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

We present the alternative derivation of the excluded volume equation. The resulting equation is mathematically identical to the one proposed in the preceding paper. As a result, the theory reproduces well the observed points by SANS (small angle neutron scattering) experiments. The equation is applied to the coil-globule transition of branched molecules. It is found that in the entire 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 the exponents $\kappa$ and $\nu$ change abruptly from phases to phases; there are no intermediate values between them, for instance between $\nu_{dilute}$ and $\nu_{melt}$.