Modeling diffusion in a slab

We investigated diffusion in a one-dimensional slab with various initial conditions (ICs) and boundary conditions (BCs) using finite difference (FD) methods in Excel. The slab was divided into 5 boxes. The BCs were maintained by setting the left-hand edge of box 1 and the right-hand edge of box 5 to constant concentrations. The ICs were implemented by setting the concentration of each of the 5 boxes to the desired values. The concentrations in each of the boxes were determined using FD methods and plotted at selected times. We discovered that the concentration profiles always tend towards a linear steady-state profile that depends only on the BCs and not the ICs. When the slab starts out with a constant initial concentration profile and zero-concentration BCs, the diffusing molecules are eliminated from the edges of the slab, and the initial rectangular profile is smoothed out by the diffusion process. After an initial transition period, the profiles appear to approach a sinusoidal shape that decays exponentially with time. That intuition was confirmed by the long-time linear behavior of a semi-log plot of the concentrations with time. Using separation of variables, we were able to show that an exponentially decaying sine wave is a characteristic solution to the diffusion equation. We confirmed that hypothesis by making the initial profile sinusoidal and then observed an exponential decay in all boxes, demonstrating that our 5-box FD model successfully solves the diffusion problem.