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Categories: Mathematics, Physics.

Fick’s laws

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 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 $\mathrm{m}^{-3}$):

$$\begin{aligned} \boxed{ \vec{J} = - D \: \nabla C } \end{aligned}$$

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 can also just differentiate the definition of $M$ directly:

$$\begin{aligned} \dv{M}{t} = \dv{}{t}\int_V C \dd{V} = \int_V \pdv{C}{t} \dd{V} \end{aligned}$$

Above, we have two valid expressions for $\idv{M}{t}$, which must be equal, so stripping the integrals leads to this continuity equation:

$$\begin{aligned} \pdv{C}{t} = - \nabla \cdot \vec{J} \end{aligned}$$

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:

$$\begin{aligned} C(x, 0) = \delta(x - x_0) \end{aligned}$$

By solving Fick’s second law with this initial condition, $C$’s time evolution turns out to be:

$$\begin{aligned} H(x - x_0, t) \equiv C(x, t) = \frac{1}{\sqrt{4 \pi D t}} \exp\!\Big( \!-\!\frac{(x - x_0)^2}{4 D t} \Big) \end{aligned}$$

This result is a normalized Gaussian, as a consequence of the central limit theorem: 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 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.

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