Determining Particle Size Using Mie Scattering

Joe Faella and Sam Amodeo

University of Minnesota

School of Physics

Minneapolis, MN 55455

Abstract

We conduct an experiment based on the theory of Mie scattering to determine the size of four different sets of spherical particles with different diameters, each in the micron scale. A polarized 632.8-nm laser beam is aimed at a cuvette of water containing latex spheres. The resulting spherical diffraction pattern is observed by a photodiode and used with an analysis program to calculate the average size. The results show that Mie theory could correctly determine the size of 3-micron diameter and 9.7-micron diameter spheres to be 3.0 +/- 0.3 microns and 9.7 +/- 0.3 microns, respectively.

Introduction

Mie Scattering is a name given to a certain set of solutions to Maxwell's Equations. The solutions were found by Gustave Mie, hence the name. Mie scattering refers to the scattering of light from particles larger than the wavelength of said light. This is in contrast with Rayleigh Scattering, which occurs when light scatters from a particle with a diameter smaller than the incident light. At the heart of Mie scatterring is a "simple" boundary conditions problem: How will light behave as it interacts with this object? The mathematics are quite complex (and will be simplified and summarized in the following section), but once solved it becomes easy to compare measured data to simulated data and determine particle sizes.

Mie scattering algorithms have also become very important in the creation of metamaterials, artificial structures made of microscopic elements that collectively create unusual optical properties. In this case, it is necessary to know the optical effects of electric and magnetic dipoles on microscopic dielectric rods. These effects are intrinsically related to their Mie scattering (Vynck 2009, pg. 6), since the visible light traveling through them will have a wavelength slightly smaller than the diameter of the rods.

In the medical sciences, Mie scattering is also used to discover cancerous tissues. Compared to healthy tissue, the cells in in cancerous tissue will have increased in cell diameter, enough so that Mie scattering can show the discrepancy (Phytila 2007, Figure 2).

Mie scattering is also the reason why clouds appear to be white: Unlike Rayleigh scattering, Mie scattering is not wavelength-dependent. Thus, large particles in the clouds will scatter all wavelengths of light, which will combine to appear white.

Theory

To find the size of the spheres the light is scattering off of, we would use Mie theory, which solves for Maxwell's equations with boundary conditions given by a sphere. It presents very complex calculations that would show the near-exact diffraction patterns for any given size, and the derivation of such would be challenging for most readers to follow. We instead use a relatively simple approximation, solved for in "Mie Scattering" by R.M. Drake and R.E. Gordon[1]. This derivation considers diffraction off of a homogeneous, nonmagnetic sphere of radius a and refraction index m and solves for the intensity %$I(\theta)$% at some constant radius r away.

To approximate the Mie diffraction pattern, we treat the radiation from the sphere at each angle as a plane wave. The intensity measured some distance away

and at some angle to the original wave’s direction follows the form of a sum of the diffracted radiation and the original radiation that did not interact with the sphere. The two sources of radiation are shown as plane waves in Figure 1:

Figure 1: To approximate the Mie diffraction pattern, we treat the radiation from the sphere at each angle as a plane wave. The intensity measured some distance away and at some angle to the original wave’s direction follows the form of a sum of the diffracted radiation and the original radiation that did not interact with the sphere

To find the diameter a, we treat a as a controlled variable in a simulation (example in Figure 2), calculating at what angle the maximums and minimums would appear for different diameters. Maximums and minimums will appear at the same angles at every observation radius r, and that their variability only depends on the size of the sphere a and its index of refraction m. The actual simulations that are fit to our experimental data are calculated directly from Mie Theory for the lowest uncertainty. The derivation of it can be found in Appendix A of Weiner [4].

Figure 2: A Mie scattering simulation of a diffraction pattern for a sphere of 3 micron-diameter.

Apparatus and Methods

Figure 3: Full setup of a Mie Scattering experiment. This diagram is to show the orientation. The individual parts will be explained in further figures.

We used a photodetector to measure intensity of light as a function of angle. The photodetector was mounted on the end of a metal arm, which rotates via a stepper motor (Figure 5). We used a chopper (Figure 4) and lock-in amplifier(Figure 7) to reduce background noise. The laser was a standard 5mW HeNe lab laser.

