1. Teachable Unit: Brownian Motion
Created by:
Claudia De Grandi and Katherine Zodrow
May 2013

2. Unit Summary
This unit contains materials for 2 or 3 class periods. Parts of this unit can stand alone.
Teaching Materials Include
Powerpoint slides detailing unit
Powerpoint slides to be used in the classroom
2 Matlab modules (for beginners) to help explain Brownian motion
In class quiz/questions
Homework

3. Part 1 (75 min) Introduce learning goals
Perform a 1D random walk as a class
History of Brownian motion (lecture)
Reflection and discussion
The length of time it takes to perform the 1D random walk will depend on class size, setup, etc. Part 1 may be less than 75 minutes if #2 listed here can be performed quickly.

4. Part 2 (75 min)
Present and discuss the solutions to the homework (from Part 1)
Computer lab group activity with Matlab (Module I)
Students follow instructions on handout to simulate and analyze 1D random walks
Students hand in a final lab report
Students are given final answer key sheet
TAs and Instructor available for in-class help

5. Part 3 (75 min)
Introduction to data analysis: How do researchers use Brownian motion?
Computer lab group activity with Matlab on 2D random walks (Module II)
Final question: Estimate the radius of an atom
Discuss solution of the question
Reflection, final comments on initial learning goals.
If you have extra time and availability, you can bring a microscope to class to show to the students a Brownian motion, or add extra material and information about nowadays laboratory work with Brownian particles, for instance see slide 33.

6. Assessment
Initial reflection on learning goals questions (see slide 9)
Initial multiple choice quiz about binomial distribution (see slide 11)
Homework (end of Part 1) on diffusion in different viscosities
2 Matlab Modules, to be turned in as a short lab report
In-class final problem questions about the size of an atom
Final homework/report: revised and detailed answers to learning goals questions

7. Materials needed coin to flip for each student
a relatively spacious room to implement the random walk activity or a large white board and Post-it stickers
a computer and screen to project slides
Device for students to use Learning Catalytics (formative assessment questions and class random walk activity)
Computer with Matlab for each student group
Handout for students with a copy of the slides

8. Classroom Slides

9. Brownian motion, Atoms and Avogadro’s Number
How do we know atoms exist?
What is the size of an atom?
How would you observe an individual atom?
Introduce the questions we aim to answer by the end of the teaching unit. Ask the students to think by themselves for few minutes about those questions
and write on a piece of paper their tentative answers. At the end of the unit, they will be able to correct and expand upon their initial guess.
Optional: engage the students in a group discussion where they share their answers
Estimated time: 10 minutes

10. Brownian motion, Atoms and Avogadro’s Number
How do we know atoms exist?
What is the size of an atom?
How would you observe an individual atom?
In particular: encourage students to make a sort of diary to keep track of the development of their answers to these questions during the 3 lectures time of the teaching units.
Students will first write down some tentative answers, they can mark down comments on what they have learned during some activity of the tidbit, they can correct and
change their answers. At the end of the full activity as a homework problem they will write a detailed answers to those questions, discussing and comparing what was their initial
preconception and what is the most important thing they have learned in class.
Suggestion: make a ‘diary’ to keep track of your learning process
At the end of the 3 lectures your homework will be to summarize what you have learned and give your best answers to those questions.

11. Today in class, we will Review the binomial distribution: quiz
Perform a 1D random walk as a class and extract our diffusion coefficient
Review the history of Brownian motion:
R. Brown(1827): botanist observing motion of pollen grains
Einstein’s theory and connection to Avogadro’s number (1905)
Perrin’s experiment (1908)
Give a brief introduction to the objectives and activities that will be covered during tidbit. Our main goal is to connect Brownian motion to Avogadro’s number and the existence of atoms, this will help us to answer the initial questions we posed.
Estimated time: 2 minutes

