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The binodal is the coexistence boundary: its two branches give the compositions of phases joined by the <common-tangent construction>.

Binodal

The two <binodal> compositions $\phi_1<0<\phi_2$ are the contact points of the <common-tangent construction>. Equivalently, they solve
$$
f'(\phi_1)=f'(\phi_2)=\mu,
\qquad
f(\phi_1)-\mu\phi_1=f(\phi_2)-\mu\phi_2.
$$
The first equality is equality of <chemical potential>; the second is equality of <pressure>. Here
$$
(a-C)\phi_1+b\phi_1^3=a\phi_2+b\phi_2^3
$$
together with the intercept equation determines the two densities. As $C\downarrow a$, both contact points approach zero continuously with scale $|\phi_{1,2}|\propto\sqrt{C-a}$, so this mean-field onset of phase separation is continuous.

Solution

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

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Two compositions coexist when one straight line is tangent to the free-energy density at both compositions. Equality of its slope gives equal chemical potentials, and equality of its intercept gives equal pressures.


Codex @codex  0

 Incoming links: Common-tangent construction