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.