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Spring 2011

An investigation into particle shape effects on the light scattering An investigation into particle shape effects on the light scattering

properties of mineral dust aerosol properties of mineral dust aerosol

Brian Steven Meland University of Iowa

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Part of the Physics Commons

Copyright © 2011 Brian Steven Meland

This dissertation is available at Iowa Research Online: https://ir.uiowa.edu/etd/1024

Recommended Citation Recommended Citation Meland, Brian Steven. "An investigation into particle shape effects on the light scattering properties of mineral dust aerosol." PhD (Doctor of Philosophy) thesis, University of Iowa, 2011. https://doi.org/10.17077/etd.lnrhudcw

Follow this and additional works at: https://ir.uiowa.edu/etd

Part of the Physics Commons

AN INVESTIGATION INTO PARTICLE SHAPE EFFECTS ON THE LIGHT

SCATTERING PROPERTIES OF MINERAL DUST AEROSOL

by

Brian Steven Meland

An Abstract

Of a thesis submitted in partial fulfillment of the requirements for the Doctor of

Philosophy degree in Physics in the Graduate College of

The University of Iowa

May 2011

Thesis Supervisor: Professor Paul D. Kleiber

1

ABSTRACT

Mineral dust aerosol plays an important role in determining the physical and

chemical equilibrium of the atmosphere. The radiative balance of the Earth’s atmosphere

can be affected by mineral dust through both direct and indirect means. Mineral dust can

directly scatter or absorb incoming visible solar radiation and outgoing terrestrial IR

radiation. Dust particles can also serve as cloud condensation nuclei, thereby increasing

albedo, or provide sites for heterogeneous reactions with trace gas species, which are

indirect effects. Unfortunately, many of these processes are poorly understood due to

incomplete knowledge of the physical and chemical characteristics of the particles

including dust concentration and global distribution, as well as aerosol composition,

mixing state, and size and shape distributions. Much of the information about mineral

dust aerosol loading and spatial distribution is obtained from remote sensing

measurements which often rely on measuring the scattering or absorption of light from

these particles and are thus subject to errors arising from an incomplete understanding of

the scattering processes.

The light scattering properties of several key mineral components of atmospheric

dust have been measured at three different wavelengths in the visible. In addition,

measurements of the scattering were performed for several authentic mineral dust

aerosols, including Saharan sand, diatomaceous earth, Iowa loess soil, and palagonite.

These samples include particles that are highly irregular in shape. Using known optical

constants along with measured size distributions, simulations of the light scattering

process were performed using both Mie and T-Matrix theories. Particle shapes were

approximated as a distribution of spheroids for the T-Matrix calculations.

It was found that the theoretical model simulations differed markedly from

experimental measurements of the light scattering, particularly near the mid-range and

near backscattering angles. In many cases, in the near backward direction, theoretical

2

models predicted scattering intensities for near spherical particles that were up to 3 times

higher than the experimentally measured values. It was found that better agreement

between simulations and experiments could be obtained for the visible scattering by using

a much wider range of more eccentric particle shapes.

Abstract Approved: ____________________________________ Thesis Supervisor

____________________________________ Title and Department

____________________________________ Date

AN INVESTIGATION INTO PARTICLE SHAPE EFFECTS ON THE LIGHT

SCATTERING PROPERTIES OF MINERAL DUST AEROSOL

by

Brian Steven Meland

A thesis submitted in partial fulfillment of the requirements for the Doctor of

Philosophy degree in Physics in the Graduate College of

The University of Iowa

May 2011

Thesis Supervisor: Professor Paul D. Kleiber

Graduate College The University of Iowa

Iowa City, Iowa

CERTIFICATE OF APPROVAL

_______________________

PH.D. THESIS

_______________

This is to certify that the Ph.D. thesis of

Brian Steven Meland

has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Physics at the May 2011 graduation.

Thesis Committee: ___________________________________ Paul D. Kleiber, Thesis Supervisor

___________________________________ Mark A. Young

___________________________________ Vicki H. Grassian

___________________________________ Steven R. Spangler

___________________________________ Frederick N. Skiff

___________________________________ Kenneth Gayley

ii

To Marit

iii

ACKNOWLEDGMENTS

Over this past six years, I have spent countless hours in a dark lab making light

scattering measurements, hunched over textbooks while studying for the physics

qualifying exam, and sitting in front of a computer writing thousands of lines of code or

analyzing data. It has been difficult, stressful, and tiring but at the same time exciting and

hugely rewarding. I will never forget the feelings of accomplishment when I received the

fellowship I was hoping for, getting my first paper published, or finding out that I passed

my exams. However, those feelings should not be mine alone. There have been many

people throughout the years who offered their support so I could get my degree. Let this

just be one more instance of me saying “Thank you”.

First, I would like to acknowledge my fellow graduate students and colleagues at

the Iowa Advanced Technology Laboratories. To Dr. Dan B. Curtis, thank you for

introducing me to the workings of the lab and being patient with me as I learned the basic

principles of light scattering. To Dr. Paula K. Hudson, for taking the time to answer my

many questions on aerosol particles, and for repeatedly explaining “ )log( pDddN ” to me. Dr.

Juan G. Navea, thank you for always bringing a chemist’s perspective to the lab and for

all of your stories that started with “Back when I was in grad school…”. Dr. Jonas

Baltrusaitis, thank you for always collecting “just one more” set of SEMs. To Mark

Smalley, for collecting so much scattering data that I was still analyzing it two years after

you graduated. Finally, to my friends Murat, Eric, and Paul, the daily trips to the IMU

always prepared me for the rest of the day.

There have also been a number of sources of financial support which I wish to

acknowledge. Through the University of Iowa, I received a Presidential Fellowship and a

Graduate Student Incentive Fellowship. Through NASA, I received the NASA Earth and

Space Science Fellowship. These awards have helped immensely by providing me with

ability to focus all of my efforts towards my research. This research would not have been

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possible without funding received through the National Science Foundation (Grant AGS-

096824).

There have been many professors throughout my academic career that have been

instrumental in my academic success. As an undergraduate student Dr. Ananda Shastri

and Dr. Mathew Craig taught me the fundamentals of physics, from electromagnetic

theory to quantum mechanics. You both encouraged and prepared me to continue on to

graduate school. To my graduate school academic advisors Dr. Paul D. Kleiber, Dr. Mark

A. Young, Dr. Vicki H. Grassian, thank you for all the help throughout the years.

Between helping with revisions on manuscripts, data analysis, and asking tough questions

about the latest set of measurements, I don’t know where you also found the time to teach

classes. It’s a good thing that there were three of you. I would also like to thank all of my

dissertation committee members, Dr. Paul D. Kleiber, Dr. Mark A. Young, Dr. Vicki H.

Grassian, Dr. Steven R. Spangler, Dr. Frederick N. Skiff, and Dr. Kenneth Gayley.

There have been many people outside of school to whom I owe thanks for their

support. To my friends Steve, Ellery, Jason, and Ryan, our Wednesday night games were

always a welcome break from working in the lab. To my parents, Steve and Kathy, and

my sister, Melissa, you have supported my decision to spend all these years studying

physics from the beginning. You knew I would pass my exams even when I was

uncertain and you knew I would finish even when I couldn’t see the light at the end of the

tunnel. Our Saturday morning phone calls and occasional care packages always helped

home seem not so far away. Most importantly, I wish to thank my wife Marit for her

support and patience throughout this process. You have listened intently as I rambled on

the days latest set of measurements, shared in the stress of graduate student life

(especially while I studied for the qualifying exam), and dealt with the uncertainty of

when I would finally finish writing. I will never be able to thank you enough.

v

ABSTRACT

Mineral dust aerosol plays an important role in determining the physical and

chemical equilibrium of the atmosphere. The radiative balance of the Earth’s atmosphere

can be affected by mineral dust through both direct and indirect means. Mineral dust can

directly scatter or absorb incoming visible solar radiation and outgoing terrestrial IR

radiation. Dust particles can also serve as cloud condensation nuclei, thereby increasing

albedo, or provide sites for heterogeneous reactions with trace gas species, which are

indirect effects. Unfortunately, many of these processes are poorly understood due to

incomplete knowledge of the physical and chemical characteristics of the particles

including dust concentration and global distribution, as well as aerosol composition,

mixing state, and size and shape distributions. Much of the information about mineral

dust aerosol loading and spatial distribution is obtained from remote sensing

measurements which often rely on measuring the scattering or absorption of light from

these particles and are thus subject to errors arising from an incomplete understanding of

the scattering processes.

The light scattering properties of several key mineral components of atmospheric

dust have been measured at three different wavelengths in the visible. In addition,

measurements of the scattering were performed for several authentic mineral dust

aerosols, including Saharan sand, diatomaceous earth, Iowa loess soil, and palagonite.

These samples include particles that are highly irregular in shape. Using known optical

constants along with measured size distributions, simulations of the light scattering

process were performed using both Mie and T-Matrix theories. Particle shapes were

approximated as a distribution of spheroids for the T-Matrix calculations.

It was found that the theoretical model simulations differed markedly from

experimental measurements of the light scattering, particularly near the mid-range and

near backscattering angles. In many cases, in the near backward direction, theoretical

vi

models predicted scattering intensities for near spherical particles that were up to 3 times

higher than the experimentally measured values. It was found that better agreement

between simulations and experiments could be obtained for the visible scattering by using

a much wider range of more eccentric particle shapes.

vii

TABLE OF CONTENTS

LIST OF TABLES............................................................................................................. ix

LIST OF FIGURES ............................................................................................................ x

LIST OF SYMBOLS ..................................................................................................... xviii

CHAPTER 1 INTRODUCTION ........................................................................................1

Atmospheric Aerosol ........................................................................................1 Radiative Forcing..............................................................................................6 Remote Sensing ................................................................................................9 Chapter Overview...........................................................................................11

CHAPTER 2 EXPERIMENTAL SETUP ........................................................................13

Light Scattering Apparatus .............................................................................13 Mineral Dust Samples.....................................................................................16 Aerosol Size Distributions ..............................................................................18 System Alignment and Calibration.................................................................23 From Images to Phase Functions ....................................................................29

CHAPTER 3 MODELING THE EXPERIMENTAL SCATTERING APPARATUS .....49

Model of the Experimental Apparatus............................................................49 Model Parameters ...........................................................................................54 Model Results .................................................................................................55 Discussion.......................................................................................................58

CHAPTER 4 LIGHT SCATTERING THEORY .............................................................65

Mie Theory .....................................................................................................68 T-Matrix Theory .............................................................................................71

CHAPTER 5 MULTI-WAVELENGTH LIGHT SCATTERING STUDIES ...................77

Error Analysis .................................................................................................77 T-Matrix Shape Distribution...........................................................................80 Asymmetry Parameter Calculations ...............................................................81 Non-Clay Mineral Dust Results......................................................................81 Clay Mineral Dust Results..............................................................................83 Iron Oxide Results ..........................................................................................85 Arizona Road Dust Results.............................................................................87 Discussion.......................................................................................................89

CHAPTER 6 DETERMINING PARTICLE SHAPE DISTRIBUTIONS FROM FTIR SPECTRAL FITTING ........................................................................110

Modeling.......................................................................................................110 Error Analysis ...............................................................................................113 Results for Quartz .........................................................................................117

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Discussion.....................................................................................................118

CHAPTER 7 AUTHENTIC MINERAL DUST LIGHT SCATTERING.......................129

Particle Size Distributions ............................................................................129 T-Matrix Shape Distribution.........................................................................130 Optical Constants..........................................................................................131 Error Analysis ...............................................................................................132 Authentic Mineral Dust Results....................................................................133 Shape-Fitting Palagonite Data ......................................................................135 Discussion.....................................................................................................136

CHAPTER 8 FUTURE WORK ......................................................................................151

Larger Mineral Dust Aerosol Particles .........................................................151 Particle Shapes Used in T-Matrix Calculations............................................153 Particle Coatings ...........................................................................................155 Expanding the Light Scattering Database.....................................................156

APPENDIX......................................................................................................................158

A.1 Determining the System Calibration and Angle Mapping Functions....158 A.1.1 Polarization Correction (Sub-function) .......................................164 A.1.2 Angle Mapping (Sub-function) ...................................................165 A.1.3 Calibration Curve Fitting (Sub-function) ....................................166

A.2 Splicing APS and SMPS Data ...............................................................167 A.2.1 Fitting SMPS Data with a Log-normal Distribution (Sub-function) ................................................................................................171 A.2.2 Optimizing Overlap Between APS and SMPS Data (Sub-function) ................................................................................................172

A.3 Modeling the Experimental Apparatus ..................................................173 A.3.1 Determining Aperture Collisions (Sub-function) ........................179 A.3.2 Determining Detector Collisions (Sub-function) ........................180 A.3.3 Determining Mirror Collisions (Sub-function)............................181

REFERENCES ................................................................................................................182

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LIST OF TABLES

Table 2.1 Physical properties of mineral dust samples used in this work: volume equivalent mode diameter (Dve), aerodynamic shape factor (χ), lognormal size distribution best fit width parameter (σ), and surface area weighted effective radius (Reff). ..............................................................32

Table 2.2 Calculated characteristic size parameter ( effeff RX 2 ) mineral dust samples used in this work. ..............................................................................33

Table 2.3 Optical properties of mineral dust samples used in this work. For cases where optical constants were not available for a given wavelength, a linear extrapolation of values near that wavelength was used........................34

Table 2.4 Chemical composition of the well defined mineral dust samples used in this work. ........................................................................................................35

Table 2.5 Size properties for polystyrene latex spheres (PSL).......................................35

Table 5.1 Reduced χ2 values for comparison of the experimental phase functions with simulations using Mie theory and using T-Matrix theory assuming a “Standard” shape distribution. .....................................................................95

Table 5.2 Asymmetry parameter values for experimental (gExp), Mie theory (gMie), and T-Matrix theory (gTM) phase functions for 470, 550, and 660 nm...........96

Table 6.1 Reduced χ2 values for comparison of the experimental phase functions with simulations based on different particle shape models, the moderate “SEM-based” and “Standard” models, and the extreme “Window” model. ...........................................................................................................122

Table 6.2 Asymmetry parameter values for experimental (gExp) and T-Matrix theory assuming the SEM-Based (gSEM), Standard (gStandard), and Window (gWindow) shape model phase functions for 470, 550, and 660 nm. ................................................................................................................122

Table 7.1 Reduced χ2 values for comparison of the experimental phase functions with simulations using Mie theory and using T-Matrix theory assuming a “Standard” shape distribution. ...................................................................140

Table 7.2 Asymmetry parameter values for experimental (gExp), Mie theory (gMie), and T-Matrix theory (gTM) phase functions for 550 nm................................140

x

LIST OF FIGURES

Figure 2.1 Light scattering experimental apparatus. Aerosol generated by the atomizer is directed to the focal point of an elliptical mirror that acts as the scattering region. The aerosol is then collected for real time particle sizing measurements by an Aerodynamic Particle Sizer (APS) and a Scanning Mobility Particle Sizer (SMPS). A tunable Nd:Yag pumped OPO is used as the light source. .....................................36

Figure 2.2 Detailed view of the optical setup (a) and scattering region (b) as viewed from above. Output from the OPO is directed to a telescope setup in order to decrease beam width by roughly a factor of three. A double Fresnel Rhomb prism is used to adjust the polarization of the incident laser before entering the scattering region. Scattered light reflects from the elliptical mirror and is subsequently focused through an aperture onto the CCD camera. The scattering angle is defined relative to the direction of the incident beam................................37

Figure 2.3 Measured aerosol particle size distributions (open circles) using an Aerodynamic Particle Sizer. Also shown are log-normal fits to the particle size distribution for mode diameters of 110, 220, and 440 nm. The log-normal fits are used in Mie calculations to give a range of possible scattering signals to account for uncertainty in the small diameter part of the size distribution. ........................................................38

Figure 2.4 CCD image of light scattering for 771 nm diameter PSL for parallel (a) and perpendicular incident light (b). ....................................................39

Figure 2.5 Experimental (solid line) and Mie theory (dashed line) phase functions for 771 nm diameter PSL. The experimental data has been mapped from pixels to scattering angle, but the system calibration has not been applied. Mie and experimental data has been normalized to the same amplitude at 35o. ..................................................40

Figure 2.6 Experimental (solid line) and Mie theory (dashed line) phase functions (a) and polarizations (b) for PSL with mean particle diameter of 771 nm. Experimental data has been calibrated and properly normalized. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line. ...................................................41

Figure 2.7 Calibration curve for 771 nm mean particle diameter PSL (solid line). A fit to the calibration curve is shown as well (dashed line)............42

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Figure 2.8 Experimental (solid line) and Mie theory calculations (dashed line) for phase functions (a) and polarizations (b) for 457 nm mean diameter PSL. Experimental data has been calibrated using the calibration curve generated using 771 nm mean diameter PSL. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line...................................................................................................43

Figure 2.9 Experimental (solid line) and Mie theory calculations (dashed line) for phase functions (a) and polarizations (b) for 1025 nm mean diameter PSL. Experimental data has been calibrated using the calibration curve generated using 771 nm mean diameter PSL. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line...................................................................................................44

Figure 2.10 Experimental (solid line) and Mie theory calculations (dashed line) for phase functions (a) and polarizations (b) for ammonium sulfate. Experimental data has been calibrated using the calibration curve generated using 771 nm mean diameter PSL. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line. ...............................45

Figure 2.11 CCD image of light scattering for quartz mineral dust, (a). Direct scatter has been subtracted out in (b). ........................................................46

Figure 2.12 Integrated scattering intensity of quartz mineral dust. ............................47

Figure 2.13 Phase function (a) and polarization (b) for quartz mineral dust. The phase function has been calibrated and properly normalized. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line...................................................................................................48

Figure 3.1 Ray diagram for reflection of a ray confined to the inside of an ellipse with semi-major axis length, a, and semi-minor axis length, b. The tangent and normal lines at the point of reflection, (x1, y1), are shown as dashed lines. ...............................................................................59

Figure 3.2 Schematic representation of scattering setup. A ray emitted from f1 (dashed line) passes between two solid vertical lines centered about f2 (aperture) and intercepts the detector. The extension of the ellipse has been cut down to represent the physical dimensions of the elliptical mirror in our scattering setup. The inset depicts a close up of the scattering area which has been rotated by an angle, . ...................59

Figure 3.3 Top-down view of possible scattering volumes resulting from the overlap of the incoming laser (arrow) with the aerosol dust jet (circle). The cross-hatched region represents the scattering volume. ........60

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Figure 3.4 Flow diagram of important elements of the scattering apparatus simulation code. .........................................................................................60

Figure 3.5 Histogram of the number of detector intercepts of rays (a), emitted from a rectangular scattering source located at f1, as a function of location on the detector where the interception occurred. The dark solid line at the bottom of the figure represents the physical extent of the detector. The mapping of the detector intercepts to the scattering angles (gray) along with a fit (black) is given in (b)..................................61

Figure 3.6 Histogram of the number of detector intercepts of rays for 0o (a), 10o (b), 20o (c), and 30o (d) rotations of the scattering volume. The solid black line in each figure corresponds to the extent of the detector............62

Figure 3.7 Calibration functions for the standard rectangle scattering volume (circle markers), a rotated rectangular scattering volume (square markers), and a larger rectangular scattering volume (diamond markers). ....................................................................................................63

Figure 3.8 Experimentally determined calibration function (solid line) along with model calibration function for the standard rectangle scattering volume (circle markers). The scattering ellipse has been defined to cover polar angles between +31.5o and -27.8o. (This corresponds to a range of experimental scattering angles 17 o - 172 o.) .............................64

Figure 5.1 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for calcite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ......................97

Figure 5.2 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for gypsum measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ..........98

Figure 5.3 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for quartz measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ......................99

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Figure 5.4 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for illite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ....................100

Figure 5.5 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for kaolinite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ........101

Figure 5.6 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for montmorillonite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ........102

Figure 5.7 Range of ratios of experimental phase functions to theoretical phase functions generated using Mie theory (a) and T-Matrix Theory (b). Results are shown for the non-clay samples calcite, gypsum, and quartz (dark gray) and for the clay samples illite, kaolinite, and montmorillonite (light gray). ...................................................................103

Figure 5.8 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for hematite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Optical constants used for the theoretical calculations were obtained from Longtin et al. [1988]. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ..........................................................104

Figure 5.9 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for hematite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Optical constants used for the theoretical calculations were obtained from Bedidi & Cervelle. [1993]. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. .........................................105

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Figure 5.10 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for hematite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Optical constants used for the theoretical calculations were obtained from Hematite Sokolik & Toon [1999]. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. .........................................106

Figure 5.11 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for goethite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ........107

Figure 5.12 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for Arizona Road Dust measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. .............................................................................108

Figure 5.13 Normalized phase functions (a) and linear polarizations (b) for Arizona Road Dust measured at 470 nm (left), 550 nm (center), and 660 nm (right). Empirical phase functions and polarizations (dashed line) were generated using a uniform weighting of clay (illite, kaolinite, and montmorillonite) and non-clay (calcite, gypsum, and quartz) samples. .......................................................................................109

Figure 6.1 SEM image of quartz particles with best-fit ellipses determined using the ImageJ software package..........................................................123

Figure 6.2 The left panel shows the two different moderate particle shape distributions used in the experiment: (a) “Standard” shape distribution using a uniform distribution of oblate and prolate spheroids with AR ≤ 2.4; (b) “SEM-based” shape distribution as determined from Figure 6.1 using ImageJ software package. The right panel shows the corresponding comparison of the T-Matrix simulation results (dashed lines) with experimental IR resonance extinction spectrum (solid with circles). ..................................................124

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Figure 6.3 The left panel shows the different extreme particle shape distributions as determined from IR resonance spectrum of quartz in Kleiber et al. [2009]: (a) “Unconstrained” model; (b) “Gaussian” model; (c) “Window” model. The right panel shows the corresponding comparison of the T-Matrix simulation results (dashed lines) with experimental IR resonance extinction spectrum (solid with circles)....................................................................................124

Figure 6.4 Comparison of visible scattering phase function (a) and polarization profiles (b) at 550 nm for the three different extreme model shape distributions shown in the left panel of Figure 6.3. .................................125

