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Brownian Motion of Ellipsoidal Particles
Brownian Motion of Ellipsoidal Particles
Noah Trebesch and Ben Freund
School of Physics and Astronomy
University of Minnesota - Twin Cities
Minneapolis, MN 55455
May 17, 2013
Abstract
Abstract
The rotational and translational Brownian motion of microscopic ellipsoidal particles in two dimensions was studied using video microscopy. The qualitative trends in the position and orientation of the particle versus time, the displacement in the position and orientation of the particle versus time, and the distributions the the sizes of the displacements were found to agree with theory. Two methods were used to quantify the agreement of experimental results with theory by estimating Boltzmann's constant. The best estimate differed from theory by
, while the worst differed by . The quality of the quantitative data is questionable, however, due to a lack of precision in the experiment and the possible presence of a number of systematic errors.
Introduction
Introduction
Brownian motion is a physical phenomenon that can be observed in microscopic particles. When suspended in a fluid, microscopic particles travel along random paths with random velocities (called translational Brownian motion), and they rotate in random directions with random angular velocities (called rotational Brownian motion). Albert Einstein hypothesized that Brownian motion was caused by the roughly
collisions per second of the Brownian particle with the molecules that make up the fluid the particle is suspended in [1]. From this standpoint, he was able to derive an equation for the RMS translational or rotational displacement a Brownian particle of any shape should experience in one dimension. Since Einstein's equation was published in 1906, many experiments with spherical particles have been done to confirm his predictions, but very little experimental or theoretical work has been done with non-spherical particles [1], [2], [3].
In biology, large molecules are common. These macromolecules experience Brownian motion, but they almost always have globular shapes that cannot be represented well by spheres. An ellipsoidal shape can provide a better analog to such molecules, so understanding the Brownian motion an ellipsoid should experience is helpful in understanding and predicting the Brownian motion of macromolecules [3]. This, in turn, is useful for predicting and understanding the diffusion of macromolecules [3]. Diffusion is a process caused directly by Brownian motion in which high concentrations of particles suspended in a fluid spread throughout the fluid and across permeable or semipermeable membranes until their concentration is uniform. Diffusion is a common process in the biological world, which is why it is important to understand it [3]].
Theory
Theory
Brownian Motion
Brownian Motion
Einstein's equation for the RMS displacement in a single rotational or translational degree of freedom due to Brownian motion is given below.
In this equation,
is the displacement, is Boltzmann's constant, is the absolute temperature, and is the time elapsed (also called the time step). The symbol is a constant of proportionality that depends on the shape of the Brownian particle and the medium in which it is suspended. Einstein also showed that the probability density function of displacements along any degree of freedom can be approximated by a Gaussian distribution:
In this equation,
is the standard deviation of the distribution. For a given particle and time step, this equation can be used to predict the probability of the particle being displaced by any given amount. These equations are very general. They make no assumptions about the shape of the Brownian particle or the medium in which it is suspended, and they are valid for both translational and rotational Brownian motion. In the derivation of these equations, Einstein also showed that Brownian motion along each degree of freedom is independent of the Brownian motion along any other degree of freedom [1], [2].
In three dimensions, an ellipsoid can translate and rotate along each of its three semi-principal axes, giving it six degrees of freedom along which Brownian motion can occur. The RMS change in each of these six quantities can be described using equation [[#EqEinstein][1]]. In the experimental setup, the movement of the particles was confined almost entirely to two dimensions, and prolate spheroids were used. (A prolate spheroid is an ellipsoid in which two of the semi-principal axes are the same length and are shorter than the third semi-principal axis.) This eliminates one translational and one rotational degree of freedom from the system. Also, it was impossible to observe the rotation of the ellipsoid about its long semi-principal axis, so this degree of freedom was not considered. This means that only three displacements were observed: two translational and one rotational. A diagram of these displacements is shown in figure [[#FigEllipsoid][1]].
In 1934, Francis Perrin derived equations for each of the six degrees of freedom that describe the motion of a prolate spheroid in three dimensions. The three equations relevant to this experiment are given below. Equation [[#EqPerrinTrans1][3]] gives the inverse value of
for translational Brownian motion along the long semi-principal axis of length
. Equation [[#EqPerrinTrans2][4]] gives the inverse value of for translational Brownian motion along either of the the other two semi-principal axes of length , and equation [[#EqPerrinRot2][5]] gives the inverse value of
for rotational Brownian motion about either of the short semi-principal axes.
