Physical Signatures of Supercritical Fluid Boundaries

TL;DR

The paper argues that the supercritical fluid (SCF) region hosts three distinct states: gas, SCF, and liquid, separated by two universal boundaries, the $L^{\pm}$ lines, rather than a single crossover. It combines molecular-dynamics simulations of argon with memory-function theory to reveal structural, transport, and dynamical signatures of crossing the $L^{\pm}$ lines, notably a sub-short-range order in the radial distribution function $g(r)$, deviations of viscosity $\eta$ and diffusion $D$ from kinetic predictions, and a stretched-exponential VACF above the gas–SCF boundary. The authors connect these observations to universal scaling along $L^{\pm}$, validate some predictions against experimental data on methane, and propose a refined phase diagram featuring three states with two measurable crossover boundaries. They further provide practical experimental indicators (e.g., $g(r)$ features, $D$ vs. $P$ behavior, and VACF dynamics) and discuss extensions to other fluids and quantum or holographic contexts. Key thermodynamic scalings along the boundaries include $\delta P^{\pm} \sim (T-T_c)^{\beta+\gamma}$ and $\delta \rho^{\pm} \sim (T-T_c)^{\beta}$, underscoring the universal character of the $L^{\pm}$ lines.

Abstract

In the supercritical fluid (SCF) region, at temperatures and pressures above the critical point, the thermodynamic singularity separating liquids and gases no longer exists. Recent arguments based on thermodynamics and critical scalings have revived the proposal that the SCF constitutes an intermediate state of matter, separated from the liquid and gas by two supercritical boundaries, the $L^\pm$ lines. However, until now, the nature of the supercritical state and the physical signatures of these boundaries have remained elusive. Here, we demonstrate that the SCF is characterized by distinct structural, transport, and dynamical behavior. Specifically, the spatial arrangement of particles-captured by the radial distribution function-as well as the diffusion coefficient, shear viscosity, and velocity autocorrelation function in the SCF regime are qualitatively different from those in both the liquid and gas states and exhibit clear physical signatures upon crossing the $L^\pm$ lines. Our theoretical predictions are validated by molecular dynamics simulations of argon and are further supported by existing experimental evidence. These results provide a clear physical foundation for a refined phase diagram of matter in the supercritical region, comprising three distinct states-gas, supercritical fluid, and liquid-separated by two crossover boundaries obeying universal scaling laws.

Figures (10)

• Figure 1: Phase diagram of argon. The critical point (CP) is located at $(T_c,P_c)\approx (151\,\text{K}, 48.55\,\text{bar})$1967Critical. The liquid-gas coexistence line (solid black line) is obtained from the NIST database NIST. The gray dotted line is the critical isochore, which represents supercritical continuation of the coexistence line. The two black dashed lines are $L^{\pm}$ lines, which are computed using the NIST data and the thermodynamic criterion proposed in li2024thermodynamic, and represent the boundaries between liquid, gas and SCF phases.
• Figure 2: Radial distribution functions (RDFs).(a-c) Typical reduced RDF, $\tilde{h}(r) = r[g(r)-1]$, of gas, SCF and liquid states measured in simulations of argon (gas: $T=195$ K, $P=20~\text{bar}$; SCF: $T=195$ K, $90 ~\text{bar}$; liquid: $T=100$ K, $P=40 ~\text{bar}$). The distance $r$ is expressed in units of $\sigma_0$, the position of the first peak. (d) Radial distribution function $g(r)$ at different pressures, zoomed in around the second peak, for $T=195\ \text{K}$. (inset) Height of the second peak $h_2$ as a function of $P$; below $P_2 \approx 55$ bar (red circle), the second peak can not be identified anymore.
• Figure 3: Transport coefficients.(a) Reduced viscosity $\eta^*=\eta\left(\sigma^2 / \sqrt{\epsilon m}\right)$, where $\sigma$ and $\epsilon$ are length and energy parameters in the Lennard-Jones potential (see Methods), and $m$ is the atomic mass. The errors bars represent the standard error of the mean. (b) Reduced diffusion coefficient $D^*=D \sqrt{\left(\frac{m}{\epsilon \sigma^2}\right)}$. In the kinetic theory expression, $D_{0} = \frac{3}{8\rho \sigma^2} \left(\frac{ k_{\rm B}T}{\pi m} \right)^{1/2}$, we have used both the ideal gas form $\rho_{\rm ideal} (P,T)= k_{\rm B} P/T$ (ideal kinetic theory), and the exact $\rho(P,T)$ from simulation data (kinetic theory). (c) Reduced diffusion coefficient as a function of the two-particle structural entropy $S_2$. The blue dashed line represents an exponential fit of the data above the $L^+$ line, $D = 1.18 e^{S_2/k_{\rm B}}$.
• Figure 4: Velocity auto-correlation functions.(a) Fitting the tail of the VACF. Assuming a stretched exponential form, $Z(t) \sim e^{-(t/\tau)^\beta}$, the exponent $\beta$ is obtained by the slope of the linear fit in the log-log plot of $-\ln[Z(t)]$ vs. $t$. (b)$\beta$ as a function of $P$. Error bars represent the standard deviation of the fitted values from ten independent fitting ranges. (c) Einstein frequency $\Omega$ (Thz) as a function of pressure $P$. The power-law fits (dashed colored lines) highlight the two distinct behavior in the gas and liquid states.
• Figure 5: Experimental diffusion coefficient for supercritical methane at $T=200$ K. Experimental data obtained by quasi-elastic neutron scattering (QENS) ranieri2024crossover and nuclear magnetic resonance (NMR) oosting1971proton are shown respectively with open circles and open diamonds. The green line is the theoretical prediction from ideal kinetic theory, with scaling $D^* \propto P^{-1}$.