^{1}, Paul M. Tyler

^{1}, Josef G. Trapani

^{1}and Ashley R. Carter

^{1,a)}

### Abstract

Brownian motion experiments have become a staple of the undergraduate advanced laboratory, yet quantification of these experiments is difficult, typically producing errors of 10%–15% or more. Here, we discuss the individual sources of error in the experiment: sampling error, uncertainty in the diffusion coefficient, tracking error, vibration, and microscope drift. We model each source of error using theoretical and computational methods and compare the model to our experimental data. Finally, we describe various ways to reduce each source of error to less than 1%, improving the quantification of Brownian motion.

The authors would like to thank Amy Wagaman for careful reading of the manuscript. This work was supported by a Howard Hughes Medical Investigator undergraduate fellowship (MAC), a SOMAS-URM grant (JGT), and Amherst College.

I. INTRODUCTION

II. EXPERIMENTAL MATERIALS AND METHODS

III. THEORETICAL AND COMPUTATIONAL MODELING OF ERROR

A. Sampling error

B. Uncertainty in the diffusion coefficient

C. Tracking error

D. Drift or vibrational noise

E. Multiple sources of error

IV. REDUCING ERROR

A. Experimental methods to reduce error

B. Computational methods to reduce error

V. CONCLUSIONS

### Key Topics

- Diffusion
- 33.0
- Brownian motion
- 19.0
- Random walks
- 7.0
- Undergraduates
- 7.0
- Error analysis
- 6.0

##### G09B

## Figures

(Color online) A simple implementation of the Brownian-motion experiment produces data with 15%–30% error. (a) Cartoon showing a bright field microscope that images micron-sized polystyrene beads undergoing Brownian motion in water. (b) Image taken at 100× magnification; beads (labeled 1 and 2) are just below the focal plane and appear black. (c) Post-processing produces a binary image. (d) The position of each particle in x and y is extracted for each frame and placed into a track. (e) In Method 1, the diffusion coefficient is found by making a histogram of the displacements in both x and y (gray) and fitting a Gaussian curve (black line) to the data; the standard deviation (dashed black lines) is used to calculate a diffusion coefficient of 0.512 μm2/s, a difference of 15.1% from the nominal value. (f) In Method 2, the mean squared displacement (MSD) of each particle (gray) as a function of time is averaged together (black) and the slope of the line is used to find a diffusion coefficient of 0.575 μm2/s, a difference of 29.2%. Labeled MSD traces correspond to the particles shown in B through D.

(Color online) A simple implementation of the Brownian-motion experiment produces data with 15%–30% error. (a) Cartoon showing a bright field microscope that images micron-sized polystyrene beads undergoing Brownian motion in water. (b) Image taken at 100× magnification; beads (labeled 1 and 2) are just below the focal plane and appear black. (c) Post-processing produces a binary image. (d) The position of each particle in x and y is extracted for each frame and placed into a track. (e) In Method 1, the diffusion coefficient is found by making a histogram of the displacements in both x and y (gray) and fitting a Gaussian curve (black line) to the data; the standard deviation (dashed black lines) is used to calculate a diffusion coefficient of 0.512 μm2/s, a difference of 15.1% from the nominal value. (f) In Method 2, the mean squared displacement (MSD) of each particle (gray) as a function of time is averaged together (black) and the slope of the line is used to find a diffusion coefficient of 0.575 μm2/s, a difference of 29.2%. Labeled MSD traces correspond to the particles shown in B through D.

Simulated data show that the largest sources of error in the experiment are sampling error, tracking error, and vibration. (a) and (b) The diffusion coefficients (D) from 5,000 simulated data sets are plotted (gray circles) as a function of the number of tracks in a data set for either Method 1 or Method 2. The standard deviation of the diffusion coefficient, σ (thick black line), follows the theoretical sampling error for each method (yellow dashed line). The mean value of the diffusion coefficient (thin black line) tracks the nominal diffusion coefficient , as expected. (c) through (f) Same data as in A and B (black line) except with the following added sources of error: uncertainty in the bead radius and temperature (magenta line); tracking error (thick red line); drift (blue filled circles); vibration (green line); and all of the above (gray dashed line). In (c) and (d) the fractional sample standard deviation remains virtually unchanged by the sources of error, and all of the traces are on top of the theoretical sampling error (yellow line). In (e) the fractional difference between the mean and nominal value of the diffusion coefficient shows that the simulated data with tracking error and vibration are above the noise floor set by sampling error in Method 1. In Method 2 (f), the sampling error causes large fluctuations in .

