Alternative Approach to the Excluded Volume Problem The Critical Behavior of the Exponent $ν$
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 ($T<Θ$), the exponent $κ=d\logα\,/\,d\log N\, (N\rightarrow\infty)$ takes the value $\frac{1}{12}$, showing that contrary to the case of linear molecules ($κ=-\frac{1}{6}$), 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 $0<\barϕ\le 1$, the gradients $κ$ seem to converge on a common value lying somewhere from $κ=\frac{1}{12}$ to $0.1$. Since $ν_{dilute}=\tfrac{1}{2}$, $ν_{melt}=\tfrac{1}{3}$, and $0.33\cdots\leν_{conc}\,(=ν_{0}+κ) <0.35$ for $0<\barϕ\le 1$, the simulation results suggest that the exponents $κ$ and $ν$ change abruptly from phases to phases; there are no intermediate values between them, for instance between $ν_{dilute}$ and $ν_{melt}$.