AI 中文总结
本研究通过零位移概率与均方位移的模型无关关系测量分形维数，发现聚合物熔体中分形维数不符合蠕动模型预测，更接近类渗流场景，揭示了动态涌现的有限尺寸分形几何。
AI 中文摘要
蠕动模型假设缠结聚合物在分形管内滑动。本文采用零位移概率与均方位移之间的模型无关关系，该关系适用于随时间变化的分形结构，可直接测量单体运动所经历几何结构的分形维数$d_\mathrm{f}$。对于二维障碍物阵列和slip-link（滑移链）模型，$d_\mathrm{f}$与蠕动预测的$d_\mathrm{f}=1/ν$（其中$ν$为Flory指数）一致。然而在聚合物熔体中，我们发现$d_\mathrm{f}\approx2.6$——该值接近渗流簇的分形维数，而非蠕动理论的$d_\mathrm{f}=2$。这与蠕动图像形成鲜明对比，后者中Rouse链在$d_\mathrm{f}=2$、谱维数$d_\mathrm{s}=1$、行走维数$d_\mathrm{w}=4$的分形结构中滑动；我们的结果反而指向类渗流场景，其特征为$d_\mathrm{f}\approx2.6$、$d_\mathrm{s}\approx1.3$和$d_\mathrm{w}\approx4$——揭示了一种动态涌现的有限尺寸分形几何，与静态管不同。
英文摘要
The reptation model postulates that entangled polymers slide within a fractal tube. Here we employ a model-independent relation between the zero-displacement probability and the mean-square displacement that applies to time-dependent fractal structures, enabling direct measurement of the fractal dimension $d_\mathrm{f}$ of the geometry experienced by monomer motion. For two-dimensional obstacle arrays and in the slip-link model, $d_\mathrm{f}$ agrees with the reptation prediction $d_\mathrm{f}=1/ν$ (where $ν$ is the Flory exponent). In polymer melts, however, we find $d_\mathrm{f} \approx 2.6$ --- a value close to the fractal dimension of percolation clusters, not the reptation value $d_\mathrm{f}=2$. This contrasts sharply with the reptation picture, in which a Rouse chain slides in a fractal structure with $d_\mathrm{f}=2$, spectral dimension $d_\mathrm{s}=1$, and walk dimension $d_\mathrm{w}=4$; our results point instead to a percolation-like scenario, characterized by $d_\mathrm{f}\approx 2.6$, $d_\mathrm{s}\approx 1.3$, and $d_\mathrm{w}\approx 4$ --- revealing a dynamically emergent, finite-size fractal geometry distinct from the static tube.
