含义Meaning
球形粒子在黏性流体中的扩散系数与温度成正比、与黏度和粒子半径成反比。由此可从扩散系数（动态光散射、DOSY NMR）求流体力学半径。
The diffusion coefficient of a sphere in a viscous fluid is proportional to T and inversely proportional to viscosity and radius. Measured D (DLS, DOSY NMR) gives the hydrodynamic radius.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\eta\) | 溶剂黏度（25 °C 水 0.89 mPa·s）solvent viscosity (water 0.89 mPa·s at 25 °C) | Pa·s |
| \(r\) | 流体力学半径hydrodynamic radius | m |
| \(f\) | 摩擦系数friction coefficient | kg·s⁻¹ |
推导Derivation
爱因斯坦（1905）：平衡时扩散通量与外力驱动的漂移通量抵消，得 \(D = k_\mathrm BT/f\)（涨落–耗散关系）。
Einstein (1905): balancing diffusion against drift gives \(D = k_\mathrm BT/f\) (fluctuation–dissipation).
斯托克斯定律给出低雷诺数下球的摩擦系数 \(f = 6\pi\eta r\)（无滑移边界）。
Stokes’ law gives \(f = 6\pi\eta r\) for a sphere at low Reynolds number.
合并即得。
Combine.
适用条件与局限Assumptions & limitations
粒子须远大于溶剂分子；小分子（与溶剂相当）常用滑移边界 4πηr 或经验修正。非球形粒子需形状因子。
Particles must be much larger than solvent molecules; small solutes often fit slip (4πηr). Non-spheres need shape factors.
例题Worked example
直径 10 nm 的蛋白质（r = 5 nm）在 25 °C 水中：D = 1.381×10⁻²³×298/(6π×0.89×10⁻³×5×10⁻⁹) = 4.9×10⁻¹¹ m²/s。
A 10 nm protein in water at 25 °C: D ≈ 4.9×10⁻¹¹ m²/s.