Brownian Motion

In the phenomenon of diffusion a concentration of some substance – a little concentration of perfume, for example – spreads out in such a way that the radius R of the concentration grows like , where t is the elapsed time, or more simply

The constant of proportionality is 2D, where D is the so-called diffusion constant.

In a random walk back and forth along some axis, consisting of many random steps, the expected mean square displacement X in time t has the same behavior, i.e.

(Recall that E means “expected value”, a kind of idealized average.)  In his 1905 dissertation, Albert Einstein suggested that these two phenomena are the same.  That is, at the molecular level, the way diffusion works is that tiny molecules move randomly.   Since there seems no reason why this behavior, if it exists, should not apply to all particles, even those much bigger than single molecules, Einstein also suggested that the observable Brownian motion should obey this law.  This is experimentally testable:  we sample the squared displacement of many Brownian particles as a function of elapsed time t and we see if the sample mean of this quantity obeys

This we can do by plotting and looking for a straight line relationship.  If the data lie on a line, the slope of the line determines the diffusion constant D.

The sample mean < >  will  be a good approximation to the expected value E if we use enough  particles, but we must expect that our sample means will scatter about the (true) expected value with a standard deviation proportional to , where N is the number of sampled values that go into the mean.

Einstein further suggested that the diffusion constant D and the hydrodynamic resistance f of the Brownian particle must be related as

The hydrodynamic resistance of a sphere of radius a in a fluid of viscosity µ was known from the work of Stokes to be , so a measurement of D for the Brownian particle would determine Boltzmann’s constant , and hence Avogadro’s number.   Jean Perrin won the Nobel prize in 1926 for work done in 1907 carrying out this experimental program, and finding that Avogadro’s number determined in this way agrees well with other determinations, finally proving beyond doubt that molecules exist and obey statistical laws.

Procedure:

Open Tracker . Once you are at the Tracker home page, click on Web Start Tracker, then open up the Brownian motion movie:

FileŕOpenŕVideosŕBrownianMotion1.mov

Play the movie and watch them jiggle.  The jiggling is Brownian motion.  You may notice them going from a fairly sharp focus to a fuzzier focus.  The change in focus is a result of movement in the z-axis (up and down).  We only care about movement in the x and y axis, so don’t be too concerned with focus change.

Each group will be assigned a particular time interval (e.g. 5 frame steps) and will get data for as many particles as feasible.  Those assigned large time intervals should be able to acquire data on more particles than the groups with smaller intervals, but will get fewer data points per particle due to the larger time jumps:  you run out of frames sooner.  Make sense?  A group jumping through 20 frames will move through the clip and get to the end faster than a group jumping through 5 frames.

Let’s say you’ve been assigned 10 frame intervals.  Start off by adjusting the frame rate.  Click on the Clip Inspector icon (it looks like a piece of film with a little black triangle on the lower right edge), and adjust the Step Size to 10, or whatever you were assigned.  That is the only value you need to adjust for this lab, so click OK.

Now, it will probably help to place the axes in some way that divides up the screen so you can keep track of your points.  I like putting it in the middle and giving myself quadrants.  We only care about the difference in position so it really doesn’t matter where we put the origin.  Do whatever is convenient.

Once your axes are placed, click on the Track Control icon (looks like fireworks).  The Track Control panel should now be open.  Click on New and choose Point Mass.  A box labeled mass A (or whatever letter you are on) will appear, click on it and choose Name.  Name the point mass whatever you want.   Something descriptive of the particle you will be tracking might be useful.

Choose your first particle.  Zoom in on the area containing the particle by right clicking near the particle, choosing Zoom, and picking a magnification.  I find that 2.0x usually works fairly well.

Hold down Shift, which should turn the cursor into a little box with a cross in it.  Then left click as close to the center of the particle as you can.  You will automatically be advanced to the next frame.  You will want to remove the trail left by your previous clicks by clicking on the Show Trails icon (two red rounded diamonds).  Once the trails are off, you should be able to see the particle clearly from frame to frame. Keep clicking until you get to the last frame.  If at any time you want to see the trail, click the Show Trails icon again.  You will see something that is relevant in biology:  a Brownian path “explores” its neighborhood pretty thoroughly!

