含义Meaning
弯曲液面两侧存在压强差（附加压力），凹侧压强高。液滴、气泡越小内部压强越大；它也解释了毛细管中液面的上升或下降。
A curved interface sustains a pressure difference, higher on the concave side. Smaller drops and bubbles have higher internal pressure; it also explains capillary rise and depression.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\gamma\) | 表面张力surface tension | N·m⁻¹ |
| \(R_1, R_2\) | 主曲率半径principal radii of curvature | m |
| \(h\) | 毛细上升高度capillary rise | m |
| \(\theta\) | 接触角contact angle | ° |
推导Derivation
半径 r 的气泡膨胀 dr：表面能增加 \(\gamma\,d(4\pi r^2) = 8\pi\gamma r\,dr\)。
Expanding a bubble by dr raises surface energy by \(8\pi\gamma r\,dr\).
这等于压强差做的功 \(\Delta p\,d(\tfrac43\pi r^3) = 4\pi r^2\Delta p\,dr\)，相等得 \(\Delta p = 2\gamma/r\)。
Equate to pressure work \(4\pi r^2\Delta p\,dr\).
毛细管：弯月面半径 \(r/\cos\theta\)，附加压力与液柱静压 \(\rho gh\) 平衡，得茹林定律。
In a capillary the Laplace pressure balances \(\rho gh\) (Jurin’s law).
适用条件与局限Assumptions & limitations
连续介质描述，纳米尺度（几个分子直径）下表面张力本身随曲率变化（托尔曼长度）。
Continuum description; at nanometre scales γ itself depends on curvature.
例题Worked example
25 °C 水（γ = 0.072 N/m）中半径 1 μm 的气泡：Δp = 2×0.072/10⁻⁶ = 1.44×10⁵ Pa ≈ 1.4 atm。半径 0.1 mm 玻璃毛细管中水上升约 15 cm。
A 1 μm bubble in water has Δp ≈ 1.4 atm; water rises ~15 cm in a 0.1 mm radius capillary.