{\left(\Delta W\right)}^2⟩/⟨{\left(\Delta U\right)}^2⟩}$.

Figure 9

(Color online) The reduced diffusion coefficient, $D^{\ast }\equiv {\left(N/V\right)}^{1/3}{\left(k_{B}T/m\right)}^{-1/2}D$, for the asymmetric dumbbell model with $q=\pm 0.5\text{e}$ scaled according to Eq. (1), with three different scaling exponents: $\gamma =6.0$ (Upper set of curves, $D^{\ast }$ multiplied by 10), $\gamma =4.9$ (Middle set of curves), and $\gamma =4.0$ (Lower set of curves, $D^{\ast }$ divided by 10).