Learning Scenario - Diffusion Limited Aggregation (AgentSheets)

Basic Model:

Description

This is an agent-based model that depicts the diffusion-limited aggregation of particles. The model begins with one seed particle and almost 200 molecules moving randomly around the screen. When a particle touches the seed particle, it becomes another sedentary seed particle. The user can change the "stickiness" of the seed particles (their likelihood to turn other particles into seed particles) as well as the initial number and location of normal particles and seed particles. A graph plots the number of regular and "sticky" particles throughout the simulation.

Background Information

Diffusion limited aggregation is the tendency of small particles, such as dust particles, to form clusters. Because such particles come close to random movement, these clusters cause the mathematics of their movement to become more complex. As the particles touch and cling to one another, they form shapes that exhibit characteristics of fractals.

Science/Math

The fundamental principle behind this model is HAVE = HAD + CHANGE where the number of particles you HAVE (whether regular/sticky) depend upon the number of particles you HAD in the previous iteration plus the CHANGE (regular becoming sticky). Each round, the new location of each particle is dependent on their location relative to another source particle. If the particle is next to a source particle, there is a % chance, determined by the user-set variable "sticky", that the particle will become a source particle and no longer move. Otherwise, the particle moves randomly.

Teaching Strategies

An effective way to introduce students to the concept of diffusion-limited aggregation is to begin by simulating the effect of clumping on random movement. First, ask students to walk randomly around the room. A good way to define random movement is to have the students walk 3 steps, close their eyes and turn an unplanned amount, open their eyes and walk another 3 steps in the direction they are facing.

Next, have a student volunteer act as the source particle and stand still in the middle of the room with their arms stretched out to the sides. When another student moving randomly comes into contact with the source particle, he/she stops moving and stretches out his/her arms to become another source particle. Once the majority of the students have stopped moving, begin a discussion about the activity. Ask the following questions:

1. In the movement you just did, what pattern did you form? (It might be helpful to draw the pattern on the board.
2. How many people are you touching? What is the maximum number of people that anyone is touching? Why do you think that is?
3. How large of a space is the pattern taking in the room? Is the pattern larger or smaller than you expected? Why do you think that is?
4. Does the pattern repeat at all? Why or why not?
5. Can you think of examples of anything in the environment that moves like this, in random movement and then connecting with those around it? Explain. What are the similarities and differences between the types of movement?

Implementation:

How to use the model

This model has two parameters that can be manipulated to produce different results:

1. The "sticky" parameter describes the likelihood that a moving particle adjacent to an unmoving seed particle will stop moving and become a seed particle.
2. The initial number and placement of moving particles and seed particles in the model.

To change the "sticky" parameter, click on the down facing arrow to the top right of the model and select Simulation Properties. To increase or decrease the percentages, press the up and down arrows to the right of the default percentages.

To add an agent or background, first pause the model. Next, click the agent you wish to add, click the pencil to the left of the model, and click the location on the model where you would like to add the agent. To select the moving or source molecule, you may have to right click on the molecule image (a red or blue dot). To choose moving particles, click "person". To choose source particles, click "sickperson". To take away an agent, first pause the model, and then click the eraser to the left of the model. Click the agent you wish to "erase".

To run the model, click the "Play" button below the bottom left-hand corner of the model. When you click the "Play" button, the model will automatically generate a line graph with a line showing the number of moving molecules in the simulation. The exact numbers of moving and seed molecules is updated throughout the simulation and can be found in Simulation Properties (see how to change the "sticky" parameter).

For a complete tutorial on how to use AgentSheets, please go here.

Learning Objectives

1. Examine the relationship between the "sticky" parameter and the patterns produced.
2. Consider diffusion-limited aggregation as an example of fractals.
3. Examine the roll that diffusion plays in the patterns of this model.

Objective 1

To accomplish this objective, have students run the model several times with the default settings until all the particles have stopped moving. Then allow students to change the "sticky" parameter. Ask the following questions:

1. What patterns appear when the moving particles have a 100% chance of sticking to source particles? What do they look like? Explain. Make an axes out of balls and draw a function on top of it (see picture). How does the function seem to change when the graph bounces off of the walls? What line does the function seem to be reflected across?
2. Do the patterns change when the "sticky" parameter decreases? How? Why do you think this is? Explain.
3. How does the "sticky" parameter affect the number of branches in the pattern? How does it affect the thickness of the branches? Explain.

Objective 2

To accomplish this objective, have students complete Objective 1 so they have the opportunity to fully observe the patterns in the model. Next, lead a discussion about fractals and fractal patterns. Ask the following questions:

1. What is a fractal? If you do not know, look for the answer on the Internet, but make sure to choose a trustworthy source.
2. What do fractal patterns look like? What characteristics do they often have? Explain.
3. Do the frozen particles in this model exhibit any fractal patterns? Why or why not? Explain.
4. If they do exhibit fractal patterns, are there any examples of fractals in nature or art that they resemble? Explain.

Objective 3

To accomplish this objective, have students complete Objective 1 so they have the opportunity to fully observe the patterns in the model. Next, lead a discussion about diffusion and its role in the model. Ask the following questions:

1. What is diffusion? In what ways is diffusion present in the natural world? Explain.
2. Does diffusion play a role in this model? If so, what role? Explain.
3. How does diffusion help produce the pattern formed by the frozen particles? Why?
4. What would happen if diffusion were not an element in this model? How would the pattern produced be different? Explain.

Extensions and Related Models:

1. Discuss applications for the concept of diffusion-limited aggregation in the natural and unnatural world.
2. Compare the growth of the pattern in this model with plant growth.

Extension 1

Have students consider where diffusion-limited aggregation takes place in the world. Ask the following questions:

1. Are there any examples in life where diffusion-limited aggregation takes place? Where? Explain. Dust balls, dendrite, soot, coral reef.
2. How do those examples compare to this model? What are the similarities and differences?
3. How could this model be improved to better simulate those situations?
4. Is this model useful in understanding the movement of those particles? Why or why not?
5. Is this model an accurate representation of the types of patterns those particles form? Why or why not? How could it be better? Explain.

Extension 2

Have students consider the relationship between the growth of the pattern in this model and plant growth. Ask the following questions:

1. Are there any similarities between this model and the growth of a tree? If so, what are they? If not, what are the differences?
2. What is the probability that a tree's growth will occur on the end of a branch? What is the probability that it will occur on the trunk, or on the side of a branch? Is the probability higher for one place than the other? Why?
3. What is the probability that a particle in this model will stick to a particle at the end of a "branch"? What is the probability that it will land on side of a "branch" or closer to the initial source particle? Explain.
4. What is the likelihood that a particle will land on the initial source particle at the beginning of the simulation as opposed to near the end of the simulation, when more particles have attached to the initial source particle? When is the probability higher? Why?
5. 5) Does the growth pattern of the particles look at all similar to the growth pattern of a tree's branches? Why does this take place? How does it relate to the probabilities we just talked about?

Related Models

Precipitate Model

This model simulates a precipitate reaction of molecules in a solution. Students should discuss how the relationship of the particles in this model is functionally similar to the relationship of the particles in the Diffusion Limited Aggregation model.

Disease Model

This model allows the user to model a population of people susceptible to a disease, infected with the disease, and recovered from the disease. Students should compare and contrast the application of diffusion in both models.

Multi-Function Data Flyer

This model allows the user to step through the process of building the Sierpinski's Triangle. Students should discuss the fractal elements in the Diffusion Limited Aggregation model.