Simulated data show that the largest sources of error in the experiment are sampling error, tracking error, and vibration. (a) and (b) The diffusion coefficients (D) from 5,000 simulated data sets are plotted (gray circles) as a function of the number of tracks in a data set for either Method 1 or Method 2. The standard deviation of the diffusion coefficient, σ (thick black line), follows the theoretical sampling error for each method (yellow dashed line). The mean value of the diffusion coefficient (thin black line) tracks the nominal diffusion coefficient , as expected. (c) through (f) Same data as in A and B (black line) except with the following added sources of error: uncertainty in the bead radius and temperature (magenta line); tracking error (thick red line); drift (blue filled circles); vibration (green line); and all of the above (gray dashed line). In (c) and (d) the fractional sample standard deviation remains virtually unchanged by the sources of error, and all of the traces are on top of the theoretical sampling error (yellow line). In (e) the fractional difference between the mean and nominal value of the diffusion coefficient shows that the simulated data with tracking error and vibration are above the noise floor set by sampling error in Method 1. In Method 2 (f), the sampling error causes large fluctuations in .

Reduction of experimental noise produces measurements of D within 1%. Either bright field microscopy (a) or epifluorescence microscopy (b) can be used; however, fluorescence images have a higher contrast and do not require background subtraction. (c) and (d) Measurements of the diffusion coefficient D using Method 1 are found by fitting a Gaussian (black line) to a histogram of the tracked displacements (gray) to find the standard deviation (black dashed line). For bright field microscopy in (c), D is (mean ), within error of the nominal value of 0.415 μm2/s. For epifluorescence microscopy in (d), D is , also within error of the nominal value of 0.430 μm2/s. (e) and (f) Measurements of D using Method 2 are found by fitting the slope of an averaged ensemble MSD (black) made up of individual MSDs for each track (gray). Here, D is found to be using bright field microscopy in (e), while fluorescence microscopy in (f) gives a D of , both within error of the nominal values.

Reduction of experimental noise produces measurements of D within 1%. Either bright field microscopy (a) or epifluorescence microscopy (b) can be used; however, fluorescence images have a higher contrast and do not require background subtraction. (c) and (d) Measurements of the diffusion coefficient D using Method 1 are found by fitting a Gaussian (black line) to a histogram of the tracked displacements (gray) to find the standard deviation (black dashed line). For bright field microscopy in (c), D is (mean ), within error of the nominal value of 0.415 μm2/s. For epifluorescence microscopy in (d), D is , also within error of the nominal value of 0.430 μm2/s. (e) and (f) Measurements of D using Method 2 are found by fitting the slope of an averaged ensemble MSD (black) made up of individual MSDs for each track (gray). Here, D is found to be using bright field microscopy in (e), while fluorescence microscopy in (f) gives a D of , both within error of the nominal values.

Tracking with IMAGEJ or with a MATLAB program produces similar results. (a) We track a single bead undergoing Brownian motion in water with IMAGEJ MTRACK2 (gray closed circles), and a MATLAB tracking program with an improved centroid-finding algorithm 27 (black open circles). (b) We also track a bead stuck to the surface with both programs (colors same as in a).

Tracking with IMAGEJ or with a MATLAB program produces similar results. (a) We track a single bead undergoing Brownian motion in water with IMAGEJ MTRACK2 (gray closed circles), and a MATLAB tracking program with an improved centroid-finding algorithm 27 (black open circles). (b) We also track a bead stuck to the surface with both programs (colors same as in a).

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