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 J\vec{J} be the diffusion flux (with unit m2s1\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 CC (with unit m3\mathrm{m}^{-3}):

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

Where DD (with unit m2/s\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 VV contains an amount MM of diffusing matter, distributed in space according to C(r)C(\vec{r}), such that:

MVCdV\begin{aligned} M \equiv \int_V C \dd{V} \end{aligned}

Over time tt, matter enters/leaves VV. Let SS be the surface of VV, and J\vec{J} the diffusion flux, then MM changes as follows, to which we apply the divergence theorem:

dMdt=SJdS=VJdV\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 MM directly:

dMdt=ddtVCdV=VCtdV\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 dM/dt\idv{M}{t}, which must be equal, so stripping the integrals leads to this continuity equation:

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

From Fick’s first law, we already have an expression for J\vec{J}. Substituting this into the continuity equation yields the general form of Fick’s second law:

Ct=(DC)\begin{aligned} \boxed{ \pdv{C}{t} = \nabla \cdot \Big( D \: \nabla C \Big) } \end{aligned}

Usually, it is assumed that DD is constant with respect to space r\vec{r} and concentration CC, in which case Fick’s second law reduces to:

Ct=D2C\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 DD, where the initial concentration C(x,0)C(x, 0) is a Dirac delta function:

C(x,0)=δ(xx0)\begin{aligned} C(x, 0) = \delta(x - x_0) \end{aligned}

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

H(xx0,t)C(x,t)=14πDtexp ⁣( ⁣ ⁣(xx0)24Dt)\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 2Dt\sqrt{2 D t}, meaning that the distance of a diffusion is proportional to t\sqrt{t}.

This solution HH is extremely useful, because any initial concentration C(x,0)C(x, 0) can be written as a convolution of itself with a delta function:

C(x,0)=(Cδ)(x)=C(x0,0)δ(xx0)dx0\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)C(x, t) is the same combination of fundamental solutions HH:

C(x,t)=(CH)(x)=C(x0,0)H(xx0,t)dx0=14πDtexp ⁣( ⁣ ⁣(xx0)24Dt)C(x0,0)dx0\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.


  1. U.F. Thygesen, Lecture notes on diffusions and stochastic differential equations, 2021, Polyteknisk Kompendie.