Now, pull out the Data Table and Graph.  Recall, that it’s over on the right hand side.  Put the cursor over the spotted white space next to the scroll bar, when it turns into a double headed arrow, left click, hold and drag out the data table.

RIGHT CLICK and choose Copy Data, then paste the data into an Excel spreadsheet.  Remember, you don’t need to highlight the headers (t, x, y), they come along for the ride with the data.  Once you are in Excel, label the units in the header.  Be aware that the x and y values are in pixels.

Repeat this process for as many particles as you can.  Be sure to start a new TRACK (with a new name, like mass B for the 2nd track)  for each new particle.  When you paste data into Excel for a new particle, be sure it is in its own region in the spreadsheet, separated from the tables for other particles.

Analysis

Make sure your data is well laid out and labeled, something like columns A, B, and C in Fig. 1 is recommended.

Figure 1: Layout of Brownian motion data.

First you need to start two new columns.  These columns will contain the amount the particle moved from frame to frame.  The data you pasted in tell you where the particle was at time 0, 0.415 s, 0.83 s and so forth (the times will be larger or smaller depending on what frame number you use).  This does not tell you how far the particle traveled in the 0.415 s between your recordings of its position.  In order to obtain that information you need to make a column where you subtract the first data point from the second data point.  So if you found the particle was at x=0 pixels when t=0 s, and at x=0.368 pixels when t=0.415 s, your first entry in your new column would be 0.368 pixels.  You don’t need to do this by hand though; Excel will do the work for you.  See the Calculations section of the Excel handout for directions on how to make Excel do the math.  Repeat the process for the Y direction also.

Now, you need to square the values from your new columns, which are D and E if Fig. 1 and make two new columns.  As you did when getting the difference, make Excel do the math.  Repeat the process for the Y direction also.

Once you have two columns containing the square of the difference in position between frames, take the average of these numbers.  In the empty cell at the bottom of the column of squares, which are columns F and E in Fig. 1, you would type =Average(F4:F27), and this will give you the mean of the squares.  Similarly for the y direction.  (Of course the cells you use will undoubtedly be different from these, so use the appropriate range for your table.)

Repeat this for all your data points until you have means of the squared displacements in each direction for all the particles.

Once you have the mean squares, make a column of the time interval you were using in seconds, and place the means of squares from the x and y direction next to the time interval, as in Fig. 2.

In the cell below the mean squares of x and y, calculate the mean of each column.  Then take the standard deviation of each column.  The formula for standard deviation is STDEV, so you will need something like =STDEV(BE15:BE23) in the next available cell (see Fig. 2).

Fig. 2:  Layout of Brownian Motion Data

We determined the mean square displacement over and over, for many particles:  our best estimate is the sample mean of all these determinations, found at the bottom of Fig. 2.  How good is it?  We know roughly how the individual determinations scatter around the mean:  that is the observed sample standard deviation, also computed at the bottom of Fig. 2.  Since we used 9 samples to get the sample mean, the standard deviation of this sample mean should be that sample standard deviation divided by the square root of 9, or 3, which is about 1.  That is, we seem to have determined the mean x squared for this time interval to be about 8.3, plus or minus 1.

Since we have determined  for only one value of t, we can’t know if these quantities are proportional.  For that, we need to see what the other groups found!  To finish, we will have all groups report their values.  Together we will make a plot, and if all goes well, determine the slope 2D.

From this measurement it is possible to extract Avogadro’s number, using Einstein’s relation (1.4) between D and the hydrodynamic resistance f.  It implies

Here R is the gas constant, T is the absolute temperature (about 297 K when the video was made), is viscosity of water (about  0.00091 Pa-s at this temperature) and a, the radius of the little polystyrene spheres we are watching, is 0.93 microns.  Finally, the digitized microscope images have a scale 4.165 pixels per micron.  Get a class value for Avogadro’s number.