The latex spheres were suspended in a solution of water inside a cuvette (Figure 5), which was mounted above the rotating arm. This cuvette was thin enough (~1mm) so that we could account for Snell's Law refraction only, and ignore other sources of refraction (such as from the sides of the cuvette). The range of measured angles was -90 to 90 degrees. This corresponds to 400 steps from the motor, and so we were able to measure the intensities in .45 degree increments.

Figure 4: Laser and chopper. The chopper's apparatus is the smaller one on top of the laser's apparatus. The number on the screen is the frequency at which the chopper is spinning. This frequency is input into the lock-in amplifier via the cord plugged in to the right of the screen.

Figure 5: View of the stepper motor, cuvette and rotating arm. The photo diode would normally rest on top of the post on the end of the arm. The radius used for our data was 37 cm, meaning the pole was set one hole inward from where it is in this figure. The cuvette is set just above the center of the spinning arm to make sure that the angle, and not the radius (kept constant), is the only controlled variable.

Figure 6: A view down the laser's path. It passes through the chopper's outside ring and continues through to the center of the cuvette. The arm then rotates around the cuvette to catch the diffraction from each angle.

Figure 7: The lock-in amplifier takes an input reference frequency (left plug-in) from the frequency the chopper is rotating at. It then takes a reading input (right plug-in) from the photo diode on the end of the rotating arm and tracks only signals that are chopped at that frequency (the laser).

To analyze the data, we will compare our measured data to simulated data. We use a program authored by Scott Prahl from Oregon Medical Laser Center [5]. This program takes in a series of parameters, including the sphere’s size and index of refraction. The program then generates tables of data, consisting of intensity values as a function of angle, for angles of 0 to a user-specified angle (+/- 90 degrees).

Results and Analysis

The measurements with subtracted background for each sphere diameter are shown next to their simulated data in figures 8-11. In each data set, the points around the center peak are omitted because its intensity is at least two magnitudes above the other peaks. The key to finding the size of the spheres is comparing the angles at which the maximums and minimums occur, but there will always be a large peak at zero degrees. Because of this, we increased the precision to get a zoom in on the smaller peaks, and let the sensor overload on the middle peak. The units for intensity are omitted in the simulated data because of this angle dependance as well, as it only shows the intensity as a relative way to compare the peaks, no matter what the base intensity of the source may be.

While data was taken from -90 to 90 degrees approximately, using Snell's law to account for the flat side of the cuvette reduces that range to just inside +/- 55 degrees. In increasing order of diameter, I present the data collected for each sphere size and compare the pattern to the simulated data above it, which uses the diameter of the sphere given by the manufacturer. If the data does not seem centered on zero degrees, it is because the cuvette's face was not perpendicular to the laser beam.

Figure 8: Experimental and simulated data sets for the Mie scattering pattern off of micron-diameter spheres. The data from the center peak is omitted because the precision level overloads the sensor. Original figure.

The diffraction pattern we collected for the micron-diameter spheres (Figure 8) should show sizable peaks at around +/- 45 degrees and minimums around +/- 32 degrees. While there is indeed a plateau where the minimum should be, the presence of other peaks suggests that the sample was contaminated with something of a different sphere size. This may have also been a problem of too little sphere concentration, preventing the diffraction pattern from surfacing above the background.

Figure 9: Experimental and simulated data sets for the Mie scattering pattern off of 3 micron-diameter spheres. The data from the center peak is omitted because the precision level overloads the sensor. Original figure.

The diffraction pattern we collected for the 3 micron-diameter spheres (Figure 9) shows maxima and minima that closely follows the simulated pattern. The innermost peaks are inside +/- 15 degrees, with the rest of the peaks following suit. We can explore the actual size of the 3 micron spheres in the analysis section.

Figure 10: Experimental and simulated data sets for the Mie scattering pattern off of 5.5 micron-diameter spheres. The data from the center peak is omitted because the precision level overloads the sensor. Original figure.

