Deguchi Tetsuo, Uehara Erica
Department of Physics, Faculty of Core Research, Ochanomizu University, Ohtsuka 2-1-1, Bunkyo-ku, Tokyo 112-8610, Japan.
Polymers (Basel). 2017 Jun 28;9(7):252. doi: 10.3390/polym9070252.
We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connectivity expressed by graphs "topological polymers". Graphs with no loop have only trivial topology, while graphs with loops such as multiple-rings may have nontrivial topology of spatial graphs as embeddings in three dimensions, e.g., knots or links in some loops. We thus call also such polymers with nontrivial topology "topological polymers", for simplicity. For various polymers with different structures in chemical connectivity, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of the renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer⁻Grest model) expressed with complex graphs. We then address topological properties of ring polymers in solution. We define the knotting probability of a knot by the probability that a given random polygon or self-avoiding polygon of vertices has the knot . We show a formula for expressing it as a function of the number of segments , which gives good fitted curves to the data of the knotting probability versus . We show numerically that the average size of self-avoiding polygons with a fixed knot can be much larger than that of no topological constraint if the excluded volume is small. We call it "topological swelling".
我们回顾了近期关于化学连接具有非平凡结构的聚合物以及具有非平凡拓扑结构的聚合物（如溶液中的打结环聚合物）的统计和动力学性质的理论研究。我们将通过图形表示化学连接具有非平凡结构的聚合物称为“拓扑聚合物”。无环的图形只有平凡拓扑，而具有环（如多环）的图形作为三维空间中的嵌入可能具有非平凡的空间图形拓扑，例如某些环中的结或链环。因此，为简便起见，我们也将这种具有非平凡拓扑的聚合物称为“拓扑聚合物”。对于化学连接具有不同结构的各种聚合物，我们通过模拟系统地数值评估了回转半径的均方和流体力学半径。我们评估了回转半径与流体力学半径的比值，从重整化群的角度来看，我们预期该比值是通用的。此外，我们表明，对于用复杂图形表示的实际拓扑聚合物（Kremer⁻Grest模型），短程链内相关性大大增强。然后，我们讨论了溶液中环聚合物的拓扑性质。我们通过给定的具有(n)个顶点的随机多边形或自回避多边形具有该结的概率来定义结的打结概率。我们给出了一个将其表示为链段数(n)的函数的公式，该公式能很好地拟合打结概率与(n)的数据曲线。我们通过数值表明，如果排除体积较小，具有固定结的自回避多边形的平均尺寸可能比没有拓扑约束时大得多。我们将其称为“拓扑膨胀”。
