Strong Pressure-Energy Correlations in van der Waals Liquids

Strong correlations between equilibrium fluctuations of the configurational parts of pressure and energy are found in computer simulations of the Lennard-Jones liquid and other simple liquids, but not for hydrogen-bonding liquids such as methanol and water. The correlations that are present also in the crystal and glass phases reflect an effective inverse power-law repulsive potential dominating fluctuations, even at zero and slightly negative pressure. In experimental data for supercritical argon, the correlations are found to be approximately 96%. Consequences for viscous liquid dynamics are discussed.

Figure 1

Results from equilibrium molecular dynamics simulations of 864 particles interacting via the Lennard-Jones potential studied in the $NVT$ ensemble where “argon units” were used ($\sigma =0.34\mathrm{nm}$, $ϵ=0.997\mathrm{kJ}/\mathrm{mol}$). (a) Normalized fluctuations at $T=80\mathrm{K}$ and zero average pressure ($\mathrm{\text{density}}=34.6\mathrm{mol}/\mathrm{l}$) of the virial, $W$, $\Delta W(t)/\sqrt{⟨(\Delta W{)}^2⟩}$, and of the potential energy, $\Delta U(t)/\sqrt{⟨(\Delta U{)}^2⟩}$. The equilibrium fluctuations of virial and potential energy are strongly correlated, as quantified by the correlation coefficient: $R\equiv ⟨\Delta W\Delta U⟩/\sqrt{⟨(\Delta W{)}^2⟩⟨(\Delta U{)}^2⟩}=0.94$, where averages are over the full length of the simulation (10 ns) after 10 ns of equilibration. (b) Configurational parts of pressure and energy—virial versus potential energy—for several state points of the Lennard-Jones liquid. Each color represents simulations at one particular state point where each data point marks instantaneous values of virial and potential energy from a 10ns simulation. The black dashed lines mark constant density paths with the highest density to the upper left (densities: 39.8, 37.4, 36.0, 34.6, $32.6\mathrm{mol}/\mathrm{l}$). State points on the blue dashed line have zero average total pressure. The plot includes three crystallized samples (lower left corner).

Figure 2

$⟨\Delta W\Delta U⟩/(k_{B}TV)$ plotted as a function of $(⟨(\Delta W{)}^2⟩⟨(\Delta U{)}^2⟩{)}^{1/2}/(k_{B}TV)$ for several liquids. If the correlation is perfect ($R=1$) the data fall on the diagonal. The region above the diagonal corresponds to $R>1$ and is thus forbidden. LJ: Lennard-Jones results from the simulations reported in Fig. 1. Similar results were found in the $NVE$ ensemble, for larger samples, and using Langevin dynamics (results not shown). Exp: 500 particles interacting via a pair potential with exponential repulsion; $U(r)=\frac{ϵ}{8}[6e^{-14(r/\sigma -1)}-14(\sigma /r{)}^6]$, simulated with Metropolis dynamics. Kob-Andersen binary LJ: The Kob-Andersen binary Lennard-Jones liquid, $N=1000$ [11]. This includes data for the less-viscous liquid, the highly viscous liquid, as well as the glass. Asymmetric dumbbell: 512 asymmetric “dumbbell” molecules [12]. OPLS-UA Toluene: A 7-site united-atom model of toluene, $N=1000$ [13]. GROMOS Methanol: 512 methanol molecules [14]. SPC/E water: 4142 SPC/E water molecules [15]. Except for the two systems with Coulomb interactions (Methanol and water), all systems studied have strong correlations between fluctuations in virial and potential energy; correlation coefficients are above 0.9 for all state points shown, except those with negative pressure, where they are slightly smaller. The correlation coefficients increase with increasing density and temperature.

Figure 3

Data for supercritical argon at 3 different densities covering the temperature range 200–660K [17] showing a $W,U$ correlation of 96% [$K_T\equiv -V(\partial p/\partial V{)}_T$, $p^{\mathrm{conf}}\equiv p-N{k}_{B}T/V=W/V$, ${\beta }_V^{\mathrm{conf}}\equiv (\partial p/\partial T{)}_V-N{k}_B/V$, and $c_V^{\mathrm{conf}}\equiv C_V/V-(3/2)N{k}_B/V$]. The full line corresponds to perfect correlation between virial and potential energy [16]. The inset shows the prediction for an exact inverse power-law interatomic potential (full line): $B(T)=A(T)$. The poor fit shows that the fact that fluctuations are well described by an effective inverse power law does not imply that this is the case also for the equation of state.