Segment Distribution around the Center of Gravity of Branched Polymers

Mathematical expressions for mass distributions around the center of gravity are derived for branched polymers with the help of the Isihara formula. We introduce the Gaussian approximation for the end-to-end vector, $\vec{r}_{G\nu_{i}}$, from the center of gravity to the $i$th mass point on the $\nu$th arm. Then, for star polymers, the result is \begin{equation} \varphi_{star}(s)=\frac{1}{N}\sum_{\nu=1}^{f}\sum_{i=1}^{N_{\nu}}\left(\frac{d}{2\pi\left\langle r_{G\nu_{i}}^{2}\right\rangle}\right)^{d/2}\exp\left(-\frac{d}{2\left\langle r_{G\nu_{i}}^{2}\right\rangle}s^{2}\right)\notag \end{equation} for a sufficiently large $N$, where $f$ denotes the number of arms. It is found that the resultant $\varphi_{star}(s)$ is, unfortunately, not Gaussian. For dendrimers \begin{equation} \varphi_{dend}(s)=\sum_{h=1}^{g}\omega_{h}\left(\frac{d}{2pi\left\langle r_{G_{h}}^{2}\right\rangle}\right)^{d/2}\exp\left(-\frac{d}{2\left\langle r_{G_{h}}^{2}\right\rangle}s^{2}\right)\notag \end{equation} where $\omega_{h}$ denotes the weight fraction of masses in the $h$th generation on a dendrimer constructed from $g$ generations, so that $\sum_{h=1}^{g}\omega_{h}=1$. To be specific, $\omega_{1}=1/N$ and $\omega_{h}=(f-1)^{h-2}/N$ for $h\ge 2$. These distributions can be described by the same grand sum of each Gaussian function for the end-to-end distance from the center of gravity to each mass point. Note that for a large $f$ and $g$, the statistical weight of younger generations becomes dominant. As a consequence, the mass distribution of unperturbed dendrimers approaches the Gaussian form in the limit of a large $f$ and $g$. It is shown that the radii of gyration of dendrimers increase logarithmically with $N$, which leading to the exponent, $\nu_{0}=0$. An example of randomly branched polymers is also discussed.