SISSA-Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136 Trieste, Italy.
Univ Lyon, Ens de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique and Centre Blaise Pascal, F-69342 Lyon, France.
Phys Rev E. 2017 Jan;95(1-1):012117. doi: 10.1103/PhysRevE.95.012117. Epub 2017 Jan 12.
While Flory theories [J. Isaacson and T. C. Lubensky, J. Physique Lett. 41, 469 (1980)JPSLBO0302-072X10.1051/jphyslet:019800041019046900; M. Daoud and J. F. Joanny, J. Physique 42, 1359 (1981)JOPQAG0302-073810.1051/jphys:0198100420100135900; A. M. Gutin et al., Macromolecules 26, 1293 (1993)MAMOBX0024-929710.1021/ma00058a016] provide an extremely useful framework for understanding the behavior of interacting, randomly branching polymers, the approach is inherently limited. Here we use a combination of scaling arguments and computer simulations to go beyond a Gaussian description. We analyze distribution functions for a wide variety of quantities characterizing the tree connectivities and conformations for the four different statistical ensembles, which we have studied numerically in [A. Rosa and R. Everaers, J. Phys. A: Math. Theor. 49, 345001 (2016)1751-811310.1088/1751-8113/49/34/345001 and J. Chem. Phys. 145, 164906 (2016)JCPSA60021-960610.1063/1.4965827]: (a) ideal randomly branching polymers, (b) 2d and 3d melts of interacting randomly branching polymers, (c) 3d self-avoiding trees with annealed connectivity, and (d) 3d self-avoiding trees with quenched ideal connectivity. In particular, we investigate the distributions (i) p_{N}(n) of the weight, n, of branches cut from trees of mass N by severing randomly chosen bonds; (ii) p_{N}(l) of the contour distances, l, between monomers; (iii) p_{N}(r[over ⃗]) of spatial distances, r[over ⃗], between monomers, and (iv) p_{N}(r[over ⃗]|l) of the end-to-end distance of paths of length l. Data for different tree sizes superimpose, when expressed as functions of suitably rescaled observables x[over ⃗]=r[over ⃗]/sqrt[〈r^{2}(N)〉] or x=l/〈l(N)〉. In particular, we observe a generalized Kramers relation for the branch weight distributions (i) and find that all the other distributions (ii-iv) are of Redner-des Cloizeaux type, q(x[over ⃗])=C|x|^{θ}exp(-(K|x|)^{t}). We propose a coherent framework, including generalized Fisher-Pincus relations, relating most of the RdC exponents to each other and to the contact and Flory exponents for interacting trees.
尽管弗洛里理论[J. Isaacson 和 T. C. Lubensky, J. Physique Lett. 41, 469 (1980)JPSLBO0302-072X10.1051/jphyslet:019800041019046900; M. Daoud 和 J. F. Joanny, J. Physique 42, 1359 (1981)JOPQAG0302-073810.1051/jphys:0198100420100135900; A. M. Gutin 等人, Macromolecules 26, 1293 (1993)MAMOBX0024-929710.1021/ma00058a016]为理解相互作用的随机支化聚合物的行为提供了一个极其有用的框架，但该方法本质上是有限的。在这里，我们使用缩放论证和计算机模拟的组合来超越高斯描述。我们分析了各种数量的分布函数，这些数量特征化了树的连接性和构象，这些数量是我们在[Rosa 和 Everaers, J. Phys. A: Math. Theor. 49, 345001 (2016)1751-811310.1088/1751-8113/49/34/345001 和 J. Chem. Phys. 145, 164906 (2016)JCPSA60021-960610.1063/1.4965827]中数值研究的四种不同统计系综中得到的：（a）理想随机支化聚合物，（b）相互作用的随机支化聚合物的 2d 和 3d 熔体，（c）具有退火连接性的 3d 自回避树，（d）具有淬火理想连接性的 3d 自回避树。特别是，我们研究了分布函数（i）p_{N}(n)，它是通过随机切断树中质量为 N 的分支而切割的分支的重量 n；（ii）p_{N}(l)，它是单体之间的轮廓距离 l；（iii）p_{N}(r[over ⃗])，它是单体之间的空间距离 r[over ⃗]，以及（iv）p_{N}(r[over ⃗]|l)，它是长度为 l 的路径的末端到末端的距离。当表示为适当缩放的可观测量 x[over ⃗]=r[over ⃗]/sqrt[〈r^{2}(N)〉]或 x=l/〈l(N)〉时，不同树大小的数据会叠加。特别是，我们观察到分支重量分布（i）的广义克拉默斯关系，并发现所有其他分布（ii-iv）都是雷德勒-德克洛兹类型的，q(x[over ⃗])=C|x|^{θ}exp(-(K|x|)^{t})。我们提出了一个连贯的框架，包括广义费舍尔-平卡斯关系，将大多数 RdC 指数相互关联，并将其与相互作用的树的接触和弗洛里指数联系起来。
