Excluded Volume Effects of Branched Molecules

The expansion factor, $\alpha^{2}=\langle s_{N}^{2}\rangle/\langle s_{N}^{2}\rangle_{0}$, of branched molecules in the melt state is estimated. The equilibrium expansion factor is determined as the point in which all the inhomogeneity terms of the osmotic potential, $\Delta G_{osmotic}$, go to zero. Numerical analysis shows that $\log\,\alpha=0.082\,\log N+\text{const.}$ for $10^{3}\le N\le 10^{7}$, giving $\alpha\cong \text{const.}\,N^{1/12}$ so that $\langle s_{N}^{2}\rangle^{1/2}\propto N^{1/3}$ which coincides with the value for the critical packing density.