12. Binomial Distribution
reminder
probability of k successes in n trials
p = probability of one success
Briefly review the main results about the binomial distribution.
Initial assessment Quiz [6 minutes]
(with Learning Catalytics): two multiple choices questions that students answer individually quickly.
Note: they can refer to the slide on the screen.
Questions 1:
A 1D (One-dimensional) random walk can be described by a binomial distribution with p=1/2, each success is a step forward(+1) and each failure a step backward (-1) along a line.
After n steps, what is the average position of a 1D random walker that starts at x=0?
A. <x>=1/2
B. <x>=n/2
C. <x>=0
Questions 2:
After n steps what is the average mean displacement squared (variance) σ^2=⟨(x−⟨x⟩)^2⟩?
A. σ^2∝ \sqrt(n)
B. σ^2∝ n
C. σ^2∝ n^2
Estimated time: 8 minutes
average number of successes :
variance :

13. extract the diffusion coefficient D
Note: each student was asked to bring a coin to flip to class.
Ask the students to take their coin and stand up and form a line, standing shoulder to shoulder and facing the part of the classroom where they can see a screen (where the slide is projected) and the board (that will be used later). There should be enough space for them to make few steps forward and backward (move the tables around if necessary), it would be ideal if the floor has some sort of tiles or pattern that can be considered as a unit for a single step (otherwise you can decide that one step if for instance 2 of their own feet long, or you could have marked in advance some lines on the floor with some tape for instance).
Alternatively, if the classroom space does not allow the students to stand and form a line, you can use Post-it stickers, each student sticks one on a similar line drawn on the white-board and move it back and forth to represent his or her random steps.
According to the size of the class and the space available in the classroom, this can be implemented in different ways: if the class is reasonable small (<20), the students
can all participate in the random walk, either standing and “1d-random-walking” or moving a Post-It on the board that represents their position.
If the class is big (~100) the students will be split in half: the “walkers”, i.e. people implementing the random walk by standing and “1d-random-walking”, and the “data-collectors”, the ones that record the walker’s position and send it through Learning Catalytics. So the students will be asked to form pairs of 1 walker and 1 collector.
The instructor explains how the students will realize an ensemble of 1D random walkers.
Each student is constrained to move on a straight line, each step is decided by the flip of the coin.
Assuming all the coins are fair, the random walk will be an unbiased one with p=1/2.
Engage students curiosity by mentioning some examples of 1d random walk in nature or real life (a protein that is searching its target up and down on a line of DNA; a single channel in an electronic device as an amplifier, subjected to external white noise; the price of a stock-market, that everyday goes up and down unpredictably).
Perform three diffusion processes, for time t=2,4,6(each time add 2 steps to the previous position). Count out loud each time step, the students will move of one step each time accordingly to the outcome of their own coin.
Calculate the average position and variance for each time t: <x>, sigma^2. On the board make a histogram for the final positions of the ensemble of walkers, and quickly calculate average and variance (for big class size this can be easily implemented by asking the students to submit their position through Learning Catalytics, which instantaneously will show an histogram with mean and standard deviation)
At the first random walk, t=2, when the mean has been calculated, let them argue and discuss about what they would have expected, show on the screen the results from Question 1 of the Assessment Quiz, clarify which one is the right answer, and let them speculate why the average just found may deviate from the expected value (due to small number of people, possible trajectories probed are too few respect to all possible trajectories, 2^t, therefore it is only a partial mean, on a restricted sub-region of all the possible paths).
Make a graph of variance versus time, plot the results found for the three times on the board, show on the screen the results from Question 2 of the Assessment Quiz,
let them discuss what they observe, what they would have expected. Clarify which one is the right answer and discuss what is the meaning of a linear increase in time of the variance, random process--> diffusion, counter-example of converging processes: lollipops (if you would hold a lollipop, a cookie, a beer in front of each random walker, that will not be random anymore and will all converge to the lollipop eventually, sigma^2-->0 in time, or gaussian process: since we have tables,constrained space--> gaussian, variance cannot grow, finite width around the mean value).
If you have extra time:
Extract the Diffusion coefficient from the graph, according to \sigma^2=2Dt, extract the D for the ensemble, the expected value is D=1/2 in this case (D=L^2/(2 \tau), with L=step size, \tau=time unit), since we assumed L=1, and \tau=1.
Make sure the connection between the activity and the initial assessment quiz on the random walk distribution is clear, and the students have understood clearly which ones are the corrected answers to the quiz and why. Go over the answers again, if necessary.
Estimate time: minutes
Learning goals:
extract the diffusion coefficient D
understand how the variance
depends on time