Figure 6.5 Comparison of experimental scattering phase functions (a), ratios of experimental to theoretical phase functions (b), and polarization profiles (c) at 470 nm with T-Matrix simulations based on different particle shape models: the moderate “SEM-based”, and “Standard” models, and the extreme “IR-Based” model. Phase functions in (a) for different shape models are offset by factors of ten for clarity. ..........126

Figure 6.6 Comparison of experimental scattering phase functions (a), ratios of experimental to theoretical phase functions (b), and polarization profiles (c) at 550 nm with T-Matrix simulations based on different particle shape models: the moderate “SEM-based”, and “Standard” models, and the extreme “IR-Based” model. Phase functions in (a) for different shape models are offset by factors of ten for clarity. ..........127

Figure 6.7 Comparison of experimental scattering phase functions (a), ratios of experimental to theoretical phase functions (b), and polarization profiles (c) at 660 nm with T-Matrix simulations based on different particle shape models: the moderate “SEM-based”, and “Standard” models, and the extreme “IR-Based” model. Phase functions in (a) for different shape models are offset by factors of ten for clarity. ..........128

Figure 7.1 Measured size distributions obtained by splicing Aerodynamic Particle Sizer and Scanning Mobility Particle Sizer measurements (dotted line) along with log-normal fits to the size distributions (solid line) for the authentic mineral dusts used in this work. .................141

Figure 7.2 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for diatomaceous earth measured at 550 nm. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ..........................................................142

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Figure 7.3 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for Iowa loess measured at 550 nm. Optical constants used for the theoretical calculations were obtained from Cuthbert [1940].Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ....................143

Figure 7.4 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for Iowa loess measured at 550 nm. Kaolinite optical constants used for the calculations and were obtained from Egan & Hilgeman [1979].Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. ........144

Figure 7.5 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for palagonite measured at 550 nm. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. .............................................................................145

Figure 7.6 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for Saharan sand measured at 550 nm. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity. .............................................................................146

Figure 7.7 Normalized phase functions (a), linear polarizations (b), and best fit shape distribution (c) for palagonite measured at 550 nm. The shape distribution was determined by optimizing the fit to the polarization. ....147

Figure 7.8 IR spectral data (solid line) and T-Matrix simulations using shape distribution fits to the polarization (solid line with filled circles) and to the IR spectral data (dashed line) (a), and best fit shape distribution (b) for palagonite. The shape distribution in (b) was determined by optimizing the fit to the IR spectral data..........................148

Figure 7.9 Particle shape distributions for palagonite (a) and Saharan sand (b). Processed samples were ground using a mortal and pestle followed by mechanical grinding using a Wig-L-Bug............................................149

xvii

Figure 7.10 Scanning electron micrographs of the authentic mineral dust samples. SEMs are shown for diatomaceous earth (a), Iowa loess (b), palagonite (c), palagonite post-processing (d), Saharan sand (e), and Saharan sand post-processing (f). .....................................................150

xviii

LIST OF SYMBOLS

AR Axial ratio

)(C System calibration function

CS Cunningham slip correction factor

AD Aerodynamic particle diameter

Dp Particle diameter

MD Mobility particle diameter

veD Volume equivalent particle diameter

f1 First focal point of elliptical mirror (light scattering region)

f2 Second focal point of elliptical mirror (location of aperture)

)(F Light scattering phase function

g Asymmetry parameter

)(// I Scattered light intensity for parallel polarized incident light

)(I Scattered light intensity for perpendicular polarized incident light

k Wavenumber

m Complex refractive index

n(Dp) Particle size distribution

)(P Light scattering polarization profile

Reff Effective particle radius

S(AR) Particle shape distribution

X Characteristic size parameter

xix

χ Aerodynamic shape factor

λ Wavelength

θ Scattering angle

ρ Density

1

CHAPTER 1

INTRODUCTION

Mineral dust aerosol plays an important role in determining the physical and chemical

equilibrium of the atmosphere. Mineral dust affects climate forcing through the direct

scattering and absorption of both incoming visible solar radiation and outgoing terrestrial IR

radiation. Dust particles can also serve as cloud condensation nuclei and provide sites for

heterogeneous chemical reactions involving important trace gas species, which, in turn,

indirectly affect the earth’s radiation balance. Unfortunately, many of these processes are

poorly understood due to our incomplete knowledge of the physical and chemical

characteristics of the particles, including dust concentration and global distribution, as well as

aerosol composition, mixing state, and size and shape distributions. Field measurements of

aerosol properties are often carried out by remote sensing using satellite or ground based

instruments. However, dust retrieval algorithms can depend critically on the optical

properties of the dust. Because of uncertainties in aerosol optical properties, dust loading and

other characteristics inferred from the field data such as dust composition, and size and shape

distributions can be highly uncertain. In this work, the light scattering properties of several of

the major mineral components of atmospheric dust aerosol have been studied in order to

gauge the accuracy of different light scattering theories in modeling the optical properties of

mineral dust aerosol. Particular emphasis has been placed on the effect of particle shape on

the analysis.

Atmospheric Aerosol

Before beginning a discussion of mineral dust aerosol, it is important to understand

some of the basic principles and definitions relating to atmospheric aerosol in general.

Aerosol is defined to be any solid or liquid particulate matter suspended in a gaseous

medium. In many treatments of the subject, the term aerosol is also interchangeably used to

refer specifically to the particle itself. The particles in a typical atmospheric aerosol generally

2

comprise only a small fraction of the overall mass and volume of the aerosol, on the order of

10-6 [Hinds, 1999]. Aerosol can be generated either through natural processes or through

human activity. Common examples of natural aerosol include clouds of water droplets or ice

particles, large salt particles generated from ocean spray, pollen, ash from volcanic eruptions,

and windblown soil. Anthropogenic aerosol includes smoke and soot from industrial

emissions such as carbon particles from incomplete combustion of fossil fuels, urban smog,

aerosol generated during surface mining, and an increase in the emissions of windblown soil

due to agricultural practices, deforestation, and the desertification of land areas. Total global

emissions of natural aerosol are on the order of 3.1x109 metric tons per year, whereas

anthropogenic aerosol emissions are about 0.46x109 metric tons per year [D’Almeida et al.,

1991; Andreae, 1995]. Anthropogenic aerosol makes up a relatively small fraction of total

emitted atmospheric aerosol, ~15%, but is concentrated in the industrialized regions [Hinds,

1999; Sokolik et al., 2001; Satheesh & Moorthy, 2004]. On a regional scale these

anthropogenic aerosol emissions can exceed those of natural aerosol.

A number of physical properties are used to classify aerosol, or more accurately,

classify the characteristics of the particulate phase of the aerosol. These properties include

particle concentration, composition, shape, and size distributions. Concentration is a measure

of either the mass or number density of the particles of the suspended medium, and is often

expressed in units of mg/m3 or number per m3 respectively. Particle composition directly

affects many of the secondary properties of the aerosol (such as optical properties).

Depending on particle composition and the method of aerosolization (mechanical grinding,

combustion, etc.), the shape of an aerosol particle can vary significantly. While liquid

droplets tend to be spherical, solid aerosol particles tend to have more complicated shapes.

Some, such as asbestos, exhibit long fibrous shapes. Smoke or soot resulting from incomplete

combustion can also form long chain aggregates of many smaller primary particles. Many

dust particles, such as quartz and silicate clays, are highly irregular in shape and can contain

sharp edges, points, and internal voids.

3

Particle size is one of the most important aerosol properties. Atmospheric aerosol

may contain particles with sizes that span many orders of magnitude, ranging from tens of

nm to hundreds of µm, provided there are sufficient forces present, such as strong winds, to

keep the particles suspended [Hinds, 1999]. It is useful to define a number of size ranges or

“modes”, to describe aerosol particles that exhibit similar physical properties such as settling

velocity or atmospheric residence time. The smallest mode is the nucleation mode, which

includes particles with diameters of roughly 10-100 nm. Nucleation mode particles tend to

quickly coagulate with each other and with particles in the accumulation mode, especially

near source regions where aerosol concentrations are high. This leads to relatively short

atmospheric lifetimes for nucleation mode particles. The accumulation mode consists of

particles with diameters between 0.1 and ~ 2.0 µm. Coagulation of accumulation mode

particles is too slow for the particles to reach the coarse mode and other removal

mechanisms, including rainout (where the aerosol particles serve as nucleation sites for

raindrops) and washout (removal by falling rain or snow), are very weak. Particles in the

accumulation mode can have atmospheric residence times on the order of weeks. During this

time, the particles can be transported by wind action over intercontinental distances

[Prospero, 1999]. The largest size mode, the coarse mode, consists of particles with

diameters larger than ~2 µm. These particles are quickly removed from the air due to

gravitational settling and impaction; atmospheric lifetimes for these particles are generally

only a few hours or days. Dust particles generated during strong winds, salt particles from

sea spray, and particles generated during agricultural or mining practices can fall into the

coarse mode.

As mentioned above, the term “atmospheric aerosol” covers a wide spectrum of

particulate matter that is suspended in the Earth’s atmosphere. Mineral dust aerosol is just

one component of atmospheric aerosol, though an important one, and is the focus of this

work. Mineral dust aerosol itself refers to a large class of naturally occurring elements or

compounds formed by geological processes and comprised mainly of crustal minerals. Strong

4

mineral dust source regions exist throughout northern Africa, Middle East, and central Asia

[Prospero, 1999]. On a global scale, mineral dust composition is highly inhomogeneous and

depends strongly on the source region. For instance, the soil of the Gobi desert typically has

less iron but more aluminum and calcium content than the global average earth crust

composition [Petrov, 1976], and dust aerosol generated from the Gobi desert will share these

traits. Due to relatively strong visible absorption, minerals containing high concentrations of

iron minerals, such as hematite and goethite, play an especially important role in light

scattering processes. In studies of the relative iron oxide content of a number of aerosol

samples from northern Africa and Eastern Asia, Lafon et al. [2006] found that goethite was

present in high concentrations in all samples investigated. Sokolik & Toon [1999] have

compiled data of mineralogical composition of atmospheric dust collected from a number of

regions. They show that samples collected near Central America can have clay compositions

as high as 70% by bulk mass, which agrees with measurements of aerosol composition by

Reid et al. [2003] who found that silicate clays such as illite, kaolinite, and montmorillonite

make up the majority of mineral dust aerosol in that region. Samples collected near Nigeria,

show a much lower clay fraction (<25%) and are composed primarily of quartz. This is due

to much of the Nigerian sample originating from the nearby Saharan desert. Some of the

mineral types found in abundance in ground soils include quartz, calcite, gypsum, dolomite,

mica, feldspars, kaolinite, illite, montmorillonite, palygorskite, chlorite, and organic matter

[Pye, 1987]. Due to long residence times in the atmosphere, mineral dust composition can

also be altered through chemical aging and mixing with other atmospheric aerosol [Sokolik &

Toon, 1999].

Mineral dust aerosol can play several important roles in atmospheric processes. Dust

particles can serve as nucleation sites to promote cloud formation. Mineral dust can also

provide a heterogeneous reaction site for reactive trace gases in the atmosphere [Dentener et

al., 1996; Bates et al., 2004; Sullivan et al., 2006]. Arguably one of the most important

processes, the absorption and scattering of light by mineral dust, has been the focus of

5

numerous investigations over the past several decades. These include experimental and

theoretical studies of light scattering and absorption throughout the visible and infrared

regions of the electromagnetic spectrum. Due to the irregular shape of mineral dust particles,

the scattering process can be quite complex. A number of theories have been developed to

model light scattering by small particles including Mie theory, discrete dipole approximation

(DDA) methods, Rayleigh Debye Gans theory, geometrical optics methods (GOM), and T-

Matrix theory. Mie theory is the simplest and fastest way to calculate light scattering, though

it is strictly limited to spherical particles. While Mie theory can provide a good first order

approximation to the scattering, significant errors can arise in using Mie theory for

irregularly shaped particles, particularly for near back scattering angles. DDA methods allow

a great deal of freedom in specifying particle shapes. However, the method is

computationally intensive and so is restricted to relatively small particles. T-Matrix theory is

computationally efficient and allows for modeling the scattering from a wider range of

particle sizes and shapes. In addition to particle shape, light scattering theories also require

estimates of the optical constants (refractive index), as well as the particle size, composition,

and morphology.

The net radiative effect of dust in the atmosphere remains highly uncertain due a

number of factors. Of the natural aerosol types, mineral dust is among the most poorly

characterized. Even though there have been studies which include characterization of mineral

dust in the atmosphere, such as the series Aerosol Characterization Experiments (ACE)

[Huebert et al., 2003; Seinfeld et al., 2004], there is still uncertainty as to how to best model

mineral dust aerosol, including the composition, size, and shape distributions, mixing state,

and particle morphology [West et al., 1997; Sokolik et al., 2001]. There is also limited

understanding of the chemical and physical processes that govern mineral dust aerosol

generation, transport, and atmospheric aging, all of which can affect global light scattering

calculations [Sokolik et al., 2001].

6

Particle shape effects introduce particularly significant uncertainties into scattering

calculations. Even for scattering theories that allow flexibility in particle shape, there is still

uncertainty in how to specify a range of shapes that are truly representative of the mineral

dust being modeled. Mineral dusts are made up of a number of components. Even if the

refractive indices are known for each component, the best way to determine the overall

average index value is not always clear as it depends strongly on the mixing state and

morphology of the dust. Optical properties tend to vary nonlinearly with refractive index.

This makes it difficult to estimate the magnitude of any errors that may result due to

uncertainty in the optical constants.

A precise understanding of the properties and behavior of aerosol is necessary for a

broad range of applications. Aerosol can have a direct impact on human health. Small

particles, those with diameters < 2.5 µm (respirable size range), can penetrate deeply into the

lungs. Aerosol transport governs the deposition of highly fertile loess soil to regions

throughout the world. Aerosol can have a both a regional and global effect on climate by

scattering and absorbing incoming solar and outgoing terrestrial radiation. Satellite

measurements of surface properties, such as earth surface temperatures, depend on

corrections to account for the optical depth of the aerosol. The effect of mineral dust aerosol

on satellite remote sensing measurements and on climate forcing as they relate to light

scattering is discussed in more detail below.

Radiative Forcing

The main two processes that control the overall state of the Earth’s climate system are

heating through absorption of incoming solar radiation and cooling by terrestrial emission of

long-wave infrared radiation. Radiative forcing refers to any system that alters this radiative

balance. Positive forcing corresponds to heating of the atmosphere, while negative forcing

corresponds to cooling. Greenhouse gases, which are largely transparent to incoming solar

radiation, absorb IR radiation emitted by the Earth resulting in positive forcing. Soot, from

7

industrial burning of fossil fuels or volcanic eruptions, contains high levels of black carbon,

and is the strongest absorbing component of atmospheric aerosol. Estimates of warming due

to atmospheric soot are generally within the range 0.15 - 0.2 W/m2 [Haywood et al., 1997;

Myhre et al., 1998]. Sea salts and sulfates, which are non-absorbing, generally lead to

negative forcing by scattering incoming solar radiation back into space. The magnitude of the

forcing for sulfates is roughly -0.3 - -0.5 W/m2 [Chuang et al., 1997; Feichter et al., 1997;

Myhre et al., 1998].

Radiative forcing due to mineral dust is complicated and depends on many factors.

Mineral dust can both directly and indirectly affect the radiative balance and the sign of that

forcing can be positive or negative. Directly, mineral dust causes negative forcing through

the scattering of incoming solar radiation and positive forcing by absorbing outgoing long-

wave terrestrial IR radiation. Indirectly, particles within the nucleation size mode can serve

as cloud condensation nuclei. The increase in cloud cover increases the albedo (reflectivity)

of the Earth’s surface, having a cooling effect. Some estimates put the magnitude of the

radiative forcing due to tropospheric mineral dust to levels comparable to, but opposite in

sign, to that of greenhouse gases [Hansen & Lacis, 1990, Penner et al., 1994]. However, the

net radiative effect of mineral dust aerosol is still highly uncertain, both in magnitude and

sign [Penner et al., 2001, Forster et al. 2007]. Myhre and Stordal [2001] estimate the total

forcing from mineral dust to be within the range -1.4 – +1.0 W/m2 based on simulations

using different size and spatial distributions of dust. They found that increasing

concentrations of hematite leads to higher positive forcing due to increases in absorption. A

similar range of values, -2.0 – +0.5 W/m2, was found by Satheesh & Moorthy [2004].

Forcing from mineral dust aerosol of anthropogenic origin may be as large as +0.09 W/m2

[Tegen et al., 1996], though this number is uncertain due to difficulties separating

contributions from natural sources of mineral dusts from those that are man-made since both

components are highly mixed away from source regions. [Satheesh & Moorthy, 2004]

8

Radiative forcing calculations depend on many factors. Estimates of the aerosol

temporal and 3-dimensional spatial distribution must be made on a global scale. However,

data on these distributions are scarce and soil transport, production, and removal processes

are poorly understood [Myhre & Stordal, 2001]. Accurate estimates for a number of key

optical properties of the aerosol must also be made. These properties include the single

scattering albedo (ratio of scattering to extinction), the asymmetry parameter (ratio of

forward to backward scattering), and the optical depth (measure of total scattering and

absorption) of the aerosol. Many of these properties can not be measured directly, and are

instead estimated from other measured or modeled scattering properties such as extinction

spectra or the angular distribution of scattered light (phase function). Accurately modeling

the optical properties of mineral dust aerosol can be very difficult, however, and depends on

accurate knowledge of aerosol size, composition, refractive index, shape, and mixing state. A

large source of error in forcing calculations may come from poor knowledge of the

refractive index, though the effects of improper modeling of particle shapes may be equally

large [Kahnert & Nousiainen, 2006; Kahnert & Kylling , 2004]. Significant error can be

introduced into forcing calculations from assumptions of how the mineral dust particles are

mixed. For example, Sokolik and Toon [1999] found that for an external mixture of clays

(75%), quartz (20%), and hematite (5%) that the overall forcing was negative, -27.9 W/m2. In

contrast, they found that by aggregating a small (2%) amount of hematite with quartz (18%)

which was then externally mixed with clays (80%), the overall forcing became positive,

+11.4 W/m2. Similar results were found by Myhre et al [1998] for mixtures of sulfates with

soot, -0.16 W/m2 (external mixture), compared to +0.10W/m2 (internal mixture). Estimates of

the asymmetry parameter for dry mineral dust aerosol generally fall within the range 0.6 - 0.8

[D’Almeida et al., 1991; Andrews et. al., 2006; Kahnert & Nousiainen, 2006]. It is believed

that the asymmetry parameter is relatively insensitive to uncertainties in the refractive index

and particle shape, and that differences in the scattering tend to average out over various

angle ranges, though particle size plays a much stronger role [Mishchenko et al., 1995].

9

Changes in the asymmetry parameter can have significant effects on the overall forcing

though, a 10% decrease in the asymmetry parameter can result in a ~20% reduction in

radiative forcing [Andrews et. al., 2006].

Remote Sensing

Remote sensing applications, including measurements of surface and oceanic

temperatures, trace gas concentrations, and measurements of the total dust loading in the

atmosphere, offer a signficant advantage in that they are able to make measurements over

large areas in relatively little time. However, these applications require an accurate

assessment of the optical and physical properties of atmospheric aerosol in order to make

corrections to the measurements. For surface temperature measurements, detectors onboard

satellites monitor the emitted terrestrial IR radiation. The raw IR data must be corrected to

account for absorption and scattering of the signal as it passes through the atmosphere.

Aerosol measurements, such as those performed by the Multi-angle Imaging

SpectroRadiometer (MISR) and by the Moderate Resolution Imaging Spectro-radiometer

(MODIS) sensors onboard the Terra and Aqua satellites rely on scattered solar radiation at

numerous wavelength bands to determine concentration and composition of dust in the

atmosphere [Diner et al., 1998; Ichoku et al., 2004; Bruegge et al., 2007]. Many of these

measurements will later be used in climate forcing calculations. Therefore, any errors arising

from the remote sensing algorithms will carry over to the forcing calculations as well.

Much like the radiative forcing calculations, remote sensing measurements require

some a priori knowledge of the aerosol optical properties. For instance in order to determine

aerosol composition, the MISR satellite uses a predetermined lookup table of calculated

optical properties for a number of different aerosol types (sulfate, sea spray, mineral dust,

biogenic particles, and urban soot) and uses a linear combination of those properties to fit the

observed data, thus yielding the the relative amounts of each aerosol type [Diner et al.,

10

1998]. This approach requires many assumptions to first be made about average aerosol

composition, shape, size, and complex refractive index values.

Determining aerosol loading and composition is difficult and requires multi-angle

measurements of intensity and polarization [Chowdhary et al., 2001; Veihelmann et al.,

2004]. These measurements, especially of the polarization, depend strongly on particle shape

however [Veihelmann et al., 2004; Dubovik et al., 2006]. For roughly spherical aerosol, such

as marine sulfate aerosol, the use of Mie theory is sufficient to accurately describe light

scattering [Masonis et al., 2003]. Mineral dust aerosol, however, is generally irregular in

shape and requires more advanced scattering theories to calculate the scattering properties.

Approximating mineral dust as spheres can lead to overestimation in predicted backscattering

[Mishchenko et al., 1995; Kalishnikova & Sokolik, 2002; Curtis et al., 2008] which in turn

leads to an underestimation of the optical thickness. Spherical approximations can also lead

to errors in satellite IR spectral measurements. Hudson et al. [2008a, 2008b] found

significant errors in both IR spectral line shape and peak position for a number of mineral

dust samples when the particles shapes were approximated as spheres. The algorithm

employed by MISR instead uses a distribution of spheroidal shapes for mineral dust

calculations [Kahn et al., 1997]. There is still some uncertainty as to whether this

approximation to the particle shapes creates appreciable errors in the scattering. Some work

suggests that using a spheroidal approximation for particle shape introduces negligible errors

[Kahnert & Kylling, 2004], while others have found that the neglect of sharp edges could

lead to appreciable errors [Kalashnikova & Sokolik, 2002].

It is clear that there is a high level of uncertainty in the current understanding of the

light scattering properties of mineral dust aerosol. Since aerosol plays a key role in so many

atmospheric processes, it is important to assess and to mitigate this uncertainty with more

rigorous laboratory measurements of the light scattering, and through systematic testing of

the various light scattering theories as they apply to mineral dusts. That is the intent of the

current work.

