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Brownian motion is the random motion exhibited by a particle suspended in a fluid (either a liquid or a gas). This random motion results from the constant bombardment of the particle from all sides by the constituents of the fluid. Early observations of Brownian motion provided some of the first evidence of atoms. In 1905, Einstein published a paper showing that Brownian Motion could be explained by assuming that fluids were composed of atoms moving randomly with an average kinetic energy. In 1908, Jean Perrin experimentally confirmed the predictions of Einstein's theory, for which he was awarded the Nobel Prize in Physics in 1929. In this experiment, we will observe the Brownian motion of silica beads suspended in water, and use particle-tracking software to measure the mean-squared displacement of these beads over a period of time. Using a simple model from statistical mechanics, we can look at how diffusion scales with particle size.

[1] A. Einstein, "Investigations on the theory of the Brownian movement", 1926 (Reprinted by Dover Publications, 1956).

[2]  F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw - Hill, New York, 1965).

[3] TrackPy: Fast, Flexible Particle Tracking Toolkit for Python, v0.3.2. (github.com/soft-matter/trackpy and soft-matter.github.io/trackpy)

In this experiment you will observe Brownian motion of silica beads in water, and compare their motion to that predicted by statistical mechanics. Specifically, your objectives for this experiment include the following:

• to learn to mix particle dilutions and prepare microscope slides;
• to learn to focus and adjust an optical microscope;
• to use image and video capture software to record videos of random particle motion;
• to understand how the particle-tracking software TrackPy works, and to adapt the software to our individual experiment;
• to collect particle tracks and compute the mean-squared displacement of these tracks;
• to compare the observed displacements to those predicted by theory, and to investigate how particle diffusion scales with particle size; and
• to investigate how statistical and systematic effects can bias your measurements.

The theory of the Brownian motion of spherical particles can be derived from statistical mechanical considerations. (See Refs [1] and [2] for a full treatment of the problem.) Here we will simply summarize the development of the theory. The motion of the particle is assumed to be governed by the following two forces:

• a time-independent dissipative frictional force $\mu$, caused by the particle's motion through a fluid with a non-zero viscosity, and
• a time-dependent random bombardment of the particle from all directions by the atoms of the fluid.

Using the equipartition theorem, it can be shown that the mean squared displacement of the particle in one dimension, $\langle x^2 \rangle$,  is given by

where$k_B$ is Boltzmann's constant and $T$  is the absolute temperature of the fluid. Since we are assuming spherical particles, we can use Stokes' law for the frictional force on a spherical particle moving through a viscous fluid,

where $a$ is the particle radius and $\eta$ is the viscosity of water (which varies slightly with temperature). Thus, integrating we find

where $D = k_BT/6\pi\eta a$ is defined as the diffusion coefficient. In a given time interval $t$, the random collisions with the particles in the fluid will give rise to a random displacement, and the probability for any particular displacement, $x$, is given by the Gaussian distribution

where the width of the distribution is related to the diffusion coefficient,

If one experimentally measures these displacements in one dimension (e.g. along either the x- or y-axes), you can plot a normalized histogram and fit to Eq. (4) in order to extract the diffusion coefficient.

The particles we will investigate are silica spheres of (close to) uniform size suspended in water. (Particle sizes ranging from about 0.5 to 5 μm are available.) Solutions may have been prepared ahead of time, or you may need to mix your own (with help from the lab staff).

Your sample will be a microscope slide consisting of a single drop of the sample solution held between the slide and a cover slip as shown in Fig. 1. It is important that the boundaries of the sample drop not reach the edges of the cover slip, microscope slide, or the spacer tape as this will create a flow of the solution towards the point of contact. Due to evaporation, your drop will disappear over the course of a day or so. Likewise, due to gravity, the particles will slowly settle out of solution on the scale of a few hours. It is therefore likely that you will not be able to reuse your sample slide from the first day on the second day of the experiment. Preparing a new sample on the second day (from the same dilution) will not impact the results of the experiment.

Figure 1: Sample drop contained between a microscope slide and cover slip.

A compound binocular microscope will be used for viewing and recording movies of the particles undergoing Brownian motion. Several components of the microscope are illustrated in Fig. 2, and the device has the following characteristics of importance to the experiment:

• a specimen stage which translates in both x and y;
• coarse and fine focus controls;
  • The fine focus adjustment moves the eyepiece by 0.5 $\mu$m per small tick mark
• a rotating turret containing 4x, 10x, 40x, and 100x objectives;
• two 10x eyepieces, and a CCD camera for recording images and movies;
• a variable-intensity light source; and
• an Abbe condenser with adjustable iris.

(a)	brownian_microscope_lighting (b)
Figure 2: The features of the microscope are highlighted. Figure (a) shows the view below the sample stage, whereas figure (b) shows a bigger, overall view of the microscope.

The total magnification of the microscope is given by the product of the 10x eyepiece and the selected objective's magnification.

