s19HolographicVideoMicroscopy

Measuring Boltzmann's Constant with Holographic Video Microscopy

Nick Barth and Mason Hastings

University of Minnesota, Twin Cities

School of Physics and Astronomy

Minneapolis Minnesota 55455

Abstract

The value for the Boltzmann constant was found to be k_B = (1.33 ± 0.25) x 10^-23 J/K in agreement with the accepted value of 1.3806 x 10^-23 J/K by tracking the position of a polystyrene latex sphere with a diameter of approximately 3μm as it diffuses through water using holographic video microscopy techniques. The relationship between the observed motion of the particle and the thermal energy of the solution provides the means to determine the Boltzmann constant. In addition to the Boltzmann constant, the refractive index and radius of the sphere were found to be n_p = 1.611 ± 0.002 and a_p = 1.452 ± 0.007 μm respectively using the same holographic video microscopy techniques.

Introduction

In digital holographic video microscopy the interference pattern between a reference plane wave and the wave that scattered off a particle is recorded in each frame of a video. This recorded interference pattern contains the information necessary to reconstruct a three dimensional image which makes the recording a hologram by definition. From the hologram, the position of a particle in three dimensions can be determined if the scattered wave can be calculated as a function of the reference wave and the size, position, and refractive index of the particle. In this experiment, the position as a function of time is determined by using a video camera to record the holograms and calculate the position for every video frame. The motion of a sphere in a fluid with known viscosity is described by Brownian motion, from which a value for the Boltzmann constant was calculated.

Theory

What are we looking at?

Brownian motion is being observed in this project and so it is important to know that Brownian motion is the random motion of a particle caused by collisions with water molecules when in an aqueous solution. These collisions occur because of the thermal energy present in the water, and the motion itself is governed by the Stokes-Einstein equation.

In this equation, "D" is the diffusion coefficient, "k_B" is the Boltzmann constant, "T" is the temperature, "n" is the viscosity of the surrounding liquid, and "a_p" is the radius of the particle. From the diffusion coefficient, we can find the value of Boltzmann's constant.

How do we record the motion of the particle?

The particle is placed in the path of the HeNe laser, which is the reference wave, E₀(r,t), given by

where u(r) is the amplitude profile of the wave at a position r, k is the wave vector, and ω is the frequency. The particle scatters a small amount of E₀(r,t) which makes a scattered wave, E_s(r,t) given by

where f_s(k(r-r_p)) is the scattering function described using Lorenz-Mie theory, and r_p is the position of the particle.

In the focal plane of the microscope, these two waves combine to make an interference pattern with intensity I(r) given by

Apparatus

Our experimental apparatus consists of a microscope and a 19mW, 632 nm HeNe laser. Essentially, we replaced the traditional light source of the microscope with the laser, and fixed a camera to the top of the microscope to record a video of our samples. The samples were made very crudely using a microscope glass slide and multiple cover slips to create a volume for our colloidal sphere solution. A schematic of the volume can be seen in the apparatus diagram below, where the "spacers" we used were smaller bits of a standard cover slip (we broke the cover slips in half and used each half as a spacer), and the glue we used was nail polish primer.

Here is a gif of data that we took with this method (after the image was normalized), showing the observed scattering of the light around the particle in addition to the Brownian motion of the particle using our equipment.

By recording I(r), and knowing that E₀(r,t) is a simple plane wave, f_s(k(r-r_p)) can be determined, which includes the position of the particle in three dimensions, the radius of the particle, and the refractive index of the particle.

Results

We were able to obtain a value for Boltzmann's constant of k_B = (1.33 ± 0.25) x 10^-23 J/K which is in agreement with the accepted value of 1.3806 x 10^-23 J/K. The large uncertainty is large because only 5 seconds of video suitable for processing was obtained. Longer videos often had too much noise in the normalize holograms to adequately fit.

In addition to the Boltzmann constant, the refractive index and radius of the sphere were found to be n_p = 1.611 ± 0.002 and a_p = 1.452 ± 0.007 μm respectively.

Mean squared displacement

Below if a graph of the mean square displacement, Δr²_s( Δst₀), over a time interval Δst₀ where t₀ is the time interval between video frames, and s is an integer. Δr²_s is given by

This is a schematic diagram of our apparatus, showing the path of the laser on the right and the interaction between the laser and our samples on the left. As indicated, our samples were placed below the 100x objective lens.

(Note: two ± were used because the + sign in LaTeX wasn't working)

Here is our actual lab setup, with the camera fixed to the top of the microscope, cocked sideways in order to prevent a defect in the tube lens in the microscope from effecting our data, as can be seen in the second picture. The third picture is just showing our beam path before the microscope. Contrary to the schematic figure, we did not put the laser directly in the microscope, we walked it across two mirrors and two irises beforehand, to make sure the beam was parallel to the table as it entered the back of the microscope.

Radius and Refractive Index

Below is a plot for measurements of the refractive index, n_p and the radius of the particle, a_p.

Conclusions

As a proof of concept, our project was successful. While we were not able to achieve an accurate position measurement in the z-direction, we were able to show that it can be done given the proper preparation. The main problem that we faced throughout the course of this project was the slide preparation. With more time allotted to sample preparation I believe we would have been able to gather proper data for the z-axis movement and been able to gather an accurate measurement of Boltzmann's constant.