Hard spheres

FeOs provides an implementation of the Boublík-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state (Boublík, 1970, Mansoori et al., 1971) for hard-sphere mixtures which is often used as reference contribution in SAFT equations of state. The implementation is generalized to allow the description of non-sperical or fused-sphere reference fluids.

The reduced Helmholtz energy density is calculated according to

βAV=6π(3ζ1ζ21ζ3+ζ23ζ3(1ζ3)2+(ζ23ζ32ζ0)ln(1ζ3))

with the packing fractions

ζk=π6αCk,αραdαk,        k=03

The geometry coefficients Ck,α and the segment diameters dα depend on the context in which the model is used. The following table shows how the expression can be reused in various models. For details on the fused-sphere chain model, check the repository or the publication.

Hard spheres

PC-SAFT

Fused-sphere chains

dα

σα

σα(10.12e3εαkBT)

σα

C0,α

1

mα

1

C1,α

1

mα

Aα

C1,α

1

mα

Aα

C1,α

1

mα

Vα

Fundamental measure theory

An model for inhomogeneous mixtues of hard spheres is provided by fundamental measure theory (FMT, Rosenfeld, 1989). Different variants have been proposed of which only those that are consistent with the BMCSL equation of state in the homogeneous limit are currently considered in FeOs (exluding, e.g., the original Rosenfeld and White-Bear II variants).

The Helmholtz energy density is calculated according to

βf=n0ln(1n3)+n121n3+136πn2n22f3(n3)

The expressions for n12 and n22 depend on the FMT version.

version

n12

n22

references

WhiteBear

n1n2n1n2

n223n2n2

Roth et al., 2002, Yu and Wu, 2002

KierlikRosinberg

n1n2

n22

Kierlik and Rosinberg, 1990

AntiSymWhiteBear

n1n2n1n2

n22(1n2n2n22)3

Rosenfeld et al., 1997, Kessler et al., 2021

For small n3, the value of f(n3) numerically diverges. Therefore, it is approximated with a Taylor expansion.

f3={n3+(1n3)2ln(1n3)n32(1n3)2if n3>10532+83n3+154n32+245n33+356n34else

The weighted densities nk(r) are calculated by convolving the density profiles ρα(r) with weight functions ωkα(r)

nk(r)=αnkα(r)=αρα(r)ωkα(rr)dr

which differ between the different FMT versions.

WhiteBear/AntiSymWhiteBear

KierlikRosinberg

ω0α(r)

C0,απσα2δ(dα2|r|)

C0,α(18πδ(dα2|r|)+12π|r|δ(dα2|r|))

ω1α(r)

C1,α2πσαδ(dα2|r|)

C1,α8πδ(dα2|r|)

ω2α(r)

C2,αδ(dα2|r|)

C2,αδ(dα2|r|)

ω3α(r)

C3,αΘ(dα2|r|)

C3,αΘ(dα2|r|)

ω1α(r)

C3,αr2πσα|r|δ(dα2|r|)

-

ω2α(r)

C3,αr|r|δ(dα2|r|)

-