Fick's Laws - Qualitative Graphical Interpretation

Qualitative Assessment of a Concentration Profile Using Fick's First Law

In the previous sections, we discovered Fick's Laws, the first tells us that for a concentration profile $C$, the net flux can be calculated from the gradient of the concentration profile:

Let's think about what this allows me to do qualitatively. I can look at any concentration profile at some snapshot in time and determine the positions in which the net flux is highest, as well as the direction of the net flux. (At least, we can do this for systems that are "Fickian", or obey Fick's Laws. (There are, of course, non-Fickian, or systems that display anomolous diffusion.)

Before we continue, it is important to highlight that net flux tell us how many atoms or molecules are passing through some point on the graph each unit time (and in which direction). However, net flux at some point $x$ does not tell us how the concentration profile will change with time. For that we need to think about how all the fluxes add up locally. We'll see a very useful quantitative application of Fick's First law in the next section that may make this clearer.

Using Figure 7.7.1, we can compare the magnitude and direction of the fluxes at each point.

• Point (a.) has a moderate negative slope. Eq. 7.7.1 tells us we should have a moderate net flux in the positive direction. There is net movement of atoms locally "downhill" to the low concentration regions.
• Point (b.) has a slightly larger negative slope than Point (a.). There is more net flux of atoms at (b.) than (a.), and also in the $+x$-direction - from high concentration to low.
• Point (c.) has a slope magnitude even greater than Point (b.), but it has a positive slope, meaning net flux will go in the $-x$ direction.
• Point (d.) is at the top of the high concentration region. Interestingly, the slope is zero here, which tells us that the net flux is zero. At the very top (or bottom) of the concentration profile's peaks (and valleys) there is no net exchange of atoms one way or another. This makes sense - to the left of the peak, we have flux in the $-x$ direction, and to the right of the peak, flux is in the $+x$ direction, there has to be a transition where there is no flux. This simply means there isn't a preferred flow of atoms to the left or right at that precise point.
• Point (e.) simply continues our trend. The slope is a bit less steep than that of (b.) and a bit more than (a.), at the net flux is in the $+x$-direction.

One important take-away from this analysis: the way that Fick's First law is often interpreted is that atoms and molecules prefer to move to regions of lower concentration. That's an imprecise statement. In Fickian diffusion, molecules don't "prefer" to move in any direction. They move via random walk!

An atom at Point (a.) has a 50-50 chance of jumping to the right or the left! It doesn't prefer to jump to the right! However, the flux itself - the net motion of all the atoms - does go from regions of high concentration to low concentration, and the magnitude of the flux depends on the gradient of the concentration profile.

Qualitative Assessment of a Concentration Profile Using Fick's Second Law

In a similar fashion to Fick's First Law, we can qualitatively analyze concentration profiles and predict how they'll evolve in time. Recall that Fick's Second Law tells use the rate of change of of concentration, relating it to the curvature of the profile itself. Because it governs how the concentration profile changes in space and time, we often call this a governing equation.

For those of you that are interested - this is linear, second-order partial differential equation. We won't ask you to solve this equation, but we do want you to be able to 1. interpret it physically, 3. apply it qualitatively to concentration profiles, and 3. test whether some expression $C(x,t)$ satisfies the equation. We'll address 1. and 2. below, and 3. on the next page.

Let's look at the same profile as we did in Figure 7.7.1, but now considering the curvature. In Figure 7.7.2, we show arrows at each point indicating the relative magnitudes of the curvature of that point, and the sign. Larger arrow have larger second derivatives, and upward(downward)-pointing arrows have positive(negative) curvature. We can use these curvatures and Eq. 7.7.2 to predict if the concentration at each point will have increased or decreased at some time $\Delta t$ later.

• Point (a.) has a fairly large negative curvature, meaning that it will decrease in concentration with time.
• Point (b.) has almost no curvature - it is at an inflection point. Therefore, we expect the concentration at some time $\Delta t$ later to stay the same.
• Point (c.) has a very small positive slope - the inflection point is perhaps slightly tothe right - and so we expect the concentration to increase with time, although very slightly.
• Point (d.) has a fairly large negative curvature, perhaps a bit larger than that at (a.). So, at this point we expect a large negative change in concentration with time.
• Point (e.) has a modest positive curvature, a bit larger than (c.), and so we expect a modest increase in composition.