论文标题
分支聚合物重心周围的片段分布
Segment Distribution around the Center of Gravity of Branched Polymers
论文作者
论文摘要
借助Isihara公式的帮助，用于分支聚合物的重心周围质量分布的数学表达式。我们介绍了端到端向量的高斯近似，$ \ vec {r} _ {gν_{i}} $，从重心到$ν$ th arm的$ i $ th spoint。然后，对于星聚合物，结果为\ begin {equation}φ_{star}（s）= \ frac {1} {n} {n} \ sum_ {ν= 1}^{f}^{f} \ sum_ {i = 1} r_ {gν_{i}}^{2} \ right \ rangle} \ right）^{d/2} \ exp \ left（ - \ frac {d} {2 \ left \ left \ left \ langle r_ {gν_{gν_{gν_{i}}}}}^{2}^{2} {2} {2} {2} {2} \ right \ range}}对于足够大的$ n $，\ end {equation}，其中$ f $表示武器数。发现结果$φ_{star}（s）$不是高斯。对于dendrimers \ begin {qore}φ_{dend}（s）= \ sum_ {h = 1}^{g}ω__{h} \ left（\ frac {d} r_ {g_ {h}}^{2} \ right \ rangle} \ right）^{d/2} \ exp \ left（ - \ frac {d} {2 \ left \ left \ left \ langle r_ {g_ {g_ {g _ { \ end {equation}其中$ω_{h} $表示从$ g $ Generations构建的树枝状聚合物中$ h $ th代的重量分数，因此$ \ sum_ {h = 1}^{g}^{g}ω__________{h} = 1 $。具体来说，$ω_{1} = 1/n $和$ω__{h} =（f-1）^{h-2}/n $ for $ h \ ge 2 $。这些分布可以通过从重力中心到每个质量点的端到端距离的每个高斯函数的相同总和来描述这些分布。请注意，对于大型$ f $和$ g $，年轻一代的统计重量成为主导。结果，不受干扰的树枝状聚合物的质量分布以大型$ f $和$ g $的限制接近高斯形式。结果表明，树枝状聚合物的回旋半径与$ n $一起增加对数，这导致了指数，$ν_{0} = 0 $。还讨论了随机分支聚合物的一个例子。
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, $\vec{r}_{Gν_{i}}$, from the center of gravity to the $i$th mass point on the $ν$th arm. Then, for star polymers, the result is \begin{equation} φ_{star}(s)=\frac{1}{N}\sum_{ν=1}^{f}\sum_{i=1}^{N_ν}\left(\frac{d}{2π\left\langle r_{Gν_{i}}^{2}\right\rangle}\right)^{d/2}\exp\left(-\frac{d}{2\left\langle r_{Gν_{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 $φ_{star}(s)$ is, unfortunately, not Gaussian. For dendrimers \begin{equation} φ_{dend}(s)=\sum_{h=1}^{g}ω_{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 $ω_{h}$ denotes the weight fraction of masses in the $h$th generation on a dendrimer constructed from $g$ generations, so that $\sum_{h=1}^{g}ω_{h}=1$. To be specific, $ω_{1}=1/N$ and $ω_{h}=(f-1)^{h-2}/N$ 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, $ν_{0}=0$. An example of randomly branched polymers is also discussed.
