Taylor Mark P, Aung Pyie Phyo, Paul Wolfgang
Department of Physics, Hiram College, Hiram, Ohio 44234, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Jul;88(1):012604. doi: 10.1103/PhysRevE.88.012604. Epub 2013 Jul 18.
The zeros of the canonical partition functions for flexible square-well polymer chains have been approximately computed for chains up to length 256 for a range of square-well diameters. We have previously shown that such chain molecules can undergo a coil-globule and globule-crystal transition as well as a direct coil-crystal transition. Here we show that each of these transitions has a well-defined signature in the complex-plane map of the partition function zeros. The freezing transitions are characterized by nearly circular rings of uniformly spaced roots, indicative of a discontinuous transition. The collapse transition is signaled by the appearance of an elliptical horseshoe segment of roots that pinches down towards the positive real axis and defines a boundary to a root-free region of the complex plane. With increasing chain length, the root density on the circular ring and in the space adjacent to the elliptical boundary increases and the leading roots move towards the positive real axis. For finite-length chains, transition temperatures can be obtained by locating the intersection of the ellipse and/or circle of roots with the positive real axis. A finite-size scaling analysis is used to obtain transition temperatures in the long-chain (thermodynamic) limit. The collapse transition is characterized by crossover and specific-heat exponents of φ≈0.76(2) and α≈0.66(2), respectively, consistent with a second-order phase transition.
对于一系列方阱直径，已近似计算出长度达256的柔性方阱聚合物链的正则配分函数的零点。我们之前已经表明，此类链分子可经历线团-球粒和球粒-晶体转变以及直接的线团-晶体转变。在此我们表明，这些转变中的每一种在配分函数零点的复平面图中都有明确的特征。冻结转变的特征是近乎等间距根的近圆环，这表明是不连续转变。塌缩转变由根的椭圆形马蹄形段的出现发出信号，该段朝着正实轴收缩并定义了复平面无根区域的边界。随着链长增加，圆环上以及与椭圆形边界相邻空间中的根密度增加，且前沿根朝着正实轴移动。对于有限长度的链，可通过确定根的椭圆和/或圆与正实轴的交点来获得转变温度。采用有限尺寸标度分析来获得长链（热力学）极限下的转变温度。塌缩转变的特征分别是交叉指数φ≈0.76(2)和比热指数α≈0.66(2)，这与二级相变一致。
