Flory approximation ($\mathcal{F}/(k_{B}T)\sim R^2/(N{b}^2)+v{N}^2/R^d$)

The Flory approximation balances the entropic cost of expanding an ideal chain against a mean-field excluded volume interaction. For $N$ segments of length $b$ in $d$ spatial dimensions, minimizing $R^2/(N{b}^2)+v{N}^2/R^d$ gives $R^{d+2}\sim v{N}^3{b}^2$. This yields the scaling exponents $3/5$ in three dimensions and $3/4$ in two dimensions. It is a scaling approximation, not an exact numerical free energy or proof of critical exponents; confinement replaces $R^d$ by the accessible coil volume.

For a planar polymer with $N$ segments of length $\ell$, approximate its coil free energy by $F/(k_{B}T)\sim R^2/(N{\ell }^2)+v_2{N}^2/R^2$. The first term is the entropic stretching cost of an ideal chain, while $v_2$ is an excluded-area coefficient. Minimizing gives $R^4\sim v_2{\ell }^2{N}^3$ and size exponent $\nu =3/4$. For $v_2\sim {\ell }^2$, this becomes $R\sim \ell N^{3/4}$. A parameter multiplying $N^2/R^2$ must carry area units; using an unsquared segment length there would be dimensionally inconsistent.