Brownian Motion and Single Particle Tracking

Transcription

1 Brownian Motion and Single Particle Tracking 1.1. Introduction to the Brownian Motion Theory In 187, Robert Brown observed the random motion of micro-particles suspended in gases and liquids. He called this random or jiggling motion Brownian motion. However, it was only in 1905 that Albert Einstein first explained this phenomenon on the basis of kinetic theory of molecules. In a quantitative manner, Einstein connected quantities of kinetic theory such as viscosity and mobility with the Brownian motion. Einstein performed a statistical analysis of molecular motion and its effect on particles suspended in a liquid. As a result, he calculated the mean square displacement of these particles. He argued that an observation of this displacement would allow an exact determination of atomic dimensions and prove the existence of atoms, and verify the molecular kinetic theory of heat. Jean Baptiste Perrin, a brilliant experimentalist, performed a series of experiments in the first decade of the twentieth century, one of which depended on Einstein's calculation of the mean square displacement of suspended particles. His results confirmed Einstein's relation and thus the molecular-kinetic theory. Eventually a physical explanation of the phenomenon of Brownian motion led to the acceptance of the kinetic theory of matter and the kinetic atomic/molecular theory of heat. The kinetic theory of matter posits the existence of atoms and molecules, and their constant motion due to which they elastically collide with one another. The kinetic molecular theory describes temperature as the constant motion of atoms and molecules in matter. Brownian motion which is characterized by the constant and erratic movement of minute particles in a liquid or a gas is thus due to the inherently random motions of the atoms or molecules that make up the fluid in which the particles are suspended. The fluidic atoms or molecules collide with the larger suspended particles at random, thus making them move randomly. According to Einstein the Brownian motion actually arises from the agitation of individual molecules due to the thermal energy kbt they possess at a specific temperature. The collective impact of these molecules against the suspended particle yields enough momentum to create movement of the particles. The origin of Brownian motion can be understood on the basis of the theorem of equipartition of energy. Each colloidal micro particle, possessing a mass m is free to exhibit translational motion. The mean kinetic energy of the particle in three dimension is, 1 m v 3 = k T B (1) This energy, though small in value, leads to a measurable amplitude of vibration for a small micro particle. It is worth noticing that in addition to the random fluctuating force, the particles also experience a drag force (frictional force) as they are pulled through the solvent. To find a solution to the motion of the particles, we will use the Langevin equation for a v particle of mass m and velocity dv = α v + F (t) m dt () Page 1 of 11

2 From Eq. (), we can see that each colloidal particle is subject to two forces, the random molecular bombardment F( t) that causes the v Brownian motion, and the resistive force α, where α is the damping coefficient related to viscosity of the fluid or solvent. In one dimension, the scalar form of Eq. () can be written as, d m x dt + α dx F dt (t) = 0 (3) Multiplying both sides of the above equation by x, yields, d x x + αx dx dt F (t)x = 0 m dt (4) To simplify, we can use the expansion of the expression for the first term in Eq. (4), (x dx ) = ( ) d x x dt dt + dt d and for the second term in Eq. (4), d (x) dx dt = x dt The modified Eq. (4) becomes, m d (x ) dt m( dx ) + α d dt (x ) F (t)x = 0 dt (5) Now we can use the theorem of equipartition of energy to find the average energy of single particle for one degree of freedom 1 m v 1 = k T B (6a) or m ( dx ) 1 k T dt = B (6b) Since F in Eq. (5) is a random force, hence substitute Eq. (6b) into Eq. (5), we obtain xf = x F = 0. Then, we can average Eq. (5) over time and where β = dt d (x ). m dt dβ + α β k T = 0 B (7) If we integrate the Equation (7) with respect to time we can write the solution in the following form Page of 11

3 k β = B T α + Ae αt m (8) where A is the integration constant. For a reasonably long observation time ( t = τ), the factor ατ / m will be very small and hence we can ignore the second term on the right hand side. Finally integrating the modified Equation (8) over the observation time τ, we get that has solution k T τ d(x ) dt = dt (9) 0 τ B 0 α k T x = B α τ (10) For spherical particles, each of radius a, Stokes law can be used to write α, α = 6 πηa (11) where η is the viscosity of the fluid. Equation (10) hence takes the form, k T x = B 6πηa τ (1) It can be easily shown that the mean squared displacement in two dimensions can be written as 4k T r = B 6πηa τ (13) Hence, by plotting r as a function of time, we expect a straight line through the origin whose slope can be used to obtain Boltzmann s constant k B. Equation (13) is traditionally written as, where r = 4 Dτ D = k B T 6πηa = k B T α is the self-diffusion constant. Using the relation between Boltzmann's constant and molar gas constant R, we can also find Avogadros' number, N A, as N A = 1 r 3πηa RT τ (14) (15) (16) Page 3 of 11