The condenser is used to take the light coming from the light source and focus it into a cone which narrows to a point at the sample, and then expands back into a cone of light entering the objective. The condenser needs to be adjusted separately for each objective, so that the cone angle matches the numerical aperture (or acceptance angle) for that objective. Getting the best image means adjusting the position of the condenser, the opening diameter of the iris, and the brightness of the light source. 

The frames of the video are then analyzed in TrackPy, a python-based package for identifying and tracking particles. You will run TrackPy in-lab on a department computer which has been set up for you, but you are encouraged to download and install the TrackPy package on your own computer so that you have the ability to work with your video file outside of lab. TrackPy is a set of python routines which does the following: - identifies individual potential particles in each frame of the video and allow the user to specify different criteria for inclusion or exclusion; - tracks the locations of individual particles from one frame to the next, thereby reconstructing the path which the particles trace out over the course of the video; and - calculates and outputs statistical information about the trajectories of these particles for use in analyzing the behavior of individual particles or of ensemble averages of particles. TrackPy gives the user a lot of flexibility in determining the values of various parameters which are used to identify and track the particles. A large part of what you do in lab will be changing these parameters and observing the effects of the changes. ===== 3.5 Installing ImageJ for home use (optional) ===== ==== 3.5.2 TrackPy ==== To install the TrackPy package on your own computer, first make sure that you have installed |Anaconda. Open the “Anaconda Prompt”, “Terminal” (Mac only) or “Command Prompt” (Windows only) and type and execute the following commands (agreeing to proceed when prompted): conda update conda
conda install -c conda-forge trackpy
conda install -c conda-forge pims
conda install pandas=0.23.0 If there are no issues, you should be able to run any file that calls the TrackPy library. We provide a customized tutorial notebook below, but other example notebooks detailing the functionality of the package are available here. (Click “Clone or download” and select “Download ZIP”. Unzip the resulting file to a convenient location on your computer.) === 3.5.2.1 What to do if there are issues === If you are having issues (in particular if you see the error “'frame' is both an index level and a column label.”), there is a possible fix below: == Create a separate environment == You can try creating a new environment in Python for your code.  To do so, open up the Anaconda navigator and select the 'Environments' tab.  Click on the 'Import' button on the bottom of the screen, and use the following file for the specification: PHY211Config.yml After everything downloads, you should have an Anaconda environment with appropriate versions of the needed packages.  Simply select the new environment by clicking on it and then launch Jupyter notebook as normal. For the command line inclined,  conda env create -f PHY211Config.yml -n phy211
conda activate phy211 should achieve the same effect, provided you have the config file in an appropriate directory.

On the first day of the experiment, you will gain experience in performing the following tasks:

• preparing samples and using the microscope;
• recording videos;
• calibrating the plate scale of the CCD camera;
• Using Tracker to capture the motion of particles over time; and
• outputting useful statistical information about your tracks via Python.

At the end of the first day (or before returning on the second day), you should obtain a histogram of particle displacements extracted for at least one video, and you should calculate at least a preliminary estimate for diffusion coefficient.

On the second day of the experiment, you will investigate systematic effects. You will have some freedom to choose what effects you study, and the approach will be open-ended as you explore different ways to look into your chosen topic(s).

The third day should be spent either expanding your range of data from day 2, or systematically trying to identify possible confounding factors that might cause the particles to not exhibit pure brownian motion.

The particles we are using are too small to see individually with our eyes, so we need to us the microscope to magnify them. Figure 4 shows a schematic of the situation, and shows that typically particles will cluster around the middle of the drop (between the upper cover slip and lower microscope slide.

Figure 4: Particles in the sample fluid will be concentrated in a band somewhere between the cover slip and the microscope slide.

In order to find these particles, we will focus the microscope in steps.

1. Make sure that the CCD camera is connected to one of the computer USB ports.  
2. Open the ToupView application on the desktop.
3. The main window should look something like Fig. 5.

Figure 5: A screenshot of the ToupView software

You are provided with a Bausch and Lomb Standard Series Stage Micrometer (see Fig. 10) for calibrating the pixel scale of the CCD camera. The stage micrometer has rulings scribed on it in increments of 0.1 mm and 0.01 mm.

(a)
Figure 10: The Bausch and Lomb Standard Series Stage Micrometer slide contains a calibrated scale is in the center of the reflective disc in the middle of the slide.

You need to record an image of the micrometer scale for each magnification at which you record data. The distance between rulings in pixels can be measured in an image processing program and compared to the known physical distance.

After completing the above steps, you'll use the tracker software to track the positions of several particles. The following video will show you how to get started, but as a quick summary:

1. Open your video with Tracker
2. Select Track → New → Point Mass
3. Create a keyframe by control-shift clicking on a particle
   1. command-shift on a Mac
4. Click on the Search button in the Autotracker
5. Fix issues as they arise
   1. You can override positions by shift-clicking on the particle

After following several particles, you will export the particle tracks and process them via a Python script. This script can find diffusion coefficients in two ways:

• One is to plot a histogram of particle displacements over a given time step. The variance of such a distribution is proportional to the diffusion coefficient.
• The second method is to break up all trajectories into sub-trajectories of given time length and to explicitly plot $\langle x^2 \rangle$ versus time, $t$. The slope of such a plot yields the diffusion coefficient.

