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Adding $d\varphi$ to a free-energy density adds a constant at fixed total compositional order parameter. In the common-tangent construction for phase coexistence, it shifts the tangent slope by $d$ while preserving its points of contact. Thus coexistence compositions are unchanged, whereas their chemical potential shifts. This does not assert unchanged equilibrium composition at a fixed externally imposed chemical potential.

Take $b>0$ for a stable quartic Landau free energy. The cubic-term removal in a conserved quartic Landau free energy uses $\varphi =\psi -c/(3b)$ and gives, apart from a constant,For a closed binary fluid mixture, the compositional order parameter has fixed integral. Adding $d\int \varphi$ therefore adds the same constant to every allowed state. Equivalently, in the common-tangent construction for phase coexistence, a linear term changes the tangent slope, hence the chemical potential, by $d$, but leaves the coexistence compositions unchanged. This is why linear composition bias leaves coexistence compositions unchanged; it would not mean an unchanged composition at a fixed externally imposed chemical potential.

In the shifted variable, the binodal compositions are ${\psi }_{\pm }=\pm \sqrt{-a^{\prime }/b}$ for $a^{\prime }<0$. They merge atThe continuous phase transition therefore has the same scalar quartic Landau theory as the symmetric binary fluid mixture, with a shifted critical composition and control parameter. In the mean-field approximation, the composition difference vanishes with order-parameter critical exponent $1/2$. This argument presumes the critical composition lies in the physically accessible composition range; conservation affects the dynamics, not this static phase coexistence construction.

Use the stable parameters $b,\kappa >0$. The bulk stationary points satisfy $f^{\prime }(\varphi )=a\varphi +b{\varphi }^3=0$. For $a<0$, $\varphi =0$ has negative curvature, while the two minima areBoth have $f(\pm {\varphi }_B)=-a^2/(4b)$ and $f^{\prime \prime }(\pm {\varphi }_B)=-2a>0$. To establish minimization at fixed total composition, writeFor mean composition $\bar{\varphi }$ with $|\bar{\varphi }|<{\varphi }_B$, a mixture of the two minima with volume fraction $\lambda =(\bar{\varphi }+{\varphi }_B)/(2{\varphi }_B)$ in the positive phase attains this lower bound in the bulk. Macroscopic slabs have only a subextensive interfacial excess free energy, so the bound is attained per unit volume in the thermodynamic limit. Thus constrained minimization gives phase coexistence in the mean-field approximation; at the endpoints only one phase is needed.

The two equilibrium conditions also follow directly by constrained optimization. Introduce a Lagrange multiplier $\mu$ for $\lambda {\varphi }_{+}+(1-\lambda ){\varphi }_{-}=\bar{\varphi }$ in the mixture's bulk free-energy density. Varying ${\varphi }_{+}$ and ${\varphi }_{-}$ gives $f^{\prime }({\varphi }_{+})=f^{\prime }({\varphi }_{-})=\mu$; varying $\lambda$ gives $f({\varphi }_{+})-\mu {\varphi }_{+}=f({\varphi }_{-})-\mu {\varphi }_{-}$. These equate the chemical potentials and pressures, respectively.

Equivalently, their bulk chemical potentials are both zero and their pressures are equal:These are precisely the conditions of the common-tangent construction for phase coexistence: the horizontal supporting tangent touches both minima. Equal chemical potentials prevent net composition exchange, and equal pressure gives mechanical equilibrium at a flat interface.