4 1.. Calculation of the Mean Square Displacement In this introduction we want to limit ourselves to a movement in two dimensions. Consider a trace of arbitrary movements (random walk) of a particle as depicted in Figure 1a). The mean square displacement can be then calculated as follows. For each time point separated by a fixed lag time Δ t one obtains a position xi and yi. A displacement is then calculated as: x 1 = x(t 1 ) x = x(t 1 + t) x 1 ( t) = x x 1 y 1 = y(t 1 ) y = y(t 1 + t) y 1 ( t) = y y 1 (17) x i = x(t i ) x i+1 = x(t i + t) x i ( t) = x i+1 x i y i = y(t i ) y i+1 = y(t i + t) y i ( t) = y i+1 y i The displacement and the square displacement can be calculated for every step of the same trace corresponding to the same step size of stepping time (step during the time length Δ t) as ( r ( t)) 1 = ( x ( t)) + ( y ( t)) 1 1 ( r ( t)) = ( x ( t)) + ( y ( t)) (18) ( r ( t)) i = ( x ( t)) + ( y ( t)) i i The mean square displacement is obtained as an average of all steps corresponding to a single lag time Δ t : M SD = ( r( t)) 1 = (( r ( t)) n + ( r ( t)) + + ( r ( t)) ) = 1 n n r i ( t) 1 i i=1 (19) The same procedure applies to double step during a time length of t, 3 t etc. Now the mean square displacement (MSD) can be plotted to its corresponding step time interval which gives characteristic curves. If the analyzed diffusion is of isotropic nature then one would expect a linear correlation. In this case the slope of the line corresponds to the diffusion coefficient multiplied with its Page 4 of 11

5 factor (normally in the 1D case, 4 in the D case or 6 in the 3D case). Diffusion or random walk can be hindered or restricted which changes the characteristic form of the MSD plots. In the case of diffusion restricted to a confined space the MSD naturally does not exceed the diameter of this space. One of the aims of this lab is to directly reproduce the experiments of J. Perrin that led to his Nobel Prize. He used latex spheres, and we will use polystyrene spheres; otherwise the experiments will be identical. (In addition to reproducing Perrin's results, you will probe further by looking at the effect of varying solvent molecule size. In the first part of this lab, you will replicate Perrin's work with modern equipment. You will track the motion of synthetic beads suspended in by using a research-grade inverted microscope. A CCD camera will transfer images of the beads to a computer for automated particle tracking and analysis. Techniques developed in this lab include a phase contrast microscopy, pipetting, image data acquisition, Brownian motion theory and software design for image filtering and particle tracking, and data analysis in Matlab.. Data Collection The heart of this experiment is XDS-3 Inverted Biological Microscope (GXMXDS-3, GX Optical, UK). The instrument has a CCD camera (GC650C AVI GigE, Allied Vision Technologies, Exton, PA) with resolution up to pixels, but in the interest of transmission speed here we'll use pixels resolution. In this experiment you will explore the Brownian motion of microspheres in water. The goal of this experiment is to determine a fundamental constant, Avogadro s number (Boltzmann s constant) from observations of Brownian motion. Please read this section carefully before starting the experiment!.1 Floating Bead Solution To observe Brownian motion you will use polystyrene microbeads. These microbeads (or microspheres) are available in the lab in a range of different diameters (0.8 to 1.1 μ m). Refer to for the microspheres' safety data sheets and physical properties. When originally packaged, the beads are suspended in 5 or 10 ml of water. Their concentration is very high. The estimated mass of microbeads in 1 cm 3 is 1.05 grams. Due to the high sphere density the solution is diluted 10 4 times. This increases the average distance between the beads so that the spheres do not frequently collide with each other and can be independently visualized and tracked. A diluted solution has already been prepared for you and provided in the squeezed bottle. This solution should be shaken well before use, since the spheres start to settle down at the bottom with the passage of time. Page 5 of 11

