From 7c412050570ef229dd78cbcffbf80f23728a630d Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sat, 13 May 2023 15:42:47 +0200 Subject: Improve knowledge base --- source/know/concept/ficks-laws/index.md | 21 +++++++++++++-------- 1 file changed, 13 insertions(+), 8 deletions(-) (limited to 'source/know/concept/ficks-laws') diff --git a/source/know/concept/ficks-laws/index.md b/source/know/concept/ficks-laws/index.md index b205af9..8d5da7d 100644 --- a/source/know/concept/ficks-laws/index.md +++ b/source/know/concept/ficks-laws/index.md @@ -14,6 +14,7 @@ A diffusion process that obeys Fick's laws is called **Fickian**, as opposed to **non-Fickian** or **anomalous diffusion**. + ## Fick's first law **Fick's first law** states that diffusing matter @@ -21,13 +22,14 @@ moves from regions of high concentration to regions of lower concentration, at a rate proportional to the difference in concentration. Let $$\vec{J}$$ be the **diffusion flux** (with unit $$\mathrm{m}^{-2} \mathrm{s}^{-1}$$), -whose magnitude and direction describe the "flow" of diffusing matter. +whose magnitude and direction describes the "flow" of diffusing matter. Formally, Fick's first law predicts that the flux is proportional to the gradient of the concentration $$C$$ (with unit $$\mathrm{m}^{-3}$$): $$\begin{aligned} \boxed{ - \vec{J} = - D \: \nabla C + \vec{J} + = - D \: \nabla C } \end{aligned}$$ @@ -43,6 +45,7 @@ but they say more about those systems than about diffusion in general. + ## Fick's second law To derive **Fick's second law**, we demand that matter is conserved, @@ -65,8 +68,7 @@ $$\begin{aligned} = - \int_V \nabla \cdot \vec{J} \dd{V} \end{aligned}$$ -For comparison, we differentiate the definition of $$M$$, -and exploit that the integral ignores $$t$$: +For comparison, we can also just differentiate the definition of $$M$$ directly: $$\begin{aligned} \dv{M}{t} @@ -74,7 +76,8 @@ $$\begin{aligned} = \int_V \pdv{C}{t} \dd{V} \end{aligned}$$ -Both $$\idv{M}{t}$$ are equal, so stripping the integrals leads to this **continuity equation**: +Above, we have two valid expressions for $$\idv{M}{t}$$, +which must be equal, so stripping the integrals leads to this **continuity equation**: $$\begin{aligned} \pdv{C}{t} @@ -101,6 +104,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Fundamental solution Fick's second law has exact solutions for many situations, @@ -110,11 +114,12 @@ where the initial concentration $$C(x, 0)$$ is a [Dirac delta function](/know/concept/dirac-delta-function/): $$\begin{aligned} - C(x, 0) = \delta(x - x_0) + C(x, 0) + = \delta(x - x_0) \end{aligned}$$ -According to Fick's second law, -the concentration's time evolution of $$C$$ turns out to be: +By solving Fick's second law with this initial condition, +$$C$$'s time evolution turns out to be: $$\begin{aligned} H(x - x_0, t) -- cgit v1.2.3