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RandomMotion_Diffusion-Weeks7,8_final

Page history last edited by Kim Moore 10 years, 6 months ago

Phys 131, Fall 2012

 

Lab 4: Random Motion & Diffusion(Weeks 7, 8)

 

Introduction

 

During the past two weeks you have explored random motion of 2-micron beads in water, observing that the histogram of bead displacements gets wider when you increase the measurement time interval, ∆t. Nevertheless, the average displacement of all beads in the x and y direction, ∆x and ∆y, equals zero. (For every bead that randomly jiggled toward positive x, another bead jiggled toward negative x. The same is true for the y-motion.)

 

However, the average displacement magnitude, i.e. the average distance traveled by the beads (let’s call it <r>), does deviate from zero: Every bead changes its position by some amount! As you remember, the distance traveled can be calculated as r=sqrt(∆x2+∆y2), including only the squares of ∆x and ∆y—which are always positive and so add up to something bigger than zero. This equation indicates that instead of the distance traveled, r, we can measure the square of the distance traveled r2=∆x2+∆y2, also called the Mean Squared Displacement or MSD. This makes the math a bit easier, and r2 increases linearly with the measurement time interval, as you will see today.

 

The “diffusion constant,” D,is defined to be the proportionality constant between the average displacement squared, r2, of the diffusing object and the measurement time interval, ∆t,over which the diffusion occurs. There is also a factor of 4 in there (for geometry reasons—a 4 in 2-dimensions, a 6 in 3-dimensions, a 2 in 1-dimension):

r2=4D∆t

 

Investigation

 

Your overall task for the next two weeks is to twofold:

 

  1. Video 1. How does the diffusion constant, D, depend on the measurement time interval? Measure the square of the average bead displacement, r2, for different measurement time intervals, ∆t. See the table on the next page to find the conditions under which your group should observe r2as a function of time interval. Choose at least five suitable time intervals in the video (at least six frames) so beads move visibly, but not so far apart that it gets hard to distinguish nearby beads (examine at least 10 beads). For this you may find Automatic Tracking helpful (see the technical skills document).

 

  1. Videos 2 and 3. Now explore the dependence of D on the parameters of the system. Since you established with video 1 the dependence of r2on Δt, we will not need to keep proving this. Instead, collect data for only two suitable time intervals (three frames) for as many beads as possible. Analyzing this data will allow you to make claims about how varying the investigated parameter affects the diffusion constant. You will then compare the data you have collected with the data from the other groups, to see how diffusion depends on all the parameters that are tested in your lab section.

 

 

 

Group # (Parameter)

 

(with a fourth group, even more combinations can be investigated)

Video 1

 

(condition for testing r2vs. t dependence)

Video 2

Video 3

1 (bead size)

1-micron silica beads in water

2-micron silica beads in water

5-micron silica beads in water

2 (fluid viscosity)

2-micron silica beads in water

2-micron silica beads in low viscosity glycerol/water mix

2-micron silica beads in high viscosity glycerol/water mix

3 (bead mass & viscosity)

2-micron polystyrene beads in water

2-micron polystyrene beads in low viscosity glycerol/water mix

2-micron polystyrene beads in high viscosity glycerol/water mix

 

 

Interpretation

 

We did not explore the dependence of the diffusion constant, D, on temperature. Discuss with your group how you might expect D to vary as a function of temperature.

 

Using the data you have collected over the course of these two weeks, make an argument for a plausible expression for the diffusion constant, D, as a function of some (or all) of the following parameters: temperature, fluid viscosity, bead size, and bead mass. (An argument should contain a Claim, Data, and a Warrant—i.e., an explanation of how the claim is related to the data.)

 

 

 

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