14. understand how to extract the diffusion Learning goal:
coefficient from 2D images
relate the diffusion coefficient to the
Avogadro’s number
Learning goal:

15. We have discussed Brownian motion in 1 dimension, but we often observe Brownian motion in 2 or 3 dimensions. The question of the cause of Brownian motion eluded scientists for quite some time until, in 1905, Einstein proposed a link between the thermal energy of molecules and the diffusion of particles through a medium. Although Einstein explained the theory, it was not until 1908 that Perrin quantitatively verified Einstein’s theory. Perrin received the Nobel prize in 1926 for this work. Unlike scientists in the early 1900s, we can use digital cameras and image processing software to view and analyze particles undergoing Brownian motion. Here is a movie of 3.2 um latex particles suspended in water.
In this movie, we see latex particles moving as they are bombarded by water molecules.
We can use image analysis software to analyze the mean squared displacement of these particles, and thus calculate their diffusion coefficient. First, we must locate the particles, and a computer program can be written that finds points of peak intensity in each frame of the movie.

18. You can see here a random walk of a real particle
You can see here a random walk of a real particle. Later in class, we will generate a random walk using Matlab. Just like we can calculate the diffusion coefficient of this particle that actually exists, we will calculate the diffusion coefficient of our simulated particles. We will then use that diffusion coefficient to calculate Avogadro’s number.

19. Ideal gas law ideal gas constant Pressure Temperature Volume
Through the two slides here below review the following concepts:
ideal gas law, mole and Avogadro’s number, Einstein theory of Brownian motion.
Put emphasis on the historical context: at the end of the 19th century scientists did not know the value of the Avogadro’s number, but they could still confirm the ideal gas law
and quantify the amount of moles by looking at the ratio of interacting substances in a chemical reaction.
Make connection with the previous activity of the 1D random walkers: by analyzing the dependence of the variance in time (slope on the graph \variance versus time) it is possible to extract the Avogadro’s number.
Be clear in defining the friction coefficient and the viscosity as they will be used in the following Think, Pair, Share assessment.
Estimate time: 8 minutes.

20. Ideal gas law mole=as many molecules as in 12 grams of 12C
Volume
Pressure
ideal gas constant
Temperature
Ideal gas
law
# of moles
In our case: water
mole=as many molecules as in 12 grams of 12C
mole= Avogadro’s number(NA) of molecules
total number of molecules
Boltzmann constant

21. Ideal gas law Einstein’s theory of Brownian motion! historically
Volume
Pressure
ideal gas constant
Temperature
Ideal gas
law
# of moles
Einstein’s theory
of Brownian motion!
In our case: water
historically
mole=as many molecules as in 12 grams of 12C
mole= Avogadro’s number(NA) of molecules
total number of molecules
Boltzmann constant

22. Einstein’s
theory

23. Einstein’s
theory

24. Einstein’s
theory

25. Diffusion coefficient
Einstein’s
theory
Friction coefficient
Diffusion coefficient

26. Diffusion coefficient
Einstein’s
theory
Radius of
green
particles
viscosity
Friction coefficient
Diffusion coefficient
measurable!
in Brownian
motion

27. Diffusion coefficient
Einstein’s
theory
Radius of
green
particles
viscosity
Friction coefficient
Diffusion coefficient
measurable!
in Brownian
motion
time
known quantities

28. Today’s Recap Avogadro’s number! Diffusion coefficient of
1D random process
Diffusion coeff. of 2D brownian particles
Einstein’s theory
Avogadro’s number!

