DIFFUSION PURPOSE The objective of this experiment is to study by numerical simulation and experiment the process of diffusion, and verify the expected relationship between time and diffusion width. THEORY The random motion of molecules was long known as Brownian motion after botanist Robert Brown in 1827 visualized the helter- skelter motion of pollen grains. He thought this meant that they were alive, but Albert Einstein, in one of his famous papers of 1905, showed that random interactions were quite capable of producing the behavior observed. The process is known as a random walk, and with computers especially is easy to simulate. Consider such a random walk of N steps. Repeat the walk many, many times. What you will get is a distribution of the likelihood, or probability, of being a certain distance from the starting point after taking N steps. This distribution is the normal curve, or gaussian distribution. What emerges from such simulations, or their exact mathematical description, is that the spreading of this distribution is not proportional to N, but to the square root of N. Equivalently, once can say that the mean deviation squared is proportional to N. If I am taking steps in equal times, then the number of steps is proportional to the time I am walking, and distance squared is proportional to time t. A word or two of explanation is in order. For concreteness, consider a one- dimensional random walk. Note that since you are just as likely to take positive steps as negative steps, your average position would remain at zero. Thus, it isn t the average distance you move that is of interest. But consider that for the first few steps you are really likely to be rather close to start. But as time goes by, there s more of a chance you will be found a little ways off center. The distribution widens. It is that width that rises as the square root of time t, or of square root of number of steps N. It is a very direct application of this idea to say that randomly diffusing gas will spread in this manner, since we can visualize the gas molecules moving about. If you were wondering when you d smell a released gas, this is just what we want. Though the average position of the gas is centered on its source, the chance you get a whiff at some distance away would go up as square root of time (assuming diffusion is the only way it spreads, rather than air currents, say.) Likewise we can visualize bacteria swimming randomly (or indeed, we can visualize the movement of the molecules that collided with Brown s pollen grains started the whole business.) One of the interesting variations is the notion of the biased random walk. Imagine bacteria swimming around, but now some chemical signal makes them more likely to swim toward the source of the signal. Except, since the chemicals in the solution are few, the bacterium senses the chemical only intermittently. So it swims, gets a whiff and heads that way, then goes back to randomness, then back toward the signal, etc. The randomness becomes biased toward the stimulus. DIFFUSION LAB 1
Diffusion of heat (as observed by temperature) works in the same fashion as mass diffusion. After all, heat is just random thermal motion, so if some molecules bang into others, the subsequent motion, even within a solid, is due to random vibrations. Thermal energy diffusion thus behaves just like diffusion of a gas. Temperature change due to conduction will spread like the square root of t. This characteristic that some types of motion depend on the square root of time is a hallmark of diffusion and so conversely, when phenomena are observed to follow such a t law it is considered to be a strong indicator of random motion, whether the movement of bacteria, the change in length of polymers, or the variation of population (to give some biological examples.) To observe thermal diffusion in this lab, we will watch heat travel along a rod. If you change the temperature of a long rod at one end by To, then the temperature change at an point x along the rod, at a time t, will be denoted by T(x,t). Diffusion of heat can be shown to yield this equation for T(x,t) : x ΔT(x,t) = ΔT o 1 erf 2 κ t κ is the thermal diffusivity and erf is a function, like a sin, cos, exp, or log. erf stands for error function. If you imagine a normal gaussian curve of error, as shown below, the erf is the area under that curve, taken out to the same distance + and from the peak, as illustrated by the shaded region below. So erf(0) is 0 because we didn t go any distance out from the center when computing the area. As we make the argument of erf bigger, the erf goes to 1, because we now include all the area. Excel has this function built in. To get erf(1.5), for example, one can just write =ERF(1.5) in a cell in Excel and the value is returned. The interesting thing about our equation is the inverse square root of time that turns up in the denominator. This t behavior, characteristic of diffusion, creates a long, slow tail. (It is notably longer than an exponential tail, for example.) The observation of this characteristic time behavior is the object of this part of the lab. In this lab therefore we will do two things. (1) We will use a simple simulation to emulate a one dimensional random walk, to test the square- root of time dependence. (2) We will use heat conduction along a rod to test the same square- root of time dependence. PROCEDURE A. The Simulator We eventually expect to develop a physical simulation of the first step, but for now, we are using a nice computer simulation. You will need to go to: http://www.mathsisfun.com/data/quincunx.html DIFFUSION LAB 2
