  The thermodynamic stability of germanium tin $\mathrm{Ge_{1-x}Sn_x}$ alloys
is investigated across the composition range $0 \le x \le 1$ by applying
density functional theory (DFT) together with the cluster expansion formalism
(CE). It is known that GeSn alloys are immiscible and that non-equilibrium
growth techniques are required to produce metastable films and nanostructures.
Insight into the driving forces behind component segregation is gained by
investigating the equilibrium thermodynamics of GeSn systems. The alloy free
energy of mixing is computed by combining enthalpies from CE with entropy terms
for configurational and vibrational degrees of freedom. Volume deformations due
to the large mismatch in ionic radii are readily found to be the key driving
force for immiscibility at all temperatures of relevance. This leads to a study
of epitaxial stabilisation by employing latticed matched substrates to favour
growth of alloys with fractional compositions of $\mathrm{x=0}$, approximately
$\mathrm{x=0.5}$ and $\mathrm{x=1}$. Reduction of the free energy of mixing due
to epitaxial strain in thin films is quantified for each substrate leading to
indicators for growth of kinetically stable films.