In these equations,
is the viscosity of the fluid in which the ellipsoidal particle is suspended, and is a constant given by equation [[#EqPerrinConst][6]][[[#PerrinEllipsoidsI][4]], [[#PerrinEllipsoidsII][5]]]. For a sphere,
is constant in both translational directions, and it is given by the equation below:
In this equation,
is the radius of the particle. The rotation of a sphere in any direction cannot be observed [1].
Anisotropic and Isotropic Diffusion
Anisotropic and Isotropic Diffusion
The diffusion of a particle is governed by its diffusion coefficient, .
From this equation, it should be clear that diffusion and the RMS displacement due to Brownian motion are very similar concepts. It may not be immediately apparent from equations [3] and [4], but these equations show that
Figure 1: An original diagram showing the lengths of the semi-principal axes of a prolate spheroid and the three observable kinds of displacement it can experience in two dimensions.
> , which means an ellipsoid will always diffuse along its long axis more quickly than its short ones. This means that the RMS displacement along the long axis will always be greater than the RMS displacement along the short axis. This would also be true in a fixed coordinate system (i.e., the “lab frame”) if it were not for rotational Brownian motion. [4], [5].
At small time intervals, when the RMS translational displacement is smaller than the size of the particle and the RMS angular displacement of the particle is correspondingly small, the RMS displacement along the lab frame's coordinates are influenced by the initial orientation of the ellipsoidal particle and its angular displacement. It is said that the translational and rotational displacement of the particle are coupled when this is the case, and the diffusion the particle experiences is said to be anisotropic. As the time interval grows, the RMS angular displacement of the particle also grows, and the RMS displacement along the lab frame's coordinates start to become uniform. The degree to which the initial orientation and angular displacement of the ellipsoidal particle influences the RMS displacements along the lab frame's coordinates diminishes as the time interval grows. Eventually, when the degree of influence is small enough, the translational and rotational displacement of the particle are said to be uncoupled, and the diffusion of the particle is said to be isotropic [3].
#SecWall
Wall Effects
Wall Effects
In this experiment, two dimensional confinement was simulated by separating two flat, parallel surfaces (hereafter referred to as “walls”) by a distance of . This experimental setup changed the effective viscosity of the liquid because of the boundary conditions present at the walls. Near the walls, the viscosity of the fluid approached infinity because the ability of the atoms or molecules that make up the fluid to slide along the surface of the walls was extremely limited. The way fluid viscosity is affected by this seemingly simple set-up is actually a very complex problem, and an analytical solution to it has yet to be found. However, several models exist that approximate the way the viscosity of the fluid changes with distance from the two walls. The model developed by Luc Faucheux and Albert Libchaber is one of the simplest, yet it is nearly as accuracy and is sometimes more accurate than other, more complex models [6], [7], [8].
In the Faucheux and Libchaber model, a simple linear superposition of the effects from the two walls is expected. In this experiment, only movement along directions parallel to the walls was observed, so only the effective parallel viscosity is of interest. Equation [9] shows the expected viscosity experienced by a spherical particle of radius
a height above a single wall, and equation [[#EqFaucheux2][10]] shows the expected viscosity experienced by the same particle at a height above one wall in a two wall system, where the two walls are separated by a distance
[8], [9].
Equation [10] was used to approximate the change in viscosity caused by the walls for the ellipsoidal particles. The radius
appears in equation [10] because the viscosity of the fluid varies with distance from the walls, so was used in this equation for the approximation. Because the particles are ellipsoidal rather than spherical, the use of this equation introduces a systematic error that will be discussed later.
Experimental Procedure and Apparatus
Experimental Procedure and Apparatus
Particle Preparation
Particle Preparation
Ellipsoidal particles are not commercially available. To obtain ellipsoidal particles, they were instead prepared from much more widely available spherical particles. To do this, a solution of poly(vinyl alcohol) (PVA) and polystyrene (PS) microspheres was prepared. A mass of
PVA and approximately PS microspheres were added for every milliliter of water to create the solution. These values were chosen based on the recommendations of reference [10]. The diameter of the microspheres used was
. A volume of of solution was poured into a Petri dish, and the water was allowed to evaporate, creating a solid PVA film with the PS microspheres embedded in it [9], [10].