11

Chapter Overview

The current work begins with a discussion of the experimental apparatus and methods

used to measure light scattering from mineral dust aerosol in Chapter 2. There, key equations

describing aerosol physical properties and size distributions are given. In addition, the

procedure for analyzing and calibrating the data to account for the nonlinear response

function of the experimental apparatus is discussed. A list of the mineral dust aerosol

samples used in this work, and some of their relevant physical and optical properties, is given

in Tables 2.1-2.4. A mathematical model of the experimental apparatus and scattering

process has been created in the Matlab computing language. This model is used to estimate

the magnitude of the uncertainties in our measurements and assists in the optical alignment of

the setup. A description of this model and the results of the model simulations are presented

in Chapter 3. In Chapter 4 a brief overview of the two light scattering theories used in the

work, Mie and T-Matrix theories, is given. The relative strengths and limitations of each

theory are discussed.

Experimental light scattering results are presented in Chapters 5-7. A multi-

wavelength investigation of the scattering from several key mineral components of

atmospheric dust is is described in Chapter 5. Results are presented there for a number of

clay (illite, kaolinite, montmorillonite) and non-clay minerals (calcite, gypsum, quartz), iron-

oxides (hematite, goethite), and for an example of an authentic, multi-component dust

mixture (Arizona Road Dust). Experimental results are compared to model simulations using

both T-Matrix and Mie theories. Possible uncertainties due to limitations in assumed particle

shape distributions are also discussed. In Chapter 6, focus is given to one mineral dust,

quartz. Extensive analysis of particle shape effects on model scattering is performed using

results from scanning electron micrograph images as well as results from IR spectral fitting

routines. T-Matrix results for a number of particle shape distributions are considered. In

Chapter 7, scattering from a number of real-world mineral dust samples including Saharan

sand, palagonite (a Martian regolith simulant), diatomaceous earth, and Iowa loess soil is

12

investigated. Finally, in Chapter 8, a summary of the results presented in this work is given.

Possible directions for future this work are also discussed.

13

CHAPTER 2

EXPERIMENTAL SETUP

This chapter will detail the experimental setup used to collect light scattering data for

mineral dust aerosol. Information will be provided for each mineral dust sample used in these

experiments including optical properties (indices of refraction) and particle size information

(mode diameters, shape factors, etc.). The procedures for data analysis will also be presented,

including the processing of scattered light images collected on the CCD camera, phase

function normalization, and system calibration.

Light Scattering Apparatus

A diagram of the full scattering apparatus is given in Figures 2.1-2.2. Mineral dust

samples are prepared as a suspension in high purity water (Optima Water, > 99.9% purity). A

constant output atomizer (Model 3076, TSI Inc.) is then used to aerosolize the sample. The

aerosol is passed through a cyclone and multiple diffusion dryers (one commercial dryer ,

ATI Diffusion Dryer 250, and one home built dryer, each filled with silica as a desiccant) to

remove excess water vapor from the flow. Next, the aerosol enters a conditioning tube where

additional dry air is combined with the flow in order to adjust the total flow rate and

minimize relative humidity. Relative humidity is monitored at the end of the conditioning

tube via a solid state sensor (Model HIH-3602-C, Honeywell) to determine that it is within an

acceptable range, typically 10-20%. The aerosol is then directed through a nozzle to a

windowless scattering region located at one of the focal points of an elliptical mirror, f1 (see

Figure 2.2). The nozzle, located just above the scattering region, compresses the output

aerosol stream down to a narrow jet with a diameter ~1mm. A collection cup, located below

the scattering region, is connected to an auxiliary vacuum pump and two particle sizing

instruments which are pulling in air at a total flow rate of 10 lpm. Since the combined flow

rate from the atomizer and conditioning tube is much lower (~3.5 lpm), this keeps the

collection cup at a low pressure relative to ambient. This pressure gradient helps ensure that

14

the aerosol flow is efficiently captured from the scattering region for size analysis and so that

the mirror and other optics remain free of deposited dust. The aerosol nozzle is mounted on

an x-y translation stage so fine adjustments to its alignment can be made to position it

directly above f1.

Following capture in the collection tube below the scattering region, the aerosol flow

is split for analysis to a pair of particle sizing instruments. A small fraction (~2%) of the total

aerosol flow is directed to a Scanning Mobility Particle Sizer (SMPS, Model 3034, TSI Inc.).

The SMPS measures particle diameters in the range of ~ 20-500 nm. Particles from roughly

one half of the aerosol flow are collected by an Aerodynamic Particle Sizer (APS, Model

3321, TSI Inc.). This instrument is capable of measuring particle diameters in the range ~

0.5-20 µm. The measurement principles of these instruments will be explained in detail

below. The remainder of the aerosol flow is filtered (HEPA capsule Filter) and directed to an

auxiliary vacuum pump (GAST Manufacturing, Inc., Model DOA-P704-AA).

The tunable output of an Optical Parametric Oscillator (OPO, Continuum Sunlite EX)

is used as the light source in this experiment. The OPO is pumped by the third harmonic of a

Nd:YAG laser (Continuum Precision II) operating at 10 Hz with an average output power of

2.5 Watts at 355 nm. The OPO is capable of covering a wavelength range between 0.45-1.7

µm. The pulsed output of the OPO is linearly polarized and highly monochromatic

(bandwidth ≈ 0.08 cm-1). The pulse width is on the order of 7 ns. The output power of the

OPO depends on wavelength; the maximum average power near 500 nm is ~ 500 mW.

The output of the OPO is first attenuated by a factor of 100 using a series of neutral

density filters. A Keplerian telescope arrangement of lenses, with focal lengths of 500 mm

and 150 mm respectively, is used to narrow down the beam width from 5 mm to 1.5 mm. A

pinhole is placed at the focal point within the telescope to act as a spatial filter in order to

ensure a Gaussian beam profile. The smaller beam width results in a smaller scattering

volume and therefore higher angular resolution for the measured scattered light as will be

discussed further in Chapter 3. Next, the beam passes through a polarization rotator (double

15

Fresnel rhomb prism) in order to select the angle of the linear polarization vector of the

incident beam relative to the scattering plane; in these experiments polarizations parallel or

perpendicular to the scattering plane are used. A long focal length lens (f = 60 cm) is then

used to focus the beam onto one of the focal points of the elliptical mirror, f1. The

intersection of the laser with the aerosol jet in the scattering region defines the scattering

volume.

A close up of the scattering region and CCD detector is shown in Figure 2.2b. Laser

light, incident on the aerosol particles within the scattering region, will be scattered to all

angles. Scattered light collected by the elliptical mirror is focused at the second focal point,

f2, of the mirror. The elliptical mirror was custom machined from a solid aluminum block by

Opti-Forms, Inc. (Temecula, California). The ellipse defined by the mirror has a semi-major

axis length of 60 cm and a semi-minor axis length of 30 cm. The mirror is mounted on an x-y

translation stage, in the scattering plane, to assist in alignment of the setup. An aperture

located at the second focal point, f2, limits the detector’s field of view of the scattering

region. The scattered light is then imaged onto a CCD camera (Santa Barbara Instrument

Group) located approximately 3 cm behind the aperture. The CCD camera is thermo-

electrically cooled to 5o C to reduce noise as a result of dark current. The elliptical mirror and

CCD array are contained within an opaque box to limit room light from reaching the

detector. Within the box, a wall separates the detector from the scattering region in order to

cut down on background light scattering from the edges of the mirror. A small aperture in

this wall allows scattered light from the aerosol to pass to the CCD array. This optical

arrangement allows for the mapping of scattering angle to position on the detector, where a

scattering angle of 0o (forward scatter) corresponds to light scattered in the direction of the

incident light and 180o (backward scatter) is light scattered opposite the direction of the

incident light. The physical dimensions of the mirror and laser beam diameter limit the range

of measurable scattering angles to a continuous range of 17-172o. The forward direction is

more limited due to large background signals associated with intense forward scattered light.

16

The CCD camera is positioned so that this range of scattering angles covers most of the

active region of the detector, which has 2184 bins. Integration times of 100 seconds are

typically used to collect each scattering image; corresponding to an average over ~1000 laser

pulses.

Mineral Dust Samples

A number of mineral dust samples, both single and multi-component, have been

examined in this work. Single component samples include both silicate clay and non-clay

samples, including iron oxides. The clay samples include montmorillonite, kaolinite, and

illite. The non-clay samples include quartz, calcite, gypsum, hematite, and goethite. All of

these powdered samples were purchased from commercial vendors and used without further

processing, with the exception of illite which came as a rock and needed to be broken down

(see below). A number of samples that are more representative of real world mineral dusts,

which can often be internally and/or externally mixed, were also used. Saharan sand and

Iowa loess samples were both collected in the field by colleagues. Arizona Road Dust and

Diatomaceous Earth samples were obtained commercially. Palagonite (JSC Mars-1), a

complex mixture often used as a simulant for Martian soil, was obtained from NASA. A list

of all samples used and the sources for each is given in Table 2.1.

Certain samples, including some of the real-world samples, required further

processing after they were received. The illite, Saharan sand, and palagonite samples

contained particles that were too large to be aerosolized by the atomizer and/or settled out of

the aerosol flow in the experimental apparatus before reaching the scattering region. These

samples were first manually ground down using a mortar and pestle for 20 minutes. They

were then further ground mechanically using a Wig-L-Bug for another 20 minutes.

All samples were prepared by suspending 1-2 grams of the mineral dust in high purity

water (Optima Water, > 99.9% purity). Depending on sample density and settling rate, these

mineral dusts can stay in suspension for 0.5-3 hours, which is sufficiently long to perform the

17

collection of all light scattering data. For the samples with a high settling rate, such as

hematite and kaolinite, a magnetic stir bar was added to the sample in the atomizer. The

entire atomizer was then placed on a magnetic stir plate. Agitation of the sample, along with

the recirculation of the sample provided by the atomizer’s mode of operation, was sufficient

to maintain a relatively constant number density of particles throughout the experiment.

Some details of the physical properties of these mineral dusts are also given in Tables

2.1 and 2.2. Optical properties used in the scattering calculations are given in Table 2.3. The

chemical formulas for the well defined mineral dusts are given in Table 2.4. A full discussion

of the methods used to determine the physical properties of these samples is given below.

The sample optical constants given in Table 2.3 are, in most cases, obtained from

measurements on bulk material samples. It is a valid question whether bulk optical constants

are applicable to particles in the 0.1 – 2 m size range of interest here. This question is

particularly difficult to answer for irregularly shaped particles since it is difficult to unravel

particle shape effects from possible effects associated with changing optical constants.

However, for uniform spherical particles, where Mie theory is applicable, it is possible to

probe this issue. Our work on visible scattering from uniform spherical particles in the 0.1 –

2 m diameter size range, including PSL and ammonium sulfate aerosol, shows no evidence

for any consistent or significant deviation in the experimental scattering results from Mie

theory predictions based on bulk optical constants for these materials. (See for example

Figures 2.6, 2.8 – 2.10 below). In addition, work by Hudson, et al. [2007] measuring IR

resonance extinction profiles for near spherical ammonium sulfate aerosol particles in the

submicron size range also agrees quantitatively with Mie theory predictions based on bulk

optical constants for ammonium sulfate. It is clear that, in some size range, bulk optical

constants must become inappropriate. However, careful experiments by other groups on

uniform spherical SiO2 smoke particles suggests that bulk optical constants continue to be

reliable down to particle diameters of < 100 nm as discussed in detail in Bohren and Huffman

[1983].

18

Aerosol Size Distributions

When dealing with irregularly shaped particles, such as mineral dust, careful

consideration must be used when describing what is meant by the particle size. Irregularly

shaped objects don’t have a single dimension that adequately describes the size, as any

physical measurement of the particle width would depend on the particle orientation.

Fortunately, it is often possible to instead use an effective diameter when working with

mineral dusts. An equivalent diameter is the diameter of a spherical particle with the same

specified physical property as the irregularly shaped particle. Some examples of equivalent

diameters include mobility, volume, and aerodynamic diameters. These will be fully

explained below.

In general, an aerosol will contain particles with sizes that span many orders of

magnitude. Different types of aerosol will contain particles that cover different ranges of

sizes. For example, soot particles, such as from an industrial plume, typically fall within the

range of 20 nm – 1 µm. Mineral dust aerosol, however, can range from tens of nm to

hundreds of µm in size provided the source region produces sufficient wind force to keep the

larger particles aloft [Hinds, 1999]. It is useful to define a number of size ranges or “modes”,

to describe aerosol particles that exhibit similar physical properties such as settling velocity

and atmospheric residence time. The nucleation mode includes particles with diameters in the

range of about 10-100 nm. Particles in the nucleation mode have short residence times in the

atmosphere due to rapid coagulation with other particles. These particles can also serve as

cloud condensation nuclei. The accumulation mode is made up of particles between 0.1- ~2

µm. These particles are important because they are able to stay suspended in the atmosphere

for long periods of time since removal mechanisms are relatively slow. For this reason,

accumulation mode particles can be transported by wind action over great distances. For

example, Saharan dust particles have been identified in the Caribbean and southeastern US

[Prospero, 1999]. The mineral dust studied in this work falls within the accumulation mode.

The largest mode, the coarse mode, consists of particles with diameters larger than ~2 µm.

19

These particles are quickly removed from the air due to gravitational settling. Dust particles

kicked up during strong winds and particles generated during agricultural or mining practices

can fall into the coarse mode.

Since we are dealing with particles that cover a large range of sizes it is convenient

to use a size distribution function, n(Dp), rather than speaking in terms of individual particle

diameters. Here, Dp is an arbitrary equivalent particle diameter. Let N be the particle number

concentration, the number of particles per unit volume of air, often expressed in units of cm-3.

The size distribution function is the number of particles per unit volume of air with diameters

between Dp and Dp + dDp. The size distribution is then just the number concentration

normalized by the range of particle sizes:

pp dD

dNDn )( (2.1)

Since aerosol particles cover size ranges over many orders of magnitude, it is useful

to instead use a logarithmic scale when expressing diameters. The logarithmic form of the

size distribution function, n(log(Dp)), then becomes:

)log())(log(

pp Dd

dNDn (2.2)

Further references to the particle size distribution will assume this log derivative form.

The next step is to define an equivalent diameter that will later be used in the light

scattering calculations (Chapter 4). If one assumes that the particle density is constant

throughout the volume of the particle (no internal voids, etc.), it is possible to define a

volume equivalent diameter, veD , which is the diameter of a spherical particle with the same

volume as the particle in question. Since the volume equivalent diameter can’t be easily

measured directly, alternate measurement techniques must be employed in order to get an

estimate for veD .

The two equivalent diameters that are commonly measured by commercial

instrumentation are the aerodynamic diameter, AD , and the mobility diameter, MD , for a

20

particle. The aerodynamic diameter for a particle is the diameter of a sphere with a reference

density, ρ0, of 1g/cm3 that has the same settling velocity as the particle [Hinds, 1999]. The

volume equivalent diameter can be calculated from the aerodynamic diameter using the

following relation:

)()(0

VES

AS

p DCDC

AVE DD (2.3)

where χ is the aerodynamic shape factor for the particle and is a correction to the Stoke’s

equation for the drag force on a nonspherical particle. With the exception of certain

streamlined shapes, irregularly shaped particles will experience a greater drag force then an

equivalent volume sphere, therefore χ > 1 typically. For a spherical particle, χ = 1. The

reference density is ρ0 and ρP is the bulk density of the particle. The Cunningham slip

correction factors, CS, account for non-continuum effects near the particle surface that result

in a slightly faster settling rate than would be expected. The slip correction is more important

for smaller particles (D << 1 µm).

The mobility diameter, MD , is the diameter of a spherical particle which follows the

same path as the irregularly shaped particle in a known electrical field. The volume

equivalent diameter can similarly be written in terms of the mobility diameter:

)()(

MS

VESDC

DCMVE DD (2.4)

By setting equations 2.3 and 2.4 equal to each other, a relationship between the

mobility and aerodynamic diameters is obtained:

23

23

)()()(0

VES

ASMS

pAM DC

DCDCDD

(2.5)

Measurements of the mobility diameter or the aerodynamic diameter are

straightforward using commercially available instruments. The volume equivalent diameter

can be calculated using either equation 2.3 or 2.4 provided the particle shape factors are

21

known and slip corrections can be calculated. These can be determined by simultaneously

measuring both diameters for a given sample then using equation 2.5 to solve for the shape

factor as originally outlined in Khlystov et al. [2004]. Provided the slip correction factor is

relatively constant over a range of diameters, the aerodynamic and mobility diameters will

have a linear relation. In that case, the shape of the size distribution will be the same for

measurements of either MD or AD . If both diameters can then be measured over the same

region, a least squares fitting algorithm can be used to overlap the distributions. The relative

shift between the two size distributions will be the product of the shape factors and

Cunningham slip correction factors.

Shape factors for the clay (montmorillonite, kaolinite, and illite) and non-clay

samples (calcite, quartz, and gypsum) used in our experiments were determined in

independent measurements [Hudson et al., 2008a and 2008b]. It was found that the

Cunningham slip corrections for the range of particle sizes studied (roughly 0.1 to 10 µm)

were very close to 1 and could therefore be neglected in equation 2.5. Many of the non-clay

samples were found to have shape factors near 1 (0.98-1.05). The clay samples were seen to

have much higher values however, with shape factors ranging between 1.11-1.30.

Measurement of the entire size distribution of a typical mineral dust aerosol is not

possible using currently available commercial instruments based on a single physical

property (e.g. aerodynamic or mobility diameter) since the dynamic range of diameters for a

typical sample can span four orders of magnitude. As outlined above, our setup uses an APS

to measure aerodynamic diameters within the range ~0.5-20 µm. Later experiments also

incorporated an SMPS which is capable of measuring mobility diameters in the range 0.02-

0.5 µm. For these latter experiments, the full size distribution was obtained by splicing

together the APS and SMPS data. However, in our apparatus there is no region of overlap

between these two instruments making it more difficult to determine the magnitude of the

shape factor and hence the shift between the aerodynamic and mobility diameter

distributions, as was done in the previous work of Hudson et al. [2008a and 2008b]. Instead,

22

the small diameter portion of the size distribution measured by the SMPS was first fit by a

lognormal distribution. Using the resultant lognormal fitting parameters, it was possible to

extrapolate the distribution to larger particles to overlap the data collected by the APS. The

shape factor was then calculated by overlapping the APS data with the lognormal fit to the

SMPS data. The raw data from the SMPS and APS were next converted to volume equivalent

diameters using this calculated shape factor and equations 2.3 and 2.4. The data were then

spliced together at volume equivalent diameter of 500 nm.

The SMPS is a relatively recent addition to our apparatus. Prior to its addition, the

full size distribution was determined in a different way. In those measurements, we were only

able to measure the large particle part of the size distribution. These aerodynamic diameters

were first converted to volume equivalent diameters using shape factors that were determined

by Hudson et al. in separate measurements. In order to determine the contribution of small

particles to the size distribution, it was necessary to extrapolate the distribution to smaller

particles using a lognormal fit to the APS data. The mode diameters of the log-normal fits

were constrained to agree with those measured by Hudson et al. [2008a, and 2008b] using a

similar aerosolization method. In order to account for possible errors resulting from

uncertainty in the small particle contribution to the light scattering calculations, separate

lognormal fits of the size distribution were performed using mode diameters that were up to a

factor of 2 larger or smaller than the previously measured mode diameters of these particles

(See Figure 2.3). All three fits for the size distributions then could be used as inputs to light

scattering calculations. The magnitude of the standard deviation between the resulting

scattering predictions then served as an estimate of the errors in the theoretical scattering

resulting from uncertainty in the small diameter part of the size distribution that was not

directly measured. The total light scattering intensity is mainly dominated by large particles.

Therefore, errors due to uncertainty in the small diameter part of the size distribution are

small as will be seen below.

23

Before leaving the discussion of aerosol particle size, it will be useful to first define

two additional commonly used methods of reporting particle size, the effective radius and the

characteristic size parameter. Light scattering intensity by particles typically scales with the

projected surface area of the particle. As such, it can be convenient to define an average

particle size for the distribution using an effective radius, Reff, which is defined as the

projected surface area weighted average, i.e. the ratio of the third moment of the size

distribution to the second moment of the size distribution:

0

2

0

3

)(

)(

dRRnR

dRRnRReff (2.6)

where R is the radius of the particle and n(R) is the size distribution. The effective radii for

the samples used in this study are included in Table 2.1 and fall in the range ~200-500 nm.

This corresponds to particles in the accumulation mode.

The characteristic size parameter, Xeff, is defined as:

eff

eff

nRX

2 (2.7)

where λ is the wavelength of incident radiation, and n is the refractive index of the external

medium. For scattering from particles in air, n = 1 to a good approximation. Values for the

size parameter are included in Table 2.2 for the samples used in this study. As will be seen in

Chapter 4, the functional form of the light scattering theories used in this work depend on X

rather than the D or λ individually.

System Alignment and Calibration

Once the experimental apparatus was roughly aligned using pinholes placed

throughout the OPO beam path, it was necessary to perform a finer alignment and to generate

24

an angle mapping function (scattering angle to pixel number map) and a system response

calibration function for the scattering apparatus. Scattering data collected by the CCD camera

is a function of camera position, in pixels (Figure 2.4). It is therefore necessary to generate an

angle mapping function to convert between pixels and scattering angles. In addition, the

system detection efficiency as a function of angle is not flat. This will be discussed in more

detail in Chapter 3. The calibration function corrects the scattering data for the angle

dependent system response.

System calibration and alignment is done using a sample that is both highly spherical

in shape and that has a nearly monodisperse size distribution. Spherical particles are used for

the system calibration since there exists an exact analytical solution to Maxwell’s equations

for light scattering for spheres, Mie theory. Further details of Mie theory, along with other

light scattering theories will be given in Chapter 3. A monodisperse size distribution was

desirable to eliminate any possible errors associated with measuring the full size distribution

and to reduce computation time. Polystyrene latex spheres fit both of these requirements and

are commercially available (PSL, Polysciences Inc.) in a number of different diameters in the

submicron range (see Table 2.5). Since the optical constants for polystyrene are well known

[Boundy and Boyer, 1952], an accurate theoretical prediction of the scattering for PSL

particles was possible.

System alignment was performed using PSL particles with a diameter of 771 nm.

This size was chosen due to the structure of the theoretical phase functions and polarizations.

