# 菲克定律

## 菲克第一定律

${\displaystyle {\bigg .}J=-D{\frac {\partial \phi }{\partial x}}{\bigg .}}$

• ${\displaystyle J}$ 為“擴散通量”（於某單位時間內通過某單位面積的物質量），例如${\displaystyle ({\tfrac {\mathrm {mol} }{\mathrm {m} ^{2}\cdot \mathrm {s} }})}$ ${\displaystyle J}$ 量度在一段短時間內物質流過一小面積的量。
• ${\displaystyle \,D}$ 擴散係數擴散度，其量綱為[長度2 時間−1]，例如${\displaystyle ({\tfrac {\mathrm {m} ^{2}}{\mathrm {s} }})}$
• ${\displaystyle \,\phi }$  為濃度（假設為理想混合物），其量綱為[(物質的量) 長度−3]，例如${\displaystyle ({\tfrac {\mathrm {mol} }{\mathrm {m} ^{3}}})}$
• ${\displaystyle \,x}$  為位置長度，例如${\displaystyle \,\mathrm {m} }$

${\displaystyle J=-D\nabla \phi }$

${\displaystyle J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu _{i}}{\partial x}}}$

## 菲克第二定律

${\displaystyle {\frac {\partial \phi }{\partial t}}=D\,{\frac {\partial ^{2}\phi }{\partial x^{2}}}\,\!}$

• ${\displaystyle \,\phi }$ 為濃度，其量綱為[(物質的量) 長度−3]，例如${\displaystyle ({\tfrac {\mathrm {mol} }{m^{3}}})}$
• ${\displaystyle \,t}$ 為時間[s]
• ${\displaystyle \,D}$ 為擴散係數，其量綱為[長度2 時間−1]，例如${\displaystyle ({\tfrac {m^{2}}{s}})}$
• ${\displaystyle \,x}$ 為位置[長度]，例如${\displaystyle \,m}$

${\displaystyle {\frac {\partial \phi }{\partial t}}=-\,{\frac {\partial }{\partial x}}\,J={\frac {\partial }{\partial x}}{\bigg (}\,D\,{\frac {\partial }{\partial x}}\phi \,{\bigg )}\,\!}$

${\displaystyle {\frac {\partial }{\partial x}}{\bigg (}\,D\,{\frac {\partial }{\partial x}}\phi \,{\bigg )}=D\,{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\,\phi =D\,{\frac {\partial ^{2}\phi }{\partial x^{2}}}}$

${\displaystyle {\frac {\partial \phi }{\partial t}}=D\,\nabla ^{2}\,\phi \,\!}$

${\displaystyle {\frac {\partial \phi }{\partial t}}=\nabla \cdot (\,D\,\nabla \,\phi \,)\,\!}$

${\displaystyle \nabla ^{2}\,\phi =0\!}$

### 例：一維解（擴散長度）

${\displaystyle n\left(x,t\right)=n(0)\mathrm {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right)}$

${\displaystyle n\left(x,t\right)=n(0)\left(1-2\left({\frac {x}{2{\sqrt {Dt\pi }}}}\right)\right)}$

## 應用

### 生物學上的應用

${\displaystyle J={-P\cdot A\cdot (c_{2}-c_{1})}\,\!}$

• ${\displaystyle \,J}$ 為通量；
• ${\displaystyle \,P}$ 滲透率，量度某氣體於某溫度時通過膜的導率的實驗量；
• ${\displaystyle \,A}$ 為擴散的表面積大小；
• ${\displaystyle \,c_{2}-c_{1}}$ 為流動方向下兩邊氣體的濃度差（從${\displaystyle c_{1}}$ ${\displaystyle c_{2}}$ ）。

## 注釋

1. ^ A. Fick, Pogg. Ann. (1855), 94, 59, doi:10.1002/andp.18551700105（德文）.
2. ^ A. Fick, Phil. Mag. (1855), 10, 30.（英文）
3. ^
4. ^
5. ^ D. Brogioli and A. Vailati, Diffusive mass transfer by nonequilibrium fluctuations: Fick's law revisited, Phys. Rev. E 63, 012105/1-4 (2001) [1]

## 參考來源

• W.F. Smith, Foundations of Materials Science and Engineering 3rd ed., McGraw-Hill (2004)
• H.C. Berg, Random Walks in Biology, Princeton (1977)