A wide-range temperature-dependent deep potential for sodium with near-experimental accuracy from melting to the critical point

Abstract

We construct temperature-dependent deep potentials for sodium using finite-temperature DFT data obtained with the PBE, AM05, and $r^2$SCAN exchange-correlation functionals and investigate the thermophysical properties of sodium from room temperature to the critical region. The predicted critical parameters show a dependence on the exchange-correlation functional. The model based on the $r^2$SCAN functional gives a critical temperature $T_c=2.508(8)$ kK, density $ρ_c=0.203(4)$ g/cm$^{3}$, and pressure $P_c=0.249(6)$ kbar, in close agreement with the recommended values. This model is then used to reconstruct the normal-pressure and critical isobars and to calculate the enthalpy, heat capacities, thermal expansion coefficient, bulk moduli, Grüneisen parameter, and speed of sound. The calculated normal-pressure bcc density differs by only 0.06% from the experimental value. Direct solid-liquid coexistence simulations give a melting temperature of $346(2)$ K, 25 K below the recommended value, consistent with the sensitivity expected from meV/atom free-energy errors. Liquid-vapor coexistence simulations reproduce the binodal and yield a surface tension that approaches zero near the critical point. The calculated self-diffusion coefficient and shear viscosity extend the available transport-property data into the expanded-liquid and near-critical regions, where direct experimental information is sparse. We also demonstrate the thermodynamic consistency of the results by comparing the speed of sound obtained from direct acoustic simulations with that calculated from the equation of state.