Physics 111 Laboratory 7
Brownian Motion
Prior to lab: Read pages 352-353 in Giancoli and read through the lab write-up. Prior to lab answer the following questions: 1) Sucrose has a diffusion coefficient of 9.610-11 m/s2 in blood. What is the average distance that a typical sucrose molecule travels in 1 hr? 2) Why doesn’t e. coli move with Brownian motion?
Lab goals:
-To become acquainted with the appearance of Brownian motion via direct observation and measurement of the positions of micron-sized spherical particles in water
-To become acquainted with the statistical distribution of particle displacements, and use this to calculate Boltzmann’s constant kB.
-To calculate kB by measurements of the mean square displacement as a function of time.
I. Introduction
In 1827 Robert Brown, a botanist, was studying pollen grains suspended in water and noticed that the pollen grains were in constant motion. While Brown could not explain the observed motion of the pollen grains (or the inorganic particles he later examined), it was eventually explained by Albert Einstein and termed “Brownian Motion.”
Einstein starts his paper with “It will be shown in this paper that, according to the molecular-kinetic theory of heat, bodies of microscopically visible size suspended in liquids must, as a result of thermal molecular motions, perform motions of such magnitude that these motions can easily be detected by a microscope.” The idea is that there are many small particles constantly bombarding a large particle. (If water molecules were the size of baseballs, the pollen molecule would have a diameter of several miles rather than 5 µm.) Fluctuations in the force due to these collisions (impulse!) cause the random zig-zagging motion.
The mathematics of this random motion (and diffusion in 1-D in general) can be described by a random walk. Suppose that you start at the origin and take equal length steps in either the positive or negative x-direction with equal probability; after many steps your average position is at the origin (since you are just as likely to take a positive step as a negative step), but as time goes on it becomes more and more possible that you will be found further away from the origin. This motion can be calculated by averaging the squares of the displacements (Δx2, with the indicate taking an average value) and setting it equal to two times the diffusion coefficient (D, which is a property of the size and shape of the particle and the fluid it is in) multiplied by the time (t, elapsed time): Δx2= 2Dt. This is for 1-dimension.In 2D the result is Δr2 = 4Dt and in 3D it isΔr2 = 6Dt.
This experiment has four parts. In part I, you take photographs of a single particle at a set time interval (~2s), using a Celestron microscope with a digital camera. In part II, you use a software package, Image J (perhaps familiar to some of you as it is used in other fields, too), to record the position of the particle. In part III, you analyze the position data, with the help of Excel for manipulating and graphing the data. In part IV, you compare the motion that you just analyzed to the motion of a living organism.
II. Part 1
Sample Preparation: Place a sheet of lens paper (or paper towel) on the table, and place a dimpled microscope slide, dimple side up, on top. Using a small syringe, transfer a drop of the latex sphere solution onto the dimple in the dimpled slide. Cover the sample with a cover slide. Avoid getting air bubbles in the sample—these may create drift currents that make measurements more difficult. Record the particle size and the ambient temperature.
Camera/microscope preparation:The microscope you will be using has an LCD display that can be used to record and transfer images to your computer. The first step is to calibrate your system so that you will be able to measure the distances that the spheres move. Your instructor will provide a calibration slide. Insert this slide into your microscope, sliding it underneath the two small arms. You should see a series of parallel lines. You may find it easiest to start with 4x magnification and gradually increase it to 40x, adjusting the focus after each increase. The center-to-center distance between the white lines is 60 µm.
If your microscope already has pictures saved on it, you'll want to clear them. Use the “escape" button to scroll through the options. If you see a screen full of photos, clear them by pressing “menu" then following the steps to “delete all." Then set your microscope to take a photograph. A camera icon will appear in the upper right hand corner of the display when it is set for photographs. If you don't see this icon, scroll through the different options by pressing the “escape” button. Once it is in photograph mode, press “snap” to take a photograph.
