Department of Chemistry, Chemical Biology, and Biomedical Engineering, Stevens Institute of Technology, Hoboken, New Jersey 07030, USA.
J Chem Phys. 2010 Jul 28;133(4):044903. doi: 10.1063/1.3457160.
We report calculations on three classes of knotted ring polymers: (1) simple-cubic lattice self-avoiding rings (SARs), (2) "true" theta-state rings, i.e., SARs generated on the simple-cubic lattice with an attractive nearest-neighbor contact potential (theta-SARs), and (3) ideal, Gaussian rings. Extrapolations to large polymerization index N imply knot localization in all three classes of chains. Extrapolations of our data are also consistent with conjectures found in the literature which state that (1) R(g)-->AN(nu) asymptotically for ensembles of random knots restricted to any particular knot state, including the unknot; (2) A is universal across knot types for any given class of flexible chains; and (3) nu is equal to the standard self-avoiding walk (SAW) exponent (congruent with 0.588) for all three classes of chains (SARs, theta-SARs, and ideal rings). However, current computer technology is inadequate to directly sample the asymptotic domain, so that we remain in a crossover scaling regime for all accessible values of N. We also observe that R(g) approximately p(-0.27), where p is the "rope length" of the maximally inflated knot. This scaling relation holds in the crossover regime, but we argue that it is unlikely to extend into the asymptotic scaling regime where knots become localized.
（1）简单立方晶格自回避环（SAR），（2）“真正的”theta 态环，即在简单立方晶格上生成的具有吸引力的最近邻接触势的 SAR（theta-SAR），以及（3）理想的高斯环。对大聚合指数 N 的外推表明，所有三类链中都存在纽结定位。我们的数据外推也与文献中发现的假设一致，即（1）对于受限在任何特定纽结状态的随机纽结系综，包括无纽结，R(g)~AN(nu)渐近；（2）对于任何给定类型的柔性链，A 是通用的；（3）nu 对于所有三类链（SAR、theta-SAR 和理想环）都是标准的自回避行走（SAW）指数（与 0.588 一致）。然而，当前的计算机技术不足以直接采样渐近域，因此对于所有可访问的 N 值，我们仍然处于交叉标度区域。我们还观察到，R(g)大约等于 p(-0.27)，其中 p 是最大膨胀纽结的“绳长”。这种标度关系在交叉标度区域中成立，但我们认为它不太可能扩展到渐近标度区域，在该区域中纽结会被定位。