The light scattering phase function is proportional to the total scattered light intensity as a

function of scattering angle. The polarization is a measure of the degree of linear polarization

of the scattered light as a function of angle. For 550 nm scattered light, 771 nm particles gave

phase functions and polarizations with a large number of interference maxima and minima,

as can be seen in Figures 2.4-2.6. This proved to be helpful in the determination of the

system calibration and angle mapping functions, as will be discussed below.

25

Prior to experimental measurements of mineral dust, light scattering for 771 nm PSL

is collected for both perpendicular and parallel polarized incident light (as measured from the

scattering plane). A double Fresnel rhomb prism is used to rotate the polarization of the

incident light. The scattered light intensities for perpendicular, )(I , and parallel, )(// I ,

incident light polarizations are then measured. Here, θ is the scattering angle as measured

from the laser axis. Multiple measurements of each polarization are taken (usually three) and

averaged in order to reduce signal noise. The phase function, )(F , and the

polarization, )(P , are determined using the following relations:

)()()( // IIF (2.8)

)()()()(

)(//

//

IIIIP

(2.9)

These measurements are compared against Mie theory predictions for the scattering.

Small adjustments to the system alignment are then performed in order to get the best

agreement with Mie theory. Once the system is aligned, the angle mapping function, )(y ,

and the calibration curve, )(C , are generated in a multi-step code written using the Matlab

computing language. Relevant sections of this code are included in the appendix but the basic

approach is described below.

The first step is to make corrections to the relative intensity of the parallel and

perpendicular polarization data sets. These data sets are generated a few minutes apart. This

relative intensity correction is necessary to correct for possible fluctuations in the particle

number concentration, which can change by + 2-3%, as well as variations in the output

power of the OPO, which can be as large as +10%, during data collection. This is done by

assuring that the polarization data goes to 0 near 0o and 180o. An optimization routine applies

a scalar factor, a, to I until the magnitude of the polarization is minimized near the

endpoints.

26

//

//

IaIIaIP

(2.10)

Once the scaling factor is determined, both the measured phase function and

polarization are corrected. The magnitude of this correction tends to be less than ~ + 7%.

Since the polarization data tend to be noisy near 0o and 180o due to decreased collection

efficiency of the mirror, there is some uncertainty in the scalar correction factor, a. In order

to gauge the magnitude of this uncertainty, for each sample three polarization profiles were

generated, one for the optimal value of a (as determined by a chi squared fitting routine used

to set the polarization to zero at the endpoints) and on each for a+0.07 and a-0.07. The

average deviation from the mean polarization profile was found to be on the order of ~4%.

This uncertainty has been included in the error bars of all experimental polarization profiles

presented.

Next, the routine determines the angle mapping function, )(y . This is done by first

assuming a sinusoidal form for )(y :

)sin()( DCyBAy (2.11)

where y is the CCD camera position in pixels and A, B, C, and D are fitting parameters. This

form is chosen as a simple functional form that well describes the observed angle

dependence. Simulations of the scattering apparatus geometry have resulted in similar

predictions for the form of )(y (see Chapter 3). Another least squares fitting routine varies

the fitting parameters, and thus varies the scattering angle mapping of the measured data,

until the residual between the measured and theoretical polarization profile is minimized.

Since the fit is done by varying the scattering angle mapping, it is necessary to have a

polarization with a large number of peaks and valleys in order to ensure a unique solution.

Once the phase function and polarization pixel data are converted to functions of

scattering angle, the system calibration function is then determined. A comparison of the un-

calibrated phase function for PSL as well as that determined by Mie theory is shown in

27

Figure 2.5. The calibration function is defined as the ratio of the theoretically calculated

phase function, TheoryF )( , to the experimentally measured phase function, MeasuredF )( :

Measured

Theory

FF

C)()(

)(

(2.12)

As can be seen in Figure 2.7, )(C tends to exhibit a number of sharp peaks and

valleys. These can be explained by limitations in the angular resolution of the scattering

apparatus as discussed in more detail in Chapter 3. Since the location of these peaks and

valleys will be different for different samples, it is not useful to use the raw calibration curve

to correct for these sharp structures in the phase functions. Instead, by performing a least

squares fit to )(C with an appropriately chosen function, many of the features due to the

angular resolution can be smoothed out while retaining the important features associated with

the geometry of the setup, as discussed in Chapter 3 Visual inspection of )(C suggests a

function of the following form for the fitting function, )(' C :

)()()( 2

)(' JIGFDC HeEeBeAC (2.13)

where A, B, C, D, E, F, G, H, I, and J are all variable parameters in the fitting routine. In

addition, the fitting routine excludes scattering angles that exhibit the pronounced valleys or

peaks due to angular resolution limitations, such as those occurring near 45o and 125o in

Figure 2.7. As will be seen below, the quality of the fits to )(C for scattering angles below

17o and above 172o are not relevant since the experimental data from those regions is not

used due to a high degree of background laser scatter in those regions. Since the fit to )(C

is not perfect, there is some uncertainty in the experimental results due to this calibration

procedure. The magnitude of this uncertainty, σCali, is estimated by calculating the average

deviation of the fitting function, )(' C , from the raw calibration curve, )(C :

)()(')(1

CCC

NCali (2.14)

28

where N is the number of points in )(C . Uncertainties were estimated from a number of

calibration functions and fits from many different days (>10) of data collection and is found

to be ~10%. This uncertainty has been included in the experimental results in Chapters 5-7.

Once )(' C has been determined, it is possible to generate the calibrated experimental phase

function, )(' F :

)()(')(' FCF (2.15)

This process of generating and applying the calibration and angle mapping functions

has been thoroughly tested both in earlier work by [Curtis et al., 2007], and in the work

developed for this thesis. Tests were performed using a number of samples which were also

made up of spherical particles so that Mie theory could be used with confidence. Samples

included PSL with particle diameters different from those used to generate the calibration

function, and thus having very different phase function and polarization profiles. Diameters

of 457 nm and 1025 nm were used for these tests. A solution of ammonium sulfate was also

used as a test sample. This sample was prepared by adding 0.05 g of ammonium sulfate to 20

mL Optima water. Droplets of the solution formed by the atomizer have a near spherical

shape.

Calibration and angle mapping functions generated using 771 nm PSL were applied

to the scattering data for each of these samples (see Figures 2.6 and 2.8-2.10). Mie theory

was used to generate theoretical phase functions and polarizations for the PSL and

ammonium sulfate. For the PSL samples, the particle size distribution was assumed to be

monodisperse with a diameter as reported by the sample manufacturer (Polysciences Inc.).

Full particle size distributions were measured for the ammonium sulfate sample using both

an APS and an SMPS and were found to have a mode diameter of roughly 60nm. In the Mie

calculations, we used the known optical constants for ammonium sulfate, m = 1.448 +

i(7.94x10-6) [Egan, 1982]. Previous measurements by Hudson et al. [2007] have found that

the ammonium sulfate aerosol generated using a similar aerosolization method is very

29

spherical, χ = 1.01 + 0.02, for small particles (Dve < 200 nm). Since the particles used to test

the calibration procedure were all spherical, Mie theory should give an accurate prediction

for the light scattering.

Comparisons of the calibrated experimental data with Mie theory is given in Figures

2.8-2.10. Experimental data includes error bars representative of the day-to-day variation in

measured values as well as uncertainties in the calibration curve (for the phase function

profiles) and uncertainties in the relative scaling between the parallel and perpendicular

intensities (for the polarization profiles). Excellent agreement between both the phase

functions and polarizations over the entire range of scattering angles was obtained for all

three test samples. The most significant deviations between measured and theoretical

scattering tends to be near regions with very sharp dips in the data, such as near 20o and 135o

in the 1025 nm PSL data. We believe this is mainly due to the limited angular resolution of

the scattering apparatus. Attempts to model the behavior of the scattering region and

elliptical mirror (Chapter 3) have shown that the extended size of the scattering volume at f1,

as well the diameter of the pinhole located at f2, are the main factors that determine the

angular resolution. Unfortunately, decreasing the pinhole diameter also has the effect of

decreasing total signal intensity. For the mineral dust samples used in this work, the phase

function and polarization are slowly varying functions of angle. Thus, the limits on angular

resolution exhibited in these test cases are expected to play a negligible role in the mineral

dust data.

From Images to Phase Functions

As was explained above, light scattering from mineral dust aerosol is imaged onto a

CCD camera through use of an elliptical mirror with the scattering located at one focal point

of the mirror. Once this scattering image is collected, there are many steps required to

process the data before a comparison with theoretical predictions for the scattering are

possible. A typical image for scattering from mineral dust particles is shown in Figure 2.11a.

30

This image was collected for quartz mineral dust with parallel polarized incident light. A

background image, collected without any mineral dust present in the scattering region, has

been subtracted from the image in order to eliminate light scattering from the edges of the

mirror. Near the center of the scattering band (y ≈1000, z ≈ 800), a small bright spot can be

seen at what corresponds to a scattering angle of 90o. This is due to direct scattering that

passes through the pinhole at f2 without first being reflected by the mirror. This direct scatter

is removed by subtracting out an image collected with the mineral dust present in the

scattering region and with the elliptical mirror blocked. The final scattering image is shown

in Figure 2.11b.These 16-bit grayscale images are next exported as text files for further

processing.

The image is integrated along the z-axis in order to obtain the total scattering intensity

as a function of camera position. To cut down on signal noise, this integration is performed

only near the scattering signal band, corresponding to pixels z = 750-950. The integrated

signal is shown in Figure 2.12 for parallel polarized light, //I . This entire process is then

repeated for perpendicular polarized incident light in order to generate I .

The polarization and phase functions are then generated using equations 2.8 and 2.9.

An optimization routine is again used to determine the relative scaling factor, a, for I by

minimizing the magnitude of the polarization near 0o and 180o. At this point, the calibration

function, obtained using a fit to the ratio of the measured to Mie theory prediction for

scattering from PSL (see above), is applied to the phase function. Since the polarization is

defined as a ratio of the difference in the scattering intensities to the sum of scattering

intensities (see Equation 2.9), the application of a calibration curve is unnecessary since it

will cancel out. The angle mapping function is then used to convert the phase functions and

polarizations to functions of scattering angle rather than pixels.

In order to compare the measured phase function directly to theoretical phase

functions, it is necessary to normalize the data. In keeping with the normalization convention

presented in Bohren & Huffman, the following condition is used:

31

1)()sin(21

0

dF 2.16

Due to the geometry of the scattering apparatus and finite width of the laser beam, it

is not possible to reliably measure the light scattering over the entire range of scattering

angles and our data is limited to a practical range of 17-172o. In order to properly normalize

the phase function, we must first extend the phase function in the forward and backward

directions using the methods outlined by Liu et al. [2003]. At near forward scattering angles,

light scattering is somewhat less sensitive to particle shape effects [Bohren & Huffman,

1983] and so we can use Mie Theory or T-Matrix theory to define the forward scattering

signal. Theoretical scattering data for angles less than 17o are first spliced onto the

experimentally measured phase function. The experimental data is then extrapolated out to

180o by using a linear fit of the data between 160o and 172o as the phase functions are

relatively monotonic over this range of scattering angles. A plot of the final calibrated phase

function and polarization for scattering from quartz dust at 550 nm is given in Figure 2.13 as

an example.

32

Mineral Dust Source Dve (nm)† χ σ Reff (nm)

Illite** Source Clay Repository 15075150

1.30 ± 0.04 1.86 ± 0.02 203 ± 8

Kaolinite Alfa Aesar 31070140

1.20 ± 0.06 2.18 ± 0.03 300 ± 41

Montmorillonite Source Clay Repository 210100210 1.11 ± 0.05 1.81 ± 0.06 252 ± 24

Calcite Omya Inc. 18090180

1.05 ± 0.03 2.11 ± 0.01 358 ± 8

Gypsum Alfa Aesar 18090180

0.98 ± 0.06 1.88 ± 0.01 243 ± 5

Quartz Strem Chemicals 220110220 1.00 ± 0.03 2.10 ± 0.03 431 ± 19

Hematite Sigma Aldrich 150150300 1.0* 1.75 ± 0.01 330 ± 2

Goethite Alfa Aesar 4070320

0.95 ± 0.07 1.40 ± 0.03 211 ± 11

Arizona Road Dust Powder Technology Inc. 100

150300 1.00 ± 0.05 1.73 ± 0.01 318 + 4

Palagonite** NASA JSC Mars-1 180 1.02 ± 0.02 1.87 ± 0.01 240 ± 7

Saharan Sand**† Field Sample 35 150

1.02 ± 0.02 1.84 ± 0.06 1.87 ± 0.05

188 ± 9

Iowa Loess Field Sample 80 1.05 ± 0.10 2.06 ± 0.14 150 ± 18

Diatomaceous Earth† Alfa Aesar

30 120

1.36 ±0.11 1.72 ± 0.08 2.46 ± 0.19

241 ± 15

† Also shown is the range over which the mode diameters were varied to gauge the uncertainties in the theoretical scattering calculations. *No calculations of the shape factors were made for this sample. A shape factor of 1.0 was assumed based on similar samples. **The values given for illite, palagonite, and Saharan sand are for the post-processed sample (see text). † For the diatomaceous earth and Saharan sand samples, a bi-modal log-normal distribution was required in fitting the size distribution data. Results are presented for each mode.

Table 2.1 Physical properties of mineral dust samples used in this work: volume equivalent mode diameter (Dve), aerodynamic shape factor (χ), lognormal size distribution best fit width parameter (σ), and surface area weighted effective radius (Reff).

33

Mineral Dust Xeff

470 nm Xeff

550 nm Xeff

660 nm

Illite 2.7 2.3 1.9

Kaolinite 4.0 3.4 2.9

Montmorillonite 3.4 2.9 2.4

Calcite 4.8 4.1 3.4

Gypsum 3.2 2.8 2.3

Quartz 5.8 4.9 4.1

Hematite 4.4 3.8 3.1

Goethite 2.8 2.4 2.0

Arizona Road Dust 4.3 3.6 3.0

Palagonite 3.2 2.7 2.3

Saharan Sand 2.5 2.1 1.8

Iowa Loess 2.0 1.7 1.4

Diatomaceous Earth 3.2 2.8 2.3

Table 2.2 Calculated characteristic size parameter ( effeff RX 2 ) mineral dust samples used in this work.

34

Mineral Dust Refractive Index (470 nm)

Refractive Index (550 nm)

Refractive Index (660 nm) Reference

n k n k n k

Illite 1.42 10.66e-4 1.41 7.73e-4 1.40 11.39e-4 Egan and Hilgeman [1979]

Kaolinite 1.49 1.22e-4 1.49 0.48e-4 1.50 0.921e-4 Egan and Hilgeman [1979]

Montmorillonite 1.53 1.01e-4 1.52 0.38e-4 1.52 0.93e-4 Egan and Hilgeman [1979]

(o-ray) 1.67 1.0e-4 1.66 1.0e-4 1.65 1.0e-4 Calcite

(e-ray) 1.49 1.0e-4 1.49 1.0e-4 1.49 1.0e-4 Ivlev and Popova [1973]

Gypsum 1.54 1.0e-4 1.53 1.0e-4 1.53 1.0e-4 Ivlev and Popova [1973]

1.55 1.0e-4 1.55 1.0e-4 1.55 1.0e-4 Quartz

1.56 1.0e-4 1.56 1.0e-4 1.53 1.0e-4 Longtin et al. [1988]

2.96 0.30 3.10 0.0925 2.97 0.0041 Longtin et al. [1988]

3.26 0.30 3.26 0.21 2.98 0.17 Bedidi & Cervelle [1993] Hematite*

3.12 0.82 3.17 0.46 2.94 0.089 Sokolik & Toon [1999]

Goethite 2.39 0.074 2.27 0.094 2.19 0.098 Bedidi and Cervelle [1993]

Arizona Road Dust** 1.54 1.0e-4 1.54 0.69e-4 1.53 0.96e-4 Spectral Average

Palagonite 1.54 0.0091 1.53 0.0078 1.53 0.0059 Johnson et al. [2001], Clancy et al. [1995],

Saharan Sand 1.48 1.09e-4 1.47 0.75e-4 1.46 0.556e-4 Egan [1985]

Iowa Loess*** --- --- 1.58 1.0e-4 --- --- Cuthbert [1940]

Volten et al. [2001]

Diatomaceous Earth 1.45 2.04e-4 1.45 1.33e-4 1.44 0.997e-4 Egan [1985]

PSL 1.61 0.0 1.60 0.0 1.59 0.0 Boundy & Boyer, [1952]

Ammonium Sulfate 1.45 7.12e-6 1.45 7.94e-6 1.45 1.13e-5 Egan [1982] *Light scattering calculations for hematite were performed using values from Longtin et al. [1988], Bedidi & Cervelle [1993], and Sokolik & Toon [1999]. ** Optical constants for Arizona Road Dust are based on an average of clay (montmorillonite) and non-clay (quartz) optical constants and will be discussed in more detail in Chapter 5 *** Optical constants for Iowa Loess are for 589 nm as presented in Cuthbert et al [1940]. Additional calculations were performed using kaolinite optical constants from Egan and Hilgeman [1979] as discussed in Chapter 7.

Table 2.3 Optical properties of mineral dust samples used in this work. For cases where optical constants were not available for a given wavelength, a linear extrapolation of values near that wavelength was used.

35

Mineral Dust Chemical Composition

Illite (KH3O)(AlMgFe)2(SiAl)4O10[(OH)2(H20)]

Kaolinite Al2Si2O5(OH)4

Calcite CaCO3

Gypsum CaSO4·2(H2O)

Quartz SiO2

Hematite α-Fe2O3

Goethite Fe3+O(OH)

Diatomaceous Earth SiO2*7H2O

Table 2.4 Chemical composition of the well defined mineral dust samples used in this work.

Particle Diameter (nm) Standard Deviation (nm)

457 11

771 25

1025 10

Table 2.5 Size properties for polystyrene latex spheres (PSL)

36

Figure 2.1 Light scattering experimental apparatus. Aerosol generated by the atomizer is directed to the focal point of an elliptical mirror that acts as the scattering region. The aerosol is then collected for real time particle sizing measurements by an Aerodynamic Particle Sizer (APS) and a Scanning Mobility Particle Sizer (SMPS). A tunable Nd:Yag pumped OPO is used as the light source.

37

Figure 2.2 Detailed view of the optical setup (a) and scattering region (b) as viewed from above. Output from the OPO is directed to a telescope setup in order to decrease beam width by roughly a factor of three. A double Fresnel Rhomb prism is used to adjust the polarization of the incident laser before entering the scattering region. Scattered light reflects from the elliptical mirror and is subsequently focused through an aperture onto the CCD camera. The scattering angle is defined relative to the direction of the incident beam.

38

Figure 2.3 Measured aerosol particle size distributions (open circles) using an Aerodynamic Particle Sizer. Also shown are log-normal fits to the particle size distribution for mode diameters of 110, 220, and 440 nm. The log-normal fits are used in Mie calculations to give a range of possible scattering signals to account for uncertainty in the small diameter part of the size distribution.

39

Figure 2.4 CCD image of light scattering for 771 nm diameter PSL for parallel (a) and perpendicular incident light (b).

40

Figure 2.5 Experimental (solid line) and Mie theory (dashed line) phase functions for 771 nm diameter PSL. The experimental data has been mapped from pixels to scattering angle, but the system calibration has not been applied. Mie and experimental data has been normalized to the same amplitude at 35o.

41

Figure 2.6 Experimental (solid line) and Mie theory (dashed line) phase functions (a) and polarizations (b) for PSL with mean particle diameter of 771 nm. Experimental data has been calibrated and properly normalized. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line.

42

Figure 2.7 Calibration curve for 771 nm mean particle diameter PSL (solid line). A fit to the calibration curve is shown as well (dashed line).

43

Figure 2.8 Experimental (solid line) and Mie theory calculations (dashed line) for phase functions (a) and polarizations (b) for 457 nm mean diameter PSL. Experimental data has been calibrated using the calibration curve generated using 771 nm mean diameter PSL. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line.

44

Figure 2.9 Experimental (solid line) and Mie theory calculations (dashed line) for phase functions (a) and polarizations (b) for 1025 nm mean diameter PSL. Experimental data has been calibrated using the calibration curve generated using 771 nm mean diameter PSL. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line.

45

Figure 2.10 Experimental (solid line) and Mie theory calculations (dashed line) for phase functions (a) and polarizations (b) for ammonium sulfate. Experimental data has been calibrated using the calibration curve generated using 771 nm mean diameter PSL. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line.

46

Figure 2.11 CCD image of light scattering for quartz mineral dust, (a). Direct scatter has been subtracted out in (b).

47

Figure 2.12 Integrated scattering intensity of quartz mineral dust.

48

Figure 2.13 Phase function (a) and polarization (b) for quartz mineral dust. The phase function has been calibrated and properly normalized. Theoretical Mie data has been spliced to the phase function for scattering angles below 17o and a linear extrapolation has been used for angles greater than 172o. Those regions are denoted with the dotted line.

49

CHAPTER 3

MODELING THE EXPERIMENTAL SCATTERING APPARATUS

The detection system used in these experiments has an instrumental response function

that is angle dependent and nonlinear. The instrument response is characterized by a number

of factors including lower collection efficiency for scattering near the edges of the collection

mirror, loss of angular resolution due to finite imaging aperture size, and finite light

scattering volume effects. In order to calibrate the instrument (i.e. determine the mapping

function from scattering angle to CCD array pixel number) and to account for the angle

dependence of the instrument response, calibration functions are derived each day for use in

processing the experimental data. The calibration procedure is described in detail in Chapter

2. Briefly, this is done by collecting light scattering data from monodisperse polystyrene

latex spheres (PSL, Polysciences Inc.). PSL was chosen due to the individual particles being

spherical and because highly monodisperse samples can be obtained with precisely defined

particle diameters. This allows for the use of Mie theory in calculating the expected light

scattering phase function and polarization profiles. By taking the ratio of the measured phase

function of the PSL to that calculated using Mie theory, it is possible to calibrate the system

by determining both the angle mapping function and the angle dependent response function

for the apparatus.

Model of the Experimental Apparatus

Early investigations into the structure of the instrument response function showed a

number of features that were not easily explained, such as a distinct dip in the system

response for scattering angles near 90o. In order to understand and quantify these features, a

code was written in the Matlab computing language (included as an Appendix) that attempts

to model the relevant parameters of our light scattering experimental apparatus, including the

geometric light scattering off of the elliptical mirror surface from an extended light scattering

volume, and including the effects of transmission through the imaging aperture in the setup.