To make the embedded microspheres ellipsoidal, the PVA film was stretched. When heated, polymers like PVA and PS behave like glass. That is to say that above a certain temperature (called the glass transition temperature), PVA and PS become semi-molten and can be stretched without breaking. The glass transition temperature of PVA is
, and the glass transition temperature of PS is [11]. To create the ellipsoidal particles, an oil bath was heated to using a hot plate, and a
by strip of the film was immersed in the bath. The film was held at its ends by two needle nose pliers and stretched by pulling the pliers apart from one another. A diagram of this stretching technique is shown in figure [2]. To generate ellipsoids with semi-principal axis lengths of
and , the film was stretched times its original length. The specific draw ratio was determined by measuring the length of the strip of film before and after it was stretched [9], [10].
Once the film was stretched, the ellipsoidal particles were recovered from the film. To do this, a solution of 7:3 parts water to isopropyl alcohol (IPA) was created. The film strips were immersed in this solution, and it was heated to
. This solution dissolved the PVA while leaving the PS particles intact. The particles were then recovered from the solution and placed in deionized (DI) water. This was done by centrifuging the solution at
for half an hour, causing the PS particles to settle at the bottom of the containment vial. The solution of water, IPA, and PVA was then removed, the vial was refilled with DI water, and the particles were mixed into the water. This entire process was then repeated once more to further remove any residue of IPA and PVA from the particles [9], [10].
Slide Preparation and Video Microscopy
Slide Preparation and Video Microscopy
Figure 2: An original diagram showing the stretching apparatus and oil bath.
Once the particles were suspended in DI water, slides were prepared to view the particles under a microscope. To do this, spherical PS particles with a diameter of were added to every milliliter of particle solution to create a new solution. Additionally,
spherical PS particles with a diameter of were also added to every milliliter of the new solution. A thin ring of adhesive was traced out on a cover slip, and
of the new solution was placed in the center of the ring of adhesive. Another cover slip was placed on top of the first, they were pressed together, and the adhesive was cured using an ultraviolet light. The adhesive prevented evaporation of the water while the cell was being observed. A diagram of a finished slide is shown in figure [3]. The large spherical particles acted as spacers and allowed the distance
between the cover slips to be controlled. The chosen distance
was small enough that the ellipsoidal particles did not tumble in three dimensions over the course of data acquisition. Though it was possible for them to do so, the effective viscosities at different heights in the cell impeded such motion. The smaller spherical particles were added to allow the Brownian motion of spherical particles to be observed and compared to the Brownian motion of the ellipsoidal particles. Observing both particles within the same cell prevented differences in the cells from creating differences in the spherical and ellipsoidal Brownian motion.
Figure 3: An original diagram of the microscope slides that were prepared. Note that this image is not drawn to scale, and the ellipsoidal particles observed were actually far from any spherical spacers. The labeled height and diameter in this diagram correspond to the variables used in equations [[#EqFaucheux1][9]] and [10]].
The prepared slides were observed using a microscope at 100 times magnification. To obtain quantitative information on the Brownian motion of the spherical and ellipsoidal particles, a series of about five hundred of images of a single magnified particle was taken using
Manager [12]. Each image was taken one half second after the last image, making each image equivalent to a single frame in a video of the motion of the particle over about four minutes. To calibrate the microscope, an image of a microspherical spacer (with a known diameter) was also captured to determine how many meters were represented by one pixel in any of the video frames. Ultimately, it was found that a length of one pixel was equivalent to
.
Raw and Transformed Data
Raw and Transformed Data
Results and Analysis
Results and Analysis
A custom script was written in MATLAB to automatically determine the position (in pixels) and orientation of the ellipsoidal particle in each frame of its video. This script was also used to determine the position of the spherical particle in each frame of its video. To work, the approximate location of the particle in the first frame of the video must be manually input. The program then finds the outlines within the small section of the image that contains the particle. It is only able to detect the two long edges of the ellipsoidal particle, as shown in figure [5]. For the ellipsoidal particle, the program finds the center and orientation of the two edges of the ellipsoid and averages them to give the final location and orientation of the particle in the video frame. For the spherical particles, the program can detect the entire continuous outline of the particle, so no averaging is performed. The particles move slowly enough that the center of the particle in the previous frame of the video can be used as the rough estimate of the center of the particle in the next frame, which allows the program to proceed automatically after the first frame.
Figure 4: An original diagram of the microscope that was used in this experiment.
