  The free-energy lattice Boltzmann (LB) model is one of the major multiphase
models in the LB community. The present study is focused on a class of
free-energy LB models in which the divergence of thermodynamic pressure tensor
or its equivalent form expressed by the chemical potential is incorporated into
the LB equation via a forcing term. Although this class of free-energy LB
models may be thermodynamically consistent at the continuum level, it suffers
from thermodynamic inconsistency at the discrete lattice level owing to
numerical errors [Guo et al., Physical Review E 83, 036707 (2011)]. The
numerical error term mainly includes two parts, one comes from the discrete
gradient operator and the other can be identified in a high-order
Chapman-Enskog analysis. In this paper, we propose an improved scheme to
eliminate the thermodynamic inconsistency of the aforementioned class of
free-energy LB models. The improved scheme is constructed by modifying the
equation of state of the standard LB equation, through which the discretization
of $\nabla \left( {\rho c_{\rm{s}}^{\rm{2}}} \right)$ is no longer involved in
the force calculation and then the numerical errors can be significantly
reduced. Numerical simulations are subsequently performed to validate the
proposed scheme. The numerical results show that the improved scheme is capable
of eliminating the thermodynamic inconsistency and can significantly reduce the
spurious currents in comparison with the standard forcing-based free-energy LB
model.