You will see the balls falling down the decision tree, in which they can go to right or left equally likely. Notice the slider called Left probability. Slide it over and notice what happens. Put it back to 0.5 for now. What we are going to do is to allow some total number of balls to fall. Say 500. (That s the yellow number at the top.) We are looking for the width of the distribution as a function of the number of rows. Start with 5 rows, and collect 500 trials. While the simulator is running, start a spreadsheet in Excel. In the top ROW write the label (i) in column A. (Write it exactly like that, with the parentheses.) In column B of the top row put (i^2) as the label. In column C of the top row put DATA, 5 ROWS just like that. In column A below the label, put the numbers 1 to 20 going down the page, one per line. Use the autofill option in Excel to help. If you don t know how to do this, your instructor can help. In column B, square each of the numbers in column A. Now wait and collect data. When you hit 500, click the pause button. When you get enough data points, we are going to enter them going down column C. (DATA, 5 ROWS). DO NOT RESET THE SIMULATOR YET- - - we want you to look at the picture and the analysis. But, we are going to center the data vertically. In other words, you will not put the first value in row 2. Since you have 6 data points, with 20 possible entry spaces start entering at row 9. Put the number of balls - - as shown across the bottom of the simulator- - down column 3. In column D, you will put column C times column A, all the way down, top to bottom. You will do this by typing: =$A2*C2 and filling down In column E you will put column C times column B, again top to bottom by typing =$B2*C2 (The dollar sign will be useful momentarily. It is used to keep a column from incrementing when you copy.) Now below column D (this is in row 23) put the SUM of column D divided by the total number of balls you used. Type =SUM(D2:D21)/500. (That assumes you used 500. If you use a different number then divide by it) This value should be the mean position of our distribution! Check the picture to see if your data found a reasonable value. DIFFUSION LAB 3
And finally, we compute the thing we were looking for: SUM column E at the bottom (next to Col D s entry), and subtract the square of the Mean value you just got. (Don t forget to square it.) Type into cell E23 =SUM(E2:E21)/500- D23^2 What you just calculated is called the standard deviation (but right now it s squared.) So we take the square root. In E24, put =SQRT(E23). Now you have the standard deviation. It tells you how wide the distribution is. Look at your picture, and figure out how wide this value is on the picture. Now, you want to repeat this protocol for at least 3 more values of row numbers. Let the last value of row numbers be 19, the largest you can get. Column F is your next set of data. Copy column C, paste into Column G. Copy D into H. You should have your results immediately, with no painstaking retyping! When you are all done, you will have standard deviations for each of the quincunx row numbers. Make a graph of row number as the x axis, and standard deviation as y. The total time is proportional to the total row number, since the balls spend the same time on each row. So row number your x axis is just like time t. Let s test if it is square root. Take the square root of the row number, and make that an x axis. Then plot your standard deviation as a function of that. Is the graph linear? That s our hypothesis. B. Thermal Diffusion. We now want to watch heat diffuse. We are making the problem one- dimensional by watching heat diffuse down a long rod. For various reasons of convenience in setup, we are going to watch heat diffuse out of the rod, but, whether out or in, the mathematics and physics is the same. We are covering the rod with insulation because an exposed rod would not only transport heat along its length, it would lose heat out the sides. We are going to cool the rod by putting it in contact with an ice bath, so that we know the starting temperature is 0 C. Well, the water is at 0, but the ice is actually colder, so we want to trap the ice so only water contacts the rod. So we will use an aluminum plate with holes to keep the ice below the surface and away from contacting the rod. Put ice into the aluminum container about ¾ full, then secure the plate that traps the ice. Finally, pour water into the container till the water is almost at the top. The temperature on the rod is going to be measured on a device called a thermistor, which is short for a thermal resistor. We have not covered resistors yet, but you don t DIFFUSION LAB 4
