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For two species of chains with the same polymerization degree $N$, the number of independently translating objects per occupied site is reduced by $N$. The symmetric Flory-Huggins polymer blend hasThe symmetry still places the thermodynamic critical point at ${\varphi }_c=1/2$, but nowThe entropy of mixing per monomer is much smaller than for simple liquids, so even a weak unfavorable interaction can cause phase separation for long chains. If the energetic interaction parameter is $\chi =A/T$ with $A>0$, this model gives $T_c=AN/2$; the conclusion relies on that specified temperature law.

For a symmetric Flory-Huggins polymer blend, the mixing free energy per lattice site isThe critical conditions $f^{\prime \prime }=0$ and $d^{3}f/d{\varphi }^3=0$ give ${\varphi }_c=1/2$ and ${\chi }_c=2/N=0.002$. ThereforeSince $d\chi /dT<0$, heating improves miscibility: this is an upper critical solution temperature in the stipulated model. The very high value is an extrapolation of the supplied interaction-parameter fit, not evidence that an intact experimental blend can be brought reversibly to this temperature; chemical stability and the fitted temperature range must be checked before interpreting it as an accessible transition.

At $293.15\mathrm{K}$, $\chi =0.132421$ and $N\chi \simeq 132$, well into strong segregation. For the polymer-blend interface width, state the convention explicitly. In the equal-segment, long-chain square-gradient approximation,Its first integral gives ${\varphi }^{\prime }=\sqrt{24\chi }\varphi (1-\varphi )/a$, soThe inverse-central-slope width is consequentlyA convention calling the hyperbolic-tangent scale itself the width instead quotes $0.752\mathrm{nm}$; these describe the same profile. The predicted interface spans only a few statistical segments, so the continuum approximation for gradients is quantitatively marginal. Intrinsic composition width and experimentally broadened width also differ because roughness and finite spatial resolution can broaden a measured profile.

The Flory interaction parameter $\chi$ measures the excess cost of unlike segment contacts. In a simple lattice with coordination number $z$, contact energies ${ϵ}_{AA},{ϵ}_{BB},{ϵ}_{AB}$ and temperature $T$,A positive Flory interaction parameter penalizes unlike contacts; a negative value favors them. Mixing still depends on the balance with entropy of mixing, rather than the sign of $\chi$ alone.

For two polymers of degrees of polymerization $N_A,N_B$, let $\varphi$ be the volume fraction of species $A$. The binary Flory-Huggins polymer blend has mixing free energy per lattice siteThe logarithmic terms favor mixing. Their factors $1/N_A$ and $1/N_B$ arise because an entire chain translates as one object; connecting many segments greatly reduces the mixing entropy per segment. Hence long polymers can demix for a much smaller positive Flory interaction parameter than small molecules.

For composition-independent $\chi$, the local stability criterion isPositive curvature means stability to small composition fluctuations, zero curvature locates the spinodal, and negative curvature means spontaneous growth of fluctuations. Actual two-phase coexistence is determined by the common-tangent construction, or equality of both species' chemical potentials; its boundary is the binodal. A locally stable but globally unfavourable homogeneous mixture can lie between the binodal and spinodal.

Setting the second and third derivatives of $f$ to zero givesFor the symmetric Flory-Huggins polymer blend, ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. For a polymer solution with $N_B=1$ and long chains of species $A$, ${\chi }_c\to 1/2$, consistent with the theta solvent limit. An effective Flory interaction parameter can depend on temperature and composition; in the simple energetic case $\chi \propto 1/T$, heating encourages mixing, but that temperature trend is not universal.

The original illustration uses stipulated $N_A=N_B=100$. At $\chi =0.01$ the mixing free energy is convex, whereas at $\chi =0.03$ a central unstable region appears. The horizontal common-tangent construction in the latter case joins the symmetric coexisting compositions; its contacts differ from the spinodal boundaries. These are model curves, not supplied experimental data.