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Mackenzie Endres
An Experimental Exploration into Brownian Motion:
Size Dependence and the Boltzmann’s Constant
Berea College Senior Seminar
25 April 2014
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Abstract:
The purpose of this research experiment was to further my understanding of Brownian motion
through experimental observation of the motion of small beads suspended in fluid. The
variable being examined war size (diameter) of the beads, and the analysis addresses the
average displacement for each time interval, the root mean squared displacement over a
twelve second interval, and qualitative comparison of the different size particle samples.
Background:
The phenomena of Brownian motion, initially observed in 1828 by botanist Robert
Brown, is nothing new to physics, yet it is not easily understood or quantitatively examined.
What Robert Brown actually observed was that pollen dropped into water dispersed into a
large number of particles of very small size, of the order micron. The dispersion of the pollen
seemed be continuous. The initial thought on this motion was that it was a sign of life in the
pollen molecules. Testing this theory Brown conducted a similar experiment with inorganic
material with similar motion observed. Explanation of this phenomena became an interest to
the scientific community, yet it was not until Einstein1 attempted to examine the theory of
molecules. Einstein argued that the existence of discrete molecules colliding with small
particles suspended in a fluid would agitate the particles causing motions due to energy
transfer. The Einstein’s calculations showed resultant motion in micrometer sized particles
should be observed if fluids are made up of many discrete particles. Einstein suggested the
Brownian motion may be caused by discrete particles and that this natural phenomena
supported the existence of molecules (Ramaswamy 16-18).
Brownian motion is a perfect example of how observation of nature can produce
profound physics theories and develop mathematical techniques. Brownian motion and
random walking are integral to this day in mathematics, natural sciences, engineering,
1
Einstein was not searching for an explanation for Brownian motion; he was working on figuring a way to find
direct evidence for the existence of discrete molecules. When he began this work he was unaware of the
observations of Robert Brown or the research and theory that followed.
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linguistics, finances, economics, and, (as surprising as it sounds) social sciences. Brownian
motion is physical evidence of the existence of molecules. It has been used to obtain
Avogadro’s number.2 A basic understanding of the physics and mathematics of Brownian
motion can be obtained through the understanding and manipulation of the ideal gas law,
partial pressure, kinetic theory, terminal velocity, combinations and permutations, vector
calculus, and statistics3 (Ramaswamy 18) (Y. K.) (Kelvin).
Experimental Procedure:
Theory:
Brownian motion was observed with respect to the variable, microsphere size. There
are other variables, temperature, and viscosity that effect the motion of particles affected by
Brownian motion, but those will not be addressed beyond theory in this paper. The variable,
microsphere size will be examined through analysis of both average displacement and average
root mean squared displacement versus time. These forms of analysis will be used to find the
average displacement with uncertainty, and to show the particles are moving away from their
origin. The root mean squared analysis will be used to find a value for Boltzmann’s constant
and check the accuracy of the experiment.
In order to understand and interoperate Brownian motion it is vital to comprehend the
magnitude of collisions occurring per second between molecules. A water molecule for
instance undergoes 1011 collisions in a single second (Nakroshin et al.). This makes finding the
result of a single collision with a water molecule and a micron sized particle impossible. The
fastest video cameras available today, developed in 2011 by MIT, is capable of recording 1011
frames per second, but can only track high intensity light and is still being tested. Also, the
2
Avogadro’s Number- 6.02*1023 This is the number of atoms or molecules per mole of a specific material
The experimental purpose of this project will be to investigate the properties of Brownian motion through
variation in temperature, viscosity of the fluid, and size of the particles. These variations will test the thermal,
kinetic theory, and ideal gas laws directly. This will also investigate and improve my understanding of all of the
concepts listed in the above paragraph.
3
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price tag on this equipment is a quarter of a million dollars (Hardesty). Because there is no way
to conduct analysis on individual interactions, a statistical approach must be used for analysis.
Setup
For my purposes the setup was similar to figure 1, fig. 1. The sample cell will be placed
on the microscope stand and observed by recording single sphere through frames and tacking
the motion by hand using Logger Pro. The data collection will take place over twelve seconds. 4
fig. 1
There are several important notes to be made about data collection and analysis. The
particle’s initial position during frame one will be used as the origin for reference in analysis.
