\[\frac{\partial c}{\partial t} = D\frac{\partial^2 c}{\partial x^2},\qquad c(x, t) = \frac{n_0}{A\sqrt{4\pi Dt}}\,e^{-x^2/4Dt}\]
恒定表面浓度下向半无限介质扩散（如掺杂、渗碳）constant surface concentration into a semi-infinite medium (doping, carburising)
\[c(x,t) = c_s\,\mathrm{erfc}\!\left(\frac{x}{2\sqrt{Dt}}\right)\]含义Meaning
浓度随时间的变化由浓度分布的曲率决定。瞬时点源的解是随时间展宽的高斯分布，特征扩散距离 \(\sqrt{2Dt}\)。
The rate of change of concentration is set by the curvature of its profile. An instantaneous point source spreads as a broadening Gaussian with characteristic distance \(\sqrt{2Dt}\).
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(n_0\) | 初始释放的物质的量amount released | mol |
| \(A\) | 截面积cross-sectional area | m² |
| \(c_s\) | 表面浓度surface concentration | mol·m⁻³ |
推导Derivation
在 x 到 x + dx 薄层取物料平衡：\(\partial c/\partial t = -\partial J/\partial x\)（连续性方程）。
Mass balance on a slab: \(\partial c/\partial t = -\partial J/\partial x\).
代入菲克第一定律（D 为常数）得扩散方程。
Insert Fick’s first law with constant D.
验证高斯函数满足方程；t → 0 时趋于 δ 函数（初始点源）。
The Gaussian satisfies it and tends to a delta function as t → 0.
适用条件与局限Assumptions & limitations
D 为常数、无对流和反应；有反应时加源项（反应–扩散方程）。
Constant D, no convection or reaction; reactions add source terms.