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

The Maxwell construction replaces a nonconvex homogeneous free-energy density by its coexistence envelope. If $U_0(M)$ excludes the source, the coexisting values $M_1,M_2$ satisfyThus one straight line is tangent at both endpoints. Equivalently,the equal-area form of the common-tangent construction. At fixed average $\overline{M}$ between the endpoints, the lever rule gives $\overline{M}=p{M}_1+(1-p)M_2$. Such a mixture has the tangent-line bulk energy, below the energy of an unstable homogeneous state.

The two phases occupy spatial domains separated by domain walls. The gradient term gives a positive wall energy, but a macroscopic interface has area of order $L^{D-1}$ rather than bulk volume $L^D$, so its cost per volume vanishes in the thermodynamic limit. For the symmetric quartic example a scalar quartic domain wall obeys $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$ and interpolates between $\pm M_0$ as $\varphi =M_0\tanh (x/\ell )$, with $M_0^2=|r|/u$ and $\ell =\sqrt{2\kappa /|r|}$. Multiplication of the differential equation by ${\varphi }^{\prime }$ gives the first integral $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$, explaining the finite positive interface cost. A uniform unconstrained system need not form both domains: phase separation is particularly relevant when the average order parameter is fixed or boundaries enforce competing phases.

Hysteresis occurs when the system follows a metastable local minimum after another minimum has become the equilibrium one. For $r<0,u>0$, the equation of state is $h=rM+u{M}^3$. The hysteresis spinodals of a scalar quartic potential follow from $r+3u{M}^2=0$:Following local minima until these limits gives different switching fields on increasing and decreasing $h$. Equilibrium instead switches at $h=0$. Actual switching can occur before a mean-field spinodal through nucleation, and depends on time scales; hysteresis is not a property of a globally minimized equilibrium potential. The second panel below distinguishes the local branches and their limiting switches.

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.

For a Landau-Ginzburg theory with positive gradient stiffness $\kappa$, $r<0$ and quartic coupling $u>0$, the degenerate minima are $\pm M_0$, $M_0^2=|r|/u$. A planar domain wall solves $\kappa {\varphi }^{\prime \prime }=r\varphi +u{\varphi }^3$. Multiplying by ${\varphi }^{\prime }$ and using the limiting minima gives $\kappa ({\varphi }^{\prime }{)}^2/2=u({\varphi }^2-M_0^2{)}^2/4$. Separating this equation yields the displayed profile, and integrating its excess energy gives the positive surface tension $2\sqrt{2\kappa }|r{|}^{3/2}/(3u)$. Its area cost is subextensive compared with bulk volume, permitting phase-separated common-tangent construction states when the average order parameter is constrained.