Non-local phase field revisited

The phase field model is revisited as an approximation of a full non-local approach. We focus on van der Waals fluids, where the non-local interaction potential consists of short-range repulsion and long-range attraction, characterized by distances d (i.e. a typical molecular size d ≈ 10−9 m) and ℓ >> d. The first non-local correction to the thermodynamic limit introduces a density square gradient free energy, expressed in terms of a characteristic length, a. Imposing that the line integral of the non-local free energy functional across an interfacial region must equal the surface tension, we find that a is up to two orders of magnitude larger than the molecular size, in agreement with the local equilibrium assumption. Instead, by finding a directly from the interaction potential, when the attractive force follows a power-law, as in the Lennard-Jones potential, then a ≈ d, which contradicts that a ≈ 10−7 m as from surface tension measurements. Conversely, when the attractive force follows a Debye-like exponentially decaying potential of range ℓ, then a ≈ ℓ >> d, in agreement with surface tension measurements and the mean field theory assumption. The other point that we have addressed is determining when the mean field model can be applied. By directly simulating the phase separation of a far-from-critical van der Waals fluid mixture, we find that the growth laws of the mean nuclei size are not modified when higher-order gradient terms are retained. An exponentially decaying attractive potential must be used since the higher-order gradient terms diverge when power-law potentials are considered, confirming that a Lennard-Jones interaction potential is not compatible.