Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 344 / 1 / d

With ${\varphi }_o=0$,For $C<a$, both branches are locally convex at the origin. Atthe negative-side curvature vanishes; for $C>a$ the interval near $0^{-}$ has $f^{\prime \prime }=a-C+3b{\varphi }^2<0$, so a homogeneous composition there is unstable and the equilibrium free energy is its convex envelope. The sketch therefore has an ordinary upward quartic on the positive side and, beyond $C_c$, a negative-curvature shoulder and a minimum on the negative side.

The two binodal compositions ${\varphi }_1<0<{\varphi }_2$ are the contact points of the common-tangent construction. Equivalently, they solveThe first equality is equality of chemical potential; the second is equality of pressure. Heretogether with the intercept equation determines the two densities. As $C\downarrow a$, both contact points approach zero continuously with scale $|{\varphi }_{1,2}|\propto \sqrt{C-a}$, so this mean-field onset of phase separation is continuous.