双语化学辞典Chem Lexicon
本科Undergraduate 聚合反应与高分子物理 Polymerisation & polymer physics

高分子链均方末端距Mean-square end-to-end distance

\[\langle R^2\rangle_0 = Nb^2\ (\text{自由连接链}),\qquad \langle R^2\rangle_0 = Nl^2\frac{1 - \cos\theta}{1 + \cos\theta}\ (\text{自由旋转链}),\qquad \langle R^2\rangle_0 = C_\infty Nl^2\]\[\langle R^2\rangle_0 = Nb^2\ (\text{freely jointed chain}),\qquad \langle R^2\rangle_0 = Nl^2\frac{1 - \cos\theta}{1 + \cos\theta}\ (\text{freely rotating chain}),\qquad \langle R^2\rangle_0 = C_\infty Nl^2\]

含义Meaning

把高分子链看作无规行走：末端距的均方值与链节数成正比，链尺寸 ∝ \(\sqrt N\)。特征比 \(C_\infty\) 反映链的刚性（聚乙烯约 6.7）。

A polymer chain as a random walk: the mean-square end-to-end distance is proportional to N, so size ∝ √N. The characteristic ratio \(C_\infty\) measures stiffness (≈6.7 for polyethylene).

符号说明Symbols

符号Symbol含义 Meaning单位 Unit
\(N\)链节（键）数number of links (bonds)1
\(b, l\)链节长度、键长link or bond lengthm
\(\theta\)键角补角bond-angle supplement°
\(C_\infty\)特征比characteristic ratio1

推导Derivation

  1. \(\mathbf R = \sum_i\mathbf b_i\)，\(\langle R^2\rangle = \sum_i\sum_j\langle\mathbf b_i\cdot\mathbf b_j\rangle\)。

    \(\langle R^2\rangle = \sum_{ij}\langle\mathbf b_i\cdot\mathbf b_j\rangle\).

  2. 自由连接：不同链节方向无关，交叉项平均为零，只剩 \(Nb^2\)。

    Freely jointed: cross terms vanish, leaving \(Nb^2\).

  3. 固定键角时相邻键有关联 \(\langle\mathbf b_i\cdot\mathbf b_{i+k}\rangle = l^2\cos^k\theta\)，求和得自由旋转链公式；受阻内旋转进一步增大。

    Fixed bond angles correlate neighbours; summing gives the freely rotating chain; hindered rotation increases it further.

  4. 中心极限定理：N 大时 R 呈高斯分布，熵 \(S = -\frac{3k_\mathrm BR^2}{2Nb^2}\) + 常数，是橡胶弹性的熵弹性来源。

    For large N, R is Gaussian; its entropy is the origin of rubber elasticity.

适用条件与局限Assumptions & limitations

  • 理想链忽略排除体积；良溶剂中实际 \(R \propto N^{0.588}\)（弗洛里指数约 3/5）。

    Ideal chains ignore excluded volume; in good solvents \(R \propto N^{0.588}\).