The limits on accuracy in measurements will include 0.5 microns of displacement, 0.03 seconds
time, and uncertainty caused by external factors such as vibrations caused by the ventilation
system, any tilt in the setup, drift caused by the cover sheet, and energy added to the system
from the light of the microscope heating the slides . Motion in the z-direction will have a
defocusing effect resulting in the need for several sets of data to be collected in order to
produce a data set with as little defocusing as possible. The motion in the z-direction cannot be
accounted for, and it is unavoidable with the dilution of the sample leading to fewer spheres. It
was not possible to select against this displacement. The Square distance (eq. 1) will be used to
calculate displacement, and the time average will be found using equation 2 (eq. 2).
Knowing/using these will be important in measuring parameters.
4
The time interval between frames should be one second.
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𝑟 2 = (𝑥 − 𝑥𝑜 )2 + (𝑦 − 𝑦𝑜 )2
〈𝑟 2 〉
=
2
(∑𝑁
𝑛=1 𝑟𝑛 )
N is the number of frames
𝑁
(eq.1)
(eq. 2)
The five measurable parameters for this experiment are temperature T, time t, viscosity ƞ,
radius, 𝛼, and time average of squares of the displacement 〈𝑟 2 〉. For the purposes of this
experiment we can treat the motion in the same way we would treat a one dimensional
scenario, and the one dimensional mean squared displacement follows the form of equation 4,
eq. 4, which has a dependence on a drag factor, μ. This is the linear drag coefficient, and
because of Stoke’s law we know it is proportional to the viscosity of the liquid and the radius of
the sphere (eq. 5) (Dongdong et al.).
𝑑〈𝑟 2 〉
𝑑𝑡
=
2𝑘𝑇
(eq. 3)
μ
μ = 6πƞα
〈𝑟 2 〉 =
(eq. 4)
2𝑘𝑇
3𝜋ƞ𝛼
𝑡
(eq. 5)
Theory suggests a linear time dependence at each temperature.5 This is intuitive
because of what we know about classical mechanics and thermodynamics. Viscosity and radius
size on the other hand have an inverse relationship with displacement. Plotting the mean
squared displacement, eq. 5, versus time should result in a straight line; the slope of which can
be used to find the Boltzmann’s constant, k. This is the first of the two methods of analysis
executed in this experiment. The second form of analysis examines the step displacement of
several spheres. This analysis will use the probability density of displacement6, equation 6, (eq.
6).
𝑃(𝛥) = √
5
1
2𝜋𝜎 2
𝑒
−𝛥2
2𝜎2
(eq. 6)
Note the viscosity of the liquid is a function of time as well and should be considered in temperature dependent
analysis.
6
𝛥 = 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
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The probability depends on the same parameters, temperature, viscosity, and particle size, as
2𝑘𝑇
well as the time interval between measurements. In equation 6, eq. 6, 𝜎 = √ 3𝜋ƞ𝛼 𝜏 and τ is
the time interval equal to the frame speed of the video camera. To calculate a value of
Boltzmann’s constant fit equation 6, eq.6, to a histogram of the step displacement of the
squares.
To investigate the size dependence I will place microspheres of varying size in a liquid
solution held at a constant temperature twenty-five degrees Celsius. The largest reference radii
from research was five micron and the data was noisy. The largest I observed was three micron
in radius, and these were moving too small of increments to be tracked by hand. They still
underwent Brownian motion, and a computer tracking program could have done this had I had
time to investigate this option. The sizes I worked with were three-quarter one, and two
microns in radii. In examining the three sizes it was clear from the raw video that there was an
inverse relationship between size of the particles and the step displacement.
Equipment:
Polystyrene microspheres of radii of three-quarter one, and two micron, optical
microscope equipped with a 40X object lens, a CCD, and a data collecting microscope camera
with a frame grabber able to record a stream of frames for analysis, liquids to dilute the
microsphere solution, , glass slides, pipets, and LoggerPro or other tracking software for
analysis.
Experimental Setup:
Most difficult obstacle of this experiment was finding a setup that would minimize uncalculated variables such as drift and parallax. Several setups were tested involving both well
slides and flat slides with and without cover slips. The well slides were the same slides used in
the research conducted at the University of Sothern Maine (Nakroshin et al.) and the University
of Pennsylvania (Dongdong et al.). The theory with the flat slides is there might be better at
removing parallax as the spheres would have less space to move in the z-direction. These set
ups were tested using milk samples. First I used shallow well slides with a cover slip. The milk
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was too concentrated to focus, so I diluted it, but the solution was still difficult to focus because
the particles were moving too quickly and there was too much parallax.
