F20VideoMicroscopy

Tracking the Brownian Motion of a Diffusing Microsphere Using Video Microscopy

Adam Markowicz and Scarlett O'Donovan

University of Minnesota - Methods of Experimental Physics II

Fall, 2020

Introduction

The random motion of particles suspended in a fluid due to thermal fluctuations was first observed in 1827 by Robert Brown and given the eponymous name Brownian motion [1]. However, it was not until the early 20th century that Albert Einstein made the connection between Brownian motion and kinetic theory. He argued that the macroscopic observation of a particle moving was due to microscopic collisions between the molecules of the fluid in which the particle was suspended and the the particle itself [2]. Those collisions were due to thermal agitation and he believed that the motion was evidence for the "atomic nature of matter" [1]. Specifically, he proposed that the the mean squared displacement of the observed motion and would be proportional to the thermal energy k_BT, where k_B is the Boltzmann constant is T was the temperature of the fluid. Jean Perrin confirmed Einstein's predictions, for which he (Perrin) was awarded the Nobel Prize in Physics in 1926 [1]. The purpose of this experiment was to use video microscopy to examine and random path of a microsphere as it diffused through a room-temperature aqueous solution. By analyzing the mean-squared displacement as a function of time of the microsphere, the estimate for the Boltzmann constant was determined to be k_B = (3.72+/-0.23)x10^-23 J/K: a value is more than ten standard deviations away from the accepted value of k_B = (3.72+/-0.23)x10^-23 J/K. Further study of our experimental method and analysis would have to be completed to determine whether the disagreement is due to a systematic error or an underestimated uncertainty.

Theory

As a particle diffuses is exhibits a random walk. The two key characteristics of a random walk is that its trajectory cannot be predicted and must be continuous [3] . Furthermore, a particle undergoing a random walk can only take discrete steps, the movement of the particle in one dimension does not influence the movement in other dimensions, and the movement of a previous step does not effect the movement in subsequent steps. An example of a random walk in three dimensions is shown in Figure 1.

Figure 1: The trajectory of a random walk of a 1μm microsphere in three dimensions. In this example, it appears that the microsphere moving down the optical axis. That is due to sedimentation [2].

Einstein and Smoluchowski predicted that the average of the mean squared displacement for a particle that diffuses as it drifts is (equation 1)

where d is the dimensionality of the trajectory and D is the diffusion coefficient [2]. The diffusion coefficient for a spherical particle with radius a moving through an unbounded Newtonian fluid with dynamic viscosity η is given by the Stockes-Einstein diffusion coefficient (equation 2):

From the Einstein-Smoluchowski equation and the Stockes-Einstein diffusion coefficient the Boltzmann constant is given by (equation 3)

Experimental Set-up

Figure 2: Set-up of the experiment

The set-up consisted of a Zeiss 1,25 microscope and a 20mW, 633nm class IIIB helium neon (HeNe) laser. The intent of the original experiment was to use analyze the motion of a microsphere in three dimensions using holographic video microscopy, which required a laser as the light source.

Through the microscope, 3 micron polystyrene microspheres were observed. The microspheres had been chemically treated to ensure they have a negative surface charge. The negative surface charge ensured the spheres repelled each other and prevented them from sticking to glass surfaces. To allow the microspheres to diffusive freely, a well was created using a microscope slide, coverslips, and nail polish as an adhesive. The microspheres were diluted using distilled water.

The trajectory of a microsphere as it diffused was recorded using a Thorlabs DCX-1545 CMOS camera (1280x1024), which was mounted on top of the microscope. A 44 AWG wire imaged under the microscope to relate the distance traveled on the screen in pixels to the distance traveled by the sphere in the well.

Data Analysis

The trajectory of a particle was manually analyzed using the Manual Tracking program available in ImageJ.

Figure 3: A raw image of the diffusing microsphere. The distance between the concentric circles produced by interference can be used to determine the position of the particle along the optical axis.

A plot of the displacement squared as a function of time of a diffusing microsphere is shown on the left in Figure 4. Given that the displacement of the particle can be tracked starting at any time - not just th time we designated to be t = 0s - we determined extra displacment-squared value for each time step size and found the mean of those values. Furthermore, we chose to eliminate the data after turbulence developed in the sample. The plot of the mean-displacement squared before turbulence develop along with the best fit line is shown on the right in Figure 4.

Figure 4: A plot of the displacement squared of a diffusing microsphere as a function of time is shown on plot on the left. On the right, the plot along with the best fit line of the mean displacement squared with only the pre-turbulent data is shown.

From the slope of the of the mean-displacement squared plot and using equation 3 we calculated the Boltzmann constant to be (3.72+/-0.23)x10^-23 J/K. Further study of our experimental method and analysis would have to be completed to determine whether the disagreement is due to a systematic error or an underestimated uncertainty. Possible sources of error include human error in manual tracking, incorrect values or underestimated uncertainties of the values in the diffusion coefficient, or an error in the code used for computing the Boltzmann constant.

Conclusion

We measured the value of the Boltzmann constant to be k_B = (3.72+/-0.23)x10^-23 J/K by tracking the trajectory of microspheres using video microscopy. Our value for the Boltzmann constant does not agree with the scientifically accepted value of 1.38 10^-23 J/K. More analysis must be completed to determine where there exists a systematic error, underestimated uncertainty, or an error in our calculation.

References

[1] P. Nakroshis, M. Amoroso, J. Legere, and C. Smith. "Measuring Boltzmann's constant using video microscopy," American Journal of Physics 71(6), 568-573 (2003).

[2] B. Krishnatreya, A. Colon-Landy, P. Hasebe, B. Bell, J. Jones, A. Sunda-Meya, and D. Grier. "Measuring Boltzmann's constant through holographic video microscopy of a single sphere," American Journal of Physics 82(1), 23-31 (2014).

[3] D. Jia, J. Hamilton, L. Zaman, and A. Goonewardene. "The time, size, viscosity, and temperature dependence of the Brownian motion of polystyrene microspheres," American Journal of Physics 75(2), 111-115 (2007).