On the basis of the Poisson–Boltzmann equation in cylindrical coordinates, we calculate the conductivity of a single charged nanotube filled with electrolyte. The conductivity as a function of the salt concentration follows a power-law, the exponent of which has been controversially discussed in the literature. We use the co-ion-exclusion approximation and obtain the crossover between different asymptotic power-law behaviors analytically. Numerically solving the full Poisson–Boltzmann equation, we also calculate the complete diagram of exponents as a function of the salt concentration and the pH for tubes with different radii and pK_a values. We apply our theory to recent experimental results on carbon nanotubes using the pK_a as a fit parameter. In good agreement with the experimental data, the theory shows power-law behavior with the exponents 1/3 at high pH and 1/2 at low pH, with a crossover depending on salt concentration, tube radius and pK_a.

中文翻译：

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碳纳米管电导率随盐度变化的幂律指数的交叉

根据圆柱坐标中的Poisson–Boltzmann方程，我们计算了填充有电解质的单个带电纳米管的电导率。电导率作为盐浓度的函数遵循幂律，在文献中有争议地讨论了幂指数。我们使用co-ion-exclusion逼近并通过分析获得不同渐近幂律行为之间的交叉。通过数值求解完整的Poisson–Boltzmann方程，我们还计算了不同半径和p K _a值的管的指数随盐浓度和pH值的变化的完整图。我们将理论应用于使用p K _a的碳纳米管的最新实验结果作为拟合参数。与实验数据非常吻合，该理论显示了幂律行为，指数在高pH下为1/3，在低pH下为1/2，并且交叉取决于盐浓度，管半径和p K _a。