The next thing I tried was a flat slide with a diluted sample. This was exposed to the air
and the result was drifting of the particles all in the same direction. I tried adding a cover slip,
but this forced the particles toward the edges where there was less pressure and increased the
drift. I tried waiting for the drift to dissipate, but the samples dried out before that occurred.
Switching back to the well slides I tried using a smaller droplet of the sample, but that did not
improve the results because surface tension in the droplet prevented the sample from
spreading and thinning out. Finally I tried the flat slide again, but this time I surrounded the
edges of the cover slip with petroleum jelly. This acted like a barrier similar in function to the
use of oil in the University of Pennsylvania experiment I based my initial research off of. The
petroleum jelly stopped evaporation and cut down on drift significantly. The addition of the
barrio resulted in virtually no drift after a matter of minutes. The way I prepared my samples
way to line the edge of the sample window with petroleum jelly, drop a small diluted sample
into the middle, and slowly lower the cover slip. Then let the sample rest on the microscope
stand for five minutes after the system has been focused before recording video of the motion
of the sample.
Brownian Motion in Nature:
Milk, pollen, and blood slides were made to demonstrate the presence of Brownian
motion outside the laboratory. The pollen had the greatest variability in size this made
observing the dependence of displacement on size. The milk was the sample used most
extensively. I conducted qualitative experiments with the milk to observe the temperature
dependence. I cooled the sample to just above the freezing point, created slides, using a
variety of techniques, and observed the motion as the sample warmed to room temperature.
In doing so I was able to observe an increase in motion of the system as temperature increased.
I also compared the motion with skim, two percent, and whole milk. This was an attempt to
qualitatively observe the relationship with viscosity, but it was unsuccessful because the
different samples had different sizes and concentrations. The observation of Brownian motion
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in blood came through an experiment with osmosis where I made a slide of bovine blood and
observed the same type of drift I had observed with the flat slide samples of milk. I repeated
this experiment using the slide setup I chose to use in the actual experiment, and I was able to
observe Brownian motion of blood.
I determined the flat slides were the best option for my set up. The well slides had too
many layers of spheres to be able to focus on or to track a single sphere through an entire
video. Even using the flat slides tracking a single three-quarter micron sphere could not be
done for the duration of a video. Several smaller time increments had to be used, and this
prevented a root mean squared analysis of the three-quarter micron sample. The step
probability of displacement was able to be conducted on all three sizes, and the results will be
discussed later on in the paper.
Results:
In this section I will address the three different micron size results first individually and
then compare the samples to each other. For the one and two micron size sample I completed
both the step analysis with the histogram and the average root means squared analysis. The
root means square analysis did not work out well for this experiment because there were not
enough data points to track. I was only seven to twelve track-able spheres per twelve second
video. The three-quarter micron spheres moved in and out of the xy-plane enough to exit the
viewing plane. Because of this an average root mean squares analysis was not possible. The
time frame for the samples did not overlap for more than three full seconds.
.5, .75, 1, 2, 3 micron (left to right)
2 Micron Spheres
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The step displacement histogram, fig. 2, follows the one dimensional displacement of 21
spheres for twelve seconds and is the addition of the step displacement in both the x and y
direction independently. By doing the analysis in this way twice as many points of
displacement were collected because each new position was a displacement in both the x and y
directions. This was also chosen to ease analysis because the calculation of two-dimensional
displacement gives a scalar magnitude. The one-dimensional motion allows for analysis by
fitting the probability displacement, eq. 6, to the histogram. The fit function gave a best fit in
the form of equation 7, eq. 7.
𝑃(𝛥) = 𝐴𝑒
(−𝑥−𝐵)2
)+𝐷
𝐶2
(
eq. 7
Here C is the constant of interest because it contains the Boltzmann Constant in the
easiest to analyze form. 𝐶 2 = 2𝜎 2
𝐶2 =
4𝑘𝑇
3𝜋ƞ𝛼
𝜏
eq. 8
From eq.8 you can extract an equation for the Boltzmann constant, k, eq. 9.