For the ellipsoidal particle, the data output by the image processing script is in the lab frame of reference. It was transformed to the particle frame for analysis. To do this, the displacement of the particle in each pair of frames was measured in the lab frame. The origin of the lab frame coordinate axes was then set at the location of the particle's center in each frame, and a simple coordinate frame rotation, given by
Figure 5: An original image showing the outline of an ellipsoid determined by the image processing script on the left along with the center of the ellipsoid, denoted by a '+', and a line representing the orientation of the particle. On the right is the center and orientation line superimposed over the original image.
was then used to transform the displacement in the lab frame to the displacement in the particle frame. In these equations,
and represent the particle's orientation in the two frames. Their average was used as the coordinate rotation angle because no information is known about the orientation of the particle between the two frames. A diagram of this transformation is shown in figure [[#FigTransform][6]].
With this rotation, the location of the particle in its own frame of reference can be determined as a function of time by taking a sum of the displacements. Ultimately, this results in three data sets that must be analyzed: there is the data from the spherical particle, the lab frame data from the ellipsoidal particle, and the particle frame data from the ellipsoidal particle. The raw data can be visualized on two or three dimensional plots, where each dimension represents a translational or rotational degree of freedom. These plots for each of these data sets is given in figure [7]. In these plots, the variation of color along the particles' paths represents the progression of time. There is not much quantitative information on these plots that can be used to assess the agreement with theory, but they are provided because similar plots are common in the literature on Brownian motion [3], [13]. These plots can be compared with the plots in the literature to provide a qualitative comparison with previous results.
Analysis of Temporal Plots and Displacement Distributions
Analysis of Temporal Plots and Displacement Distributions
The motion along each degree of freedom is independent of the motion along any other degree of freedom, so it is easier to perform an analysis considering each degree of freedom separately. The first way this may be done is by considering the position of the particle along each degree of freedom as a function to time. These plots are given in figures [[#FigTimeX1][8]]A through [[#FigTimeB][14]]A. Each plot should appear random. There should be no correlation between time and position. Qualitatively speaking, this is true for each of the degrees of freedom considered.
Next, the displacement of the particle along each degree of freedom as a function of time can be considered. These plots are given in figures [[#FigTimeX1][8]]B through [[#FigTimeB][14]]B. By equation [[#EqEinsteinProb][2]], the distribution of these displacements should be approximately Gaussian, and each distribution should be centered at zero. For all of these plots, the highest density of points appears to be centered on zero, and the point density appears to decrease as the displacement gets farther and farther from zero. This is what is expected of a Gaussian distribution, so the qualitative properties of these plots agree with theory. Also, the distribution of the displacements should not depend on time, and these plots can be used to verify that this was the case for each degree of freedom. This can be done by observing that the density of points for any given displacement is approximately equal at all times.
With the temporal consistency of the distribution qualitatively verified, the actual distribution of the displacements can be plotted. This is done for each considered degree of freedom in figures [8]C through [14]C. Using MATLAB's
, and .
Figure 6: An original diagram of the lab frame coordinate axes (in red), the particle frame axes (in blue), and some relevant symbols from equations [11] and [12]. In these equations,
function, a Gaussian distribution can be fit to the displacement data [14]. This function can also provide a user-specified percentage confidence interval on the estimates of the mean and standard deviation of the fit Gaussian distribution. The uncertainty of a value with random error should represent a 68.27% confidence interval, so the
function was set to provide estimates of Gaussian distribution parameters and 68.27% confidence intervals on these parameters. In this way, estimates of the mean and standard deviation of each fit were generated with estimates of uncertainties. These values for each degree of freedom are given on the plots.
These distributions can be used to quantitatively verify the theory behind Brownian motion using equation [2]. The standard deviation of each degree of freedom should be equal to the square root of a number of parameters that, with one exception, are all known. Bolzmann's constant is known, the temperature was measured, and the time step was measured. By equations [3] through [7], it is known that mobility depends on the lengths of the effective viscosity of the water the particles were suspended in and the semi-principal axes for the ellipsoid or the radius for the sphere. The lengths of the semi-principal axes can be measured, the radius of the sphere is known, and the effective viscosity is given by equations [9] and [10]. The effective viscosity of the fluid depends on the free viscosity of the fluid, which is known, the effective radius of the particle, which is also known, and the vertical distance
Figure 7: The degrees of freedom of the particle plotted against one another. Color varies from blue to red with time. A: Spherical data. B: Lab frame ellipsoidal data. C: Particle frame ellipsoidal data.