50

Careful measurements were taken of the physical dimensions of the experimental apparatus

to ensure an accurate model of the setup was obtained.

As was described in Chapter 2, a jet of aerosol particles is directed to one focal point,

f1, of an elliptical mirror. The output from a tunable Nd:YAG pumped OPO is also focused

onto f1, and crosses the dust jet at right angles. Since the scattered light from the aerosol

sample originates from one focal point of the elliptical mirror, it will be refocused by the

mirror to the second focal point, f2. At f2, the scattered light passes through an aperture,

which functions as a field stop, and the light is then imaged onto a CCD array. However,

finite scattering volume effects significantly alter this simple picture. In order to account for

these effects we will first treat the case of a scattering source with an infinitesimal volume

located at an arbitrary point (x0, y0) near the focal point, f1 (see Figure 3.1).

We define an ellipse with semi-major and semi-minor axial lengths of a and b,

respectfully. We then define a coordinate system with the origin located at the center of the

ellipse partially circumscribed by the elliptical mirror as shown in Figure 3.1. A point

scattering source located at (x0, y0) emits a ray (for this model, each emitted ray corresponds

to light scattered off an aerosol particle) at an angle θ measured from the optical axis of the

elliptical mirror. In Cartesian coordinates f1 will be located at the point (|f1|, 0) and f2 will be

located at the point (-|f2|, 0), and the equation for the ellipse in parametric form will be:

12

2

2

2

by

ax

(3.1)

The equation for the initial scattering ray will then be:

00 ))(tan( yxxy (3.2)

By letting )tan(m and solving for x and y one can determine the intersection of the ray

with the ellipse (x1, y1):

)]}2{1()([ 21

20

20

200

22

20022

2

1 yxmymxba

mbymxmmb

ax

(3.3)

51

0011 )( yxxmy (3.4)

Once the point of intersection is known, the reflected ray can be determined by

reflecting the incident ray off of the tangent to the ellipse at (x1, y1):

2' (3.5)

11 ))('tan( yxxy (3.6)

Here θ’ is the angle the reflected ray makes with the x-axis and γ is the angle between the

incident ray and the normal to the tangent line to the ellipse at that point (Figure 3.1). The

reflection of the ray can be repeated as many times as necessary by repeated applications of

equations (3.1) through (3.6). If the origin of the emitted ray were at f1, the reflected ray

would then pass through f2.

In order to make this general model of geometric scattering applicable to our

apparatus, a number of additional elements must be introduced. The first of these is to limit

the range of angles over which the ellipse is defined to polar angles, Θ, between + 34.3o in

accordance with the physical dimensions of the mirror in the experimental setup (see Figure

3.2). This is done by the condition that any ray striking the ellipse outside this angle range is

discarded. This range of polar angles corresponds to scattering angles in the range 3-177o for

a scattering source located exactly at f1.Also included in the model is a simulated aperture

(field stop) located at f2. In the code, the aperture consists of two straight lines oriented

parallel to the y-axis. There is a small gap between the two lines to define the aperture

opening through which reflected rays may pass. Any ray that intersects either of the aperture

lines is ignored. The model CCD detector is located a small distance behind the aperture.

Any ray intersecting this detector line is counted as a successful “hit”. The initial emission

(i.e., scattering) angle and the final position of the corresponding detector hit are then logged.

This process is repeated many times in order to build up accurate statistics for the

mapping function from scattering angle, θ, to detector position, yDet. The system response

function, S(yDet), can be determined through mapping the number of hits per unit area on the

52

detector from a uniform distribution of initial scattering angles. A uniform distribution of

hits would imply a flat instrument response function whereas a non-uniform distribution

would imply some variation in collection efficiency across the detector. The system

calibration function, C(yDet), is then just the inverse of the system response function:

)(1)(

DetDet yS

yC (3.7)

By mapping the relationship between initial scattering angles and final detector

position (the angle mapping function), it is possible to rewrite this as a function of scattering

angle:

)(1)(

S

C (3.8)

In the above development, it was assumed that all rays were emitted from the same

point, (x0, y0). However, in our experimental apparatus, laser light scatters from a finite

volume of dust particles. This scattering volume is determined by the intersection of the

aerosol jet with that of the laser beam, as can be seen in Figure 3.3. Depending on the relative

sizes of the jet and the laser beam profile, the cross sectional area of overlap between the two

when viewed from above can take on different shapes. For example, if the aerosol jet

diameter, dJet, is much larger than that of the diameter of the laser beam profile, dLaser, the

area of overlap (i.e. the two-dimensional projection of the intersection volume) would best be

described as rectangular with a width approximately equal to dLaser and length on the order of

dJet. For the case of the laser beam profile having a much larger diameter than the aerosol jet,

the intersecting area would be circular with a diameter equal to that of dJet. In our current

configuration the width of the aerosol jet is approximately 1.5 mm and the diameter of the

beam profile is approximately 1 mm. For this arrangement, the area of intersection would

best be described as rectangular.

The next step is to incorporate an extended scattering volume into the model. Instead

of choosing a point from which all rays are emitted, we can choose the center point of an

53

extended area, (x0, y0). The space within this area is then randomly populated with point

emitters as above in a Monte Carlo averaging scheme. For each point within the area (where

a point represents a single scattering source from a mineral dust aerosol particle), a large

number of rays are then emitted, and the corresponding scattering angles and detector hits are

logged. This process is repeated for all of the generated source points in the volume.

As defined above, the scattering angle is measured from the center of the scattering

volume. A ray emitted at an angle, from a point near the edge of the scattering volume will

not necessarily intercept the detector at the same point as a ray with the same scattering angle

but originating from the center of the scattering volume. By considering the range of

scattering angles that map to a given detector position, a rough estimate of the uncertainty in

initial scattering angle resulting from these finite scattering volume effects can be obtained,

This angular uncertainty defines an effective angular resolution for the apparatus. This point

will be discussed in more detail below. The angular resolution of the apparatus can be

improved by decreasing the size of the aperture or by decreasing the size of the scattering

volume. Doing so, however results in a decrease in overall collection efficiency. It is possible

to obtain a monotonic relationship between initial scattering angle and detector position by

using an appropriate best least-squares fit mapping function to the data. As with the

experimental data (see Chapter 2), a sinusoidal function tends to fit the angle mapping very

well:

)sin( DCyBA DetEmission (3.9)

where A, B, C, and D are fitting parameters and yDet is the position of a ray terminating on

the detector (See Figure 3.5 (b) below). It is convenient when comparing to experimental

data to first express the detector position in terms of pixel number before performing the fit.

This is done by dividing the length of the detector into a number of evenly spaced bins. The

number of bins (2184) corresponds to the number of pixels that run the length of the CCD

camera in the experimental setup.

54

Additional parameters have been included in the model to allow for rotation of the

scattering volume about its center point by a set angle, (see insert to Figure 3.2). This

allows for exploring the effects of laser misalignment in the setup. If the laser is incident on

the scattering area at some angle, this is equivalent to simply rotating the scattering volume.

In addition, parameters in the code can control the orientation of the detector. Slight rotations

in the detector can help determine misalignment of the CCD camera in the experimental

setup.

To summarize the steps involved in modeling the experimental apparatus, a section of

an ellipse is first defined between the polar angles = + 34.3o. A point located near the first

focal point of the ellipse is chosen as the center point for a rectangular scattering area that

contains a large number of ray emitting source points. The code loops through the source

points, each of which emits a large number of rays over a range of scattering angles. These

rays are then reflected off of the ellipse (provided they intercept the ellipse within its defined

range of polar angles, ). The reflected rays that intercept either of the aperture lines are

discarded. Those that pass between the aperture lines strike the detector line. The detector

position of the intercept, together with the corresponding scattering angle, is then logged for

each ray. Finally, the mapping between scattering angle and detector position, and the angle

dependent instrument response function can be determined. Figure 3.4 shows a flow diagram

of the key steps in the simulation.

Model Parameters

In order to obtain statistically significant results for this model, it is necessary to

track a large number of emitted rays from a large number of scattering points. For a given

simulation, the number of randomly distributed points within the scattering volume is

generally set to 20,000 points. For each point, the code loops through 180 emitted rays from

that point. The emission angles from the particle are evenly distributed within the range of 0o

to 180o, where 0o corresponds to forward scattering and 180o corresponds to backward

55

scattering in the experimental setup. The minor and major axes of the ellipse are set to

correspond to the physical dimensions the elliptical mirror, 17.5 cm and 30 cm respectively.

The detector has been placed 2.5 cm behind f2 (i.e. the field stop aperture).

By visual inspection of the scattering region in the experimental setup, it is most

appropriate to choose a rectangular extended area for these simulations. The length (oriented

perpendicular to the optical axis of the ellipse) is set to1.5 mm and the width is set to 1 mm.

This corresponds to the case of the incoming laser beam profile having a smaller diameter

than that of the aerosol jet (see Figure 3.3).

Even though a large number of rays are generated in a single simulation (3.6x106),

only a fraction of these will make it through the aperture to the detector. The fraction of

emitted rays that finally intersect the detector is proportional to the width of the aperture

opening in this two dimensional model. In the physical setup, this attenuation is proportional

to the aperture area. For these simulations, a 1 mm aperture opening was used. This results in

only ~10% of all emitted rays making it to the detector.

Model Results

Preliminary simulations were performed assuming the center of the scattering volume

was located directly at the first ellipse focus, f1. The scattering volume was oriented with its

length running parallel to the y-axis. The detector was also oriented parallel to this axis. By

plotting a histogram of the number of detector hits as a function of detector position, as

shown in Figure 3.5 (a), we can see a symmetric buildup of hits with a distinct dip near the

center. These features are the results of the finite nature of the scattering volume. Changing

the shape of the scattering volume will result in slightly different peaks and dips in the

histogram. For example, increasing the length of the scattering volume along the laser axis

will result in a much wider dip. Alternately, increasing the width of the scattering volume

along the mirror optical axis will result in a less pronounced dip. Even though the area itself

56

is centered directly on the focal point, scattering points away from the center do not originate

from f1 and therefore do not pass directly through f2.

A plot of the scattering angle as a function of the detector position for the

corresponding hit is given in Figure 3.5 (b). Here, the detector position has been divided into

a number of bins which correspond to the pixel number on the CCD array. It can be seen that

this is not a one-to-one mapping. This is again due to scattered rays originating from source

points in an extended volume. The width of this band is linearly dependent on the width of

the aperture used in the simulation and gives a rough estimate for the angular resolution for

the instrument. In this case the width of the band is ~ 7o. Also included is a fit to the angle

mapping (using equation 3.9) which will later be used to generate the system angle

calibration function.

Model simulations were also performed to test the effects of rotating the scattering

volume and the detector. Due to the symmetry involved in the setup, rotating the scattering

volume by an angle produces very similar results to rotating the detector by an angle -.

For this reason, here only the effects of rotating the scattering volume will be discussed.

Figure 3.6 depicts how of the number of detector hits as a function of detector position

changes for a range of rotation angles. Even for angles as small as 10o the relative width of

the two peaks near the edge of the detector can be observed to change noticeably, with

significantly more hits on one side of the detector. This can have a significant effect on the

system calibration, as will be discussed below. As the ratio of the length to the width of the

scattering volume increases, the effects due to rotation of the scattering volume become more

pronounced. It is important to note that the large rotations used in this simulation would

correspond to a gross misalignment of the laser in the experimental apparatus that would

easily be detected by alignment pinholes in the setup and corrected. It has been included for

informative purposes only and helps to highlight subtle changes which might otherwise be

overlooked by limiting the model simulations to small rotations in the scattering volume.

57

These simulation results are used to generate a model angle-dependent instrument

response function (see equations 3.7 and 3.8). Typical results are shown in Figure 3.7 for a

number of different model inputs. The model calibration functions are averages over three

consecutive model simulations for each set of input parameters. There will be variations in

the model results from one simulation to the next due to the randomly populated scattering

volume, though these variations are quite small due to the high number of scattering points

that were chosen as inputs. Averaging multiple simulations together assures a more uniform

distribution of scattering points within the scattering volume. Calibration curves are included

for three sets of inputs; the standard rectangular scattering volume (length = 1.5 mm, width =

1.0 mm, no rotation), a rotated rectangular scattering volume (length = 1.5 mm, width = 1.0

mm, = 15o), and a larger rectangular scattering volume (length = 3.0 mm, width = 1.0 mm,

no rotation). As the length of the scattering volume increases, the central peak in the

calibration broadens significantly and increases in amplitude. Rotations in the scattering

volume will result in a shift of the central peak away from 90o towards larger or smaller

scattering angles depending on the direction of the rotation.

In our experimental measurements, the range of scattering angles in which reliable

data can be collected is limited to the range 17-172o. This is due to strong forward and

backward light scattering from optical elements along the beam path (even in the absence of

aerosol particles) onto the mirror. Due to the high background signal intensities in the

extreme forward and backward directions measured by the CCD, it is difficult to subtract this

signal out. As a result we exclude data for scattering angles less than 17o and greater than

172o from the results. This is equivalent to having a mirror which is defined over a range of

angles which are not symmetric about a 90o. In order to account for this, the ellipse in the

simulation was redefined to cover a range of polar angles between +31.5o and -27.8o (i.e.

scattering angles in the range 17-172o for a scattering source located exactly at f1).

The standard rectangular scattering volume (length = 1.5 mm, width = 1.0 mm, no

rotation) was again used for this simulation. An averaged experimentally determined

58

response calibration function, obtained using Mie theory and experimental data collected for

polystyrene spheres, is included for comparison. The experimental calibration function is an

average of 6 calibration functions derived from multiple days of collection. The model

calibration function has been vertically scaled to the experimental calibration function in

order to more accurately compare the structure between the two. It can be seen that the

model calibration functions and those obtained experimentally, as described in Chapter 2,

are in good agreement for all scattering angles for which we are able to reliably measure the

light scattering (Figure 3.8).

Discussion

A model simulation has been developed, using the Matlab computing language, for

the scattering and detection process in our experimental apparatus. This simulation includes

inputs for the relevant parameters in the scattering including an extended scattering volume

of chosen size and shape, incoming laser beam orientation relative to the elliptical mirror

(rotating the scattering volume), adjustable aperture width, and CCD array (detector) position

and orientation. The effects adjustments to these parameters may have on the resulting

system response have been extensively explored. This has provided insight both into the

workings of our experimental setup and into the structure of the system calibration and

response functions that are generated experimentally. This has in turn allowed for more

accurate alignment procedures to improve day-to-day reliability and accuracy in our data.

59

Figure 3.1 Ray diagram for reflection of a ray confined to the inside of an ellipse with semi-major axis length, a, and semi-minor axis length, b. The tangent and normal lines at the point of reflection, (x1, y1), are shown as dashed lines.

Figure 3.2 Schematic representation of scattering setup. A ray emitted from f1 (dashed line) passes between two solid vertical lines centered about f2 (aperture) and intercepts the detector. The extension of the ellipse has been cut down to represent the physical dimensions of the elliptical mirror in our scattering setup. The inset depicts a close up of the scattering area which has been rotated by an angle, .

60

Figure 3.3 Top-down view of possible scattering volumes resulting from the overlap of the incoming laser (arrow) with the aerosol dust jet (circle). The cross-hatched region represents the scattering volume.

Figure 3.4 Flow diagram of important elements of the scattering apparatus simulation code.

61

Figure 3.5 Histogram of the number of detector intercepts of rays (a), emitted from a rectangular scattering source located at f1, as a function of location on the detector where the interception occurred. The dark solid line at the bottom of the figure represents the physical extent of the detector. The mapping of the detector intercepts to the scattering angles (gray) along with a fit (black) is given in (b).

62

Figure 3.6 Histogram of the number of detector intercepts of rays for 0o (a), 10o (b), 20o (c), and 30o (d) rotations of the scattering volume. The solid black line in each figure corresponds to the extent of the detector.

63

Figure 3.7 Calibration functions for the standard rectangle scattering volume (circle markers), a rotated rectangular scattering volume (square markers), and a larger rectangular scattering volume (diamond markers).

64

Figure 3.8 Experimentally determined calibration function (solid line) along with model calibration function for the standard rectangle scattering volume (circle markers). The scattering ellipse has been defined to cover polar angles between +31.5o and -27.8o. (This corresponds to a range of experimental scattering angles 17 o - 172 o.)

65

CHAPTER 4

LIGHT SCATTERING THEORY

This chapter will focus on the theoretical background necessary to perform light

scattering calculations. Emphasis will be placed on the two scattering theories used

throughout this work, Mie theory and T-Matrix theory. Both theories provide classical

solutions to Maxwell’s equations, though they differ in their applicability to certain types of

problems. Mie theory is only strictly valid for scattering from spherical shaped particles

whereas a much wider range of particles shapes can be treated with T-Matrix theory. Both

theories are routinely used in the calculation of light scattering from mineral dust aerosols,

though T-Matrix theory has become increasingly popular as more powerful computers are

able to reduce calculation times [Dubovik et al., 2002, 2006; Kalashnikova et al., 2005;

Veihelmann et al., 2004; Mishchenko et al., 1997].

Both Mie theory and T-Matrix theory require a relatively small number of inputs for

scattering or absorption calculations. For a given mineral dust aerosol, the complex index of

refraction, the particle size, and the wavelength of incident light must all be specified. For the

case of a polydisperse aerosol (i.e. one that is not defined by a single particle diameter but,

rather, a distribution of particle diameters), separate calculations can be made for each range

of particle diameters and the resulting scattering can be linearly combined with appropriate

weighting factors based on the particle size distribution. One more important input parameter

is required for T-Matrix theory; the specification of the particle shape. The functional form

governing how these shapes are specified will be discussed below.

Prior to an in depth discussion of either light scattering theory, it is convenient to first

define the characteristic size parameter, X, as:

pnd

X (4.1)

66

where dp is the diameter of the particle, λ is the wavelength of incident radiation, and n is the

refractive index of the external medium. For scattering from particles in air, n = 1 to a good

approximation. As will be seen later, the functional form of both scattering theories depend

on X rather than either dp or λ individually. The size parameter can also serve as a useful tool

in determining the specific light scattering theory most appropriate for a given situation.

When the particle is very large relative to the wavelength of incident radiation, 1X ,

geometric scattering will often be sufficient in modeling the scattering from that particle.

While in this regime, the scattering intensity will be independent of wavelength. In the

opposite limit, 1X , Rayleigh scattering is more appropriate. In this range, the scattering

intensity is proportional to 4 . It is this strong wavelength dependence that is responsible for

the blue color of the sky during the day and the red and orange hues near a sunset. As the

particle diameter approaches that of the incident wavelength, neither Rayleigh nor geometric

scattering accurately predicts the light scattering. As 1X , it is therefore necessary to turn

to more robust theories such as Mie and T-Matrix theories. Though these are often used only

in the region where the 1X , each theory is fully valid for a wide range of values of X, and

converge in the appropriate limits to geometric or Rayleigh scattering.

Both T-Matrix and Mie theories can be used to calculate a number of important light

scattering properties including the angular light scattering intensity or phase function, the

linear polarization of the scattered light, absorption, backscattering ratio, and others. Though

the details of how these properties are calculated will differ for different scattering theories

and will be discussed below, it will be helpful to first discuss some of these basic properties.

The results of this work, presented in Chapters 5-7, will focus strongly on the scattering

phase functions, F(θ), and the linear polarizations, P(θ) . The scattering angle, θ, is measured

relative the direction of the incident light. The phase function is proportional to the sum of

the perpendicular and parallel polarized scattered light intensities from an unpolarized light

source. Due to the symmetry in the scattering matrix, this is equivalent to the sum of the

scattered light intensities from incident light polarized perpendicular to the scattering plane,

67

I , and incident light polarized parallel to the scattering plane, //I . The scattering plane is

defined to be the plane containing both the incident and scattered light signal.

//)( IIF (4.2)

The linear polarization, which is a measure of the degree of polarization of the

scattered light, can then be written as follows:

//

//)(IIII

P

(4.3)

where positive values for P(θ) correspond to scattered light that is partially polarized

perpendicular to the scattering plane and negative values correspond to light which is

partially polarized parallel to the scattering plane.

The asymmetry parameter, g, is a measure of the degree of scattering in the forward

(near 0o) or backward (near 180o) and defined as the cosine weighted average of the phase

function over all scattering angles:

dFg )sin()()cos(02

1 (4.4)

For particles that scatter isotropically, or in cases where the scattering is symmetric about a

scattering angle of 90o, g = 0. Strong forward scattering relative to that in the backward

direction will result in positive vales of g, while stronger backward scatter will result in

negative values of g.

The remainder of this chapter will provide of brief overview of the mathematical

framework of both Mie and T-Matrix theories. A rigorous derivation is beyond the scope of

this work and is already given in a number of other sources. Bohren and Huffman [1983]

provides an in depth derivation of Mie theory. For more information on T-Matrix theory, one

should see Waterman [1965, 1971] and Mischenko et al. [1996]. Some of the computational

methods and strengths and limitations of each theory will also be provided below.

68

Mie Theory

Formulated by Gustav Mie in 1908, Mie theory is the solution to the wave equation

for plane electromagnetic waves scattering from a sphere. Though the strict requirement of

spherical particles places significant limitations on Mie theory, it is still commonly used to

predict the scattering properties of aerosols which are roughly spherical, such as sea-salt or

ammonium sulfate aerosols which are generally in droplet form at relevant atmospheric

relative humilities [Masonis et al., 2003]. Application of Mie theory to non-spherical

particles can result in significant errors in prediction of the scattering properties of those

particles [Mishchenko et al., 1995; Curtis et al., 2008]. As will be seen in chapters 5-7, the

mineral dust aerosol samples used in this work include particles that are highly irregular in

shape. Nonetheless, Mie theory still provides a first order approximation to the scattering

from such particles.

