Naji Ali, Netz Roland R
Physics Department, Technical University of Munich, Garching, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 May;73(5 Pt 2):056105. doi: 10.1103/PhysRevE.73.056105. Epub 2006 May 8.
Counterions at charged rodlike polymers exhibit a condensation transition at a critical temperature (or, equivalently, at a critical linear charge density for polymers), which dramatically influences various static and dynamic properties of charged polymer solutions. We address the critical and universal aspects of this transition for counterions at a single charged cylinder in two and three spatial dimensions using numerical and analytical methods. By introducing a Monte Carlo sampling method in logarithmic radial scale, we are able to numerically simulate the critical limit of infinite system size (corresponding to the infinite-dilution limit) within tractable equilibration times. The critical exponents are determined for the inverse moments of the counterionic density profile (which play the role of the order parameters and represent the mean inverse localization length of counterions) both within mean-field theory and within Monte Carlo simulations. In three dimensions (3D), we demonstrate that correlation effects (neglected within mean-field theory) lead to an excessive accumulation of counterions near the charged cylinder below the critical temperature (i.e., in the condensation phase), while surprisingly, the critical region exhibits universal critical exponents in accordance with mean-field theory. Also in contrast with the typical trend in bulk critical phenomena, where fluctuations become more enhanced in lower dimensions, we demonstrate, using both numerical and analytical approaches, that mean-field theory becomes exact for the two-dimensional (2D) counterion-cylinder system at all temperatures (Manning parameters), when the number of counterions tends to infinity. For a finite number of particles, however, the 2D problem displays a series of peculiar singular points (with diverging heat capacity), which reflect successive delocalization events of individual counterions from the central cylinder. In both 2D and 3D, the heat capacity shows a universal jump at the critical point and the internal energy develops a pronounced peak. The asymptotic behavior of the energy peak location is used to determine the critical temperature, which is also found to be in agreement with the mean-field prediction.
带电棒状聚合物上的抗衡离子在临界温度（或者等效地，对于聚合物而言在临界线性电荷密度时）会呈现凝聚转变，这会极大地影响带电聚合物溶液的各种静态和动态性质。我们使用数值和解析方法研究二维和三维空间中单个带电圆柱上抗衡离子这种转变的临界和普适方面。通过在对数径向尺度上引入蒙特卡罗采样方法，我们能够在可处理的平衡时间内对无限系统尺寸的临界极限（对应于无限稀释极限）进行数值模拟。在平均场理论和蒙特卡罗模拟中，都确定了抗衡离子密度分布的逆矩（其充当序参量并代表抗衡离子的平均逆局域长度）的临界指数。在三维空间（3D）中，我们证明相关效应（在平均场理论中被忽略）会导致在临界温度以下（即在凝聚相）抗衡离子在带电圆柱附近过度聚集，而令人惊讶的是，临界区域呈现出与平均场理论一致的普适临界指数。同样与体临界现象中的典型趋势相反，在体临界现象中低维波动会增强，我们使用数值和解析方法都证明，当抗衡离子数量趋于无穷大时，二维（2D）抗衡离子 - 圆柱系统在所有温度（曼宁参数）下平均场理论都是精确成立的。然而，对于有限数量的粒子，二维问题会显示出一系列特殊的奇点（热容发散），这反映了单个抗衡离子从中心圆柱的连续离域事件。在二维和三维中，热容在临界点都呈现出普适跳跃，内能会出现明显的峰值。利用能量峰值位置的渐近行为来确定临界温度，发现其也与平均场预测结果一致。