𝑘=
𝑐 2 3𝜋ƞ𝛼
2√2𝑇𝜏
eq.9
fig. 2
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fig. 3
This fit was found by graphing the values for the bins from each sphere’s displacement
(indicated by the different colors) added together and fit using LoggerPro, fig. 3. Knowing:
Boltzmann’s constant; the temperature, 293 K; radius, 2 μm; dynamic viscosity, 8.90*10-4 N•s
•m-2; and time increment, 0.087s. The value of C or the standard deviation was used to find an
experimental value for the Boltzmann Constant, and this was compared to the accepted value,
k=1.38*10-23 m2•kg•s-2•K-1. Comparing this to the experimental value kE value from the fit,
kE=1.508*10-23, and the percent difference is 9.28%. This is a reasonable value for a difference,
and some of the uncertainty for this and all of the experiment include the hand tracking, the
parallax, drift in the slide, vibration causes by the air circulation system, and drift caused by the
cover slip on the slide.
Figure 4, fig. 4, is the average root mean squares displacement of eight spheres versus
time. The experimental limitations included the number of track-able spheres and the time.
The results should have been linear with a slope. The fit for this graph was compared to eq. 5
where the slope can be used to find an experimental value for the Boltzmann constant.
2𝑘𝑇
𝑚 = 3𝜋ƞ𝛼
eq. 10
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y = 4.2839x - 7.6252
R² = 0.6592
<R^2> of 2 micron
Displacement^2 (micron^2)
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
-10
-20
Time (seconds)
fig.4
From equation a theoretical value for Boltzmann constant can be found. However ne
significant conclusions can be drawn from this analysis. The linear fit is clearly not accurate for
this data. This analysis requires a large volume of data points, or spheres to make an accurate
assessment. This is because it is a statistical interpretation of the random motion. With just
seven values of displacement from the origin for each sample the analysis is not valid.
1 and 0.75 Micron Spheres
Figure 5, fig. 5, is the step displacement for one dimensional motion of the one micron
sphere size sample. Figure 6, fig. 6, is the graph of the frequency of the bins form the
histogram, fig. 5. Note: the 0.15 displacement point was excluded because through chance it
had a value of zero, and this distorted the fit The approach to this analysis was the same as the
analysis of the two micron sample. Here the columns are solid blue, but the histogram is a
stack of twenty-two spheres with data for the movement in both the x and y directions. The
experimental value for the Boltzmann Constant from this data is, kE=1.68*10-23. Comparing this
to the accepted value you get a percent difference of 21.6%. The is not a great value, but the difficulty
tracking and added displacement in the z-direction add a lot of error to the experiment
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Fig. 5
fig. 6
Fig. 6
The root mean squared analysis of this sample was no better than the previous sample, so it was
excluded from this report.
Figure 7, fig. 7, is the step displacement of the three-quarter microns, and was analyzed in the same
manner as the first two samples. Figure 8, fig. 8, is the graph of the frequency of the bins form
the histogram, fig. 7. Note: that several points were excluded because through chance it had a
value of zero, and this distorted the fit drastically. Here it is clear that the fit to the graph is not the
best, and further analysis shows the sample has a 82.0% difference in the values of the Boltzmann
Constant for the theory and the experimental data, KE=2.03*10-25 The reason for this is likely a result of a
lack of data. While figure 5 is a stacked histogram of twenty-two spheres, none of the samples could be
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tracked for the complete twelve seconds as the other samples were. This resulted in significantly less
data points than in the other samples. The video was also difficult to track by hand.
Fig. 7
Fig. 8
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Qualitative Comparison
The histograms on this page show the
displacement in two dimensions. From these it is
clear that the three-quarter micron sample had less
data to account for. This sample was clearly moving
more when observed, but the movement did not
translate into the data for the histogram.
The histograms for the one and two micron
samples are more enlightening. Both these samples
have the same number of data points and are scaled
the same. Looking from a qualitative stand point
the relationship between the displacement and the
size of the radius of the spheres can be
hypothesized. The spread of the base in each
histogram is of interest. This can be used to defend
the statement, “a one micron sized particle has more
probability of having a greater displacement.” The
histograms are half Gaussian distributions because
the displacement is a scalar calculation meaning the
values are all positive. The wider the base the more standard deviation. This deviation
indicates the regions means the particles can move more. The results of these histograms show
the end of any points of statistical relevance for the two micron sample is around three microns
while the 1 micron spheres have points of statistical relevance out to six microns of
displacement.
