 Solution

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.