Following the discussion of Bohren and Huffman [1983], we will first examine the

relationship between the light incident on a particle, Ii, to the light scattered from that

particle, Isr, through use of the scattering matrix, S.

is ISI ˆ (4.5)

Above, the subscript s denotes the scattered light and the subscript i denotes the incident light

on the particle. This relationship can be expanded in terms of the standard Stoke’s parameters

I, Q, U, and V:

i

i

i

i

s

s

s

s

VUQI

SSSSSSSSSSSSSSSS

VUQI

44434241

34333231

24232221

14131211

(4.6)

Written using the basis of the electric field vector oriented parallel, //E , or perpendicular, E ,

to the scattering plane the Stoke’s parameters are defined as:

EEEEI //// (4.7)

69

EEEEQ //// (4.8)

//// EEEEU (4.9)

//// EEEEiV (4.10)

Alternately, the scattering matrix can be written using the basis of the electric field vectors:

i

iikr

s

s

EE

SSSS

ikre

EE //

14

32// (4.11)

where k is the wavenumber, k= and S1 and S2 are given below. The distance from the

scattering particle to the observation point is given by r. This formulation of the light

scattering has the advantage that all of the optical properties of the scattering particle are

contained within S. Each element of S is a function of the scattering angle. Therefore, once S

is known for a given particle, it is relatively simple to determine the behavior of the scattered

light at all scattering angles from that particle. In particular, the light scattering intensities for

incident light polarized parallel and perpendicular to the scattering plane can be expressed in

terms of the scattering matrix elements:

221211// SSSI (4.12)

211211 SSSI (4.13)

The next step is to use a suitable method to calculate the scattering coefficients. This

has been done in Bohren and Huffman [1983] by first representing the scattered wave as a

series of spherical harmonics. The plane wave incident on the particle is then also expanded

in spherical harmonics. By choosing appropriate boundary conditions at the surface of the

particle and setting the expansion coefficients equal to each other, the following relations for

the scattering elements can be obtained:

n

nnnn bannnS )(

)1(12

1 (4.16)

70

n

nnnn bannnS )(

)1(12

2 (4.17)

where the functions πn and τn are defined as:

sin

1n

nP

(4.14)

ddPn

n

1

(4.15)

where 1nP are the associated Legendre functions of the first kind of order 1. Also in the

above, an and bn are the scattering coefficients and represent weighting factors for each of the

normal scattering modes of the sphere. For a spherical scattering particle, the scattering

coefficients can be determined analytically and can be written in terms of the Riccati-Bessel

functions, ψn.

)()()()()()()()(

''

''

mXXXmXmmXXXmXma

nnnn

nnnnn

(4.18)

)()()()()()()()(

''

''

mXXmXmXmXXmXmXb

nnnn

nnnnn

(4.19)

The primes denote differentiation with respect to the argument in parentheses. Here X is the

size parameter and m is the complex index of refraction:

Though the mathematical form of the scattering coefficients and scattering matrix

elements provide a rather abstract view of light scattering from a particle, there are a few

important aspects of Mie theory that should be pointed out. Due to the behavior of the π and

τ functions, as the diameter of the scattering particle increases there will be an increase in the

ratio of forward to backward scattering (asymmetry parameter → +1). Larger particles will

also produce scattering phase functions with narrower forward scattering peaks. It has also

been found that large non-spherical particles scatter similarly to area-equivalent spheres near

the forward direction [Mishchenko & Travis, 1994]. As the scattering angle increases, the

71

difference in scattering between spherical and non-spherical particles increases. This factor is

very important in our normalization procedure for experimental data (see Chapter 2).

A code has been written in the Mathematica programming language, based on that of

Hung & Martin [2002], to calculate the scattering phase function and polarization. As

mentioned above, Mie theory calculations require the full size distribution and the complex

index of refraction for the scattering particles as inputs. For cases where the size distribution

is polydisperse, it is necessary to calculate the light scattering for each size range, and then

add the resulting scatter contributions together using the number concentration of particles in

that size range as a weighting factor. This code is able to generate scattering properties

relatively quickly on a standard personal computer (Pentium 4 2.8 GHz processor, 1.5 GB

RAM,). Calculations of F(θ) and P(θ) are carried out at 75 evenly spaced scattering angles

over the range of 0-180o. Processing times are on the order of 15 seconds per particle size

range investigated. These low computational requirements have helped make Mie theory a

popular choice for light scattering calculations when the particles are expected to be

relatively spherical.

T-Matrix Theory

T-Matrix theory is an exact numerical solution to Maxwell’s equations for scattering

from an object of arbitrary shape. The use of a matrix approach to the determination of the

light scattering properties was first given by Waterman [1965] for perfectly conducting

spheres and later for scattering from objects of arbitrary shape and including dielectrics

[1971]. This formulation of the scattering problem was based on solutions to the integral

formulation of Maxwell’s equations. Due to the linearity of those equations, it is possible to

expand both the incident and scattered waves into spherical harmonics and then relate the

scattering coefficients of those expansions through use of a transformation matrix. A full

derivation of the T-Matrix approach is given in a number of sources [Waterman, 1965, 1971;

Barber and Yeh, 1975; Mishchenko et al., 1996] and will not be repeated here, though a brief

72

summary and a few key equations will be presented below using the notation of Mishchenko

et al.

Let an arbitrarily shaped particle be located at the origin of a spherical coordinate

system with a plane wave incident on the particle. The incident, Ei, and scattered, Es, fields

can than be expanded as a series of spherical harmonics:

n

n

nmmnmnmnmni krRgNbkrRgMaE )]()([ (4.20)

n

n

nmmnmnmnmns krNqkrMpE )]()([ (4.21)

where Mmn and Nmn are proportional to the spherical Hankel functions as given explicitly in

Mishchenko et al. [1996], and Rg denotes the regular solution. Due to the linearity of

Maxwell’s equations and the boundary conditions, it is possible to relate the incident field

coefficients, amn and bmn, to the scattered field coefficients, pmn and qmn, using the following

relation:

1'

'

''''

12''''

11'' ][

n

n

nmnmnmnmnmnmnmmn bTaTp (4.22)

1'

'

''''

22''''

21'' ][

n

n

nmnmnmnmnmnmnmmn bTaTq (4.23)

or, in matrix form:

ba

TTTT

ba

Tqp

2221

1211

(4.24)

where T is the transformation matrix (i.e. the T-Matrix). The T-Matrix does not depend on

the incident or scattered fields, only on the physical (particle shape, size, and orientation) and

optical (refractive index) properties of the scattering particle. Once the transformation matrix

is known for a given particle, it is possible to calculate the scattered light for any orientation

of incident radiation. The determination of the T-Matrix can be accomplished by use of the

73

extended boundary condition method [Waterman, 1965]. This is done by first expanding the

internal field, Eint , of the particle in terms of vector spherical functions as was done for the

incident and scattered fields above (see equations 4.20 and 4.21):

n

n

nmmnmnmnmn mkrRgNdmkrRgMcE )]()([int (4.25)

where the complex index of refraction is given by m. It is necessary to only use the part of

Mmn that is regular (i.e. finite), denoted by Rg in 4.25, over the scattering region being

investigated. For example, as was mentioned above Mmn and Nmn are proportional to the

spherical Hankel functions, H(1) ~ ( Jn ± i Nn ), where j is the spherical Bessel function, and n

is the spherical Neumann function. Since nN at the origin, the Neumann functions must

be excluded from Mmn and Nmn in that region. It is then possible to relate the expansion

coefficients of the internal field to those of the incident and scattered fields through the

transformation matrix Q:

dc

QQQQ

ba

2221

1211

(4.26)

dc

RgQRgQRgQRgQ

qp

2221

1211

(4.27)

The T-Matrix can then be related to Q by the following.

1 RgQQT (4.28)

The elements of Q are integrals over the particle’s surface and depend again on the physical

and optical properties of that particle. In general, T and Q contain an infinite number of

elements. However, for practical purposes, the summations (equations 4.20, 4.21, and 4.25)

are truncated after certain convergence criteria are reached thus allowing T to be determined

explicitly. In addition, the calculation of T can be greatly simplified by the assumption of an

ensemble of randomly oriented particles [Mishchenko et al., 1996]. In that case, T reduces to

a diagonal matrix in the indices m and m’ (see equation 4.22 and 4.23). This is a reasonable

74

assertion when dealing with aerosol particles. Particles with rotational symmetry also greatly

simplify these calculations.

All T-Matrix calculations of light scattering within this work were done using the

extended precision T-Matrix Code of Mishchenko et al., publicly available through the

NASA web site [Mishchenko & Travis, 1998;

http://www.giss.nasa.gov/staff/mmishchenko/t_matrix.html ]. This code has been used

extensively to model the light scattering from aerosol particles [West et al., 1997; Nousiainen

and Vermeulen, 2003; Veihelmann et al., 2004; Dubovik et al., 2006]. The T-Matrix code can

be used to calculate the full scattering matrix as well as characteristic dust optical properties

such as total extinction and scattering albedo for a randomly oriented distribution of particles

of specified shape.

Required input to the code include the wavelength of incident light, the particle’s

index of refraction (the optical constants), and information about the particle size and shape

distributions. The size distributions are input as lognormal distribution parameters (mode

diameter and width parameter) based on fits to experimental data as was discussed in Chapter

2. Particle shapes were modeled with a series of ellipsoids. A spheroid is defined here as an

ellipse of revolution about the minor (oblate spheroid) or major (prolate spheroid) axis whose

shape can then be characterized by a single parameter, the axial ratio (AR), the ratio of major-

to-minor axis lengths. A sphere will have AR = 1. Though there are considerable limitations

on the allowable particle shapes by restricting ourselves to spheroids (i.e. no edges), there is

still a wide range of shapes that can be successfully modeled; from flat plate-like structures

to needle shapes by using appropriate choices for the axial ratio. Separate calculations of the

light scattering for each sample were made for a range of axial ratios. A shape distribution, to

be discussed further in Chapters 5, 6, and 7, was then used to weight the scattering results for

each particle shape together.

It is important to note that there is still disagreement within the aerosol community as

to whether using a spheroidal distribution to approximate the particle shape of mineral dust

75

aerosol, which are generally highly irregular in shape and may include sharp edges and

internal voids, or carbon particles which are highly fractal in nature,can be used to accurately

calculate the scattering properties of such particles. Mishchenko et al. [1997] and Veihelmann

et al. [2004] have both used a uniform distribution of spheroid to model the scattering of

mineral dust aerosol though the range of axial ratios differed in each case. Nousiainen and

Vermeulen [2003] also found that the spheroidal approximation of particle shape resulted in

good agreement to many elements of the full scattering matrix for feldspar particles provided

a wide range of axial ratios were used. Alternate choices for particle shape include finite

circular cylinders or spheres deformed by means of a Chebychev polynomial [Mugnai &

Wiscombe, 1980] though these were not examined in this work. More advanced shape

models are possible, though this greatly increases calculation time and limits the range of

particle sizes that can be practically modeled [Dubovic et al., 2006].

A significant limitation of this T-Matrix code is the finite maximum particle diameter,

or more appropriately the maximum size parameter, X, for which the code will converge.

This value is dependent both on the axial ratio of the particle and the optical constants at the

wavelength being investigated. For example, for a spheroidal particle with size parameter,

Xa, defined by the length of the semi-major axis, and a refractive index n = 1.31, the code will

converge for Xa = 17 for axial ratios of 20, but can converge for Xa > 100 if the axial ratio is

decreased to ~2. These values of X will decrease significantly as the value of n increases

[Mishchenko & Travis, 1998]. For cases where larger axial ratios were considered, it was

sometimes necessary to cut down the large diameter portion of the size distribution. Errors

associated with the omission of the large diameter have been estimated for all T-Matrix

calculations and will be discussed in detail in Chapter 5.

The T-Matrix code is much more computationally demanding than the Mie code

discussed earlier and was not able to be run on a typical personal computer due to the need

for extended precision variables. Calculations were instead run on a 4-CPU IBM RS/6000

workstation (2.5 GB RAM, four 375-MHz 2-Way Power3-11 64 Bit Processors).

76

Computation times varied significantly and were largely dependent on the maximum size

parameter and the axial ratios of the particles being modeled. For nearly spherical particles,

computation times can be as low as a few minutes to generate the full T-Matrix and

scattering matrix. For highly eccentric particles, AR > 8, computation times longer than 5

hours were observed. Results from both Mie and T-Matrix simulations will be compared with

experimental data in Chapters 5 – 7.

77

CHAPTER 5

MULTI-WAVELENGTH LIGHT SCATTERING STUDIES

In this chapter, phase function and polarization profiles for a number atmospherically

relevant components of mineral dust aerosol including silicate clay (montmorillonite,

kaolinite, and illite), non-clay (calcite, gypsum, and quartz), and iron-oxide (hematite and

goethite) samples are given. In nature, mineral dust aerosol is composed of a mixture of

individual minerals. Atmospheric mineral dust may be internally (aggregates of different

minerals) or externally (each particle is composed of one mineral) mixed. For an external

mixture, it should be possible to treat each mineral component separately in scattering

calculations and then perform a weighted sum of the resulting scattering matrices assuming

the relative concentrations of each are known [Sokolik & Toon, 1999]. For internally mixed

samples the results can be quite different [Bohren & Huffman, 1983] Results will also be

presented for Arizona Road dust, a commonly used test aerosol that is a mixture of a number

of minerals and is more representative of a real world dust sample. For this sample, the

applicability of using weighted-average scattering properties based on an assumed

mineralogy will be investigated. Data will be presented at three visible wavelengths, 470,

550, and 660 nm, which were chosen to coincide with a number of wavelength bands used by

remote sensing satellites such as MISR and MODIS.

Error Analysis

Since (with the exception of Arizona Road Dust) these are well characterized, single

component mineral samples, wavelength dependent optical constants are available from

published sources (see Table 2.3). In some cases it was necessary to linearly interpolate

between tabulated values of the optical constants to the wavelengths investigated here, but

the optical constants were (in most cases) slowly varying functions of wavelength and the

interpolations were performed over a short wavelength range. The uncertainties in the optical

constants from the interpolation result in negligible errors in the scattering calculations. The

78

notable exceptions to this are the iron oxides, hematite and goethite, which have optical

constants that vary dramatically across the visible. The optical constants for the iron oxides

should be considered as highly uncertain, with a correspondingly large uncertainty in the

scattering calculations.

All measurements of the light scattering from the mineral dust aerosol samples were

performed simultaneously with the measurement of the aerosol size distribution as discussed

in Chapter 2. The size distributions were measured using a TSI Inc. Aerodynamic Particle

Sizer (APS) which is able to measure particle diameters within the range ~ 0.5-20 µm. At the

time this data was taken we did not have an integrated Scanning Mobility Particle Sizer

(SMPS), which covers the small particle diameter range, in our aerosol flow system. Since

the samples investigated here contain a significant number of particles with diameters below

this range it was necessary to extend the distribution to smaller diameters (D < 0.5 m) by

using a log-normal fit to the APS data at large diameters (D > 0.5 m). The log-normal

distribution was constrained to have mode diameters consistent with those made in earlier

measurements of these samples using both an APS and SMPS under similar flow conditions

[Hudson et al., 2008a and 2008b]. However, this approach results in significant uncertainty

in the aerosol size distribution for use in the theoretical scattering calculations for both Mie

theory and T-Matrix theory.

In order to gauge the magnitude of the errors in the theoretical scattering profiles

resulting from uncertainty in the input size distribution, multiple log-normal fits were

generated for each measured size distribution, all consistent with the APS data for D > 0.5

m but with varying mode diameters at small diameters. These fits were constrained to have

mode diameters that differed by a factor of two (higher and lower) than the mode diameter

measured by Hudson, DExp, in earlier work. In some cases, the distributions using mode

diameters of ExpD2 or ExpD21 resulted in very poor fits to the APS data for D > 0.5 m. In

those cases, the largest and smallest values of the mode diameter that would still give good

fits to the APS data were used instead. The values for the range of mode diameters used for

79

each sample are included in Table 2.1. Light scattering calculations for the phase functions

and polarizations were then performed using Mie theory for all three size distributions. Error

bars for the theoretical phase function and polarization profiles were then determined by the

range of the variation between the different Mie theory simulations. We did not carry out a

similar error analysis for the T-matrix theory simulations since those calculations are more

computationally demanding and require a time-consuming loop over particle shape

parameters as discussed below. It seems reasonable to assume that the errors resulting from

variation in the input size distribution should be of comparable magnitude for the T-matrix

calculations. Since light scattering intensity typically scales with the projected particle

surface area, the errors in the simulated scattering that result from uncertainty in the small

particle part of the distribution are often small. This is because the scattering signal is

dominated by scattering from the large diameter part of the size distribution directly

measured by the APS.

Errors in the experimental scattering data result from day-to-day variation in the

measured scattering data (random errors) and uncertainty in the instrument calibration

function (systematic errors). For each day of data collection, three experimental phase

function and polarization profiles were measured. The scattering phase function and

polarization for each sample were measured for 3-6 days, depending on the repeatability of

the measurements. Results are presented as an average over all the data sets, and the standard

deviation from the mean of the different data sets gives an estimate of the measurement

variability. In addition to measurement variability, error bars for the phase function also

include an estimate of the systematic error resulting from uncertainty in the instrument

calibration function. Error bars for the polarization profiles include estimates of the

uncertainties in the relative scaling factor, a, between the parallel and perpendicular

intensities. Both systematic errors are discussed in detail in Chapter 2.

For the polarization profiles, the magnitude of the error bars represents the standard

deviation from the mean of the different experimental data sets as well as uncertainty in the

80

relative intensity between the incident parallel and perpendicular polarized beams (see Eq.

2.10). The calibration error doesn’t affect the polarization because the polarization involves a

ratio of two measured values (see Eq. 2.9) and any systematic calibration effects would

cancel.

In order to quantify the goodness of fit between the theoretical and experimental

phase function profiles, a reduced chi-square factor was calculated:

})(/)]()({[1 22

1

2ii

N

ii TF

N

(5.1)

In (5.1) F is the measured phase function, T is the corresponding model prediction, i

is the estimated experimental uncertainty, all at a given scattering angle i, and N is the

number of angle data points. The reduced chi-square values for the Mie and T-Matrix theory

simulations are given in Table 5.1 for all wavelengths investigated.

T-Matrix Shape Distribution

In addition to the wavelength dependent optical constants and full size distribution, T-

Matrix theory also requires assumptions about the particle shape. Throughout this chapter a

uniform distribution of spheroidal particle shapes will be assumed. The use of spheroids

allows the particle shape to be characterized by a single parameter; the axial ratio (AR), the

ratio of major-to-minor axis lengths.

Separate T-Matrix calculations have been performed for ARs ranging between 1.0 ≤

AR ≤ 2.4 in steps of 0.2 for both prolate and oblate spheroids. The resulting phase function

and polarization profiles were summed assuming a uniform (constant) distribution of AR

values over this range. This shape distribution is a common model used for mineral dust

aerosol [Mishchenko et al., 1997] and is based on electron microscope images of mineral dust

aerosol particles collected in the field. This model will be referred to as the “Standard” shape

distribution model. As will be shown in chapter 6, this range of ARs is also roughly

81

consistent with electron microscope images collected in the lab for the samples used in this

work.

Asymmetry Parameter Calculations

As was discussed earlier (Chapter 4), the asymmetry parameter, g, is a measure of the

relative scattering in the forward (near 0o) to backward (near 180o) directions. For mineral

dust aerosol with characteristic size parameters in the range X ~ 2-5, as appropriate here (see

Table 2.3), the forward scattering tends to dominate the overall scattering signal, resulting in

positive values for g [Andrews et. al., 2006; Kahnert & Nousiainen, 2006]. For each sample,

asymmetry parameters have been calculated using equation 4.4 for the experimental (gExp),

Mie theory (gMie), and T-Matrix theory (gTM) phase function profiles for all three

wavelengths, and the results are given in Table 5.2. For the experimental asymmetry

parameter, the value given is an average over consecutive days of data collection. For the

Mie theory results, the asymmetry parameter is an average over the results for the three size

distributions that were used in the calculations and the errors are representative of the

standard deviation about the mean value of the results. As discussed above, we expect the

errors in the T-Matrix and Mie results to be of comparable magnitude.

Non-Clay Mineral Dust Results

Shown in Figures 5.1-5.3 are the scattering phase function and polarization profiles

for a number of non-clay samples including calcite, gypsum, and quartz for incident light at

470 nm, 550 nm, and 660 nm. The particles span an effective (surface area weighted) size

parameter range from Xeff = 2.8 for gypsum to Xeff = 4.9 for quartz for 550 nm incident light

(see Table 2.3). For these dust samples, the imaginary part of the refractive index is small

and the real part lies within the range n ≈ 1.4-1.6 throughout the range of visible wavelengths

investigated. The iron oxide samples differ markedly from these in that they exhibit strong

absorption in the visible range and they will be discussed separately below. Calcite and

quartz are both birefringent materials, which must be taken into account in the light

82

scattering calculations. Assuming that these particles are randomly oriented in the aerosol

flow, light will scatter from each of the three optical axes (two ordinary, 1 extraordinary)

with equal probability. For birefringent samples, it is therefore necessary to average the o-ray

(ordinary) and e-ray (extraordinary) theoretical scattering results in a 2:1 ratio. We have

found for quartz that averaging the optical constants in a 2:1 ratio prior to running the

scattering calculations produced negligible changes in the resulting theoretical scattering.

This is due to the very small difference between the o-ray and e-ray optical constants (~1%).

In order to reduce calculation times, these averaged optical constants for quartz were used.

This was not done for the calcite as the differences in the optical constants were much larger

between the e-ray and o-ray axes.

As can be seen for the non clay minerals in Figures 5.1-5.3 (a), Mie theory tends to

over predict the scattering in the backward direction, scattering near 180o, at all three

wavelengths. An better way of viewing the differences between the experimental and

theoretical phase functions is to examine the ratio of the experimental phase function to that

generated using either T-Matrix theory (TM Ratio) or Mie theory (Mie Ratio) as shown in

Figure 5.1-5.3 (b). It is easy to see in Figures 5.1 – 5.3 (b) that the experiment-to-theory ratio

for the T-Matrix simulations tend to be flatter and closer to unity than those for the Mie

theory simulations, indicating significantly better agreement with experiment.

The tendency for Mie theory to overestimate the backscattering signal results in

theoretical asymmetry parameters, gMie, that are slightly lower, ~ 4-6%, than those

determined experimentally (see Table 5.2). Though T-Matrix theory results, assuming the

“Standard” shape distribution, are in better agreement with the experimental phase function

overall (Table 5.1), the calculated asymmetry parameters for T-Matrix theory, gTM,, are still

very similar to those for Mie theory.