6 . Observation Slide Preparation A custom observation slide of dimensions 76 mm x 5 mm x 1 mm and coverslip 18 mm x 18 mm is used for sample preparation. Take the following steps to make the observation slide:..1. Fix a tip to the pipette's narrower end, and set micropipette volume by rotating, smoothly and gently, the volume adjustment knob to a volume of ~ 50 μ L.... Take a drop of the diluted microsphere solution by pressing the head of the pipette gently and released by pressing the head with a little force of thumb, and place the drop on the slide. Now, the observation slide is ready for use..3 Recording Brownian Motion The CCD GC650C camera is connected to the computer by an Ethernet cable. After booting up the computer, log on to the machine under the account Brownian motion ; you won't need a password Turn the microscope on with the power switch located on the lower left hand side of the microscope..3.. Click the "NI MAX" icon on the desktop to startup the image taking software. Be sure that NI MAX recording area appears at the center of the program window. Look under Devices and Interfaces and click on NI-IMAQdx Devices. Now, click on cam0 : Allied Vision TechnologiesGC650C. If NI MAX recording area does not appear, immediately contact a TA or the instructor for assistance Maximize the NI MAX program window by clicking Maximize button in upper-right corner of the program. This step is very important. It will allow MATLAB program to capture the bead image frames precisely from NI MAX recording area Click the MATLAB icon on the desktop. The MATLAB Command Window will appear Change the size of the MATLAB window and move it down so it does not overlap with NI MAX recording area Lower the stage by turning the large knob on the lower left hand side of the microscope clockwise, so that there is room under the objective lens Put the slide you made on the stage. The measurement volume contains water with suspended polystyrene spheres. The sample volume is about 100 μm deep. Make sure you cover the slide with the transparent plastic provided to ensure the microsphere solution is not affected by the surrounding environment Click the GRAB button above the recording area and you should now be able to observe a live updated picture on the computer screen. If not, contact a TA or the instructor for assistance. Right click on the recording area and switch to Pan in Viewer Tools. Page 6 of 11

7 .3.9. While looking from the side and observing the distance between objective and slide, slowly turn the platform up until the objective lens almost touches the slide. Do this very carefully. If you go too far you may break the slide and/or the objective. This microscope is equipped with spring loaded objective lenses, which means there is some leeway, so that touching the slide will not immediately destroy it. However, there is a limit to this, so be very cautious! Slowly adjust the height until the bottom of the slide comes into focus. Play around with the X-Y positioning stage, and focus so that you systematically go through the different layers (top of cover slip etc.) of the slide. This way you'll develop a feeling for how much movement of the adjustments is required Now, in particular, find a layer where you can observe a number of dark spots, which consists of slightly larger circles with light centers. The dark spots are spheres that are in the focal plane. Observe the motion of the particles. Try to follow an individual one over time. You will likely see it come in and out of focus, and it may disappear altogether. The spheres are floating in water, and they are free to move in three dimensions. If you are focusing on the bottom of the slide, you may see spheres that do not move because they are stuck to the bottom. You are now ready to start recording Brownian Motion. During recording (very important): (a) The microscope or slide should not be subjected to any movement, and so do not touch the table during recording; (b) The stage and the slide shouldn't be exposed to any air currents; (c) The intensity of surrounding light and light coming from the lamp of the microscope must not change; (d) Make sure there is only one bead in focus at a time in the recording area. (e) Make at least 10 records. The bead that you found must move erratically. If it does not, it means that the bead you found sticks to the glass surface, and you must find another one. The bead floats randomly in 3D, so it will move in and out of focal plane over time. Observe the bead for a while and if the bead moves out of focal plane, try to bring the bead back to focal plane (light center of the spot should be visible) by readjusting the position of the objective lens with the fine adjustment knob. Spend 5 10 minutes to get experience with bead holding in focal plane before you will record the bead s motion. When you are ready to capture the bead s motion:.3.1 Now, with the pointer on the bottom left corner of the recording area, type p=get (0, PointerLocation ) in MATLAB and press <Enter>, you will get the pixel coordinates of the bottom left corner (x1,y1). Do the same for the top right corner of the recording area to get the pixel coordinates of the the top right corner (x,y) of the recording area. Make sure not to including any white spaces outside of this recording frame, this will result in the algorithm not being able to identify the bead Find a bead and bring the bead into focus near the center of recording window. Make sure there are no other beads in the recording area except the one in focus. Page 7 of 11