This experiment provides a large parameter space to investigate. Recall that we expect the mean squared displacement (msd) $\langle x^2\rangle$ to be equal to $\frac{2k_BT}{6\pi\eta a}t$. Thus, immediately at hand we expect that:

• msd will scale linearly with time $t$
• msd will scale inversely with particle size $a$
• msd will scale linearly with temperature $T$
• msd will scale inversely with viscosity $\eta$

As important as the things we expect will affect the msd are the things that we can control that should not affect it:

• msd does not have a term for position within the sample
• msd does not have a term for time since the slide was prepared
• msd does not have a term for particle-particle interactions
• msd does not have a term for video parameters such as
  • framerate
  • exposure time
  • length of video
  • magnification

Note that just because a term doesn't appear in the equation doesn't mean it can't have an effect; it may be that a more complex model is needed to properly characterize the system. Your job here is not to come up with a new model, but rather to determine the limits of the existing one.

These ideas are not an exhaustive listing of everything that can be investigated, but more of a starting place for you. Some (such as testing temperature dependence) are not particularly feasible with our apparatus.

By its nature, this is an open ended task. There is no “correct” answer to compare your results to. You may find that the diffusion coefficient you get from the data changes very little, or not at all as a function of the parameter you choose. In this case you can say that the method we are using here is insensitive to that particular effect. Or you may find that at, above or below a certain value the analysis effected in some way which you should elaborate upon. In a real experiment you would spend considerable time investigating as many of these types of systematic effects as possible in order to better understand your final results.

Your written analysis that you submit to be graded should be built around your final conclusions. Everything in your analysis should support your final result and conclusions. For this experiment your final result(s) may end up being the measured diffusion coefficients for various microsphere solutions. Your conclusions would be your evaluation of how well your measured values did or did not match a theoretical prediction and a discussion of anything you may have discovered about how the results depend on factors such as particle size, density or settling time.

You need to make clear things you did, decisions you made in the lab which are important to understanding how you arrived at your results and conclusions. This might include:

• Information about what particles were studied, what parameters such as number density of particles in solution were varied.
• It should be made clear how any measured quantities were arrived at, for example were particle radii taken from the data on the vials or were they measured directly from the microscope images.
• Any calibration of the image scale should be described, including images, plots and calculations.
• Parameters used in the particle selection and any anomalies that were observed should be noted and discussed.
• How you assessed, quantified and propagated uncertainties should be discussed and made clear including examples of how calculations were done where appropriate.
• Details of how diffusion constants were obtained should include calculations, plots of particle step size distributions and/or lagtime should be provided.

The above list is not intended to be complete, nor should it be treated as a checklist of what should go into your written analysis. Your analysis needs to make clear to the reader what your results and conclusions are, show how your data support those conclusions, demonstrate how you processed the data, etc.

For this quarter we are focusing on developing your skills in data analysis and drawing appropriate conclusions from your data. Your analysis should focus on these things. You should not include sections on the apparatus, background theory, historical significance, and things like this. This is not to say that these things are unimportant, they are just not part of a report on your analysis and results.

Your analysis will be evaluated based on the following rubric. The rubric is not a format for your analysis, you are not expected to have a specific section on Data Handling or Presentation of Data. Elements of the different rubric categories will appear at different points through out your analysis writeup. For example you will be presenting data in your discussion of the calibration, your discussion of diffusion constants and likely in your final results. Your writeup of your analysis should be structured in a way that is clear and readable, there should be a logic to the flow of it.

Each item below is graded on a 0-4 point scale:

• 4 – Good (A): completes all listed tasks and provides appropriate context; thinks carefully about data and analysis; addresses all concerns raised by the results (where appropriate).
• 3 – Adequate (B): misses one or more minor element or lacks appropriate context; leaves a problem or ambiguity unaddressed; does not present analysis clearly enough.
• 2 – Needs improvement (C): omits or mishandles one or more item which renders the analysis fundamentally incorrect or incomplete; presents results in an incorrect or unclear way.
• 1 – Inadequate (D): omits or mishandles multiple items or treats them at an insufficient level; presentation is overall muddled or inaccurate; flaws in logic or process.
• 0 – Missing (F): omits all elements or makes no meaningful attempt.

All rubric items carry the same weight. The final grade for the analysis will be assigned based on the average (on a 4.0 scale) over all rubric items.

Our samples are from Bangs Labs (Functionalized silica spheres).

Example data sheet

Bangs Labs Technote

For our spheres the maximum settling velocity in water at room temperature should go as

$V_{m} = 5.448\times10^{-2} (\rho_{s} - 1) d^{2}$  where: 

• $V_{m}$  = maximum settling velocity in (μm/sec).
• $\rho_{s}$  = sphere density (g/cm3).
• $d$  = mean diameter of sphere (μm).