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diff --git a/source/know/concept/ficks-laws/index.md b/source/know/concept/ficks-laws/index.md new file mode 100644 index 0000000..234b180 --- /dev/null +++ b/source/know/concept/ficks-laws/index.md @@ -0,0 +1,165 @@ +--- +title: "Fick's laws" +date: 2021-09-05 +categories: +- Physics +- Mathematics +layout: "concept" +--- + +**Fick's laws of diffusion** govern the majority of diffusion processes, +where a certain "impurity" substance redistributes itself through a medium over time. +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 +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. +Formally, Fick's first law predicts that the flux +is proportional to the gradient of the concentration $C$ (with unit $\mathrm{m}^{-3}$): + +$$\begin{aligned} + \boxed{ + \vec{J} = - D \: \nabla C + } +\end{aligned}$$ + +Where $D$ (with unit $\mathrm{m}^{2}/\mathrm{s}$) +is known as the **diffusion coefficient** or **diffusivity**, +and depends on both the medium and the diffusing substance. + +Fick's first law is a general physical principle, +which was discovered experimentally, +and thus does not have a general derivation. +Proofs for specific systems do exist, +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, +i.e. the diffusing species is not created or destroyed anywhere. +Suppose that an arbitrary volume $V$ contains an amount $M$ of diffusing matter, +distributed in space according to $C(\vec{r})$, such that: + +$$\begin{aligned} + M + \equiv \int_V C \dd{V} +\end{aligned}$$ + +Over time $t$, matter enters/leaves $V$. +Let $S$ be the surface of $V$, and $\vec{J}$ the diffusion flux, +then $M$ changes as follows, to which we apply the divergence theorem: + +$$\begin{aligned} + \dv{M}{t} + = - \int_S \vec{J} \cdot \dd{\vec{S}} + = - \int_V \nabla \cdot \vec{J} \dd{V} +\end{aligned}$$ + +For comparison, we differentiate the definition of $M$, +and exploit that the integral ignores $t$: + +$$\begin{aligned} + \dv{M}{t} + = \dv{}{t}\int_V C \dd{V} + = \int_V \pdv{C}{t} \dd{V} +\end{aligned}$$ + +Both $\idv{M}{t}$ are equal, so stripping the integrals leads to this **continuity equation**: + +$$\begin{aligned} + \pdv{C}{t} + = - \nabla \cdot \vec{J} +\end{aligned}$$ + +From Fick's first law, we already have an expression for $\vec{J}$. +Substituting this into the continuity equation yields +the general form of Fick's second law: + +$$\begin{aligned} + \boxed{ + \pdv{C}{t} + = \nabla \cdot \Big( D \: \nabla C \Big) + } +\end{aligned}$$ + +Usually, it is assumed that $D$ is constant +with respect to space $\vec{r}$ and concentration $C$, +in which case Fick's second law reduces to: + +$$\begin{aligned} + \pdv{C}{t} = D \: \nabla^2 C +\end{aligned}$$ + + +## Fundamental solution + +Fick's second law has exact solutions for many situations, +but the most important one is arguably the **fundamental solution**. +Consider a 1D system (for simplicity) with constant diffusivity $D$, +where the initial concentration $C(x, 0)$ is +a [Dirac delta function](/know/concept/dirac-delta-function/): + +$$\begin{aligned} + C(x, 0) = \delta(x - x_0) +\end{aligned}$$ + +According to Fick's second law, +the concentration's time evolution of $C$ turns out to be: + +$$\begin{aligned} + H(x - x_0, t) + \equiv C(x, t) + = \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big) +\end{aligned}$$ + +This result is a normalized Gaussian, +as a consequence of +the [central limit theorem](/know/concept/central-limit-theorem/): +the diffusion behaviour is a sum of many independent steps +(i.e. molecular collisions). +The standard deviation is $\sqrt{2 D t}$, +meaning that the distance of a diffusion is proportional to $\sqrt{t}$. + +This solution $H$ is extremely useful, +because any initial concentration $C(x, 0)$ can be written as +a convolution of itself with a delta function: + +$$\begin{aligned} + C(x, 0) + = (C * \delta)(x) + = \int_{-\infty}^\infty C(x_0, 0) \: \delta(x - x_0) \dd{x_0} +\end{aligned}$$ + +In other words, any function is a linear combination of delta functions. +Fick's second law is linear, +so the overall solution $C(x, t)$ is the same combination of fundamental solutions $H$: + +$$\begin{aligned} + C(x, t) + = (C * H)(x) + &= \int_{-\infty}^\infty C(x_0, 0) \: H(x - x_0, t) \dd{x_0} + \\ + &= \int_{-\infty}^\infty \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big) \: C(x_0, 0) \dd{x_0} +\end{aligned}$$ + +This technique is analogous to using +the [impulse response](/know/concept/impulse-response/) +of a linear operator to extrapolate all its inhomogeneous solutions. +The difference is that here, we used the initial condition +instead of the forcing function. + + + +## References +1. U.F. Thygesen, + *Lecture notes on diffusions and stochastic differential equations*, + 2021, Polyteknisk Kompendie. |