A comparison of the polarization profiles for the non-clay samples is given in Figures

5.1-5.3 (c). For all three samples, the polarization is positive over the entire range of

scattering angles with a peak near 110o. The magnitude of the experimental polarization

83

increases with increasing wavelength, ranging from 31-55% for quartz. In contrast, the

theoretical polarization curves tend to be negative over the entire range with minimums

between 150o and 160o. It can easily be seen that even though the quantitative agreement

between the T-Matrix and experimental polarizations is poor, qualitatively, it is a significant

improvement over that of Mie theory.

Overall, T-Matrix theory using a commonly applied particle shape distribution model,

a uniform distribution of spheroids of moderate axial ratios, gives a significant improvement

over Mie theory in modeling both the phase function and polarization profiles for the non-

clay mineral samples, quartz, calcite, and gypsum. However, the agreement with experiment

is still not especially good, particularly for the polarization. This could indicate a limitation

of the spheroid approximation in T-Matrix theory or it could result from a difference between

the assumed and actual particle shape distribution function. These points will be discussed in

more detail in the next chapter.

Clay Mineral Dust Results

Analogous results for the scattering from the silicate clay minerals, illite, kaolinite,

and montmorillonite, are given in Figures 5.4-5.6. For the clay samples, the effective size

parameter range of Xeff = 2.3 for illite and Xeff = 3.4 for kaolinite for 550 nm incident light

(see Table 2.3), which is very similar to that of the non-clay samples. As with the non-clay

samples, the variation in the real part of the refractive index for the clays is relatively small

over the visible range, n ≈ 1.4-1.5. The illite sample is unique among the clays in that it

exhibits a greater degree of absorption within the visible range. The imaginary part of the

refractive index for illite is an order of magnitude higher than those of the other two clays

(though still small, particularly when compared to hematite or goethite).

The silicate clays show much greater variability in the scattering results than the non-

clay samples discussed above. The observed scattering profiles are more variable from day-

to-day and week-to-week, and show much greater variation between the different samples

84

and at different scattering wavelengths (Figures 5.4-5.6 (a) and (b)). For example, Mie theory

tends to over predict the scattering for scattering angles greater than ~125o for kaolinite

measured at 470 nm, but the over prediction becomes much more pronounced for the

kaolinite phase function measured at 660 nm where it can be seen for all scattering angles

above ~30o. The situation is reversed for the illite sample where Mie theory actually under

predicts the scattering for near backscattering angles at all three wavelengths. The overall

agreement between experiment and T-Matrix theory assuming a “Standard” shape

distribution is similar to that of Mie theory for all samples. There is a trend that the

asymmetry parameters calculated for both Mie and T-Matrix theories tend to be slightly

lower than those determined experimentally, i.e. a higher predicted backscatter-to-forward

scatter ratio than that seen experimentally, as was also the case for the non-clays.

An alternative way of viewing the differences between the clay and non-clay results

is to plot the range of experiment to theory ratios for the different clay (light gray band) and

non-clay (dark gray band) minerals on the same plot. This is done in Figure 5.7 for both the

experiment-to-Mie and the experiment-to-T-Matrix ratios. As is evident from both the Mie

and the TM ratios, the scattering from the clay samples seems to be inherently different from

that of the non-clay samples even though both have similar size distributions and optical

constants. The ratios for the non-clay samples tend to be constrained to a much narrower

band indicating very similar scattering behavior for the quartz, calcite, and gypsum samples.

The ratios for the clays are much more varied, corresponding to a much wider band. Possible

reasons for the greater variability in the scattering results for the silicate clays are discussed

below. However, it is clear that this increased variability makes it much more difficult to

draw consistent and general conclusions for scattering from the silicate clays than for the

non-clay mineral samples.

As was seen for the non-clay samples, the experimental polarization profiles for the

clays, shown in Figures 5.4- 5.6 (c), tend to be positive over the entire range of scattering

angles, though the peak of the polarization signal tends to occur at smaller angles for these

85

samples; ~90o for montmorillonite, ~95o for illite, and ~ 100o for kaolinite. The peak

magnitude of the polarization again increases with increasing wavelength, varying from 55-

70% for montmorillonite. The Mie polarization is again negative over all scattering angles

for the kaolinite and montmorillonite with minimums near 160o. For the illite sample, Mie

theory comes closest to agreeing with experiment in that it is positive for scattering angles

less than ~140o and the predicted peak position near 99o. The overall magnitude of the peak

is still lower than that seen experimentally, however. T-Matrix theory performs better in

predicting the peak magnitude of the polarization, but tends to place the peak of the profile at

larger scattering angles. As the imaginary part of the refractive index increases, both Mie and

T-Matrix theory predict a polarization that peaks at more positive values. This may explain in

part why the illite polarization shows better agreement with theory.

Iron Oxide Results

The effective size parameters for the hematite and goethite samples are Xeff = 3.8 and

2.4, respectively, for 550 nm incident light (see Table 2.3), which is similar to the other

samples. These samples are treated separately from the other non-clay samples because of the

relatively large index of refraction values (both real and imaginary) for both samples. The

optical constants for hematite, as reported by Longtin et al. [1988], also have a stronger

dependence on wavelength than is seen in any of the other samples, with the imaginary part

of the index changing by nearly two orders of magnitude over the wavelength range

presented here. It bears repeating that the optical constants for the iron oxide samples are

highly uncertain. The indices of refraction reported by three different sources, Longtin et al.

[1988], Bedidi and Cervelle [1993], and Sokolik & Toon [1999], give dramatically different

values for both the real and imaginary parts (see Table 2.3) which has significant effects on

the theoretical scattering calculations. For this reason, theoretical phase and polarization

profiles were generated for hematite for each set of optical constants (Figures 5.8-5.10). At

470 nm, the experimental phase function for hematite (Figure 5.8-5.10 (a)) is relatively flat

86

for scattering angles above roughly 70o which may be due to the large imaginary index of

refraction at that wavelength. This flatness results in an asymmetry parameter that is

relatively high compared to other samples investigated, gExp ≈ 0.72. Both Mie and T-Matrix

theory predict this flattening of the phase function though the overall agreement with

experiment for scattering angles within the range 30-90o is quite poor for any of the three sets

of refractive index values. The more gently sloping phase functions predicted by theory leads

to an under prediction for the asymmetry parameter, gMiey ≈ 0.68 for the Longtin optical

constants, gMie ≈ 0.66 for the Bedidi optical constants, and gMie ≈ 0.67 for the Sokolik & Toon

optical constants. As the incident wavelength increases, the experimental phase functions for

hematite become more symmetric about 90o, with a corresponding decrease in the magnitude

of g. The theoretical phase functions follow this general trend though the absolute agreement

with experiment does not improve and the asymmetry parameter is again underestimated at

all wavelengths with the exception of the 660 nm calculations based on the Sokolik & Toon

or the Bedidi refractive indices which lead to an over-prediction of the asymmetry parameter

by about10% or 30% respectively.

Compared to the hematite sample, the goethite experimental phase functions (Figure

5.11 (a)) tend to show a weaker wavelength dependence, the most notable variation occurring

at large scattering angles (>140o). This may be explained because the imaginary part of the

refractive index is less dependent on wavelength for goethite. These phase functions are also

have much higher backscatter relative to forward scatter, and therefore have asymmetry

parameter values which are lower than any of the other samples, gExp ≈ 0.56. The changes in

the theoretical phase functions are more pronounced with wavelength. The most notable

difference between the experimental and theoretical phase functions is for near backward

scattering. These general trends are correctly predicted by the theoretical simulations. As the

wavelength increases, the experimental scattering increases for scattering angles greater than

140o. However, the theoretical phase functions decrease in this region with increasing

87

wavelength. The theoretical phase functions also take on a more linear form for mid-range

scattering angles, 45-135o. A similar trend is hardly evident in the experimental data.

The hematite, and to a lesser extent the goethite, polarizations given in Figures 5.8-

5.11 (c), tend to be flatter for all scattering angles then was seen for the other non-clay

samples. Though the goethite polarization follows the trend of increasing peak magnitude

with increasing wavelength as was seen before, the hematite actually exhibits the opposite

behavior and is near zero (un-polarized scattering) for 660 nm incident light. For both

samples, the theoretical polarizations are in relatively good agreement with each other, but

with T-Matrix theory again giving slightly better fits to the experimental data. However, both

theories predict polarization values that are too low. In addition, polarization profiles

generated using the Sokolik & Toon optical constants agree with experiment much better

than those generated using either the Bedidi or the Longtin optical constants.

Arizona Road Dust Results

Arizona Road Dust (ARD) serves as a model for the complex mixtures that more

closely represent authentic mineral dust aerosol in that it is an inhomogeneous mixture of

different minerals. ARD is a commercially available sample (Powder Technology Inc.) and is

provided in a number of particle size ranges. For these measurements, ARD in the “fine” size

range was used, corresponding to particles with an effective radius, Reff ≈ 320 nm (Xeff = 3.6

for 550 nm) (see Tables 2.2 and 2.3). This is similar to other samples investigated. The ARD

sample was characterized separately in our laboratory by Cwiertny et al. [2008]. Elemental

analysis and FTIR data collected for this sample are consistent with a model composition

consisting roughly equal weight mix of clays (probably primarily montmorillonite) and non-

clays (primarily quartz). Based on this analysis, optical constants were generated for the

ARD sample by averaging those of the quartz and montmorillonite in a 1:1 ratio (see Table

2.3). For an external mixture, it is more accurate to generate separate scattering matrices for

each mineral component (i.e. each set of optical constants), then average the resulting phase

88

functions and polarizations [Sokolik & Toon, 1999]. However, for quartz and

montmorillonite, the real and imaginary parts of the refractive index are quite close for all

wavelengths examined. Since we assume the same size distribution for each constituent

mineral dust, the use of optical constant average method results in a negligible error and was

used to reduce computation time when running the T-Matrix simulations.

The experimental and theoretical phase function and polarization profiles for ARD

are given in Figure 5.12. As seen for the non-clay samples, Mie theory tends to over-predict

the scattering in the background direction (>120o) at all three wavelengths. T-Matrix theory

using the “Standard” shape distribution does a much better job fitting the experimental phase

function and agrees over the entire range of scattering angles. In contrast, the T-Matrix

polarization profiles are still negative for most angles though the theory does predict a

slightly positive polarization over the mid range scattering angles, 100-130o, it is still

negative for most angles and again fails to properly model the polarization for this sample.

As an alternative to theoretical calculations for modeling complex mixtures, it may be

possible to use a set of empirically measured phase functions and polarizations as a basis set

for the scattering of the mixture, provided the mineralogical composition of the mixture is

known. This approach has been used previously by Mishchenko et al. [2003] using an

experimentally measured quartz phase function in an aerosol retrieval algorithm for satellite

data. Average scattering phase functions and polarizations were first generated for the clay

samples (illite, kaolinite, and montmorillonite) and the non-clay samples (calcite, gypsum,

and quartz). The average clay and average non-clay scattering data were then averaged in a

1:1 ratio (consistent with mineralogical analysis for ARD). These empirical phase functions

and polarizations are shown in Figure 5.13 along with the experimentally measured ARD

scattering data. Not surprisingly, the agreement between the empirically generated and the

experimentally measured phase functions and polarizations is much better than that obtained

using either Mie or T-Matrix theory.

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Discussion

As was seen for nearly all the samples studied, Mie theory fails to accurately model

the scattering properties of irregularly shaped mineral dust aerosol for wavelengths

throughout the visible spectral region. The failure of Mie theory is most clearly evident in the

results for the non-clay minerals (calcite, gypsum, & quartz) and for the Arizona Road Dust

sample. In these samples the most pronounced deviations from experimental measurements

occur for scattering angles >150o where Mie theory consistently and significantly over-

predicts the phase function, by up to a factor of three. Near mid-range scattering angles, there

seems to be a wavelength dependent deviation from experimental values as Mie theory

under-predicts scattering in this region for the 470 nm data, but over-predicts the scattering

for the 660 nm data.

For the silicate clays (illite, kaolinite, & montmorillonite) and the iron oxides

(hematite and goethite) the deviations between the experimental phase function and Mie

theory simulations are less consistent but still appear to generally hold, particularly (and

notably) for the kaolinite sample. As noted above, there is great variability in the

experimental results among the clay samples and, for a given sample, at different

wavelengths; this variability makes it difficult to draw general conclusions from the clay

scattering data.

Here it should be noted that, from a “model testing” perspective, the results for the

non-clay minerals (particularly calcite and quartz) are much more significant than the results

for the clay or iron-oxide samples. The many reasons for this will be discussed below but

essentially it is due to the fact that the optical constants (the index of refractions) for quartz

and calcite are known at a very high confidence level while the optical constants for the other

samples have significant uncertainties.

The failures of Mie theory are most consistent and most pronounced in the

polarization data for all of the samples studied, where the experimental polarization is

generally opposite in sign to the Mie theory predictions. This might be expected since the

90

scattering polarization is much more sensitive to particle shape effects than the phase

function.

Failure to properly model the scattering phase function for mineral dusts will

ultimately result in errors in calculating the back scattering cross section, and the asymmetry

parameter, which are used in a number of aerosol retrieval and climate forcing algorithms

[Kahn et. al., 1997; Myhre & Stordal, 2001]. For most cases studied here, it was seen that the

theoretical asymmetry parameters were consistently lower than those seen experimentally, in

some cases up to ~10% lower. As noted above a 10% error in asymmetry parameter can

result in a 20% error in the climate forcing effect of dust [Andrews et. al., 2006].

The use of T-Matrix theory with the assumption of a uniform distribution of oblate

and prolate spheroids of moderate asymmetry parameter (AR ≤ 2.4,“Standard” shape model)

results in a significant improvement in the agreement between the experimental and

theoretical phase function and polarization profiles. This is most evident in the non-clay and

ARD scattering results where the reduced chi-square factors of Table 5.1 show a dramatic

and consistent improvement with T-Matrix theory. It is also readily apparent in the Figures

5.1 – 5.3 (b) and 5.7 where the TM ratios are much flatter and closer to unity than the Mie

ratios. The improvements in the model results using T-Matrix theory for the silicate clays

and iron oxides are less consistent and obvious, but the general conclusion appears to hold as

well, particularly for kaolinite. As noted above, from a model-testing perspective, the results

for the non-clay samples are deemed to be much more significant and reliable.

While T-Matrix theory offers a significant improvement over Mie theory in

simulating the scattering profiles, the agreement is still not great and, for the polarization

data, is quite poor for nearly all of the mineral samples and scattering wavelengths

investigated. This might be a result of uncertainty in the mineral optical constants, but even

for quartz and calcite where the optical constants are known very accurately, there are large

discrepancies between the measured and calculated polarization profiles. It seems most

likely that the discrepancy is a result of particle shape effects, and could indicate that the

91

“Standard” shape distribution model may not be appropriate for our samples. This issue will

be explored in detail in the next chapter.

It is also interesting to see how the theoretical simulations do in predicting the

wavelength dependence in the scattering data for a given sample. The most pronounced

wavelength variations are seen for hematite. In this case the curvature of the phase function

increases significantly, and the polarization decreases markedly as the wavelength changes

across the visible from 470 nm to 660 nm. T-Matrix theory (and to a somewhat lesser

degree, Mie theory) does very well in predicting these dramatic changes with wavelength for

either set of optical constants used in the calculations.

We also note the significant differences in scattering properties of hematite and

goethite. Iron oxides are a common constituent of mineral dust aerosol and play a critical

role in determining aerosol optical properties because of their strong visible absorption

bands. It has been generally assumed that iron oxide in mineral dust appears most commonly

in the form of hematite. However, recent analyses by Lafon et al. [2006] call this assumption

into question. The results here show that this issue is extremely important since goethite and

hematite have very different scattering properties, and very different wavelength

dependences.

We have commented that the results for the non-clays are more significant than those

of the silicate clay samples or the iron oxides for the purpose of testing theoretical models for

the scattering. This is largely due to uncertainty in the optical constants used for these

calculations. The optical constants for calcite and quartz have been measured by multiple

groups over many years and the results are known with a high level of confidence. This is

not so for the other samples under study here, particularly for the clay and iron oxide

samples. For the silicate clays there is only a single data set for the visible index of

refraction values, Egan and Hilgeman [1979]. The accuracy of this data set is not known.

Since optical constants for the clays were all obtained from this source, any errors in those

measured values would carry over into these calculations. Furthermore, the silicate clays

92

represent classes of compounds with significant variation in mineral form among samples

obtained from different regional sources. This is especially true of “montmorillonite”, which

refers to a broad class of smectite clays with significant mineralogical variability. In

addition, many clays (and especially the smectites) often have inclusions of iron-containing

impurities, which can have a very significant impact on optical constants. Thus, even if the

optical constants of Egan and Hilgeman were very accurately determined for their samples,

they may not be applicable to our clay mineral samples.

In addition, some clays (such as montmorillonite and illite, but not kaolinite) are

swellable, i.e., they absorb water with the water molecules moving into interstitial sites

between the silicate layers in the clay compound, causing the particles to swell. Because the

water is absorbed into the volume of the particle, it is very difficult to remove. Our dust

samples are aerosolized from slurry of mineral dust in water. As a result it is very likely that

the illite and montmorillonite clay samples are impregnated with water. The size and shape

of the particles, and the optical constants may change from day-to-day as a result of different

water loading. Interestingly, kaolinite, which shows the most consistent agreement with the

scattering characteristics of the non-clay samples, as noted above, does not absorb water.

The optical constants for the iron oxide samples are also highly uncertain. For

hematite, one set of optical constants used were obtained from Longtin et al. [1988], who

reports the refractive index for hematite to be n = 3.10 + 0.093i at 550 nm. However, Bedidi

& Cervelle [1993] give slightly higher values for the real part and significantly higher values

for the imaginary part of the refractive index values for hematite, n = 3.26 + 0.21i at 550 nm.

A third set of optical constants was obtained from Sokolik & Toon [1999], who report the

highest imaginary part of the imaginary part of the refractive index, give a value of n = 3.17

+ 0.46i at 550 nm. The light scattering calculations, both Mie and T-Matrix, based on the

Bedidi & Cervelle optical constants agree significantly better with the experimental

polarization profiles than those generated using the Longtin refractive indices, but result in

overall poor fits to the phase functions (particularly for the 660 nm data). The overall best fit

93

to the phase function and polarization profiles is obtained using the Sokolik & Toon optical

constants. However, at 660 nm, both sets of results using the Bedidi & Cervelle or the

Sokolik & Toon values lead to a significant over-prediction of the scattering in the backward

direction. At 660 nm, better results were obtained using the Longtin refractive indices which

suggests that the imaginary part of the optical constants from the other two sources is too

high for this particular sample. For hematite, the T-Matrix and Mie theory simulations are in

close agreement for both the polarization and phase function profiles. This degree of

concurrence suggests that moderate departures from spherical shape are relatively

unimportant in determining the scattering matrix for particles with high refractive index

values, like hematite. This result is agreement with the earlier studies of Munoz et al. [2006].

This illustrates the degree of uncertainty in the optical constants for hematite and the

significant impact that uncertainty can have on the scattering simulations. It is important to

note that the optical constants used in this work [Longtin et al., 1988; Bedidi & Cervelle,

1993, Sokolik & Toon, 1999 ] were chosen because they are commonly used in many climate

forcing and aerosol retrieval algorithms [Liang, 1997; Weaver et al., 2002; Nobileau &

Antoine, 2005; Park et al., 2005, Balkanski et al., 2007; Otto et al., 2009]. Any errors due to

uncertainties in the optical constants would also be present in those calculations. For goethite

we have only a single measurement for the optical constants. The accuracy of this

measurement is not known, but (based on the hematite results) we should expect that there is

significant uncertainty in this value.

Perhaps an even larger source of discrepancy between experiment and theory may be

particle shape effects. It is well known that these mineral dusts are highly irregular in shape

and may contain sharp edges, points, and internal voids. Here we have used a uniform

distribution of spheroids of moderate asymmetry (AR < 2.4) to model these particles. Some

sources claim that the neglect of sharp edges inherent in the spheroid approximation can

result in significant errors in calculated optical properties [Kalashnikova & Sokolik, 2002]

94

while others assert that these edge effects have a small effect on the scattering [Kahnert &

Kylling, 2004]. This remains an open question that will require more study.

Another question, which will be addressed in Chapter 6, is whether or not restricting

the range of aspect ratios to the relatively narrow range used here, AR ≤ 2.4, is truly

representative of the shapes of these mineral dust particles. This range is based on analyses of

SEM images of particles collected in the field [Mishchenko et al., 1997]. However, SEM data

are two dimensional image representations of three dimensional particles. This can lead to

errors; for example, a disc-shaped particle may appear circular or needle-like depending on

the viewing angle. It is known that the clay particles are highly plate-like [Nadeau, 1985], so

a more extreme range of aspect ratios with higher weighting of oblate shapes might be

expected to give a more realistic approximation to the shapes of those particles. In contrast,

goethite particles are known to be rod-like and so a more extreme range of aspect ratios with

higher weighting to prolate spheroids might give better agreement.

Due to the uncertainties in mineral optical constants and the difficulties in calculating the

scattering properties of non-spherical particles, it has been suggested that using empirical

phase function data based on experimental measurements of real dust samples may be a more

reliable approach. For example, empirical scattering data could be used in climate forcing

calculations and aerosol retrievals. We have seen that using average clay and average non-

clay phase functions and polarizations to model the scattering for an external mixture, ARD,

has worked extremely well. Of course, knowledge of the composition of the aerosol being

modeled is still necessary using this method. It is particularly important to have reliable

information about the iron oxide content and mineralogical form. Average scattering data for

the clays, non-clays, and hematite has been provided for this purpose in Curtis et al. [2008].

Our results here supply additional input for this database. It is important to reiterate here

that scattering is highly dependent on particle size. Those results were collected for particles

in the accumulation size mode, corresponding to effective particle diameters ≈1.0 µm, and

the empirical data should only be used to model particles in a similar size range.

