Predictive density gradient theory

Predictive density gradient theory (pDGT) is an efficient approach for the prediction of surface tensions, which is derived from non-local DFT, see Rehner et al. (2018). A gradient expansion is applied to the weighted densities of the Helmholtz energy functional to second order as well as to the Helmholtz energy density to first order.

Weighted densities (in non-local DFT) are determined from

nα(r)=inαi(r)=iρi(rr)ωαi(r)dr.

These convolutions are time-consuming calculations. Therefore, these equations are simplified by using a Taylor expansion around r for the density of each component ρi as

ρi(rr)=ρi(r)ρi(r)r+12ρ(r):rr+

In the convolution integrals, the integration over angles can now be performed analytically for the spherically symmetric weight functions ωαi(r)=ωαi(r) which provides

nαi(r)=ρi(r)4π0ωαi(r)r2drωαi0+2ρi(r)23π0ωαi(r)r4drωαi2+

with the weight constants ωαi0 and ωαi2.

The resulting weighted densities can be split into a local part nα0(r) and an excess part Δnα(r) as

nα(r)=iρi(r)ωαi0nα0+i2ρi(r)ωαi2+Δnα.

The second simplification is the expansion of the reduced residual Helmholtz energy density Φ({nα}) around the local density approximation truncated after the second term

Φ({nα})=Φ({nα0})+iαΦnαωαi22ρi+

The Helmholtz energy functional (which was introduced in the section about the Euler-Lagrange equation) then reads

F[ρ(r)]=(f(ρ)+ijcij(ρ)2ρiρj)dr

with the density dependent influence parameter

βcij(ρ)=αβ2Φnαnβ(ωαi2ωβj0+ωαi0ωβj2).

and the local Helmholtz energy density f(ρ).

For pure components, as derived in the original publication, the surface tension can be calculated from the surface excess grand potential per area according to

γ=FμN+pVA=ρVρL2c(f(ρ)ρμ+p)dρ

Thus, no iterative solver is necessary to calculate the surface tension of pure components, which is a major advantage of pDGT. Finally, the density profile can be calculated from

z(ρ)=ρVρc/2f(ρ)ρμ+pdρ