论文标题
三角形聚合物重心周围的片段分布
Segment Distribution around the Center of Gravity of a Triangular Polymer
论文作者
论文摘要
研究了针对特殊梳子聚合物（三角形聚合物）的重心周围的片段分布，其侧链的侧链是同一生成数，$ g $，作为主骨干。所有其他聚合物共有的径向质量分布表示为端到端向量的分布函数的总和，$ \ {\ vec {r} _ {gh} \} $，从重力中心到$ h $ h $ t th Generation the Generation;结果是，对于一个大的$ g $，\ begin {equation}φ_{\ text {triang}}}}（s）= \ frac {1} {n} {n} \ left \ sum_ {\ sum_ {h = 1} r_ {gh}^{2} \ right \ rangle} \ right）^{\ frac {d} {2}} {2}} \ text {exp} \ left（ - \ frac {d} {d} {2 \ left \ left \ langle r_ {gh}^{2} \ right \ rangle} s^2 \ right）+\ sum_ {h = 2}^{g} \ sum_ {j = 1}^{g-h}^{g-h} \ left（\ frac {d} {d} {2π\ weft \ left \ left \ left \ langle r_ {gh_ {j}}^{2} \ right \ rangle} \ right）^{\ frac {d} {2}}} \ text {exp} \ left（ - \ frac {d} r_ {gh_ {j}}^{2} \ right \ rangle} s^2 \ right）\ right \} \ notag \ notag \ end {equation}，发现旋转半径的均等正方s_ {n}^{2} \ right \ rangle_ {0} \ doteq \ frac {7} {15} {15} \，g \，l^{2} $，as $ g \ rightArrow \ rightarrow \ infty $。由于三角形聚合物的$ g \ propto \ sqrt {n} $，这导致$ \ left \ left \ langle s_ {n}^{2}^{2} \ rangle_ {0}^{0}^{0}^{1/2} {1/2} \ propto n^propto n^propto n^{1/4} $与随机分支相同的Exports。 On the basis of the present result, we put forth that all the known polymers obey the equality: $\left\langle s_{N}^{2}\right\rangle_{0}=A\, g\,l^{2}$, where $A$ is a polymer-species-dependent coefficient and also depends on the choice of the root monomer.我们讨论了这个经验方程的扩展。
The segment distribution around the center of gravity is investigated for a special comb polymer (triangular polymer) having the side chains of the same generation number, $g$, as the main backbone. Common to all the other polymers, the radial mass distribution is expressed as the sum of the distribution functions for the end-to-end vectors, $\{\vec{r}_{Gh}\}$, from the center of gravity to the monomers on the $h$th generation; the result being, for a large $g$, \begin{equation} φ_{\text{triang}}(s)=\frac{1}{N}\left\{\sum_{h=1}^{g}\left(\frac{d}{2π\left\langle r_{Gh}^{2}\right\rangle}\right)^{\frac{d}{2}}\text{Exp}\left(-\frac{d}{2\left\langle r_{Gh}^{2}\right\rangle}s^2\right)+\sum_{h=2}^{g}\sum_{j=1}^{g-h}\left(\frac{d}{2π\left\langle r_{Gh_{j}}^{2}\right\rangle}\right)^{\frac{d}{2}}\text{Exp}\left(-\frac{d}{2\left\langle r_{Gh_{j}}^{2}\right\rangle}s^2\right)\right\}\notag \end{equation} It is found that the mean square of the radius of gyration varies as $\left\langle s_{N}^{2}\right\rangle_{0}\doteq\frac{7}{15}\,g\,l^{2}$, as $g\rightarrow\infty$. Since $g\propto \sqrt{N}$ for the triangular polymer, this leads to $\left\langle s_{N}^{2}\right\rangle_{0}^{1/2}\propto N^{1/4}$, giving the same exponent as observed for the randomly branched polymer. On the basis of the present result, we put forth that all the known polymers obey the equality: $\left\langle s_{N}^{2}\right\rangle_{0}=A\, g\,l^{2}$, where $A$ is a polymer-species-dependent coefficient and also depends on the choice of the root monomer. We discuss the extension of this empirical equation.
