= Polymer blend miscibility
\b[<Polymer blend miscibility>] is the ability of different <polymers> to form one thermodynamically stable phase at specified <temperature>, <pressure> and composition. A specimen that merely looks uniform is not sufficient evidence. IUPAC distinguishes a homogeneous blend from a miscible one: the latter must satisfy thermodynamic miscibility criteria.<IUPAC Compendium of Chemical Terminology, fifth edition (Gold Book)>[Homogeneous polymer blend, entry HT07582, notes on the definition of miscible polymer blend]{ref}
The thermodynamic comparison is between the homogeneous state and all allowed phase-separated states at the same overall composition. For a binary incompressible lattice model with equal segment volumes, let $\phi$ be the first component's <volume fraction>, $N_1,N_2$ the segment counts per chain and $\chi$ the interaction parameter. <Flory–Huggins solution theory> gives a dimensionless mixing <Gibbs free energy> per lattice site,
$$
f(\phi)=\frac{\phi}{N_1}\ln\phi+\frac{1-\phi}{N_2}\ln(1-\phi)+\chi\phi(1-\phi).
$$
The mixing entropy per segment is reduced for long connected chains. The originating statistical treatment of <polymer> solutions makes molecular-size disparity explicit, rather than replacing the chains by equal-sized independent small molecules.<Thermodynamics of High Polymer Solutions by Paul J. Flory>[Opening summary, p. 51, chain-size-dependent mixing entropy and volume fractions]{ref}
At composition-independent $\chi$, direct <differentiation> gives
$$
f''(\phi)=\frac{1}{N_1\phi}+\frac{1}{N_2(1-\phi)}-2\chi.
$$
Negative curvature means that small composition fluctuations can lower the free energy: the homogeneous state is locally unstable. For an original symmetric example with $N_1=N_2=100$ at $\phi=1/2$, $f''=0.04-2\chi$. Values $\chi=0.01$ and $0.03$ therefore give positive and negative curvature, respectively. In this model the symmetric critical value is $\chi_c=0.02$.
Positive local curvature alone does not establish global stability. In a <common-tangent construction for phase coexistence>, a line touching the free-energy curve at two compositions can lie below the intervening homogeneous states, indicating favorable two-phase coexistence. The <spinodal> marks local loss of stability; the <binodal> bounds coexistence. These are different questions about the same free-energy landscape.
\Image[/polymer-blend-miscibility-landscape.png]
{title=Original symmetric polymer-blend free-energy model with one-phase and two-phase mixing landscapes}
{description=Both chains have 100 equal-volume segments. At interaction parameter 0.03, the common tangent joins calculated coexisting volume fractions about 0.0707 and 0.9293. The parameters are illustrative and do not fit a measured polymer pair.}
Experiments must also address kinetics and observation scale. A blend can retain an apparently uniform state because phase separation is slow. <Microscopy> or <light scattering> can reveal domains or composition fluctuations, but absence of a resolved signal only establishes the resolution and timescale of the observation. A stated thermal history and measurements after changing temperature help distinguish a persistent preparation from evidence about equilibrium. No particular polymer pair is assigned a phase diagram by the illustrative lattice parameters.
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