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We present the alternative derivation of the excluded volume equation. Theresulting equation is mathematically identical to the one proposed in thepreceding paper. As a result, the theory reproduces well the observed points bySANS (small angle neutron scattering) experiments. The equation is applied tothe coil-globule transition of branched molecules. It is found that in theentire region of poor solvent regimes ($T<\Theta$), the exponent$\kappa=d\log\alpha\,/\,d\log N\, (N\rightarrow\infty)$ takes the value$\frac{1}{12}$, showing that contrary to the case of linear molecules($\kappa=-\frac{1}{6}$), the expansion factor increases indefinitely as $N$increases. The theory is then applied to concentrated systems in good solvents.It is found that for the entire region of $0<\bar{\phi}\le 1$, the gradients$\kappa$ seem to converge on a common value lying somewhere from$\kappa=\frac{1}{12}$ to $0.1$. Since $\nu_{dilute}=\tfrac{1}{2}$,$\nu_{melt}=\tfrac{1}{3}$, and $0.33\cdots\le\nu_{conc}\,(=\nu_{0}+\kappa)<0.35$ for $0<\bar{\phi}\le 1$, the simulation results suggest that theexponents $\kappa$ and $\nu$ change abruptly from phases to phases; there areno intermediate values between them, for instance between $\nu_{dilute}$ and$\nu_{melt}$.

Alternative Approach to the Excluded Volume Problem The Critical Behavior of the Exponent $\nu$