8 Type CaptureFrames(x1,x,y1,y) in MATLAB command window and press <Enter>. (Here x1,x,y1,y are numbers obtained from the step.3.1) Make sure the bead stays in focus by adjusting the height of the objective and do not touch the table during recording The captured images should have good contrast between the background and the bead. For reference look at the image sample on the Desktop Repeat steps.3.13 and.3.14 several times (at least 10 times) by bringing in different beads into focus each time. Each set of captured images of the bead (see Fig) is automatically saved in C:/Users/Brownian Motion \ CollectData\DataSet# folder with varying set number, #, from 1 to 10 (max. number of sets). The pictures of the bead are saved in.png format. Information about number of captured frames and capture time interval saved in the same folder under the name capture_info in.txt format. Copy the capture_info.txt files to a memory stick or save them elsewhere every time as they are overwritten every time the CaptureFrames program is run in MATLAB..4 Image Calibration Calibration is a very important step to be performed prior to or after the recording of any data, since the image processing and particle tracking algorithm is strongly dependent on the particle size, hence any carelessness in this step can lead to erroneous data. Better the calibration, easier would it be to record the particles' trajectories. Another important thing is to make calibration by using the same objective and image resolution with which the record of the particle s images was done since each combination of objective and image resolution give different calibration values. The calibration value ( ε ) is in terms of μ m per pixel and by using this value we can convert the microspheres' diameter in pixels to microns. You will do calibration of the record window by using a counting chamber (Hauser Scientific, Horsham, PA) shown in Fig Lower the stage by turning the large knob on the lower left hand side of the microscope clockwise, so that there is room under the objective lens..5.. Replace the slide with the counting chamber on the microscope s stage.5.3. Click the GRAB button above the recording area and you should now be able to observe a live updated picture on the computer screen By adjusting the height and changing the chamber position with X-Y positioning stage, bring the chamber s grating into the focal plane with the smallest grating at the center of observation window By adjusting the distance (height) between objective and the calibration chamber bring the chamber s grating into the focal plane and by changing the chamber position with X-Y positioning stage place the grating place 50 μ m x 50 μ m grating frame within the recording window as shown in Figure. Page 8 of 11

9 .5.7. Type p=get (0, PointerLocation ) in MATLAB with the pointer on the bottom left corner of the grating square and the same for the top right corner of the grating square to get the x and y pixel coordinates of the bottom left corner (x1,y1) and the top right corner of the (x,y) of the recording area respectively Compute the calibration value ε (in μ m/pixel) with the information that the length of each side of this grating square is 50 μ m and with the measured values of (x1,y1) and (x,y). We expect the value of ε to be around 0.18 μ m/pixel. 3. Particle tracking through MATLAB The analysis consists of two stages: In the first, the images are processed automatically by a Matlab program. This program will analyze the particle tracks and export them as a.txt file. Then you will analyze the tracks, and provide a physical interpretation of the data that allows you to extract Boltzmann's constant (or Avogadro s number), as described in detail in Introduction. For data extraction from the captured frames, image processing and particle locating, you will use the MATLAB program DataAnalysis.m. You are encouraged to study the source code and convince yourself of the various data extraction, image processing and particle tracking tasks described here. To start, type in MATLAB command window and press <Enter> (all commands/dialog texts within MATLAB command window are shown in green) >> DataAnalysis The following dialog text will appear Enter experiments to be analyzed: Example: Enter experiments to be analyzed: [1:5] <Enter> will load five experiment sets from 1 to 5 Enter experiments to be analyzed: [1,3:5] <Enter> will load 1 st, 3 rd 4 th and 5 th, ignoring the nd set. In MATLAB Figure window the y-vs- x trajectories of the bead for each set of measurements will appear, something similar to the trajectories shown in Fig. To make visual separation between trajectories obtained from different sets of measurements, different colors are used. The colors assigned to trajectories according to its set number described in svk_color_set.m subprogram. The green-colored trajectory in Figure shows unusual movement of the bead. The point observed at (0, 75) results from the fact that during the frame capture time, the bead of our interest moves out of the focal plane and a new bead appears at that location. These points must be discarded before the bead s trajectory is built. Page 9 of 11

10 The next dialogue text will ask you if you want to use filter for your data points. Type Y if you need to filter the data and N otherwise. If you want to filter the data, you must enter xmin, xmax, ymin, ymax values into the command line. After entering these four values, you will see four red lines (such as in Fig). These lines will show the rectangle within which your valid data lies. After removal, the data points must shifted in order to eliminate disturbances within the trajectory. Now the figure shows the bead s trajectory after the invalid data points have been removed. Fig. C and Fig. D show the frequency of the bead s positions along x and y axes. Then, press the SpaceBar to continue. The bead s motion trajectories will appear in MATLAB Figure window like the trajectories shown in Fig. Follow further instructions from MATLAB in the DataAnalysis program to obtain the final plots and histograms shown in the next page with the information about Δx, Δy and MSD, save this plot for your report/presentation. The file containing the information about the dt, dx and dy of the particle is also saved in data_table_out.txt file. You can also find the image file BeadSize.jpg to visualize the bead size distribution. Page 10 of 11

11 Make sure to copy the data_table_out.txt and BeadSize.jpg to a memory stick or elsewhere as these files will be overwritten each time you do this experiment with a new sample. Repeat the entire process for different size beads (0.8 and 1.1 μ m). Analyze this data and write your own code to reproduce the MSD plot similar to the one shown here in the figure and compare it to the MSD plot you obtained from the DataAnalysis program which you saved previously. With the help of the theory provided in the initial part of this manual to compute the value of the Diffusion constants (D) and Boltzmann constant (k). Page 11 of 11