S13BrownianMotion

Brownian Motion of Ellipsoidal Particles

Noah Trebesch and Ben Freund

School of Physics and Astronomy

University of Minnesota - Twin Cities

Minneapolis, MN 55455

May 17, 2013

Abstract

The rotational and translational Brownian motion of microscopic ellipsoidal particles in two dimensions was studied using video microscopy. The qualitative trends in the position and orientation of the particle versus time, the displacement in the position and orientation of the particle versus time, and the distributions the the sizes of the displacements were found to agree with theory. Two methods were used to quantify the agreement of experimental results with theory by estimating Boltzmann's constant. The best estimate differed from theory by

, while the worst differed by . The quality of the quantitative data is questionable, however, due to a lack of precision in the experiment and the possible presence of a number of systematic errors.

Introduction

Brownian motion is a physical phenomenon that can be observed in microscopic particles. When suspended in a fluid, microscopic particles travel along random paths with random velocities (called translational Brownian motion), and they rotate in random directions with random angular velocities (called rotational Brownian motion). Albert Einstein hypothesized that Brownian motion was caused by the roughly

collisions per second of the Brownian particle with the molecules that make up the fluid the particle is suspended in [1]. From this standpoint, he was able to derive an equation for the RMS translational or rotational displacement a Brownian particle of any shape should experience in one dimension. Since Einstein's equation was published in 1906, many experiments with spherical particles have been done to confirm his predictions, but very little experimental or theoretical work has been done with non-spherical particles [1], [2], [3].

In biology, large molecules are common. These macromolecules experience Brownian motion, but they almost always have globular shapes that cannot be represented well by spheres. An ellipsoidal shape can provide a better analog to such molecules, so understanding the Brownian motion an ellipsoid should experience is helpful in understanding and predicting the Brownian motion of macromolecules [3]. This, in turn, is useful for predicting and understanding the diffusion of macromolecules [3]. Diffusion is a process caused directly by Brownian motion in which high concentrations of particles suspended in a fluid spread throughout the fluid and across permeable or semipermeable membranes until their concentration is uniform. Diffusion is a common process in the biological world, which is why it is important to understand it [3]].

Theory

Brownian Motion

Einstein's equation for the RMS displacement in a single rotational or translational degree of freedom due to Brownian motion is given below.

In this equation,

is the displacement, is Boltzmann's constant, is the absolute temperature, and is the time elapsed (also called the time step). The symbol is a constant of proportionality that depends on the shape of the Brownian particle and the medium in which it is suspended. Einstein also showed that the probability density function of displacements along any degree of freedom can be approximated by a Gaussian distribution:

In this equation,

is the standard deviation of the distribution. For a given particle and time step, this equation can be used to predict the probability of the particle being displaced by any given amount. These equations are very general. They make no assumptions about the shape of the Brownian particle or the medium in which it is suspended, and they are valid for both translational and rotational Brownian motion. In the derivation of these equations, Einstein also showed that Brownian motion along each degree of freedom is independent of the Brownian motion along any other degree of freedom [1], [2].

In three dimensions, an ellipsoid can translate and rotate along each of its three semi-principal axes, giving it six degrees of freedom along which Brownian motion can occur. The RMS change in each of these six quantities can be described using equation [[#EqEinstein][1]]. In the experimental setup, the movement of the particles was confined almost entirely to two dimensions, and prolate spheroids were used. (A prolate spheroid is an ellipsoid in which two of the semi-principal axes are the same length and are shorter than the third semi-principal axis.) This eliminates one translational and one rotational degree of freedom from the system. Also, it was impossible to observe the rotation of the ellipsoid about its long semi-principal axis, so this degree of freedom was not considered. This means that only three displacements were observed: two translational and one rotational. A diagram of these displacements is shown in figure [[#FigEllipsoid][1]].

In 1934, Francis Perrin derived equations for each of the six degrees of freedom that describe the motion of a prolate spheroid in three dimensions. The three equations relevant to this experiment are given below. Equation [[#EqPerrinTrans1][3]] gives the inverse value of

for translational Brownian motion along the long semi-principal axis of length

. Equation [[#EqPerrinTrans2][4]] gives the inverse value of for translational Brownian motion along either of the the other two semi-principal axes of length , and equation [[#EqPerrinRot2][5]] gives the inverse value of

for rotational Brownian motion about either of the short semi-principal axes.

