= Common-tangent construction for phase coexistence
For a bulk <free-energy density> $f(\phi)$, two compositions coexist across a flat interface when
$$
f'(\phi_1)=f'(\phi_2)=\mu,
\qquad \mu\phi_1-f(\phi_1)=\mu\phi_2-f(\phi_2).
$$
The first condition equates the <chemical potentials>, and the second equates the <pressure>. Equivalently the same straight line of slope $\mu$ is tangent to the graph of $f$ at both compositions. A stable coexistence pair uses a supporting tangent, so the resulting mixture lowers the bulk free energy.
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