Physical basis for constrained lattice density functional theory

Abstract

To study nucleation phenomena in an open system, a constrained lattice density functional theory (LDFT) method has been developed before to identify the unstable directions of grand potential functional and to stabilize nuclei by imposing a suitable constraint. In this work, we answer several questions about the method on a fundamental level, and give a firmer basis for the constrained LDFT method. First, we demonstrate that the nucleus structure and free energy barrier from a volume constraint method are equivalent to those from a surface constraint method. Then, we show that for the critical nucleus, the constrained LDFT method in fact produces a bias-free solution for both the nucleus structure and nucleation barrier. Finally, we give a physical interpretation of the Lagrange multiplier in the constraint method, which provides the generalized force to stabilize a nucleus in an open system. The Lagrange multiplier is found to consist of two parts: part I of the constraint produces an effective pressure, and part II imposes a constraint to counteract the supersaturation.

Accurate determination of the vapor-liquid-solid contact line tension and the viability of Young equation

“…2b shows the difference of the free energy density between liquid and vapor bulk phases obtained from ω l − ω v = ( Ω l − Ω v )/V , with Ω l and Ω v the grand free energy for bulk liquid and vapor phases, respectively. The coexistence chemical potential for vapor-liquid phase transition was found to μ c = −3.01819, at which ω l = ω v (see Fig. 2b).…”

“…Note that in this work we used two methods to define the position of the vapor–liquid interface. One is the Gibbs dividing surface for the vapor-liquid interfaces with the fluid density equal to 0.519, from which we could fit the interface using a spherical hypothesis and obtain the radius of a droplet. The other is a method used by Schrader et al 13, in which the droplet radius R is calculated from the volume of a liquid droplet with V l = 4 / 3πR 3 and V l = V ( ρ − ρ v )/( ρ l − ρ v ).…”

“…For a grand canonical ( μVT ) ensemble, the grand potential Ω in lattice model is expressed as a function of the fluid density distribution via16171819 where the sums are restricted to fluid sites on the lattice, and a is the vector from a site i to a nearest neighbor site. In equation (3), ρ i is the mean density at site i , represents external field, and ε ff and ε sf represent the strength of fluid-fluid interaction and that for fluid-solid interaction, respectively.…”

“…We chose lattice gas model to describe the systems, and the normal lattice density functional theory (LDFT)1617 was used here to determine the stable or metastable states. While for the unstable critical nuclei (nanodroplets), the constrained LDFT method1819 was applied here to stabilize the nuclei and determine the corresponding energy barrier, namely ΔΩ in equation (2). The physical basis of the constrained LDFT and calculation details can be found in ref 19…”

Mechanism of two-step vapour–crystal nucleation in a pore

“…Heterogeneous nucleation of liquids and crystals has been studied both theoretically and in simulation for a variety of geometries such as nucleation on planar walls [7,8] and in slit pores [9][10][11][12], on cylinders [13][14][15], patches [16,17], patterned templates [18][19][20][21], spheres [22][23][24] and in pores [25][26][27][28]. Experimental studies include colloidal crystallisation on patterned templates [4,29,30] and spherical impurities [31,32].…”