In these equations,

is the viscosity of the fluid in which the ellipsoidal particle is suspended, and is a constant given by equation [[#EqPerrinConst][6]][[[#PerrinEllipsoidsI][4]], [[#PerrinEllipsoidsII][5]]]. For a sphere,

is constant in both translational directions, and it is given by the equation below:

In this equation,

is the radius of the particle. The rotation of a sphere in any direction cannot be observed [1].

Anisotropic and Isotropic Diffusion

The diffusion of a particle is governed by its diffusion coefficient, .

From this equation, it should be clear that diffusion and the RMS displacement due to Brownian motion are very similar concepts. It may not be immediately apparent from equations [3] and [4], but these equations show that

Figure 1: An original diagram showing the lengths of the semi-principal axes of a prolate spheroid and the three observable kinds of displacement it can experience in two dimensions.

> , which means an ellipsoid will always diffuse along its long axis more quickly than its short ones. This means that the RMS displacement along the long axis will always be greater than the RMS displacement along the short axis. This would also be true in a fixed coordinate system (i.e., the “lab frame”) if it were not for rotational Brownian motion. [4], [5].

At small time intervals, when the RMS translational displacement is smaller than the size of the particle and the RMS angular displacement of the particle is correspondingly small, the RMS displacement along the lab frame's coordinates are influenced by the initial orientation of the ellipsoidal particle and its angular displacement. It is said that the translational and rotational displacement of the particle are coupled when this is the case, and the diffusion the particle experiences is said to be anisotropic. As the time interval grows, the RMS angular displacement of the particle also grows, and the RMS displacement along the lab frame's coordinates start to become uniform. The degree to which the initial orientation and angular displacement of the ellipsoidal particle influences the RMS displacements along the lab frame's coordinates diminishes as the time interval grows. Eventually, when the degree of influence is small enough, the translational and rotational displacement of the particle are said to be uncoupled, and the diffusion of the particle is said to be isotropic [3].

#SecWall

Wall Effects

In this experiment, two dimensional confinement was simulated by separating two flat, parallel surfaces (hereafter referred to as “walls”) by a distance of . This experimental setup changed the effective viscosity of the liquid because of the boundary conditions present at the walls. Near the walls, the viscosity of the fluid approached infinity because the ability of the atoms or molecules that make up the fluid to slide along the surface of the walls was extremely limited. The way fluid viscosity is affected by this seemingly simple set-up is actually a very complex problem, and an analytical solution to it has yet to be found. However, several models exist that approximate the way the viscosity of the fluid changes with distance from the two walls. The model developed by Luc Faucheux and Albert Libchaber is one of the simplest, yet it is nearly as accuracy and is sometimes more accurate than other, more complex models [6], [7], [8].

In the Faucheux and Libchaber model, a simple linear superposition of the effects from the two walls is expected. In this experiment, only movement along directions parallel to the walls was observed, so only the effective parallel viscosity is of interest. Equation [9] shows the expected viscosity experienced by a spherical particle of radius

a height above a single wall, and equation [[#EqFaucheux2][10]] shows the expected viscosity experienced by the same particle at a height above one wall in a two wall system, where the two walls are separated by a distance

[8], [9].

Equation [10] was used to approximate the change in viscosity caused by the walls for the ellipsoidal particles. The radius

appears in equation [10] because the viscosity of the fluid varies with distance from the walls, so was used in this equation for the approximation. Because the particles are ellipsoidal rather than spherical, the use of this equation introduces a systematic error that will be discussed later.

Experimental Procedure and Apparatus

Particle Preparation

Ellipsoidal particles are not commercially available. To obtain ellipsoidal particles, they were instead prepared from much more widely available spherical particles. To do this, a solution of poly(vinyl alcohol) (PVA) and polystyrene (PS) microspheres was prepared. A mass of

PVA and approximately PS microspheres were added for every milliliter of water to create the solution. These values were chosen based on the recommendations of reference [10]. The diameter of the microspheres used was

. A volume of of solution was poured into a Petri dish, and the water was allowed to evaporate, creating a solid PVA film with the PS microspheres embedded in it [9], [10].

