Past exam of the soft condensed matter course of the University of Cambridge / 2021 / Section B / 6 / b / Solution

Local compositional stability requires positive curvature of the mixing free energy. Differentiating twice at fixed temperature, with $\chi$ independent of composition as in the given model, givesThe spinodal occurs where this vanishes:Between the two spinodal compositions at sufficiently large $\chi$, $f^{\prime \prime }<0$ and a homogeneous mixture is unstable to infinitesimal composition fluctuations. The binodal is instead fixed by the common-tangent construction, and encloses a larger coexistence region. Between binodal and spinodal the mixture is locally stable but metastable.

A phase diagram with a conventional upper critical temperature additionally assumes that $\chi$ decreases with $T$, for example $\chi =C/T$ with $C>0$. Then high temperature gives a uniform mixture; cooling produces an asymmetric coexistence dome whose critical composition is on the dilute-polymer side. The spinodal lies inside the binodal dome. An arbitrary $\chi (T)$ need not give this orientation, so the free-energy formula alone does not specify a unique temperature axis.