= Linear composition bias leaves coexistence compositions unchanged
Adding $d\phi$ 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>.
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