### Diffusion Processes

Basic idea:

In its simplest form, diffusion is the transport of a material or chemical by molecular motion. If molecules of a chemical are present in an apparently motionless fluid, they will exhibit microscopic erratic motions due to being randomly struck by other molecules in the fluid. Individual particles or molecules will follow paths sometimes known as "random walks."

In such processes, a chemical initially concentrated in one area will disperse. That is, there will be a net transport of that chemical from regions of high concentration to regions of low concentration.

An analogous form of diffusion is called conduction. In this case, heat is the "chemical" that is transported by molecular motion. As in chemical diffusion, heat migrates from regions of high heat to regions of low heat. The mathematics describing both conduction and diffusion are the same.

In this laboratory, students will explore two-dimensional diffusion phenomena by configuring and running a program called Diffusion Simulator (DS). Given an initial concentration of a chemical (heat), DS calculates and displays how a chemical diffuses over time. Results are visual, but quantitative data may be obtained from the display.

DS operates within a rectangular area. For every point in this area, two things must be specified: 1) the diffusivity--which characterizes the rate at which a chemical will diffuse, and 2) the initial concentration of a chemical (heat). Since there are two variables which must be specified over the field, it is best to imagine that there are actually two fields, a "diffusivity field" and a "concentration field". To prepare a problem for simulation, students must configure both of these rectangular fields before a problem can be run.

In DS, each of these fields is configured by creating colored areas that represent different diffusivities or different types of initial chemical concentrations. Colored areas are produced with a primitive "painting" tool consisting of a drawing shape--rectangle or oval--and a color. Both elements--shape and color--must be chosen before the painting tool can be used. These choices exist as buttons within a menu and are specific to each field.

The diffusivity field begins as an area having uniformly high diffusivity (green). Subareas may be created with high (green), low (red), or zero (white) diffusivity. By the careful positioning of rectangles and ovals with different diffusivities, this field can be configured to represent many physical systems. For example, a well-insulated room with a not-well-insulated window can be represented as an inner rectangle of high diffusivity (the room) which is framed by a rectangular region of zero diffusivity (the walls) with a segment of this frame defined as low diffusivity (the window)., i.e.,

Diffusivity field:

The concentration field begins with zero chemical concentration. It must be configured with initial concentrations, before DS can be run. When DS is started, the initial concentrations will begin to diffuse. At each point, the concentration will change at a rate determined by the corresponding point on the diffusivity field. Again, by adding colored rectangles and ovals to this field, an initial field of chemical (heat) concentration can be created. Three classes of concentration are available:

1) An initial concentration C of value C=1000 (black) that will diffuse over time as the chemical spreads. Using the insulated house example above, this concentration definition would be equivalent to placing a hot rock in an initially-cold room, and following over time how the room would be heated as the rock cooled.

2) An initial concentration that is initialized with a value of C=1000 (blue) (or C=0 (green)) that will remain at that value throughout the simulation. This is equivalent to a source of heat whose temperature is maintained at its initial value. In the case of C=0, it is equivalent to a "sink" where any concentration that diffuses into this area becomes zero. To continue the insulated house example, this concentration type could be used to create a constant temperature heater in one corner and a constant temperature icebath in another.

3) An initial concentration of C=0 (red), but which increases by 10 at each timestep of the simulation. This can represent a heating element that adds heat at a fixed rate. Areas configured with this class of concentration change according to the diffusion process. But at each step, an additional concentration is added.

Suppose, there is a wall of high conductivity (diffusivity)(green), but with an inner layer of low conductivity (red). And suppose one side of the wall were maintained at a fixed high temperature C=1000 (blue) and the other side of the wall were maintained at a fixed low temperature C=0 (green), and one wants to discover how the temperature will adjust between these two extremes within the wall. Such a problem would be set up as follows:

Concentration field:                    Diffusivity field:

How to run Diffusion Simulator

1) Click on "diffusivity" or "concentration" to configure that field.

2) Click on "rectangle" or "oval" to establish a shape; click on one of the color options to establish a characteristic.

3) Move the mouse to the appropriate field, then click and drag to define the position and size of that characteristic. When you release the mouse button, a colored area will remain. (Note that an additional cursor appears in the complementary field, so that a precise placement of initial concentrations vis-a-vis diffusivity characteristics is possible.)

4) Repeat 2) and 3) until the field is properly configured.

5) Click on the button which defines the other field.

6) Repeat 2), 3), 4) until this second field is properly configured.

7) Click on "Ready to run"

8) Click on "Start/resume"

The simulation may be stopped at any time with "Stop", then restarted with "Start/resume". When the simulation is stopped, any mouse position within the concentration field will display the concentration at that position. Also, when the simulation is stopped, clicking the mouse within the vertical black stripe to the right of the concentration field will produces a graph of the concentration distribution along the horizontal at the mouse's vertical position.

Time to try Diffusion Simulator.