need to know about them to use one. It is hooked into device known as a multimeter, and you are going to set it to measure resistance (which is measured in ohms, greek letter Ω). What you need to know is that when temperature changes, the number of ohms will change on the meter. The lower the T, the higher the resistance R. Plug the thermistor into the meter, and be sure the meter is set to read resistance. The two plugs are equivalent you can t hook it up backwards. You should see a steady reading of around 3000 Ω, or, on a kω scale, 3.00. Hold the probe in your fingers and it should respond to body temperature by dropping, eventually to around 2 kω, roughly. (If nothing happens when you hold it, check that the plugs and settings are right!) Record the resistance when the thermistor is in the room air ( room temp ) and the value you read from a standard thermometer. Calculate the expected value of R from this equation: R = 0.0019T 2-0.1557T + 5.6484 Compare it with what you measured. Compute the correction ratio as R(actual)/R(expected). Save this number for analysis below. Now, very carefully touch the thermistor to the top of the ice water surface. Do not immerse it! You just want the black tip in the water. While one of you does this, avoiding immersion, someone else should read the resistance. This is your zero degree value. Now insert the thermistor into the aluminum rod, in the hole drilled for this purpose. The hole should line up with the slit in the insulation. When you insert the thermistor, let the foam insulation just secure it in place. The value should be very close if not identical to the one you got with the thermistor in air. The rod should only extend about 0.5 cm out of the bottom of the insulation. Position the rod (by moving the clamp) so the exposed end is above the aluminum container and can be inserted in the next step but read it first. Because diffusion is slow, it is hard to redo the experiment if you mess something up. One of you will need to start a stopwatch in a moment, so he or she should be ready. When the timer is ready to go, lower the rod into the water so the exposed 0.5 cm is in the water. The actual amount immersed is not all that important, but it is critical is that when the end goes into the cold water, the timer gets turned on, since that is when cooling begins. As the rod cools, and the resistance increases, we want you to note each time that the resistance increases by 0.1 kω. In other words, you put the rod into the water and the thermistor reads 3.00 kω (say). Next you write down the time when it reads 3.10, and then the time when it got to 3.20, etc. DO NOT STOP THE WATCH. Just keep recording. The first few points come very quickly the later ones are agonizingly slow. Take data for about 30 minutes. Begin an Excel spreadsheet during the run. You will have plenty of time to do this, so don t rush at the beginning when the data values are coming fast. DIFFUSION LAB 5
Row 1 is going to be devoted to labels. Data is entered in Row 2. Col A is the time in stopwatch MINUTES. Col B is the time in stopwatch seconds. A and B are directly what you read off the stopwatch. Col C is the resistance R exactly as you read it. Col D is going to be the corrected R. Take your measured value in Col C and multiply it by the correction factor you got above. Col E is the temperature (in C), after conversion from the corrected value of R. Write the formula = - 0.6484*D2^3 + 9.8296*D2^2-56.685*D2 + 123.28 and fill it down. This equation is from a calibration done previously. Col F is the initial temperature. It s a constant, whatever you measured at the start. Col G is the total time in sec. So you will write the formula =A2*60+B2. You will fill this formula down. Col H is the net change in temperature, T. For simplicity, we will keep the values positive, and you may interpret it as the amount that the temperature dropped. The formula for Col H is =F2- E2. Once you have all your data, make a graph (scatter plot) of T as a function of t, which will be col G as the x axis, and col H as y. Now to compare it with the prediction: col I is a repeat of time. Do this in I2 by writing = G2, and fill down. col J is the theory. Write it this way =Amp*(1- ERF(xoverkappa/SQRT(I2))) and fill down. Excel is unhappy with you at this point because you have introduced two constants, Amp, and xoverkappa, that have not been defined as names. But we are about to do that. We need a place to put these values. Let s put them below your data. Say row 30. So in A30 put the maximum temperature change. That s basically your initial temperature. Go up to the menu bar under INSERT and go down to NAME, and pick the choice DEFINE. Define it as Amp. This now refers to this cell, A30, any time the variable Amp turns up. DIFFUSION LAB 6
Next to it (B30) write max to remind you of what this is. In A31, put a number around 8. Again go to INSERT, go down to NAME, and DEFINE this a the variable xoverkappa. Col J should now light up with actual numbers! In B31 you may want to also remind yourself what the variable is in A31. Click on your graph. Go up to the CHART menu and Add Data. The data you are adding is from Col J. You should see the theory curve. You can make it more distinct by using lines and not points. What you want to do at this point is to vary the value of A31, the xoverkappa value. Each time you change the value, you will see your graph change. The idea is to get the value of A31 that makes the theory and data agree best. If there are thermal leaks that are significant, you may also have to vary the value of max. Check with your instructor. What we want you to observe is the extended tail in time, which the data (we hope!) displays and the theory predicts. How would you use this to determine κ, the thermal diffusivity? DIFFUSION LAB 7