s21_VideoMicroscopy

and the Boltzmann's constant to be which is about 14% within agreement of our expected value of

.

Introduction

Brownian motion, first described by Robert Brown in 1827[3], describes the seemingly random motion of small particles suspended in a fluid. This motion was ill described until 1905 when Einstein published his theory on the motion[4]. Einstein correctly assumed that the motion was caused by the fluid interacting with the particle. Because of this interaction the particle's position should statistically match fluid. Relating the mean squared displacement to be proportional to

, where is temperature and is Boltzmann's constant. Jean Perrin tested this theory[5], measuring the movement of the particles by hand in an optical microscope.

In modern microscopy[6], we use a digital camera to record a video and use software to analyze the position of the particle in the solution. This simple microscopy can analyze our x and y, but by using Mie scattering[7] and Mie-Lorenz fitting python program HoloPy. We can analyze the z position of our particle by fitting the holographic ring that scatter from the particle. Shining a laser with a wavelength close to the size of a colloidal sphere will produce an interference pattern that we can fit using Mie-Lorenz theory to get the position of the particle in the z position[1,2].

Theory

Brownian Motion

A particle floating through a fluid is affected by the thermal motion of the fluid. Thus the particle will follow the statistical qualities of the fluid. Meaning that the mean squared displacement(MSD) of the particle will follow the Einstein-Smoluchowski[11] equation giving us

is time and is our diffusion coefficient. Einstein proved that the diffusion coefficient is proportional to $k_bT$ with a drag coefficient[5]. George Stokes[12] calculated this drag coefficient giving us the Stokes-Einstein relation

Where

is the radius of the particle and $n$ is the dynamic viscosity of the fluid. This diffusion coefficient is found using the Einstein-Smoluchowki[11] knowing that our diffusion coefficient is related to slope of our mean squared displacement from equation (1).

Mie Scattering

A laser beam is coming in contact with a colloidal sphere that is about the same size as the wavelength of the laser beam. The laser beam can be expressed as

Where , and is the refractive index, is the wavelength of the laser, and is the amplitude of our laser. When the laser hits a particle it will undergo Mie-Lorenz scattering[1,2], which is similar to the inverse of our

with a Mie scattering function

Where is a Mie-Lorenz function[1](see Appendix A), and is the radius of the particle. Adding and we get the find our intensity of our image.

We can normalize this image by cutting out our background by dividing by the intensity of our laser. This hologram is our image and can then by fit using a Mie-Lorenz python program called HoloPy. HoloPy will return a value for the z axis based on the hologram.

Experimental Setup

A 17 mW, 633 nm laser reflects off two mirrors in a U-Shape and into a microscope with a camera attached, as seen in figure 1. Our sample containing colloidal spheres was prepared using cover slips and a glass slide. The cover slips were placed on top of the glass slide in box shape to create a sealable chamber. Nail polish was then used to seal the cover slips to the glass slide. The particle solution was then placed in the box after being diluted by a $10^{-5}$ volumetric ratio using deionized water.

Another cover slip is then placed over the “box” with oil on top of the slide for the objective. The sample was then placed on the microscope stage and coming in contact with our laser creating an interference pattern. Using our 100x oil immersion objective and video microscopy, our camera records the particles motion as it traverses the fluid.

Data and Results

Our fit yielded us with diffusion coefficients of ,,, for x,y and z respectfully. Taking the average of the diffusion coefficients from each direction gives us

which does not agree with the expected value of . Using our diffusion coefficients we calculated the Boltzmann constant giving us

, , and . Taking the average as our value we get .

The X,Y and the average all agree with accepted value while the Z does not. The Z measurement is off due for a couple reasons like, the images we used were small being roughly 100 pixels across, which makes the particles radius and its interference fringes only a couple a pixels across. This makes it harder for holopy to determine an accurate position measurement especially for the Z position due to the small fringes. This is also reason why the error bars for the Z plot are so large as holopy is much more uncertain in its Z position determination. We also used a slow frame rate of 10 fps to cover a wider period of time, which affected the amount of data we could collect.

Conclusion

By utilizing normal 2D microscopy and mie scattering on a colloidal spheres in a fluid, we tracked a particle in three dimensions and obtained an estimate of the Boltzmann constant. From our, images we calculated the particles radius and its mean squared displacement. Plotting the mean squared displacement and averaging the diffusion coefficients gave us a value that did not agree with with the expected value due to the Z position calculation errors. The diffusion coefficient allowed us to get an estimate for the Boltzmann constant that agrees with the accepted value within about 14 percent. For future iterations, higher resolution images at a higher frame rate would greatly improve accuracy of calculations. With more accurate calculations and better position measurements one could measure the density of the bead and the downward velocity of the sphere, which would allow for an estimation of gravity.

Acknowledgments

We would like to thank Kevin Booth for his help throughout the semester and Jacob Ritz, our advisor, for his guidance and expertise on our research.

References

[1] “Measuring Boltzmann’s constant through holographic video microscopy of a single colloidal sphere”; Bhaskar Jyoti Krishnatreya, Arielle Colen-Landy, Paige Hasebe, Breanna A. Bell, Jasmine R. Jones, Anderson Sunda-Meya, David Grier; https://aapt.scitation.org/doi/abs/10.1119/1.4827275

[2]“Characterizing and tracking single colloidal particles with video holographic microscopy”; Sang-Hyuk Lee, Yohai Roichman, Gi-Ra Yi, Shin-Hyun Kim, Seung-Man Yang, Alfons van Blaaderen, Peter van Oostrum, David G. Grier, https://physics.nyu.edu/grierlab/index10b/

[3] "A Brief Account of Microscopical Observa-tions”; Robert Brown;https://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf

[4] "Investigations on the theory of Brownian Movement”; Albert Einstein; http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf

[5] “Atoms”; Jean Baptiste Perrin; https://archive.org/stream/atomsper00perruoft#mode/1up\

[6]“Methods of Digital Video Microscopy for Col-loidal Studies”; John C. Crocker and David G.Grier;https://physics.nyu.edu/grierlab/methods3c/

[7] "Light Scattering Theory”; David W. Hahn;http://plaza.ufl.edu/dwhahn/Rayleigh%20and%20Mie%20Light%20Scattering.pd