支化聚合物的尺寸指数/Isaacson-Lubensky公式的推广及其在格树中的应用
Size Exponents of Branched Polymers/ Extension of the Isaacson-Lubensky Formula and the Application to Lattice Trees
摘要：
支化聚合物可以分为两类，它们遵循不同的公式：\begin{equation} ν= \begin{cases} \hspace{1mm}\displaystyle\frac{2(1+ν_{0})}{d+2} & \hspace{3mm}\mbox{for polymers with}\hspace{2mm}\displaystyleν_{0}\ge\frac{1}{d+1}\hspace{10mm}\text{(I)}\\[3mm] \hspace{5mm}2ν_{0}& \hspace{3mm}\mbox{for polymers with}\hspace{2mm}\displaystyleν_{0}\le\frac{1}{d+1}\hspace{10mm}\text{(II)} \end{cases}\notag \end{equation}表示在良溶剂中的稀释极限。II类包括具有完全膨胀构型的特殊聚合物。在这些等式的基础上，我们讨论了嵌套结构和格树的大小指数。特别地，我们比较了我们前面的结果，$ν_{d=2}=1/2$，对于具有$ν_{0}=1/4$的$z$ =2聚合物，与数值结果，$ν_{d=2}\doteq 0.64115$，对于在2维晶格上生成的晶格树。我们的推测是，虽然在聚合物物理和凝聚态物理的结论是正确的，产生的差异，从事实上，晶格树是由较少的分支结构比支化聚合物具有$ν_{0} = 1/4$在聚合物物理。本文的分析表明，二维格点树是具有平均理想尺寸指数$\barν_{0}\doteq0.32$的异构体的混合物。
英文摘要：
Branched polymers can be classified into two categories that obey the different formulae: \begin{equation} ν= \begin{cases} \hspace{1mm}\displaystyle\frac{2(1+ν_{0})}{d+2} & \hspace{3mm}\mbox{for polymers with}\hspace{2mm}\displaystyleν_{0}\ge\frac{1}{d+1}\hspace{10mm}\text{(I)}\\[3mm] \hspace{5mm}2ν_{0}& \hspace{3mm}\mbox{for polymers with}\hspace{2mm}\displaystyleν_{0}\le\frac{1}{d+1}\hspace{10mm}\text{(II)} \end{cases}\notag \end{equation} for the dilution limit in good solvents. The category II covers the exceptional polymers having fully expanded configurations. On the basis of these equalities, we discuss the size exponents of the nested architectures and the lattice trees. In particular, we compare our preceding result, $ν_{d=2}=1/2$, for the $z$=2 polymer having $ν_{0}=1/4$ with the numerical result, $ν_{d=2}\doteq 0.64115$, for the lattice trees generated on the 2-dimensional lattice. Our conjecture is that while both the conclusions in polymer physics and condensed matter physics are correct, the discrepancy arises from the fact that the lattice trees are constructed from less branched architectures than the branched polymers having $ν_{0} = 1/4$ in polymer physics. The present analysis suggests that the 2-dimensional lattice trees are the mixture of isomers having the mean ideal size exponent of $\barν_{0}\doteq0.32$.
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