-coordinate data. A: Position vs time. B: Displacement vs time. C: Displacement distribution. ,
Figure 14: Ellipsoidal
-coordinate data. A: Position vs time. B: Displacement vs time. C: Displacement distribution. ,
Figure 13: Ellipsoidal
-coordinate data. A: Position vs time. B: Displacement vs time. C: Displacement distribution. ,
Figure 12: Ellipsoidal
-coordinate data. A: Position vs time. B: Displacement vs time. C: Displacement distribution. ,
Figure 11: Ellipsoidal
Figure 10: Ellipsoidal -coordinate data. A: Position vs time. B: Displacement vs time. C: Displacement distribution. ,
-coordinate data. A: Position vs time. B: Displacement vs time. C: Displacement distribution. ,
Figure 9: Spherical
Figure 8: Spherical -coordinate data. A: Position vs time. B: Displacement vs time. C: Displacement distribution. ,
between the center of the particle and one of the walls, which is the one unknown parameter.
When the microscope was used to observe the particles on the prepared slides, it was found that all particles that were moving could be found along a single plane in the cell. The Brownian motion of spherical particles is well understood and is not the focus of this experiment, so the distributions of the displacements of the particle in the and
directions were used to find the best value for the height of the particles. To account for the uncertainties in all variables, this was done by using the known, calculated, and measured values discussed in the previous paragraph to calculate estimates of Boltzmann's constant
. Ultimately, the value of that produced the best values for was . Using this height, the distribution of
produced an estimate of for , and the distribution of produded an estimate of . These estimates are and away, respectively, from the accepted value of
.
The uncertainty on these estimates is quite high because of the contribution from the uncertainty in the time step. Unfortunately, the computer was not able to capture video frames at an extremely precise time interval, and the actual time interval between captured video frames was
. The uncertainty on this measurement is simply the standard deviation of the time intervals between all pairs of frames in the spherical data set. It does not make sense to manually adjust this uncertainty; doing so would produce an estimate of uncertainty that was not true to the data. Consequently, the qualitative information obtained from this analysis will necessarily be very imprecise.
With the distance between the particle and the walls estimated, the distributions generated from the , , and displacement data can be analyzed. Recall that a radius of was used in equation [[#EqFaucheux1][9]] as the radius of the particle. According to reference [3], the mobility of the ellipsoid along both translational directions should be equal to the average of
and . With these factors in mind, Boltzmann's constant can be estimated from the standard deviation of the distributions generated from the ellipsoidal
and displacement data as well. The estimates of Boltzmann's constant using the distribution of displacements along each degree of freedom are presented in table [1], and they will be discussed later.
Analysis of RMS Displacement Plots
Analysis of RMS Displacement Plots
In addition to using the distributions, there is another method that can be used to quantify the agreement of the experimental results with theory. The time step can be varied, and the resulting RMS displacement along each degree of freedom can be measured. By equation [1], the RMS displacement should be linearly related to the square root of the time step, so the slope of a line fit to such experimental data can also be used to generate an estimate of Boltzmann's constant. To generate displacement data over varying time steps from the original data sets, a summation of the displacement over multiple frames of video was performed. For the amount of data that was collected, this is problematic because the number of displacements that can be generated for a given time step is equal to the number of frames originally collected divided by the number of frames used to generate the new time step. It was not possible to use any the position and orientation data from any given frame more than once when calculating a RMS displacement at any given time point because the resulting correlation of the data points within the RMS displacement calculation results in unpredictable nonlinear trends in the RMS displacement versus square root of time step plots.
The plots of the RMS displacement along each degree of freedom for both particles as a function of time are given in figure [15]. Because each data point on this plot was calculated by reusing all of the same data, the points are correlated with one another. It is difficult to calculate the covariance between the points on the plots, but this calculation is necessary if a standard least-squares fit to the data is to be used. Consequently, a standard least-squares fit was not performed. Instead, at each time step, the available displacement data points were randomly partitioned into five disjoint sets, and the RMS displacement was calculated for each set. One RMS displacement from each time step were grouped together, and an unweighted least squares fit was performed on each of the five sets of RMS displacements. The average intercept and slope of these five fits were used for the lines plotted in figure [15], and the standard deviation of the five intercepts and slopes were used as the final estimates of the uncertainties of the fit. The points shown in these plots are the RMS displacements at each time point calculated using all of the data points available instead of the data points in just one of the five partitions. Because of the correlation between the data points in thees plots, error bars cannot be produced, which is why they do not appear on the plots. Additionally, any nonlinear trends seen in the plots could be the result of the correlation between the points, so
plots are not useful and were therefore not included for these plots.
Figure 15: Plots of the RMS displacement as a function of the square root of the time step. A: Spherical data. B: Lab frame ellipsoidal data. C: Angular ellipsoidal data. D: Particle frame ellipsoidal data.
