Lattice Boltzmann method for Lennard-Jones fluids based on the gradient theory of interfaces

In the present study we propose a lattice Boltzmann equation (LBE) model derived from density gradient expansions of the discrete BBGKY evolution equations. The model is based on the mechanical approach of the gradient theory of interfaces. The basic input is the radial distribution function, which is related exclusively to the molecular interaction potential, rather than semiempirical equations of state used in previous LBE models. This function can be provided from independent molecular simulations or from approximate theories. Evidently the accuracy of the interaction potential, and thus the radial distribution function, reflects on the accuracy of the thermodynamic properties and consistency of the derived LBE model. We have applied the proposed model to obtain equilibrium bulk and interfacial properties of a Lennard-Jones fluid at different temperatures, $T$, close to critical, $T_c$. The results demonstrate that the LBE model is in excellent agreement with gradient theory as well as with independent literature results based on different molecular simulation approaches. Hence the proposed LBE model can recover accurately bulk and interfacial thermodynamics for a Lennard Jones fluid at $T/T_c>0.9$.

Figure 1

Radial distribution functions at different densities, $T^{\ast }=1.25$.

Figure 2

Dependence of the first moment of force, $a^{\ast }$, on fluid density ${\rho }^{\ast }$, at $T^{\ast }=1.25$.

Figure 3

(Color online) Isotherm for a LJ fluid at $T^{\ast }=1.25$.

Figure 4

Dependence of the third moment of force, $u_2$, on fluid density, ${\rho }^{\ast }$, at different temperatures.

Figure 5

Time evolution of bulk densities at $T^{\ast }=1.20$. Solid line refers to initial densities equal to their vapor-liquid equilibrium values, while dotted line refers to initial densities that are 60% of their vapor-liquid equilibrium values.

Figure 6

Time evolution of maximum velocity at $T^{\ast }=1.20$. Solid line refers to initial densities equal to their vapor-liquid equilibrium values, while dotted line refers to initial densities that are 60% of their vapor-liquid equilibrium values.

Figure 7

Vapor and liquid coexisting densities as functions of temperature, for a LJ fluid.

Figure 8

Vapor and liquid coexisting densities as functions of pressure, for a LJ fluid.

Figure 9

Density profile for the case of a planar interface computed by the LBE model (filled points) and gradient theory (solid line).

Figure 10

Integrand profiles in Eqs. (44, 45) for the case of a planar interface computed by the LBE model (filled points) and gradient theory (solid line).

Figure 11

Surface tension as a function of temperature, for a LJ fluid. Solid line is used to guide the eye through the LBE results.

Figure 12

Verification of Laplace’s law for a LJ fluid using the LBE method.