The simulation of the 5.5 micron-diameter spheres' diffraction pattern (Figure 10) shows that there should be a very slight minimum at +/- 35 degrees, but otherwise peaks overall should not be visible. The collected data shows slight peaks near +/- 20 degrees, a cause possibly of contamination. This data is hard to use in the first place because the only obvious peak is at the center, which is omitted. If the simulation had been made before the data was taken, a different sphere size would have been selected if it promised more obvious peaks in its diffraction pattern.

Figure 11: Experimental and simulated data sets for the Mie scattering pattern off of 9.7 micron-diameter spheres. The data from the center peak is omitted because the precision level overloads the sensor. Original figure.

The simulation of the 9.7 micron-diameter spheres' diffraction pattern (Figure 11) shows slight peaks at +/- 20 degrees, but taken out of the logarithmic scale are easily seen on the experimental data. Another match between the two sets is just inside +/- 15 degrees, where there is a clear flattening of the slope.

At this point, I will show exactly why this method is correct within only a micron. If I increase the size of the sphere for the simulator from 3.0 microns to 3.2 microns, the peak locations would not change. This can be tested on the program. As mentioned before, any attempts to get the code to calculate data points less or more than four degrees apart was not successful, so until a major change in the size is made, say, moving from 3.0 microns to 3.5 microns, a whole new set and number of peaks will be shown. From this calculator, we are stuck with assuming that the actual size of the sphere is very close to 3 microns in diameter, no more than a few tenths of a micron off. The same is observed for the simulated data for 9.7 microns. With this method of calculation, however, the %$\chi^2$% value for 9.7 microns is meaningless because only one set of two peaks could be matched. Overall the simulations need to be improved greatly and the data collecting needs to favor sphere diameters that will give a high number of obvious peaks.

Conclusions

The results show a decent matchup of the 3.0 micron and 9.7 micron-sized spheres with their simulations. The uncertainty of the simulations ultimately barred the analysis from determining the particle sizes within 0.3 microns, which in truth is better than the original expectation of a one-micron uncertainty. If the simulations were able to give data points that were less separated, more could be done with the analysis of the sphere size and why or why not the data matched the simulations.

The sphere sizes that did not give matching data to their simulations could be analyzed again with a higher concentration. The threat of high concentration is a diffraction pattern showing interaction of the wave with multiple sphere, instead of a series of single-sphere interactions. While none of that was observed here, an improvement on the experiment would be to see at what maximum concentration the single-sphere diffraction patterns are still observable, so as to get the strongest reading possible. Other improvements could be made to the apparatus itself, reducing the step size of the stepper motor and possibly removing any of the slightest noises that may mess with the diffraction pattern, even with the lock-in amplifier. Depending on the field of study and the goal of using Mie scattering to find particle sizes, better simulations and reduced uncertainty are of peak interest.

Acknowledgements

We would like to thank Joe Sobek, Kai Zhang, and Kurt Wick for advising us throughout the semester. We would also like to thank Vincent Noireaux and his assistants for giving us extra materials.

Sources

1."All-Dielectric Rod-Type Metamaterials at Optical Frequencies." K. Vynck, D. Felbacq, E. Centeno, A. I. C ?abuz, D. Cassagne, and B. Guizal. 2009.

2. Pyhtila, John W., Kevin J. Chalut, Jeffrey D. Boyer, Justin Keener, Thomas D'amico, Marcia Gottfried, Frank Gress, and Adam Wax. "In Situ Detection of Nuclear Atypia in Barrett's Esophagus by Using Angle-resolved Low-coherence Interferometry." Gastrointestinal Endoscopy 65.3 (2007): 487-91. Web.

3. Drake, R.M. and Gordon, J.E. "Mie Scattering". American Journal of Physics. Issue 53. 7 Oct 1984. <http://dx.doi.org/10.1119/1.14011>.

4. I. Wiener, M. Rust, and T.D. Donnelly, Am. J Phys. 69 (2), 129-136 (2001).

5. Prahl, Scott. “Mie Scattering Calculator”. Oregon Medical Laser Center. 2012 <http://omlc.org/calc/mie_calc.html>

-- Main.faell004 - 14 May 2015