The nonequilibrium van der Waals square gradient model. (I). The model and its numerical solution

The van der Waals square gradient model has played an important role in the description of the properties of an equilibrium interface between a vapor and a liquid. We extend the model to describe nonequilibrium states with temperature gradients, pressure differences and the resulting evaporation or condensation fluxes. In the equilibrium model van der Waals introduced a square gradient term in the free energy density. This term describes the deviation from local equilibrium in the interfacial region. In this first paper we propose explicit expressions for all the thermodynamic densities needed in the nonequilibrium description. After deriving the entropy production rate we give the linear force–flux relation. It is found that the thermal resistance also contains a square gradient contribution. A numerical procedure to solve the resulting equations in a stationary state was developed and is discussed. A first discussion is given of the properties of the resulting numerical solutions. We discuss in particular how evaporation and condensation fluxes result as a consequence of changing the pressure away from the coexistence pressure and by thermostating the temperatures at the ends of the box in the liquid and the vapor phases away from the equilibrium value. The influence of the size of the square gradient contribution to the thermal resistance on the temperature profiles is assessed.