Fick's Laws - Quantitative Analysis

While qualitative analysis of Fick's Laws are important or useful, these powerful mathematical models are central in quantitative analysis of diffusional systems. They're useful if we want to calculate (e.g.) how many atoms are passing through a barrier per second or (e.g) how materials at an interface might mix over time. In this section we'll cover the ways we leverage Fick's First and Second Laws in engineering application to predict and control diffusion.

Fick's First Law and the Steady State

Fick's First Law tells us about flux - how many atoms or molecules are moving through some space in a unit time. Typically, we're working with moles or kilograms of a substance moving through some area of $\text{cm}^2$ per second, so the units of flux are in mol/($\text{cm}^2$-s) or kg/($\text{cm}^2$-s).

Now, flux was important in the previous sections with complex concentration profiles, but it turns out that at long times, many diffusional systems tend towards the steady state. We'll show why this happens later, but for now consider a steady state condition to be a condition in which the concentration profile does not change with time. In other words:

$$ \frac{\partial C}{\partial t} = 0 $$

and so

$$ 0 = \frac{\partial^2 C}{\partial x^2} $$

What does this say? It says that the concentration profile has no curvature. It must have a constant slope. So, the steady-state condition looks like the profile shown in Figure 7.8.1.

In the condition in Figure 7.8.1, a few things are clear:

First, there is no curvature, so if we hold the concentrations at the surfaces constant, there will be no change in concentration with time.
We can then calculate the concentration gradient from the positions and concentrations of the surfaces of the plate.
If we know the diffusion coefficient (which depends on the material and temperature), we can compute the very useful value of the flux.

Let's set this up generally first, and then do an example.

The slope of the concentration profile in Figure 7.8.1 is

$$ \begin{align} \frac{\mathrm{d}C}{\mathrm{d}x} &= \frac{\Delta C}{\Delta x}\\ \frac{\mathrm{d}C}{\mathrm{d}x} &= \frac{C_{\text{B}}-C_{\text{A}}}{x_{\text{B}}-x_{\text{A}}}\ \end{align} $$ and this is related to the flux by Fick's first law:

$$ \begin{align} J &= D\frac{\mathrm{d}C}{\mathrm{d}x}\\ J &= D\frac{C_{\text{B}}-C_{\text{A}}}{x_{\text{B}}-x_{\text{A}}}\ \end{align} $$

Where $D$ is the diffusion coefficient - which is a measure of the speed of diffusion in the material. Its units are typically $\mathrm{cm}^2/\mathrm{s}$, which makes sense, intuitively, it's a measure of the area ($\mathrm{cm}^2$) "covered" by diffusion in some time, $\mathrm{s}$.

Let's try a real problem (that might affect your health one day).

Methylene chloride is a common ingredient in paint removers. Besides being an irritant, it also may be absorbed through the skin. When using this paint remover, protective gloves should be worn. On the outside of the glove, you have a reservoir of methylene chloride in contact with the butyl rubber of the glove. Its concentration is $C_{\text{A}} = 0.44\,\text{g}/\text{cm}^{3}$. Inside the glove, on your skin, there is a concentration of $C_{\text{A}} = 0.02\,\text{g}/\text{cm}^{3}$.

If butyl rubber gloves ( $\Delta x = 0.04\,\text{cm}$ thick) are used, what is the diffusive flux of methylene chloride through the glove? The diffusion coefficient for methylene chloride in butyl rubber is $D = 110 \times 10^{-8}\, \text{cm}^2/\text{s}$.