COS 126 Programming Assignment

The Atomic Nature of Matter

Re-affirm the atomic nature of matter by tracking the motion of particles undergoing Brownian motion, fitting this data to Einstein's model, and estimating Avogadro's number.

Historical perspective. The atom played a central role in 20th century physics and chemistry, but prior to 1908 the reality of atoms and molecules was not universally accepted. In 1827, the botanist Robert Brown observed the random erratic motion of wildflower pollen grains immersed in water using a microscope. This motion would later become known as Brownian motion. Einstein hypothesized that this Brownian motion was the result of millions of tiny water molecules colliding with the larger pollen grain. particles. In one of his "miraculous year" (1905) papers, Einstein formulated a quantitative theory of Brownian motion in an attempt to justify the "existence of atoms of definite finite size." His theory provided experimentalists with a method to count molecules with an ordinary microscope by observing their collective effect on a larger immersed particle. In 1908 Jean Baptiste Perrin used the recently invented ultramicroscope to experimentally validate Einstein's kinetic theory of Brownian motion, thereby providing the first direct evidence supporting the atomic nature of matter. His experiment also provided one of the earliest estimates of Avogadro's number. For this work, Perrin won the 1926 Nobel Prize in physics.

The problem. In this assignment, you will redo a version of Perrin's experiment. Your task is greatly simplified because with modern video and computer technology (in conjunction with your programming skills), it is possible to accurately measure and track the motion of an immersed particle undergoing Brownian motion. We supply video microscopy data of polystyrene spheres ("beads") suspended in water, undergoing Brownian motion. Your task is to write a program to analyze this data, determine how much each bead moves between observations, fit this data to Einstein's model, and estimate Avogadro's number.

The data.  We provide several datasets, obtained by William Ryu using fluorescent imaging. Each run contains a sequence of two hundred 640-by-480 grayscale JPEG images, frame00000.jpg through frame00199.jpg.

Here is a movie of several beads undergoing Brownian motion. Below is a typical raw image (left) and a cleaned up version (right) using thresholding as described below.

Each image shows a two-dimensional cross section of a microscope slide. The beads move in and out of the microscope's field of view (the x and y directions). Beads also move in the z-direction, so they can move in and out of the microscope's depth of focus; this results in halos, and it can also result in beads completely disappearing from the image.

Particle identification. The first challenge is to identify the beads amidst the noisy data. Each image is 640-by-480 pixels, and each pixel is represented by a grayscale value from 0 (black) to 255 (white). Whiter pixels correspond to beads (foreground) and blacker pixels to water (background). We break the problem into two independent pieces: (i) classifying the pixels as foreground or background, and (ii) finding the disc-shaped clumps of foreground pixels that constitute each bead.

• Foreground / background separation. We use a simple, but effective, technique known as thresholding to separate the pixels into foreground and background components: all pixels with grayscale values strictly below τ are considered background, and all others are considered foreground. The picture below illustrates the above frame after thresholding, using τ = 180.0. This value of τ results in an effective cut for the supplied data. See figure above (right).

• Blobs. A polystyrene bead is typically represented by a disc-like shape of (at least) 25 connected foreground pixels. A blob or connected component is a maximal set of connected foreground pixels, regardless of shape or size. For this assignment, we will refer to any blob containing (at least) 25 pixels as a bead. The center-of-mass of a blob (or bead) is the average of the x and y coordinates of its constituent pixels. Create a data type Bead that has the following API.

  public Bead()                    // construct an empty blob
  public void add(int i, int j)    // add a pixel (i, j) to the blob
  public int mass()                // return number of pixels added = its mass
  public double distanceTo(Bead b) // return distance from the center of mass of this bead to b
  public String toString()         // return string representation of this blob
  

• Blob detection. Use depth first search to efficiently identify the blobs.

  For this part, write a data type BeadFinder that has the following API.

     // find all beads in the give picture using the given grayscale threshold
     public BeadFinder(Picture picture, double threshold)
  
     // return all beads that are at least P pixels large
     public Bead[] getBeads(int P)
  

  Include a main() method in BeadFinder.java that takes the name of a JPEG file as a command line argument and identifies all of its beads according to the method described above.

