Fick's Law: From Concentration Gradients to Fluxes

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In 1855 Dr. Adolf Fick published his work "Über Diffusion" (About Diffusion) in the Annals of Physics, where he pointed out the outstanding importance of diffusion processies for organic life. Following the work of Thomas Graham who studied diffusion through porous media, he discovered the what is today known as Fick's First Law.

Fick's First Law

The diffusion flux $J$ is proportional to the negative gradient of the concentration $c$, with proportionality constant $D$ (diffusion constant). \begin{align} J= -D \, \nabla c \end{align} where $\nabla$ is the nabla operator $\sum_i \frac{\partial}{\partial x_i} \hat{e}_i$. Molecules from regions with high concentration will flow to regions with lower concentration until the concentration is constant throughout the space.

The flux $J$ is defined as the number of molecules that pass through the unit area per unit time and has thus units of $1/(m^2s)$. Dimensional analysis shows, that the diffusion constant $D$ must have units \begin{align} \frac{1}{m^2s}= [D] \, \frac{1/m^3}{m} \quad \Rightarrow \quad [D]=\frac{m^2}{s}. \end{align}

The following examples show that the order of magnitude for the diffusion constant varies widely among molecules.

Molecule D
GFP $7 \, \mu m^2s$
Potassium (K) $2000 \, \mu m^2s$
DNA (Yeast) $5 \cdot 10^4 \mu m^2s$

The Random Walk Approach

From a microscopic perspective, the diffusion process can be treated as a Random Walk problem. To simplify the analysis we consider a one dimensional problem. Say, initially there are $n_i$ molecules in the intervall $(x_i,x_i+ \Delta x)$. The probability of a molecule jumping to the right $(x_{i+1},x_{i+1}+ \Delta x)$ shall be $r$, the probability of jumping from $(x_{i+1},x_{i+1}+ \Delta x)$ to $(x_i,x_i+ \Delta x)$ shall be $l$ and the probability that a molecule stays $s$.

The probability that out of the $n_i$ molecules $j$ molecules jump to the right and $n_i-j$ molecules stay during time $\Delta t$ is given by the binomial distribution. \begin{align} P(n_i,j)=\begin{pmatrix} n_i \\ j \\ \end{pmatrix} r^{j} \cdot s^{n_i-j} \end{align} The mean number of jumps of the binomial distribution to the right is given by \begin{align} \langle j_r \rangle = r \cdot n_i \end{align} In similar fashion we get for the mean number of jumps to the left \begin{align} \langle j_l \rangle = l \cdot n_{i+1} \end{align} If we assume a symmetric probability condition $p:=s=r=l$, then the difference in the number of mean jumps

\begin{align} \langle j_l \rangle-\langle j_r \rangle \end{align}

is directely related to the difference in the number of molecules with \begin{align} \langle j_l \rangle-\langle j_r \rangle= p \cdot (n_{i+1}-n_i)=p \cdot \Delta n \end{align} The net flux $\langle J \rangle$ from $(x_i,x_i+ \Delta x)$ to $(x_{i+1},x_{i+1}+ \Delta x)$ is given by the number of net jumps per area $A$ during time $\Delta t$ \begin{align} \langle J \rangle =\frac{\langle j_l \rangle-\langle j_r \rangle}{A \, \Delta t} \end{align} expressing the difference in the number of molecules $\Delta n$ in terms of concencentration $\Delta n=-\Delta c \, A \, \Delta x$ (gradient is defined as $-(c_{i+1}-c_i)=c_i-c_{i+1}$) we get \begin{align} \langle J \rangle =\frac{\langle j_l \rangle-\langle j_r \rangle}{A \, \Delta t}=-\frac{p \, \Delta c \, A \, \Delta x}{A \, \Delta t}\left( \frac{ \Delta x }{ \Delta x} \right) \end{align} We cancel $A$, extend and let $\Delta c=c_{i+1}-c_i, \Delta x$ as well as $\Delta t$ tend to zero \begin{align} \langle J \rangle=-\lim_{\Delta c , \Delta x , \Delta t \rightarrow 0} \left( \frac{p \Delta x^2}{ \Delta t}\frac{\Delta c }{ \Delta x } \right) =-D \, \frac{\partial c}{\partial x} \end{align} with diffusion constant $D$.

Further Reading: Über Diffusion; von Dr. Adolf Fick. Annalen der Physik. Volume 170, Issue 1 [1]