Discussion of Results and Error
Discussion of Results and Error
The estimates of Boltzmann's constant produced from the slopes of the plots in figure [[#FigRms][15]] are given in table [[#TabResults][1]]. Like the estimates of Boltzmann's constant generated from the distributions, these estimates are all very imprecise. This time, the precision of the estimates is limited by the uncertainty of the slopes. Because there are so few data points at larger time steps, the RMS displacement varies greatly at the larger time steps. This variation could be reduced by simply taking more data.
Table 1: A list of the estimates of Boltzmann's constant produced using the distributions and slopes for each degree of freedom of both particles.
With that said, the estimates of Boltzmann's constant in table [1] can be analyzed. All estimates of Boltzmann's constant generated from the probability distributions were within
of the accepted value. However, this is largely due to the large uncertainty associated with the estimates. A closer examination of the estimates reveals that all but one of the estimates for the ellipsoid are higher than the accepted value. This may indicate the presence of a systematic error, but it hard to say this with confidence due to the amount of uncertainty in the problem. If such a systematic error does exist, it is likely due to the use of equation [10] to estimate the effective viscosity experience by the particle. As previously noted, this equation is meant to be applied to a sphere.
It is also worth noting that the standard deviation of the distribution of is larger than it should be, resulting in a large value relative to the other values. This may be entirely due to the high uncertainties present in this experiment, but it is also possible that the transformation discussed earlier does not adequately capture the motion of the particle from one frame to the next. It may be that the time step between frames is so great that the orientation of the particle changes quite a bit within the time step, and the average of the initial and final orientations does not adequately describe the orientation of the particle between frames. It is worth noting here that the video in reference [3] was taken at
instead of
, which means this hypothesis is plausible.
Additionally, the estimates produced from the
and distributions are low and high, respectively, for both the ellipsoidal and spherical particle. This may be due entirely to uncertainty in the time step, but it could also indicate a systematic error. Imperfections in the optical lenses in the microscope and/or camera could produce an image with different numbers of microns per pixel in the and
directions, which could account for the differences in the standard deviation of the displacement distributions.
All but one of the estimates produced using the slopes of the RMS displacements as a function of the time step are lower than they should be. Again, this pattern may well be due entirely to the high uncertainty in this experiment, but it could also be indicative of a systematic error. As previously discussed, this systematic error could be generated through the use of equation [10]. As with the
values produced using the estimates gerated from the displacement distributions, the value generated by the estimate determined from the slope using RMS
data is higher than that determined from the slope using RMS data. This is caused by the slope associated with the RMS being higher than it should be. This result adds support to the previously discussed possibility that there may be systematic error introduced by the low frame rate and coordinate transformation equations. Finally, the
value produced using the RMS data is again more negative than the
value produced using the RMS data. This occurs because the slope of the line fit to the RMS data is greater than the slope of the line fit to the RMS data. This result adds support to the possibility that there may be a systematic error introduced by a difference in the and
directions in the captured video frames. A possible cause for this difference has previously been discussed.
Conclusion
Conclusion
All qualitative results point to agreement between the observed experimental results and the expected theoretical results. The paths and orientations of the particles appear random, the displacements are consistently distributed through time, and the distributions of the size of the displacements are approximately Gaussian and centered on zero. The quantitative results also point to agreement with theory: eight of the fourteen estimates of Boltzmann's constant are within
of the theoretical value, and an additional two are within
. However, the degree of agreement is much less certain due to the high uncertainties present in this experiment. These uncertainties are primarily due to the inability of the camera and computer system to capture video frames at a consistent time interval, and they are also primarily due to having data sets that are too small. A better camera and computer system and longer data sets would greatly improve the precision of the quantitative results in this experiment.
The lack of precision present in this experiment greatly limits the certainty of the analysis that can be performed on the quantitative results that were obtained. However, there is evidence that supports the possibility that there may be a number of systematic errors present in this experiment. The use of equation [10] on ellipsoidal particles, the low frame rate used, and unexpected differences in the and
directions in the captured video frames may all be sources of systematic error that account for some of the observed trends in the quantitative results. However, without more precise data, these predictions of systematic error are mostly speculative.
Acknowledgments
Acknowledgments
The authors would like to thank Professor Clem Pryke for his role as a mentor in this project, Professor Vincent Noireaux for the use of his microscope, lab space, and general equipment, Kurt Wick for his technical assistance, and Jon Kilgore for permitting the use of the Physics Machine Shop's fume hoods.
References
References
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