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, were working with moles or kilograms of a substance moving through some area of cm2 per second, so the units of flux are in mol/(cm2-s) or kg/(cm2-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:

1. First, there is no curvature, so if we hold the concentrations at the surfaces constant, there will be no change in concentration with time.
2. We can then calculate the concentration gradient from the positions and concentrations of the surfaces of the plate.
3. 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. It's units are typically $\mathrm{cm}^2/\mathrm{s}$, which makes sense, intuitively, its 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).

Fick's Second Law and Time Evolution

Fick's Second Law is a linear, second-order partial differential equation:

We already know how to interpret this governing equation qualitatively, but what about quantitively? To do this, we need some expression that is a function of $x$ and $t$ that tells us how the concentration profile varies in space and time. We need an expression for $C(x,)$. Now, some of you may have taking a partial differential equations or boundary problems course and learned some ways to find $C(x,t)$ in different scenarios. If you've done this, that's good, but we have no such expectations that you solve differential equations in this class.

Instead, what we'll do is provide a solution (or proposed solution) to Fick's Second Law, and your job will be to ensure the solution works. What does this mean? We'll provide a scenario in which diffusion is occurring and provide $C(x,t)$. Your job is to make sure that $C(x,t)$ satisfies Eq. 7.8.1 and, if so, use the equation to calculate $C(x,t)$ at some position and time.

Let's try an example. Say I tell you that if we have dye diffusion from skittles in water, the concentration profile of the dye can be calculated at any time and position using the function $C(x,t) = -x^2t^2 +B$, where $B$ is some constant. Could you tell me if this expression satisfies Fick's Second Law, and how would you do it?

Well, intuitively, you'd probably challenge it. This function tells us that the concentration profile as a function of $x$ is parabolic. However, it also says that the concentration when $x \neq 0$ increases with time as $t^2$ - even though the curvature of the profile throughout all $x$ is negative! This does not sound like it satisfies Fick's Second Law based on what we observed earlier in this chapter.

So, to test this systematically we simply need to plug $C(x,t) = -x^2t^2 + B$ into Eq. 7.8.1. Let's try it:

$$ \begin{align} \frac{\partial C(x,t)}{\partial t} &= D \frac{\partial^2{C(x,t)}}{\partial x^2}\\ -2x^2t &= -2Dt^2\ \end{align} $$

The result is simply not true for all time and positions. This cannot be a solution that satisfies Fick's Second Law.

Let's instead look at a solution that does obey Fick's Second Law, a bell-curve shape. Here, $C(x,t) = \frac{A}{t^{1/2}} \exp{\left(-\frac{x^2}{4Dt}\right)}$, where $A$ is a constant. Let's test the solution.

$$ \begin{align} \frac{\partial C(x,t)}{\partial t} &= D \frac{\partial^2 C(x,t)}{\partial x^2}\\ \frac{A\left(\exp{\left(-\frac{x^2}{4Dt}\right)}\right)\left(-2Dt +x^2\right)}{4Dt^{5/2}} &= \frac{A\left(\exp{\left(-\frac{x^2}{4Dt}\right)}\right)\left(-2Dt +x^2\right)}{4Dt^{5/2}}\\ \end{align} $$

While I didn't show all the steps to get us to the equivalency, it is indeed true. (You can solve this yourself or plug it into a calculator that does calculus.) This means that $C(x,t) = \frac{A}{t^{1/2}} \exp{\left(-\frac{x^2}{4Dt}\right)}$ does indeed satisfy Fick's Second Law and would model a Fickian system (with the correct boundary conditions). We can plot this solution $C(x,t)$ as a function of $x$ at various times $t$ to produce concentration profiles!

Quantative Analysis - Fick's Second Law Example

In the video below, I walk through the analysis of a Fick's Second Law problem outlined in the slides.