Download the acquired photograph onto your computer, using the cable provided. The connection port on your microscope is located just under the display area. Your microscope will go dark once you connect the cable. (You will find that when you disconnect the cable, you'll need to turn the microscope back on.) The computer will automatically pop up a window asking what you'd like to do; choose “Copy pictures to a folder on my computer.” Follow the steps to download the calibration photo onto a folder on your desktop.
Open “ImageJ” from the “all programs” listing under the “start” menu. Within ImageJ, open the calibration photo. Then use the line tool to draw a line from center-to-center of one or two sets of black and white lines. The center-to-center distance between pairs of white or black lines is 60 µm. To get the most accurate calibration, draw a line that covers as many stripes as you can. Then, under the “analyze” menu, choose “set scale.” In the window that pops up, choose the appropriate value for the “known distance.” You may have to use “um” for the length, unless you find a way to input “µm.” ImageJ should tell you that the scale is set to roughly 4 pixels per µm (do not change the pixel aspect ratio). Check the box “global” so that this calibration will be used in all your measurements.
Data acquisition: Insert the slide you prepared with the 1.01µm spheres into the microscope (after removing the calibration slide). Adjust the focus until you see the spheres moving around (they should appear to be jiggling). If you have trouble finding the right focus, try moving the slide so that you can focus on the edge of the cover slip, then center the slide and slowly lower the focus through the cover slip into the solution. Once you have found the spheres, check that what you see appears to be random movement and not collective drift of all of the particles in the water.
Find a region with several spheres in your field of view. Take 50 images (this should take ~100s). Once you have your photos, you have your raw data.
III. Part II
Data processing: Upload your photos to a single folder on the desktop. In ImageJ, choose “Import” “Image sequence.” Select the first photo in the folder, then press “open.” ImageJ will record the coordinates of points you select using the “point tool.” First double-click on the point tool to open a dialog box, and select “auto-next slice” so that it will automatically scroll between photos.
Select a promising sphere in your image and click on it with the pointer. Each time you click on a sphere, its position will be recorded in the “Results” window. Try to follow the same sphere between images. If you are not sure you are following the same sphere or it goes off the screen, note the image number and choose a new sphere. Once you have scrolled through your whole stack of images, you can save the “Results” file to your folder and open it with Excel.
IV. Part III:
Data analysis: In Excel, you will be able to calculate the squared displacement between steps and find the average squared displacement. Note that if you switched to a different sphere during the sequence, you can simply create a new column of squared displacements. With this information you can plot a map of the sphere’s random walk.
For a particle executing a random walk, its probability to move in the positive s-direction in a given step is equal to that for moving in the negative x-direction, and the step lengths are distributed in a Gaussian form (same for the y-direction). In Excel, calculate the delta-X for each step. Graph a histogram of this data.
According to our analysis of Brownian motion, Δx2, the mean squared displacement of the particle’s position in 3-D, is related to Boltzmann’s constant kB, the temperature T, the viscosity of the solution η, and the particle’s radius by:
(t is the time between photographs)
For 1-D, your equation will be: .
Calculate the mean squared displacement for your particle by averaging over all of its x-displacements squared. The viscosity of the saline solution is 1.02 10-3 Pa-s, and the sphere’s diameter is 1.01µm. Do your results verify Boltzmann’s constant is 1.38x10-23 m2kgs-2K-1?
V. Part IV – for fun, feel free to look at e. coli
Make a slide of e. coli. Insert the slide into the microscope. Repeat the procedure that you carried out for the 1.01µm sphere, including making a map of the e. coli’s random walk and a histogram of the displacements.
VI. Lab Report
Before leaving lab you will hand in an informal lab report of your results.
Your lab report this week should include:
- The lab title, date, the names of the individuals within your lab group
- An Excel table with all of your data
- Excel graph of the random walk
- Excel histogram of displacements
- Value and calculations for kB
- Answers to the following questions:
- Why does a particle move with Brownian motion?
- Why don’t we move with Brownian motion?