三维分形晶格吸附边界接枝高分子链表面临界指数的精确研究
Exact study of surface critical exponents of polymer chains grafted to adsorbing boundary of fractal lattices embedded in three-dimensional space
摘要：
本文研究了线性高分子的吸附问题，当聚合物-溶剂系统的容器是三维Sierpinski垫片（SG）分形家族的成员。SG族的成员由整数$b$（$2\le b\le\infty$）来枚举，并且假设每个SG分形的一侧是不可穿透的吸附边界。在高分子链的自回避行走模型（SAW）中，我们计算了临界指数$γ_1, γ_{11}$和$γ_s$，它们分别与在吸附不可穿透边界上接枝有一个、两个和没有末端的所有可能的SAW的数目有关。通过应用精确重整化群（RG）方法，对于$2\le b\le 4$，我们得到了这些指数的具体值，对于各种类型的聚合物构象。我们讨论了它们之间的相互关系以及它们与SG分形上SAW相关的其他临界指数之间的关系。
英文摘要：
We study the adsorption problem of linear polymers, when the container of the polymer--solvent system is taken to be a member of the three dimensional Sierpinski gasket (SG) family of fractals. Members of the SG family are enumerated by an integer $b$ ($2\le b\le\infty$), and it is assumed that one side of each SG fractal is impenetrable adsorbing boundary. We calculate the critical exponents $γ_1, γ_{11}$, and $γ_s$ which, within the self--avoiding walk model (SAW) of polymer chain, are associated with the numbers of all possible SAWs with one, both, and no ends grafted on the adsorbing impenetrable boundary, respectively. By applying the exact renormalization group (RG) method, for $2\le b\le 4$, we have obtained specific values for these exponents, for various type of polymer conformations. We discuss their mutual relations and their relations with other critical exponents pertinent to SAWs on the SG fractals.
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