Two-dimensional Flory approximation ($R\sim (v_2{\ell }^2{)}^{1/4}{N}^{3/4}$)

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.