Flux and Concentration Change
In your explorations of the random walk model on Section 7.4 you observed that high curvature regions of the concentration profile changed faster than low curvature regions. On this page we will work out an explanation of why this is the case.
One of the most important concepts in diffusion is the concept of flux, which tells how about the the rate flow of something through a unit area. It literally means flow. This could be the flow of water through a pipe, electrons through a wire, or (as we see here) atoms through some unit area.
Let's think about where flux in diffusional systems might come from based on our random walk model. Take 3-5 minutes on this exercise.
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Diffusion is often described as "the net movement of anything (for example, atoms, ions, molecules, energy) from a region of higher concentration to a region of lower concentration." In the space below, explain why will there be a net movement of atoms from the high concentration region to the low concentration region in the figure below even, though each individual atom is moving randomly.
Remember, in our model, the atoms are simply modeled to jump randomly. They do not "know" if they are in a high concentration region or a low concentration region.
bottom: a visualization of particles split into a high concentration region and a low concentration region. top: the concentration on either side of the dividing line.
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In the context of materials science, net flux $J$ is defined as the net flow of particles through a given area per unit time.
Net flux has a direction: In a 1D model, net flow to the right is positive and to the left is negative. For the concentration profiles below, assume that in the next unit of time, each atom has a 25% chance of randomly moving to the other side of the $x=0$ line. The first one, (a), is shown as an example of how one might compute the flux.
What will the net flux at $x=0$ be for scenarios (b), (c) and (d)?
Four concentration profiles for systems divided into just two parts. The first one shows how many atoms are expected to jump in each direction (assuming a 25% chance of jumping between regions) and the resulting flux.
Concentration Change
As we saw in the above questions, the net movement of randomly moving particles is from regions of higher concentration to regions of lower concentration simply because there are more particles in the high concentration region to randomly move to the lower concentration region than vice versa. No atoms "want" to move any direction, indeed, they're all moving randomly without preferred direction of motion.
We can calculate the flux between these regions based on the current concentration difference, and if there are only two regions, this will also tell us how much the concentration will change in each region. But what if there are more regions?
Take 3-5 minutes on this exercise.
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Focus on the $x=1$ regions in the three concentration profiles below. Which $x=1$ region(s) will increase in concentration at the next time step and which will decrease?
Three concentration profiles with three regions each.
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For the two concentration profiles in the previous question which will decrease in the next time step at $x=1$, which will decrease more and why?
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Now we will look at cases where the concentration is higher on one side and lower on the other. For each of the concentration profiles below, determine for the $x=1$ region how the concentration will change. Again assume a 25% chance of a particle jumping to either neighboring region
