Binary Flory-Huggins polymer blend ($f/(k_{B}T)=(\varphi /N_A)\ln \varphi +[(1-\varphi )/N_B]\ln (1-\varphi )+\chi \varphi (1-\varphi )$)

An incompressible, equal-site-volume mean-field approximation for two polymers of degrees of polymerization $N_A,N_B$. Here $f$ is mixing free energy per site and $\varphi$ is the volume fraction of species $A$. For a composition-independent Flory interaction parameter, local stability requires $1/(N_A\varphi )+1/[N_B(1-\varphi )]-2\chi >0$. The thermodynamic critical point is ${\varphi }_c=\sqrt{N_B}/(\sqrt{N_A}+\sqrt{N_B})$ and ${\chi }_c=(N_A^{-1/2}+N_B^{-1/2}{)}^2/2$.

Connecting segments into chains reduces their independent translational mixing entropy per segment by factors $1/N_A$ and $1/N_B$. Consequently the tolerable positive contact penalty tends to zero as both chains become long. Equal chain lengths have ${\chi }_c=2/N$; favorable unlike interactions can still produce a miscible polymer pair.