= Two-dimensional Flory approximation
{title2=$R\sim(v_2\ell^2)^{1/4}N^{3/4}$}
For a planar <polymer> with $N$ segments of length $\ell$, approximate its coil <free energy> by $F/(k_BT)\sim R^2/(N\ell^2)+v_2N^2/R^2$. The first term is the entropic stretching cost of an <ideal chain>, while $v_2$ is an excluded-area coefficient. Minimizing gives $R^4\sim v_2\ell^2N^3$ 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.
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