  % java BeadFinder run_1/frame00001.jpg
  29 (214.7241,  82.8276)
  36 (223.6111, 116.6667)
  42 (260.2381, 234.8571)
  35 (266.0286, 315.7143)
  31 (286.5806, 355.4516)
  37 (299.0541, 399.1351)
  35 (310.5143, 214.6000)
  31 (370.9355, 365.4194)
  28 (393.5000, 144.2143)
  27 (431.2593, 380.4074)
  36 (477.8611,  49.3889)
  38 (521.7105, 445.8421)
  35 (588.5714, 402.1143)
  

  The program identifies 13 beads in sample frame. (It does not count the blob abutting the right edge or the 1-pixel blob.) Our string representation of a bead specifies its mass (number of pixels) and its center of mass (in the 640-by-480 picture). By convention, pixels are measured from left-to-right, and from top-to-bottom (instead of bottom-to-top).

Particle tracking. The next step is to determine how far a bead moved from one time step t to the next t + Δt. For our data, Δ t = 0.5 seconds per frame. We assume the data is such that each bead moves a relatively small amount, and that two beads do not collide. (However, we must account for the possibility that the bead disappears from the frame, either by departing the microscope's field of view in the x or y direction, or moving out of the microscope's depth of focus in the z direction.) Thus, for each bead at time t + Δt, we calculate the closest bead at time t (in Euclidean distance) and identify these two as the same beads. However, if the distance is too large (greater than 25.0 pixels) we assume that one of the beads has either just begun or ended its journey. We record the displacement that each bead travels in the Δt units of time.

Write a main() method in BeadTracker.java that takes a sequence of JPEG filenames as command line inputs, identifies the beads in each JPEG image (using BeadFinder), and prints out (one per line) the radial displacement that each bead moves from one frame to the next. Note that it is not necessary to explicitly track a bead through a sequence of frames - you only need to worry about identifying the same bead two consecutive frames at a time.

% java BeadTracker run_1/*.jpg
 7.183
 4.793
 2.169
 5.529
 5.429
 4.396
...

Data analysis. Einstein's theory of Brownian motion connects macroscopic properties (e.g., radius, diffusivity) of the beads to microscopic properties (e.g., temperature, viscosity) of the fluid in which the beads are immersed. This amazing theory enables us to estimate Avogadro's number with an ordinary microscope by observing the collective effect of millions of water molecules on the beads.

• Estimating the self-diffusion constant. The self-diffusion constant D characterizes the stochastic movement of a molecule (bead) through a homogeneous medium (the water molecules) as a result of random thermal energy. The Einstein-Smoluchowski equation states that the random displacement of a bead in one dimension has a Gaussian distribution with mean zero and variance σ² = 2 D Δt, where Δt is the time interval between position measurements. That is, a molecule's mean displacement is zero and its mean square displacement is proportional to the elapsed time between measurements, with the constant of proportionality 2D. We estimate σ² by computing the variance of all observed bead displacements in the x and y directions. Let (Δx₁, Δy₁), ..., (Δx_n, Δy_n) be the n bead displacements, and let r₁, ..., r_n denote the radial displacements. Then
  

  For our data, Δt = 0.5 so this is an estimate for D as well. The radial displacements r_i are measured in pixels: to convert to meters, multiply by 0.175 * 10^-6 (meters per pixel).

• Estimating the Boltzmann constant. The Stokes-Einstein relation asserts that the self-diffusion constant of a spherical particle immersed in a fluid is given by
  where, for our data,

  • T = absolute temperature = 297 degrees Kelvin (room temperature)

  • η = viscosity of water = 9.135 * 10^-4 N*s/m² (at room temperature)

  • ρ = radius of bead = 0.5 * 10^-6 meters

  and k is the Boltzmann constant. All parameters are given in SI units. The Boltzmann constant is a fundamental physical constant that relates the average kinetic energy of a molecule to its temperature. We estimate k by measuring all of the parameters in the Stokes-Einstein equation, and solving for k.

• Estimating Avogadro's number. Avogadro's number N_A is defined to be the number of particles in a mole. By definition, k = R / N_A, where the universal gas constant R is approximately 8.31457 J K^-1 mol^-1. Use R / k as an estimate of Avogadro's number.

For the final part, write a main() method in Avogadro.java that reads in the displacements (e.g., from the output of BeadTracker) and computes an estimate of Boltzmann's constant and Avogadro's number using the formula described above.

% java BeadTracker run_1/*.jpg | java Avogadro
Boltzmann =  1.254e-23
Avogadro  =  6.633e+23

Deliverables. Submit the required components (Bead.java, BeadFinder.java, BeadTracker.java and Avogadro.java) along with any other helper files needed to compile your program (other than StdIn.java and Picture.java and Luminance.java). Also submit a readme.txt file.