29. Brownian motion, Atoms and Avogadro’s Number
How do we know atoms exist?
What is the size of an atom?
How would you observe an individual atom?
As a final reflection, go back to the initial learning goals questions, and ask the students to reflect on what they have learned in class today and make some notes comments on how their initial answers may or may not have changed since their initial guess.
Ask to share some of their thoughts in class (lead a short class-discussion to end the lecture).

30. Homework
At 20 °C, the dynamics viscosity η of water is Pa*s. Glycerol is 1 Pa*s. We place particles in these two solutions, holding everything else constant. Give a quantitative relationship for the diffusion of these particles in these two solutions.
Sketch a plot that compares <x2>vs. time for particles in each of these solutions.
Students are given at the end of the lecture a handout with the copy of all the slides. They are encouraged to review the materials for the next class and answer the two questions above as a homework problem (or as a think/pair/share class activity at the end of the class).

31. Today in class, we will Review solutions to the homework
Use Matlab software to simulate and analyze in details 1D random walks
Work in groups of 2/3 people
Follow instruction on handout
Hand in a report by the end of the lecture
Part 2
Introduce the activities for the second lecture.
This is the only slide for Part 2.
After 5 initial minutes devoted to review the solution to the homework (about diffusion in different viscosities), the full class is centered on the computer lab
activity with Matlab.

32. Today in class, we will
Use Matlab software to simulate and analyze 2D Brownian motion (like reproducing Perrin’s exp. images)
Work in groups as Module I, hand in final report
You will extract the Avogadro’s number from your data
Group problem: estimate the size of a molecule from Avogadro’s number
Final discussion on learning goals and final homework
Part 3
Introduce the activities for the third and final lecture.

33. Estimate of molecular radius
Assume Avogadro’s number NA= 6 X1023
NA is the total number of molecules in a mole
Once the students have finished their Matlab activity and handed in their lab report, ask them to work in groups to find an estimate
of the dimension/radius of a molecule.
Reminder Ideal gas law:
Work in groups to find an estimate of the size of
a molecule in a mole

34. Estimate of molecular radius
volume of a mole
at room Temp. (T=300)
and 1 atm.(100kPa)
Show the solution to question about the estimate of the radius of a molecule.
Discuss meaning of “estimate” and limitations. (E.g. here we suppose the whole volume is packed of atoms, without space between them, so we assume a high density, atoms may be instead
far away from each other with a much smaller radius, therefore this is an overestimate, indeed realistic atomic or molecular radius are usually less than one Amstrong).
Discuss the fact that the suspended particles since of micron size can be seen in a microscope, while the atoms kicking them are not visible because we now found out that are a thousand time smaller than them!
There are many ways to estimate the radius, this is only one of them!
estimate of particles
radius

35. Brownian motion, Atoms and Avogadro’s Number
How do we know atoms exist?
What is the size of an atom?
How would you observe an individual atom?
Final reflection on initial learning goals questions.
Discuss with the students what they have learned during the teaching unit and show the final homework they will hand in next time.
Final homework/report/reflection:
write down your best answers, compare with your initial guess, discuss what are the most important things you have or have not learned during this teaching unit

36. If you have extra time: illustrate an example of how Brownian motion is currently used in engineering labs to determine the size of suspended particles by using Dynamic Light Scattering techniques. Particles in a solution scatter light from the laser. The intensity of this scattered fluctuates over time as the particles move under Brownian motion. Larger particles move less, and therefore cause less fluctuation in scattered light than smaller particles. In this case, we know the viscosity of the solution, the temperature, and Avogadro’s number, so we can calculate the radius of the particles in the suspension.

37. Additional reading
Haw, M D. (2002) Colloidal suspensions, Brownian motion, molecular reality: a short history. J. Phys. Condens. Matter 14:7769.
Philip Nelson’s book: Biological physics: Energy, Information, Life (Chap. 4).
Random Walks in Biology, Howard Berg
Investigation on the theory of The Brownian Movement, Albert Einstein, Dover Publications (1956) (original Einstein’s paper on Brownian motion).