The entirety of my project involved the examination of spheres of radii, one-half, threequarter, one, two, and three microns. There is not analysis or data on all of these sizes for
several reasons. The work with the sizes not examined in the above analysis were helpful in
conduction qualitative observations. The half micron spheres moves the quickest, but the
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particles moved in and out of focus and were too small to confidently track by hand. The three
micron spheres stayed in focus and moved, but these particles were approaching the size limit
for Brownian motion for my set up. The movement was too minuscule to confidently track. By
observing and comparing all of these size it was clear that there is an inverse relationship
between size and displacement. This observation matches equation 5.
Conclusion:
The tracking and the analysis process in my research added uncertainty to my results.
By tracking the spheres by hand error was introduced to the sample for every data point. I was
not able to collect enough data by hand to complete analysis on two of my five samples. I only
completed the average root mean squared analysis for the largest of my three analyzed
samples. This analysis showed a poor fit to my data, but the trend of movement away from the
origin was present the same as the research I collected from other articles.
The set up I used was good for qualitative comparisons, and had some advantages over
the setups I read about in the lab reports and articles I read. By using a flat slide I removed
layers from the sample, and in this way I was able to bound movement in the z-direction. By
covering the slides I removed error due to evaporation and isolated the system from the
environment. The petroleum sealed the edges to complete the isolation and stop the drift
caused by the slide pressing on the sample. The cover spread out the sample, but was held up
by the petroleum. In the experiment conducted at the University of Southern Maine used
coverslips. The difference is that that experiment used well slides and did not seal off the
sample (Nakroshin et al.). The result of this was that after even a couple hours the sample still
had signs of drift in the results. The setup used for this experiment displayed an absence of
drift after a matter of minutes. Making the slides is quick and relatively easy. Using this setup
would cut down on the time taken to collect data in an undergraduate laboratory setting, and
would make this experiment a possible choice for an advanced laboratory assignment.
The tracking method needs improvement. If I had used an automatic tracker like the
ones used by the other sources I consulted I would have been able to spend more time on
analysis. Looking farther into analysis the students would be able to compare more samples
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and compare the size dependence and find the Boltzmann constant. The lab report from
Harvard looked only at one sphere size (Lab 7). Any papers I read about undergraduate
investigation of Brownian motion and compared different sets of data not just one video, did so
as extended experiments or summer projects (Dongdong et al.). By reducing the time for
collecting data and utilizing an automatic tracking system it would not be unreasonable to ask
the students to collect data for multiple sample sizes. The students could track the samples and
extract an experimental Boltzmann constant from each and compare the displacement of the
samples to get a stronger grasp of the size dependence.
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Work Cited
Dongdong Jia, Jonathan Hamilton, Lenu M. Zaman, and Anura Goonewardene, “The time, size,
viscosity, and temperature dependence of the Brownian motion of polystyrene
microspheres” Am. J. of Physics. 75 101 (2007)
Hardesty, Larry. "Trillion-frame-per-second video By using optical equipment in a totally
unexpected way, MIT researchers have created an imaging system that makes light look
slow.." MIT News. MIT, 13 Dec. 2011. Web. 30 Mar. 2014. <http://newsoffice.mit.edu
/2011/trillion-fps-camera-1213>.
"Lab 7: Brownian Motion." Lab Manual (2013): 2-1 to 2-11. Web.
Nakroshin, Paul , Matthew Amoroso, Jason Legere, and Christian Smith. "Measuring
Boltzmann's Constant using Video Microscopy of Brownian Motion." American Journal
of Physics 71 (6): 568-573. Print.
Ramaswamy , Sriram. "Pollen Grains, Random Walks and Einstein." Resonance. March (2000):
16-33. Web. 21 Jan. 2014. <http://www.ias.ac.in/resonance/Volumes/05/03/00160034.pdf>.
Y. K., Lee. "Applications Of Browian Motion –Article Two."Surprise 95. 2. (1995): n. page. Web.
22 Jan. 2014. <http://www.doc.ic.ac.uk/~nd/surprise_95/journal/vol2/ykl/article2.html>.