To make the embedded microspheres ellipsoidal, the PVA film was stretched. When heated, polymers like PVA and PS behave like glass. That is to say that above a certain temperature (called the glass transition temperature), PVA and PS become semi-molten and can be stretched without breaking. The glass transition temperature of PVA is

, and the glass transition temperature of PS is [11]. To create the ellipsoidal particles, an oil bath was heated to using a hot plate, and a

by strip of the film was immersed in the bath. The film was held at its ends by two needle nose pliers and stretched by pulling the pliers apart from one another. A diagram of this stretching technique is shown in figure [2]. To generate ellipsoids with semi-principal axis lengths of

and , the film was stretched times its original length. The specific draw ratio was determined by measuring the length of the strip of film before and after it was stretched [9], [10].

Once the film was stretched, the ellipsoidal particles were recovered from the film. To do this, a solution of 7:3 parts water to isopropyl alcohol (IPA) was created. The film strips were immersed in this solution, and it was heated to

. This solution dissolved the PVA while leaving the PS particles intact. The particles were then recovered from the solution and placed in deionized (DI) water. This was done by centrifuging the solution at

for half an hour, causing the PS particles to settle at the bottom of the containment vial. The solution of water, IPA, and PVA was then removed, the vial was refilled with DI water, and the particles were mixed into the water. This entire process was then repeated once more to further remove any residue of IPA and PVA from the particles [9], [10].

Slide Preparation and Video Microscopy

Figure 2: An original diagram showing the stretching apparatus and oil bath.

Once the particles were suspended in DI water, slides were prepared to view the particles under a microscope. To do this, spherical PS particles with a diameter of were added to every milliliter of particle solution to create a new solution. Additionally,

spherical PS particles with a diameter of were also added to every milliliter of the new solution. A thin ring of adhesive was traced out on a cover slip, and

of the new solution was placed in the center of the ring of adhesive. Another cover slip was placed on top of the first, they were pressed together, and the adhesive was cured using an ultraviolet light. The adhesive prevented evaporation of the water while the cell was being observed. A diagram of a finished slide is shown in figure [3]. The large spherical particles acted as spacers and allowed the distance

between the cover slips to be controlled. The chosen distance

was small enough that the ellipsoidal particles did not tumble in three dimensions over the course of data acquisition. Though it was possible for them to do so, the effective viscosities at different heights in the cell impeded such motion. The smaller spherical particles were added to allow the Brownian motion of spherical particles to be observed and compared to the Brownian motion of the ellipsoidal particles. Observing both particles within the same cell prevented differences in the cells from creating differences in the spherical and ellipsoidal Brownian motion.

Figure 3: An original diagram of the microscope slides that were prepared. Note that this image is not drawn to scale, and the ellipsoidal particles observed were actually far from any spherical spacers. The labeled height and diameter in this diagram correspond to the variables used in equations [[#EqFaucheux1][9]] and [10]].

The prepared slides were observed using a microscope at 100 times magnification. To obtain quantitative information on the Brownian motion of the spherical and ellipsoidal particles, a series of about five hundred of images of a single magnified particle was taken using

Manager [12]. Each image was taken one half second after the last image, making each image equivalent to a single frame in a video of the motion of the particle over about four minutes. To calibrate the microscope, an image of a microspherical spacer (with a known diameter) was also captured to determine how many meters were represented by one pixel in any of the video frames. Ultimately, it was found that a length of one pixel was equivalent to

Raw and Transformed Data

Results and Analysis

A custom script was written in MATLAB to automatically determine the position (in pixels) and orientation of the ellipsoidal particle in each frame of its video. This script was also used to determine the position of the spherical particle in each frame of its video. To work, the approximate location of the particle in the first frame of the video must be manually input. The program then finds the outlines within the small section of the image that contains the particle. It is only able to detect the two long edges of the ellipsoidal particle, as shown in figure [5]. For the ellipsoidal particle, the program finds the center and orientation of the two edges of the ellipsoid and averages them to give the final location and orientation of the particle in the video frame. For the spherical particles, the program can detect the entire continuous outline of the particle, so no averaging is performed. The particles move slowly enough that the center of the particle in the previous frame of the video can be used as the rough estimate of the center of the particle in the next frame, which allows the program to proceed automatically after the first frame.

Figure 4: An original diagram of the microscope that was used in this experiment.

