## Fickian Diffusion

### Quick Start

The simple equations decribed in Diffusion Basics can be used to show two complementary aspects of diffusion. The graph on the left shows the concentration through the sample after your chosen time t. The graph on the right shows how the total amount of solvent in the sample changes with time, and the flux rate or (when Integrated) total flux.

All you need to do is slide thickness L, concentration c and diffusion coefficient D. Play around with these values and t. Then read the explanations below to more fully understand what's going on.

One more thing. You sometimes absorb into a sample that is blocked on the other side - so choose the Blocked option. And you can start with a polymer full of solvent and want to know how it desorbs, so choose the Desorbtion option

### Fickian

The point of the Basics section was to establish that the laws of diffusion are very simple. The problem is that the simple laws don't give simple answers - they are differential equations with behaviour that can become quite complex. With modern computing power this isn't a problem. The equations can be solved step-by-step by breaking the problem into a grid of small cells and calculating the concentration differences between each pair of cells and from that and the diffusion coefficient, calculating how much flows from each cell to the next over a given time step. If the number of cells is large and the time steps are small then the calculation will be highly accurate, but too slow for an app. If the number of cells is small and the time steps are large then, unfortunately, the numerical methods break down. So a compromise has to be reached.

Using the same inputs as before, but adding a timescale over which to calculate, we can see what happens as we change our key parameters.
*Because the calculations can be relatively slow you have to click the Calculate button once you've set up your parameters with the sliders*. There is a TimeOut that defaults to 5s in case a calculation will take too long. If it exceeds that time the "time" graph shows a TimeOut alert. You can choose a longer TimeOut if you really need the answer.

Even though this is a simplified set-up, you already have a powerful diffusion modeller. You can simulate Absorption (with or without a Blocked second side) and Desorption. In addition to the calculated values shown with a blue background, you get a readout of the values by moving your mouse over any of the plots.

The left-hand plot shows how,
*at the specified time* the concentration varies across the width of the sample. Conc % is always defined based on the concentration C (Vol Fraction) specified as 100%. So if C is 0.2 (meaning for Desorption that the whole sample contains 20% or for Absorption that the outer 1nm contains 20% at t=0) then 100% means 0.2 vol fraction. As in Basics, the concentrations can be considered to be g/cc or Vol Fraction as densities of 1 are assumed for simplicity.

The right-hand plot shows how concentration in the sample varies with time. The left-hand axis is in Conc %, as before. The right-hand access shows (if relevant) the Permeation in g/cm²/s or, if the Integrated option is selected, the total amount over time in g/cm².

Where relevant, values are calculated for: flux at the given time; half-time to fill or empty the sample; breakthrough to the other side where a concentration of 0.1% of the original value is taken as being "breakthrough".

If any of the graphs look peculiar that will be because the numerics have broken down to a greater or lesser extent - so the results should be treated with caution. It requires more processing power to provide more robust numerics, but the apps must run OK even on a modest phone so they cannot be perfect for every simulation.

### Nanoparticle diffusion

The Diffusion Coefficients app now allows you to estimate diffusion coefficients for nanoparticles. To allow simulations with the amazingly low rates over very long times, the app now allows you to go up to 20,000 minutes (almost 2 weeks) with diffusion coefficients down to 10^{-25}. Turn on the Integrated option to see how much (or, rather, how little) has migrated. If you need to simulate a diffusion coefficient of, say, 10^{-35}, you will find you can just multiply the 10^{-25} by a factor of 10^{-1} for each unit below 10^{-25}. So for 10^{-35} multiply the result by 10^{-10}. Similarly, if your concentration is 0.001 vol fraction then calculate for 0.01 vol fraction and divide the answer by 10. In both cases these adjustments are numerically exact.