Symmetric Flory-Huggins polymer blend (${\chi }_c=2/N$)

For two equal-site-volume polymers of chain length $N$, the mixing free energy per lattice site is $f/(k_{B}T)=[\varphi \ln \varphi +(1-\varphi )\ln (1-\varphi )]/N+\chi \varphi (1-\varphi )$. Each chain is one translating object, reducing the entropy of mixing per monomer by $N$ relative to the regular solution model. Its curvature is $f^{\prime \prime }/(k_{B}T)=1/[N\varphi (1-\varphi )]-2\chi$. The symmetric thermodynamic critical point has ${\varphi }_c=1/2$ and ${\chi }_c=2/N$. Long chains can therefore demix under a very small unfavorable Flory interaction parameter.

For a strongly segregated, symmetric long-chain polymer blend, take the square-gradient functional $F/(k_{B}T{\rho }_0)=\int [\chi \varphi (1-\varphi )+a^2({\varphi }^{\prime }{)}^2/(24\varphi (1-\varphi ))]dz$. Its first integral equates the two terms, giving ${\varphi }^{\prime }=\sqrt{24\chi }\varphi (1-\varphi )/a$. Thus $\varphi =[1+\tanh (z/\ell )]/2$ with $\ell =a/\sqrt{6\chi }$. The inverse-central-slope width $w=1/{\varphi }^{\prime }(0)=2\ell$ differs by a factor two from the tanh-argument scale. The continuum approximation is marginal when these lengths approach the statistical segment size. An observed width can additionally include roughness and instrument broadening.