Brownian Motion (PHYS 334)

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.

 A close-up photo of the microscope used for the experiment

Prelab Research

Coming to lab you should read through the theory section of the wiki and maybe even download and read Einstein's original paper on the subject. However you do it, make sure you know:

  • What Brownian motion is.
  • What a diffusion constant and mean free path are.
  • What physical parameters influence the average step size of a particle undergoing Brownian motion?

References


[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)

Day 1 and Day 2 questions


Complete this question before coming to lab on Day 1.Late work will not be accepted.

DAY 1 QUESTION: Looking at the definition of the diffusion coefficient, we see that we need to know the temperature, the viscosity of the solution, and the size of the particles. All three of these quantities have uncertainties associated with them. According to the manufacturer, the diameter of the microspheres is 0.96 μm with about a 10% variability. The uncertainty in the particle size sets a limit on how well we can expect to measure the value of the diffusion constant, and in turn sets a practical limit on how well we need to know the other parameters. (It does not make sense to spend a lot of time and effort measuring one parameter to within 0.001% when the uncertainty in another parameter limits our ability to final uncertainty to no better than 10%.) How well do you need to know the temperature and viscosity of the water?   

Complete this question before coming to lab on Day 2.Late work will not be accepted.

DAY 2 QUESTION:  The diffusion constant you obtained on the first day was likely different from the predicted value by more than you would expect due to purely statistical fluctuations. This suggests the possibility of some sort of systematic effect which we are not accounting for in our analysis. This is a common occurrence in experimental work, and much time and effort is put into investigating, understanding and characterizing systematic biases in the data. On the second day of this experiment you will investigate one such possible systematic effect. To help set the stage, consider the following list of possible sources of systematic bias in the data. For each item in the list, decide whether you would expect the effect to make the diffusion constant larger or smaller. Why would you expect this to be the case?

  1. High density of particles resulting in particles colliding with each other in addition to interacting with the water molecules.
  2. Some attractive force between particles causing small groups ( 2 or 3 particles ) to clump together.
  3. Gravity causing particles to slowly settle to the bottom of the drop and therefore interact with the microscope slide.
  4. Heating from the light source raising the temperature of the sample.

Outline


In this experiment you will observe Brownian motion of silica beads in water, and compare their motion to that predicted by statistical mechanics.

To first order, one would expect the particles to behave according to Einstein's model of Brownian Motion. However, this turns out not to be the case when observing the behavior of colloids in a fluid which rests between two glass surfaces (in our case, the microscope slide and cover slip). Interactions of the particles with these boundaries has been a subject of recent research.

  • See here for a synopsis of a recent paper on this problem, or check out the full article here (A. Alexandre, et. al., Phys. Rev. Lett. 130, 077101 (15 February 2023)).

Your task for this lab can be broken down into two parts.

  1. Begin by performing measurements of the diffusion constant for microspheres over a range of 0.5 μm to 5 μm without taking boundary conditions into consideration. Focus on understanding how to operate the camera and analyze the particle motions using TrackPy. You should aim to complete this part in the first two days of the lab. You should observe discrepancies between your measured diffusion constants and the predictions for pure Brownian Motion, likely with some dependence on particle size.
  2. Based on the results of the preliminary investigation of the particle behavior the group (and in consultation with the course instructors and informed by the work linked above), design an experiment to investigate the boundary effects.

The rest of this wiki is taken from the undergraduate version of this experiment and can be used as a guide to learning how to work with the apparatus and particle tracking software while measuring the diffusion constants for various particle sizes.

Theory


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

$\dfrac{\partial \langle x^2 \rangle}{\partial t} = \dfrac{2k_BT}{\mu},$ (1)

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,

$\mu = 6\pi\eta a$, (2)

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

$\langle x^2\rangle = \dfrac{2k_BT}{6\pi\eta a}t=2Dt$, (3)

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

$P(x) = \sqrt{\dfrac{1}{2\pi\sigma^2}}e^{-x^2/2\sigma^2},$ (4)

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

$\sigma^2=\left< x^2\right>=2Dt.$ (5)

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

Equipment


The sample

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.

 A photograph of the sample slide.  A drop of water, containing silica microspheres, is sandwiched between two glass slides, with electrical tape used for spacing.
Figure 1: Sample drop contained between a microscope slide and cover slip.

The microscope

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) (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. 

Appendix B at the end of this wiki page describes a procedure for focusing the microscope on the particles, which can be tricky the first time you do it. Note that this procedure was originally written based on a different model of microscope which we no longer use, including different image capture software. The concepts are the same however.