For the ellipsoidal particle, the data output by the image processing script is in the lab frame of reference. It was transformed to the particle frame for analysis. To do this, the displacement of the particle in each pair of frames was measured in the lab frame. The origin of the lab frame coordinate axes was then set at the location of the particle's center in each frame, and a simple coordinate frame rotation, given by

Figure 5: An original image showing the outline of an ellipsoid determined by the image processing script on the left along with the center of the ellipsoid, denoted by a '+', and a line representing the orientation of the particle. On the right is the center and orientation line superimposed over the original image.

was then used to transform the displacement in the lab frame to the displacement in the particle frame. In these equations,

and represent the particle's orientation in the two frames. Their average was used as the coordinate rotation angle because no information is known about the orientation of the particle between the two frames. A diagram of this transformation is shown in figure [[#FigTransform][6]].

With this rotation, the location of the particle in its own frame of reference can be determined as a function of time by taking a sum of the displacements. Ultimately, this results in three data sets that must be analyzed: there is the data from the spherical particle, the lab frame data from the ellipsoidal particle, and the particle frame data from the ellipsoidal particle. The raw data can be visualized on two or three dimensional plots, where each dimension represents a translational or rotational degree of freedom. These plots for each of these data sets is given in figure [7]. In these plots, the variation of color along the particles' paths represents the progression of time. There is not much quantitative information on these plots that can be used to assess the agreement with theory, but they are provided because similar plots are common in the literature on Brownian motion [3], [13]. These plots can be compared with the plots in the literature to provide a qualitative comparison with previous results.

Analysis of Temporal Plots and Displacement Distributions

The motion along each degree of freedom is independent of the motion along any other degree of freedom, so it is easier to perform an analysis considering each degree of freedom separately. The first way this may be done is by considering the position of the particle along each degree of freedom as a function to time. These plots are given in figures [[#FigTimeX1][8]]A through [[#FigTimeB][14]]A. Each plot should appear random. There should be no correlation between time and position. Qualitatively speaking, this is true for each of the degrees of freedom considered.

Next, the displacement of the particle along each degree of freedom as a function of time can be considered. These plots are given in figures [[#FigTimeX1][8]]B through [[#FigTimeB][14]]B. By equation [[#EqEinsteinProb][2]], the distribution of these displacements should be approximately Gaussian, and each distribution should be centered at zero. For all of these plots, the highest density of points appears to be centered on zero, and the point density appears to decrease as the displacement gets farther and farther from zero. This is what is expected of a Gaussian distribution, so the qualitative properties of these plots agree with theory. Also, the distribution of the displacements should not depend on time, and these plots can be used to verify that this was the case for each degree of freedom. This can be done by observing that the density of points for any given displacement is approximately equal at all times.

With the temporal consistency of the distribution qualitatively verified, the actual distribution of the displacements can be plotted. This is done for each considered degree of freedom in figures [8]C through [14]C. Using MATLAB's

, and .

Figure 6: An original diagram of the lab frame coordinate axes (in red), the particle frame axes (in blue), and some relevant symbols from equations [11] and [12]. In these equations,

function, a Gaussian distribution can be fit to the displacement data [14]. This function can also provide a user-specified percentage confidence interval on the estimates of the mean and standard deviation of the fit Gaussian distribution. The uncertainty of a value with random error should represent a 68.27% confidence interval, so the

function was set to provide estimates of Gaussian distribution parameters and 68.27% confidence intervals on these parameters. In this way, estimates of the mean and standard deviation of each fit were generated with estimates of uncertainties. These values for each degree of freedom are given on the plots.

These distributions can be used to quantitatively verify the theory behind Brownian motion using equation [2]. The standard deviation of each degree of freedom should be equal to the square root of a number of parameters that, with one exception, are all known. Bolzmann's constant is known, the temperature was measured, and the time step was measured. By equations [3] through [7], it is known that mobility depends on the lengths of the effective viscosity of the water the particles were suspended in and the semi-principal axes for the ellipsoid or the radius for the sphere. The lengths of the semi-principal axes can be measured, the radius of the sphere is known, and the effective viscosity is given by equations [9] and [10]. The effective viscosity of the fluid depends on the free viscosity of the fluid, which is known, the effective radius of the particle, which is also known, and the vertical distance

Figure 7: The degrees of freedom of the particle plotted against one another. Color varies from blue to red with time. A: Spherical data. B: Lab frame ellipsoidal data. C: Particle frame ellipsoidal data.