Brownian Motion Lab

JHU Advanced Lab

Lucretius c. 60 B.C.

"Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses, so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible."

Introduction

1905 - Albert Einstein publishes

molecular theory of Brownian Motion. [ Also Smoluchowski (1906) and Langevin (1908)]

1908 - Jean Baptiste Perrin

confirms molecular theory by

observing quantitative behavior

of Brownian motion.

1827 - Botanist Robert Brown

(namesake) used an early

microscope to observe random

motion of pollen in water.

Brownian Motionestablishes reality of

atoms and molecules.

Theory 𝛔dx2= kbT3πrη

t

Einstein’s molecular theory of Brownian motion predicts that the mean square displacement of a particle

executing Brownian motion is proportional to time.

𝛔dx2: Variance in x travelkb : Boltzmann ConstantT : Temperaturer : Particle Radiusη : Viscosity of Mediumt : Time Step

Presentation Note!Always define

all variablesin equations.

Note: equation above is for one dimension (x).

Apparatus

Prepare solution of microscopic spheres

to observe under microscope.

Obtain sequence of images of spheres.

Use software to measuresphere locations.

Apparatus

Use microscope+camerato obtain measurementsof x & y displacements.

Obtain sequence of images of spheres.

Use software to measuresphere locations.

Apparatus

Use microscope+camerato obtain measurementsof x & y displacements.

Obtain sequence of images of spheres.

Use software to measuresphere locations.

dx

Data

Take N photos

Dataset withN location

measurements{x}, {y}

Measure location of sphere in

each photo

(N~120)

Dataset with (N-1) step measurements {dxi=xi-xi-1}, {dy}

𝛔dx2~sdx2

What is missing in the label?

Why might dxnot be zero?

sdx2

Estimate “reasonable” error on temperature and viscosity (σT, ση).

Use Monte Carlo technique (to be discussed)to estimate error (σsdx2) on step size sample variance.

(Fun!)

May need to reject outliers (to be discussed).

Modeling

𝛔dx2=kbTt

3πrη

Model Express kB as a function ofmeasured variables

kb ~ T t3πrηsdx2

Propagate uncertainties

(σsdx2, σT, ση) to find σk.

Result : kb ± σk

One result for x data and one for y data

Check consistency between kb results for x data and y data.

If inconsistent, estimate systematic error.

If consistent, combine into final estimate.

(All these steps to be discussed)

What magnification should you use?

What temporal step size between photographs should you use? (No one right answer!)

Does the stated size and size distribution of your microspheres agree with the sizes of the microspheres that you record in your images? Why or why not?

With the diffraction grating, is it better to use two lines close together, or two lines far apart, to make your pixel-to-lab units calibration? Why? (Hint: think about the uncertainty analysis)

How could you use Monte Carlo simulations to estimate the number of additional steps needed to decrease your error on kb by a factor of two?

Questions for Discussion

Measurement Time!!