===== Image capture and analysis ===== 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.

Part I: Understanding Brownian Motion


Outline

In the first part 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.

Over the next few days of the experiment, you should either expand your range of data or systematically try to identify possible confounding factors that might cause the particles to not exhibit pure Brownian motion.

The second part of the experiment will be a deeper exploration of one (or more) systematic. 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).

Familiarizing yourself with the apparatus

Preparing a sample

Video on preparing particle solutions.

Video on preparing a microscope slide.

Vials containing pre-mixed solutions of particles with doubly-deionized water may be available or you may need to prepare your own.

If pre-mixed vials are available, shake the vial by hand to loosen up stuck particles, then place the vial on the “Vorex Genie” for about 30 seconds to ensure that particles are evenly mixed. If you need to prepare your own sample, ask the lab staff for assistance. You will need to remove a small volume of particles from the dense solution, place it in a clean vial, and add water gradually until an appropriate dilution is achieved.

To prepare a drop for viewing, we need to place a drop of a given solution onto a microscope slide and top with a cover slip. This will confine the drop to a small area, but allow the particles within the water to diffuse naturally.

Start by cleaning the taped microscope slide with a kimwipe to remove any dirt or residue. Check that the micropipette tip is clean, press down the button on the micropipette and insert into the vial of particles. When you release the button, a small volume of solution will be pulled into the tip. Move the pipette over the slide, and carefully place a single drop in the center of the slide. If the drop spreads to the edges of the slide or to the tape, wipe the surface clean and place another drop. Wipe a cover slip clean with a kimwipe and genly place over the drop. There will be a little bit of spreading; if the drop reaches the edges or tape, wipe clean and repeat.

Place the slide onto the microscope slide holder and turn the microscope on.

A photograph of the tools used in sample preparation

Recording a video using TopView

Video on recording a movie with Topview software.

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.

 A figure depicting the positions of particles within the 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

Calibrating the CCD

Video on recording a movie with Topview 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 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.

Particle Tracking


The information we wish to extract from the videos are the step sizes of the particles, along the X and Y axes, from one frame to the next. The theory of Brownian motion predicts that the distribution of step sizes should follow a gaussian form whose width is related to the diffusion constant of the particles in the solution.

To accomplish this we use a particle tracking package called TrackPy. If you would like you can download and install TrackPy on your own computer, but we have it setup on the lab computer and you should be able to do all of your particle tracking in the lab.

TrackPy is a sophisticated open source python package developed for use in research labs. It is not the goal of this lab to learn the details of how TrackPy works, you will use it as a tool in order to extract the information on particle tracks which you need. A quick overview of the process is as follows.

  • Record a video of your particles, making sure to save the video as an AVI format and in the folder from which you will run TrackPy.
  • Your TrackPy script is setup to work with an image stack. An image stack is simply each frame of the video saved as a separate image file. You will use an image processing program named ImageJ to convert your AVI format movie into an image stack.
  • You can then use TrackPy to identify particles in each of the images in the stack, and track their motion from one frame to the next.
  • At the end of this process you will have a text file containing a list of particles with the sizes of each step they took along the X and Y axes. You can then write your own analysis code in python to read in this data, create histograms, perform curve fitting, etc.

Creating The Image Stack

At this point you should already have a saved video in your working folder. Within this folder create a subfolder into which the image stack files will be saved.

Open the program ImageJ (which is displays the name FIJI on startup, do not be concerned by this as FIJI is just ImageJ). From ImageJ Open your video which must be in the AVI format. When you open the video you will be presented with a set of three options for how to read in the data; Load as Image Stack, Grey Scale, and XXX. Make sure that the first two items are checked.

Now save the video using Save As and choose Image Stack. Save the images as PNG files into the subfolder which you created for them.

If you did this correctly there should be a number of PNG image files in your image stack subfolder equal to the number of frames which were in your video.

Running TrackPy

Note that it is a good idea to start with a fresh copy of the TrackPy script. Beware of running copies which you may find locally on the lab computer as you do not know what changes might have been made to the code by a previous group.

Download a fresh copy of the TrackPy script into your data folder using this link.

TrackPy Script

Not visible to students

Tweaked TrackPy script

Start up a Jupyter notebook session in the browser and navigate to your copy of TrackPy. The script which we provide has been heavily commented and for the most part you can simply step through it by executing the cells. You will be prompted in the comments for when you need to enter something into the code, such as the name of the folder containing your Image Stack files, or user variables such as particle size or mass.

