The Enskog–Vlasov equation: a kinetic model describing gas, liquid, and solid

Abstract

The Enskog-Vlasov (EV) equation is a semi-empiric kinetic model describing gas-liquid phase transitions. In the framework of the EV equation, these correspond to an instability with respect to infinitely long perturbations, developing in a gas state when the temperature drops below (or density rises above) a certain threshold. In this paper, we show that the EV equation describes one more instability, with respect to perturbations with a finite wavelength and occurring at a higher density. This instability corresponds to fluid-solid phase transition and the perturbations' wavelength is essentially the characteristic scale of the emerging crystal structure. Thus, even though the EV model does not describe the fundamental physics of the solid state, it can 'mimic' it-and, thus, be used in applications involving both evaporation and solidification of liquids. Our results also predict to which extent a pure fluid can be overcooled before it definitely turns into a solid.

Kinetic derivation of diffuse-interface fluid models

“…The Enskog-Vlasov and Boltzmann-Vlasov models, on the other hand, include a nonlocal force term and an Enskog-like or Boltzmann-like collision kernel. Such models have been used successfully to investigate phase change problems in a kinetic framework [21,22,23,24,25,26,27,28,29,30,31,32]. These models may be derived from the BBGKY hierarchy by decomposing the interaction full potential ϕ 12 in the form ϕ 12 = ϕ att 12 + ϕ rep 12 where ϕ att 12 is a long range weak attractive potential and ϕ rep 12 a strongly repulsive hard core potential [21].…”

“…In comparison with the hierarchy, information has been lost when transforming the collision term associated with the repulsive potential, when transforming the correlation function involving the full potential, and the second equation of the hierarchy governing f 2 is not anymore available for a kinetic derivation of the equation governing the potential internal energy e p . Nevertheless, Enskog-Vlasov and Boltzmann-Vlasov have many advantages like satisfying a H theorem, involving a nonlocal force, being more convenient for analytical, numerical or mathematical investigations while still keeping the main physical aspects of dense fluids [21,22,23,24,25,26,27,28,29,30,31,32]. Since the potential involved in the Vlasov term for such models is the attractive part ϕ att 12 , it is not anymore possible to proceed as in Section 3.5 in order to derive the van der Waals equation of state.…”

“…Phase changes have notably been investigated by employing linearized Boltzmann equations with condensation-evaporation boundary conditions [18,19,20]. Boltzmann-Vlasov and Enskog-Vlasov type equations have been used to investigate more deeply spatial aspects of phase transition [21,22,23,24,25,26,27,28,29,30,31,32], as well as particulate flows [33]. Detailed molecular dynamics of Lennard-Jones fluids have further been performed by Frezzotti and Barbante and compared to capillary fluid models with a general very good agreement [7].…”

Kinetic derivation of Cahn-Hilliard fluid models

“…Liquid-vapor phase changes have notably been investigated by employing linearized Boltzmann equations with condensation-evaporation boundary conditions [31,32,33]. Vlasov-Enskog type equations have also been used to investigate spatial aspects of phase transition [34,35,36,37,38,39,40,41,42,43,44,45,46]. Detailed molecular dynamics of Lennard-Jones fluids have further been performed by Frezzotti et al [47] and compared to capillary fluid models with a general very good agreement.…”

Can a liquid drop on a substrate be in equilibrium with saturated vapor?

“…It seems unlikely that another form of this term would fundamentally change the properties of the functionals involved. Hence, one could conjecture that drops on a solid substrate evaporate in any model conforming to an H-Theorem and mass and energy conservation -such as, for example, the Enskog-Vlasov kinetic equation for dense fluids [34][35][36][37][38][39] (which is generally viewed as much more accurate model than the DIM). Physical interpretation.…”