The TrackPy script steps you through the process of:

  1. Identifying candidate particles.
  2. Tracking candidate particles from one frame to the next and accounting for possible systematic motion of the ensemble of particles as a whole.
  3. Extracting just the particle step sizes in X and Y, and then saving them to a text file for you to analyze.

The particles in your video frames appear as small black dots. TrackPy selects candidate particles by searching each frame for gaussian blobs. You need to provide TrackPy with information defining which gaussian blobs are likely to be particles as opposed to other features in your frames such as a piece of dirt for example. The primary criteria you need to define are the particle size in terms of diameter in pixels, particle mass which corresponds to the integrated brightness of the blob, and the eccentricity which defines how circular the particles are.

To do this you will use a process which may seem arbitrary and unscientific. Essentially you pick some value for a parameter like particle size, type this value in for the appropriate variable in the code, and let TrackPy process one frame from your video. TrackPy then displays the image frame it just analyzed with a red circle drawn around every object it found which matched your criteria for particle size. You use your eye to judge whether or not the software did a good job of selecting what you know to be particles in the image, then change the value for particle size and reanalyze the frame to see if it did a better or worse job of selecting particles. Once you settle on a value for particle size you then proceed to repeat the process for a different criteria such as particle mass.

You are basically using a process of trial and error to find a set of parameters which cause TrackPy to mostly correctly identify and select particles. You are trusting in the ability of your eye-brain combination to correctly identify particles in the video (and your eye-brain is very very good at this) and your judgement in deciding when the parameter values you have entered are doing an adequate job. This is neither arbitrary nor unscientific, although it may feel that way.

So how do you determine when the particle selection criteria are “good enough”? That is up to your judgement as the experimenter, trust your judgement. If your selection criteria are too narrow the software might be missing some particles, but if it is only missing a few that is ok. If your criteria are too wide you could be allowing the software to select things which are clearly not particles, this would be less acceptable than missing a few particles. Developing and learning to trust your judgement is part of this course, so go with it and see what happens.

Part II: Further Exploration


Part II of the experiment is intentionally left open-ended. You may need to consult outside resources for more theory information or equipment manuals for details about the apparatus. Given your experience collecting data in Part I, use your judgement to determine effective collection and analysis strategies, and budget your remaining time in lab appropriately.

Based on the results of the preliminary investigation of the particle behavior the group (and in consultation with the course instructors and informed by the work linked above), design an experiment to investigate the boundary effects caused by the presence of the microscope slide and cover slip.

Appendix A: Silica settling times

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).

Appendix B : Focusing the microscope

Video on focusing the microscope.

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

 A figure depicting the positions of particles within the slide
Figure 4: Particles in the sample fluid will be concentrated in a band somewhere between the cover slip and the microscope slide.
  • Begin by looking at the slide with the 4x objective as in Fig. 6(a). (This is a total of 40x magnification: the 4x objective in conjunction with the 10x eyepiece.) Move the slide around until you find the meniscus (edge) of your drop and adjust until that is in focus. You may need to increase the light intensity or adjust the aperture or condenser position to make the image clearer.
  • Increase the objective to 10x (for a total magnification of 100x) as in Fig. 6(b). Refocus on the meniscus and re-adjust the light as needed. You may not see the particles clearly yet (unless you are using the biggest particles), but you may see artifacts or defects that exist on the cover slip, slide, lenses, or in the water. Note that these features (except for the ones on the lens) come into and out of focus as you adjust… you are seeing that the focal plane is changing up or down within the sample and that different things are visible at different levels.
  • Increase the objective to 40x (for a total magnification of 400x) as in Fig. 6(c). Refocus and adjust the light. Look first for the meniscus, but now focus deeper into the sample and look for particles. You may find more than one layer of particles; you are looking for the most dense layer. Move the slide stage around and look for areas where particles are not clumped together and are moving freely.
  • We typically do not need to go to the highest objective of 100x (1000x total magnification).
  • Once you have a clear and focused patch of particles, you can begin to play with image exposure settings to help make the particles more clearly visible (as in Fig. 6(d)). We will discuss this process next.

(a) View of the sample at the lowest magnification of 40x. (b) View of the sample at a magnification of 100x. The particles will not be visible at this magnification. (c) View of the sample at a magnification of 400x. The particles are now visible. You can see areas where several particles have clumped together. (d) This is the same 400x image, but with the camera gamma turned up the capture software. Notice how much darker the individual particles appear. The trade off however is that the background becomes less uniform. The background can be subtracted out through the recording software.

Figure 6 : Focusing the microscope image