95

Mineral Dust Wavelength χ2 (Mie) χ2 (T-Matrix) 470 nm 14.3 1.0 550 nm 14.9 0.9 Calcite 660 nm 12.4 4.4 470 nm 4.7 0.8 550 nm 1.1 0.6 Gypsum 660 nm 3.4 2.2 470 nm 33.6 6.2 550 nm 28.5 4.2 Quartz 660 nm 15.7 5.3 470 nm 3.0 3.1 550 nm 5.7 6.5 Illite 660 nm 1.7 2.7 470 nm 13.1 1.8 550 nm 1.3 1.2 Kaolinite 660 nm 10.0 2.7 470 nm 5.3 0.9 550 nm 1.4 1.7 Montmorillonite 660 nm 1.9 2.8 470 nm 10.5 2.7 550 nm 5.3 1.8 Goethite 660 nm 5.4 6.7 470 nm 4.0 4.8 550 nm 10.1 6.3

Hematite (Longtin et al. [1988])

660 nm 3.7 2.5 470 nm 4.8 5.2 550 nm 2.1 1.5

Hematite (Bedidi and Cervelle [1993])

660 nm 7.7 9.6 470 nm 4.4 5.0 550 nm 1.0 1.0

Hematite (Sokolik & Toon [1999])

660 nm 2.1 3.9 470 nm 19.0 1.9 550 nm 2.5 0.4 Arizona Road Dust 660 nm 5.6 1.5

Table 5.1 Reduced χ2 values for comparison of the experimental phase functions with simulations using Mie theory and using T-Matrix theory assuming a “Standard” shape distribution.

96

Mineral Dust Wavelength gExp gMie gTM*

470 nm 0.65 ± 0.01 0.63 ± 0.01 0.63 550 nm 0.65 ± 0.01 0.63 ± 0.01 0.64 Calcite 660 nm 0.69 ± 0.01 0.63 ± 0.01 0.64 470 nm 0.69 ± 0.02 0.67 ± 0.01 0.68 550 nm 0.68 ± 0.01 0.67 ± 0.02 0.68 Gypsum 660 nm 0.71 ± 0.01 0.66 ± 0.02 0.67 470 nm 0.69 ± 0.01 0.66 ± 0.01 0.67 550 nm 0.69 ± 0.01 0.66 ± 0.01 0.67 Quartz 660 nm 0.72 ± 0.01 0.67 ± 0.01 0.68 470 nm 0.70 ± 0.01 0.73 ± 0.02 0.73 550 nm 0.68 ± 0.01 0.72 ± 0.03 0.72 Illite 660 nm 0.71 ± 0.01 0.70 ± 0.02 0.70 470 nm 0.74 ± 0.02 0.69 ± 0.01 0.70 550 nm 0.71 ± 0.02 0.69 ± 0.02 0.70 Kaolinite 660 nm 0.76 ± 0.02 0.68 ± 0.01 0.69 470 nm 0.70 ± 0.01 0.68 ± 0.01 0.69 550 nm 0.68 ± 0.02 0.68 ± 0.02 0.68 Montmorillonite 660 nm 0.68 ±0.01 0.66 ± 0.03 0.67 470 nm 0.57 ± 0.01 0.46 ± 0.03 0.52 550 nm 0.55 ± 0.01 0.47 ±0.01 0.53 Goethite 660 nm 0.55 ± 0.01 0.49 ± 0.01 0.53 470 nm 0.72 ± 0.02 0.68 ± 0.05 0.67 550 nm 0.64 ± 0.01 0.53 ± 0.07 0.55

Hematite (Longtin et al.

[1988]) 660 nm 0.42 ± 0.01 0.34 ± 0.02 0.34 470 nm 0.72 ± 0.02 0.66 ± 0.05 0.65 550 nm 0.64 ± 0.01 0.59 ± 0.07 0.60

Hematite (Bedidi and

Cervelle [1993]) 660 nm 0.42 ± 0.01 0.54 ± 0.05 0.55 470 nm 0.72 ± 0.02 0.67 ± 0.04 0.66 550 nm 0.64 ± 0.01 0.63 ± 0.06 0.63

Hematite (Sokolik & Toon

[1999]) 660 nm 0.42 ± 0.01 0.47 ± 0.04 0.48 470 nm 0.69 ± 0.01 0.67 ± 0.01 0.69 550 nm 0.68 ± 0.02 0.67 ± 0.01 0.69 Arizona Road Dust 660 nm 0.71 ± 0.01 0.67 ± 0.01 0.69

*Errors in asymmetry parameters generated using T-Matrix theory, due to uncertainties in the input size distributions, are expected to be comparable to those generated using Mie theory.

Table 5.2 Asymmetry parameter values for experimental (gExp), Mie theory (gMie), and T-Matrix theory (gTM) phase functions for 470, 550, and 660 nm.

97

Figure 5.1 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for calcite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

98

Figure 5.2 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for gypsum measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

99

Figure 5.3 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for quartz measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

100

Figure 5.4 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for illite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

101

Figure 5.5 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for kaolinite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

102

Figure 5.6 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for montmorillonite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

103

Figure 5.7 Range of ratios of experimental phase functions to theoretical phase functions generated using Mie theory (a) and T-Matrix Theory (b). Results are shown for the non-clay samples calcite, gypsum, and quartz (dark gray) and for the clay samples illite, kaolinite, and montmorillonite (light gray).

104

Figure 5.8 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for hematite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Optical constants used for the theoretical calculations were obtained from Longtin et al. [1988]. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

105

Figure 5.9 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for hematite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Optical constants used for the theoretical calculations were obtained from Bedidi & Cervelle. [1993]. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 5.10 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for hematite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Optical constants used for the theoretical calculations were obtained from Hematite Sokolik & Toon [1999]. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 5.11 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for goethite measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 5.12 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for Arizona Road Dust measured at 470 nm (left), 550 nm (center), and 660 nm (right). Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 5.13 Normalized phase functions (a) and linear polarizations (b) for Arizona Road Dust measured at 470 nm (left), 550 nm (center), and 660 nm (right). Empirical phase functions and polarizations (dashed line) were generated using a uniform weighting of clay (illite, kaolinite, and montmorillonite) and non-clay (calcite, gypsum, and quartz) samples.

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CHAPTER 6

DETERMINING PARTICLE SHAPE DISTRIBUTIONS FROM FTIR

SPECTRAL FITTING

As discussed in Chapter 5, the use of T-Matrix theory for scattering calculations leads

to slightly better agreement with experimental measurements of scattering phase function and

polarization data for irregularly-shape mineral dust aerosol than that obtained using Mie

theory, which is limited to spherical shapes. However, for the range of axial ratios used in the

T-matrix simulations, AR ≤ 2.4, it was found that theoretical polarization profiles failed to

agree with experiment, both in peak position and overall magnitude, over most of the

scattering angles. Since the optical constants were known and the particle size distribution

was measured simultaneously with measurements of the light scattering, deviations of the

theoretical scattering simulations from experiment were most likely due to particle shape

effects.

In this chapter, a more rigorous treatment of quartz mineral dust, a major component

of atmospheric aerosol, will be presented. Particle shapes will again be approximated using

spheroids, though the range of aspect ratios used to define the shape distribution will be

significantly expanded to include more extreme particle shapes. In addition to the “Standard”

shape distribution considered in Chapter 5, two other particle shape distributions will be

considered here, one based on ex situ analysis of electron micrographs for our particular

quartz dust sample, and the other determined from spectral fits of the Si-O stretch resonance

absorption line for quartz dust.

Modeling

As described in Chapter 5, mineral dust particle shapes can be approximated by a

distribution of randomly oriented ellipsoids of specified axial ratios, AR. T-Matrix

calculations were again carried out using the code of Mishchenko & Travis [1998]. The size

distribution parameters and optical constants for quartz used in these calculations are given in

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Tables 2.1 and 2.2 respectively. For a given log normal size distribution, T-Matrix theory

calculations were carried out for a distribution of randomly oriented oblate or prolate

spheroids of fixed AR value. These calculations were then repeated for a range of both oblate

and prolate AR values. An assumed particle shape distribution was then used to generate

weighted average phase function and linear polarization curves for comparison to the

experimental data.

In addition to the “Standard” shape model (AR ≤ 2.4 prolate and oblate spheroids with

an average aspect ratio, AR = 1.7) that is based on electron microscope images of mineral

dust aerosol particles collected in the field [Mishchenko et al., 1997], a number of other

shape distributions are also used here. The first, the “SEM-Based” shape model, was

determined from the results of image analysis of the SEM images collected for our particular

quartz sample as seen in Figure 6.1. To determine the shape distribution from the SEMs, all

particles were first approximated as ellipses using the publicly available ImageJ software

[http://rsbweb.nih.gov/ij/ ]. The best-fit ellipses are shown as outlines in Figure 6.1. From

these ellipses, the AR is given by the ratio of the major to minor axis. Since the image is two-

dimensional, it is not possible to directly determine the AR value for a 3-D particle. A thin

disk-like particle sitting at an angle to the surface will project an image that could range from

circular to needle-like depending on the viewing angle. We have assumed that prolate and

oblate particles are equally represented in the shape distribution; thus we have simply

reflected the distribution of measured AR histogram about 1.0 (circles) to generate the full

shape distribution. The “SEM-Based” model covers a similar range of aspect ratios that is

seen in the “Standard”, but has a lower average aspect ratio, AR = 1.4. Both the “Standard”

and the “SEM-Based” shape models, shown in the left panel of Figure 6.2, make up what

will be referred to here as the moderate shape distributions.

In earlier IR spectroscopic modeling studies in the infrared by Hudson et al. [2008a],

it was found that in order to accurately predict IR resonance absorption peak position and

line profiles, more extreme particle shapes must be used. In studies of clay mineral dust

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particles within the fine aerosol mode (0.1-1.0 µm), it was found that disc-like particle shapes

approximated the IR resonance band better than that which was obtained using needle-like

particles or particle shapes represented by a continuous distribution of ellipsoids. This is

consistent with the plate-like structure of clay particles as reported by Nadeau [1985]. A

similar study of non-clay particles, including quartz, found that a continuous distribution of

ellipsoidal shapes gave the best agreement with IR spectral measurements [Hudson et al.,

2008b]. Those model results were all developed in the Rayleigh approximation for particle

diameters much smaller than the wavelength of incident light, an approximation which was

not strictly satisfied for the dust samples in the work of Hudson et al.

In a more recent study which used T-Matrix theory to model spectral line profiles for

quartz mineral dust, Kleiber et al. [2009] suggested that fitting the IR spectrum might allow a

determination of the particle shape distribution. The spectroscopic methods for determining

the particle shape distribution and the corresponding results are given here for completeness.

Simulated IR spectra were calculated using T-Matrix theory assuming spheroidal particle

shapes for a wide range of aspect ratios. These results served as “spectral basis functions”. A

least-squares fitting algorithm was used to find the best fit linear combination of basis

functions (AR shape parameters) to the observed IR line spectra. An “unconstrained” least

squares analysis was first performed to determine the best linear combination of shape

parameters to fit the data without applying any predetermined model for the shape

distribution. These unconstrained fits give useful insight into the range of AR parameters

needed to simulate the experimental line profiles. These results, however, typically showed

unphysical structure in the shape distributions. In order to superimpose a more physically

reasonable envelope onto the shape distributions they also carried out least squares analyses

where the shape distribution was constrained to fit different models, either a Gaussian shape

or a square Window function, each with adjustable AR center and width parameters. In each

case, comparable agreement between experiment and simulation was obtained between the

unconstrained and different constrained model solutions. This shows that the approach did

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not yield a unique shape distribution. Nevertheless, there are clear and significant

conclusions about the general characteristics of the particle shape distributions that could be

drawn from those analyses, including the range of AR values required to fit the spectra.

Figures 6.2 and 6.3 show the results from IR extinction studies for quartz dust in

comparison to simulations based on different particle shape models. In each case the left

panel shows the shape histogram and the right panel shows the corresponding spectral fit to

the Si-O stretch resonance IR absorption line. The spectral fits using the moderate shape

distribution models in Figure 6.2 are quite poor. In Figures 6.3 a-c, results are shown for the

different extreme shape model distributions, “Unconstrained”, “Gaussian”, and “Window”,

respectively. Clearly, fitting the IR spectral line profile for our particular quartz dust sample

requires extreme particle shape parameters with AR > 3. For the “Gaussian” model fit, the

mean AR value is AR = 2.4 (oblate) and the range extends to AR ~ 10 for oblate spheroids.

Visible scattering simulations were also generated using T-Matrix theory over the

same wide range of aspect ratios. The shape distributions determined from the fits to the IR

spectral data were then used as weighting functions for the visible scattering phase function

and polarization calculations. The use of the three different extreme particle shape models in

the visible scattering T-Matrix calculations results in phase function and polarization profiles

that are very similar to one another. In Figure 6.4 scattering results are presented for all three

extreme shape models at 550 nm. Similar results hold at the other wavelengths studied. For

simplicity, further analysis of the visible scattering will be limited to the “Window” shape

model as a representative extreme particle shape model.

Error Analysis

The comparison between experimental scattering data for quartz at different

scattering wavelengths and the T-Matrix based simulations for different model shape

distributions is shown in Figures 6.5 – 6.7. The uncertainty in the experimental data was

calculated as in Chapter 5 and represents day-to-day variability in the measured phase

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function and polarization profiles, as well as systematic uncertainties in the phase function as

a result of system calibration errors. The error bars in the theoretical T-Matrix simulations

are determined from two contributing factors. The first is due to the uncertainty in the small

diameter region of the size distribution. As discussed in Chapter 5, Mie calculations were run

for a range of size distributions, consistent with the APS data, in order to characterize the

corresponding variation in the resulting phase function and polarization profiles. The

magnitude of the error in the T-Matrix calculations was assumed to be of the same magnitude

as those in the Mie calculations. The estimate of this size distribution error was performed

using Mie theory due to the long computation times required for the T-Matrix calculations.

The second factor contributing to the errors in the T-Matrix simulations is due to

convergence limitations in the T-Matrix code and is discussed in detail below.

For a given particle shape parameter, the T-Matrix code carries out an integration

over the particle size distribution, requiring minimum and maximum endpoint values for the

assumed log normal size distribution. Since small particles do not scatter efficiently the

simulations are insensitive to choice of the minimum diameter. Determining the large

diameter endpoint is more problematic. The log normal size distribution has a long tail that

extends to infinite diameter. However, as a practical matter our aerosol generator and flow

system do not pass particles with diameters larger than ~ 3m. Indeed for these experiments

on quartz, the APS measured no particles (above background) in any size bin greater than

dAPS ~ 2.7 m. This represents an empirical maximum diameter endpoint for the integration.

The T-Matrix code used in these studies has convergence limitations for large and/or

highly eccentric particles. As a result, the calculations are limited by a maximum size

parameter for which the code will converge, XMax = πdMax/λ. The value of XMax is dependent

both on the optical constants and the eccentricity of the spheroids in the calculation. For

example, for oblate spheroid particles with AR = 1.5 and refractive index value n = 1.311, the

code will fully converge for size parameters up to ~ 160. In contrast, XMax decreases to ~17

when the aspect ratio is increased to AR = 10 [Mischenko & Travis, 1998]. For prolate

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spheroids, the maximal convergent size parameter, XMax tends to be slightly lower than that

for oblate spheroids with the same AR value.

In our T-Matrix calculations of the visible scattering for more extremely shaped

particles, there were cases where the code did not converge over the entire range of the

measured size and shape distributions. This introduces some convergence error into the

theoretical scattering results for the extreme shape distributions presented here. For both of

the moderate shape distributions (the “Standard” and “SEM-Based” models), the T-Matrix

simulations were converged for the entire measured size distribution up to the maximum

observed particle diameter (dAPS ≈ 2.7 µm). However, for the extreme shape models,

convergence up to this diameter was not obtained for some of the more highly eccentric

particle shapes. For example, for prolate shapes with AR = 6.0, convergence was only

obtained out to dMax ~2.0 µm. For prolate shape of AR = 10.0, the maximum diameter for

which the code would converge was dMax~1.1 µm.

In order to gauge the magnitude of the uncertainty due to this convergence limitation

on the maximum diameter, a series of simulations were run for each aspect ratio where the

cutoff diameter used in the T-Matrix calculations, dCutoff, was incrementally changed. As

dCutoff was increased in a series of steps toward the convergence limit, dMax, the calculated

scattering matrix elements were observed to converge. By comparing results for different

cutoff values we could estimate the rate of convergence and use that to place on upper limit

on the possible convergence error associated with the finite cutoff value. This error was

estimated as follows: For a fixed AR value, dMax was determined by stepping the integration

cutoff value dCutoff downward from dAPS in steps, Δd, of ~200 nm until code convergence was

obtained. The code was then run one more time with the cutoff diameter set one step

smaller, dMax - Δd. The difference in the calculated scattering matrix elements for these two

runs can be used to define an incremental error associated with each step d in changing the

integration cutoff diameter. An estimate for an upper limit on the resulting convergence

error can then be made by assuming this difference is constant for all of the succeeding steps

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between the maximal convergent diameter dMax and the maximum observed particle diameter

dAPS. The error, σi(AR), is then:

))()()( ddPdPdddAR MaxiMaxi

MaxAPSi

(6.1)

where Pi(d) is the calculated phase function or polarization value for an integration cutoff

diameter d, at scattering angle i. The dimensionless prefactor in Equation 6.1 essentially

gives the number of incremental steps between the maximum diameter for which

convergence is obtained and the maximum observed particle size.

The total error, Σi, for the phase function and polarization profiles, weighted by the

shape distribution, S(AR), is then obtained from:

AR

ii ARARS )()( (6.2)

There are a few aspects of this convergence error that should be stressed. First, these

errors are only present for the most extreme particle shapes (i.e. AR > 4.0). The final reported

scattering data is a weighted sum of the results over all aspect ratios (most of which have AR

< 4.0); therefore, the contribution to the total error in the data tends to be small. Second, dAPS

as reported above is the maximum particle diameter bin for which particle counts (above

background) were observed. The number density for that bin was very low. Though

scattering intensity does scale with projected surface area, the number of particles with

diameters greater than 1.0 µm only made up ~7% of the total particles measured by the APS,

and those with diameters greater than 2.0 µm only made up 0.05%. Therefore the total

contribution to the scattering due to the larger particles will be small. As such, the errors due

to the finite cutoff value tend to be much smaller than the errors due to uncertainties in the

size distribution itself. The reported error bars on the simulation results in Figures 6.5-6.7

include both contributions.

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Results for Quartz

The visible scattering phase functions and polarizations for quartz mineral dust at

470, 550, and 660 nm are given in Figures 6.5-6.7 respectively. Results are presented for the

“SEM-Based”, “Standard” (AR ≤ 2.4), and “IR-Based” shape models. As noted above, all

three shape models derived from the IR spectra give very similar results (Figure 6.4); for

simplicity here we use the “Window” shape model as the ”IR-Based” distribution. To show

more clearly the differences between the phase functions generated with each of the shape

models, the ratio of the experimental to theoretical phase function is also included in each

figure (Figures 6.5-6.7 (b)). The phase functions were first normalized in accordance with

Equation 2.13 then offset by factors of 10 for clarity. Because the experimental and simulated

phase functions are normalized, and since the simulations are based on known optical

constants, and measured particle size and shape distributions, the results shown in Figures

6.4-6.7 are absolute fits of theory to experiment with no adjustable parameters.

For all wavelengths investigated, the theoretical phase functions generated using T-

Matrix theory for the different shape distributions all appear similar to one another. However,

by looking at the ratio of experiment to theory, it is evident that using extreme particle shapes

in the calculations (“IR-Based” model) results in better agreement with experiment for mid-

range and large scattering angles than calculations which use the more moderate shape

distributions. In order to quantitatively judge the goodness of fit to the phase function data, χ2

fitting parameters (calculated using Equation 5.1) have been calculated for all models at all

three wavelengths and are given in Table 6.1. The results of the χ2 analysis shows that the

“IR-Based” model gives much better overall agreement with experiment for all wavelengths

investigated.

Particle backscattering is an important scattering property for radiative balance

calculations and remote sensing dust retrievals. The most significant deviation from

experiment for the results generated with the moderate shape distributions occurs for near

backscattering angles, >135o, where the theoretical scattering intensity can be as much as

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twice as large as measured values. Though the ”IR-Based” model comes closer to predicting

the magnitude of the scattering intensity for large angles, the moderate shape distributions do

give slightly better agreement with the overall asymmetry parameter for the phase functions

for 470 and 550 nm (see Table 6.2).

The scattered light polarization is much more sensitive to particle shape effects and

the theoretical polarization profiles given in Figures 6.5-6.7 (c) show much more variation

between the different shape models. The predicted polarizations for the “SEM-based” model

are again negative over the entire scattering angle range and look very similar to those

generated using Mie theory (Figure 5.3). This isn’t surprising as a large fraction of the

particle shapes for this model are very close to spheres. There is a slight improvement in

going to the “Standard” shape distribution, yielding slightly positive peaks near 120o, but

overall agreement with experiment is still poor. Only by including extreme particle shapes,

AR >> 3, does the theoretical polarization agree with experiment for all scattering angles.

Spectral differences in the profiles across the range of wavelengths studied here are

not large. However, it should be noted that the observed linear polarization for near right-

angle scattering, ~ 90○, increases from P(θ) = +17% to +28% as the scattering wavelength

is varied from 470 to 660 nm. This variation is very well modeled in the T-matrix simulation

for the ”IR-Based” particle shape model, which predicts a polarization increase from +18%

to +27% over this spectral range.

Discussion

As was discussed in Chapter 4, T-Matrix theory, and light scattering theories in

general, require a number of input parameters including the particle size distribution,

wavelength dependent optical constants, and particle shape distribution. In order to

accurately model the scattering, it is therefore important to use input parameters that are

representative of the physical and optical properties of the scattering medium. Errors in these

input distributions can compromise the conclusions from any theoretical scattering model.

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For this study, we have used known optical constants for quartz, measured the particle size

distribution simultaneously with the light scattering measurements, and determined the

particle shape distribution using ex situ analysis of quartz mineral dust at multiple

wavelengths. Therefore, all results presented here are absolute comparisons of experiment

with theory, with no adjustable parameters. The mode diameter of our particular quartz

sample is d < 3 m, corresponding to the size range important for long-range atmospheric

transport [Prospero, 1999].

T-Matrix theory has been used here because of its computational efficiency and

because it is readily adapted to the range of particle sizes important in our experiments.

Particle shapes were represented using a spheroidal approximation for a range of aspect

ratios. The spheroid shape approximation has been previously used in a number of reported

studies [Kahn et al., 1997; Nousiainen & Vermeulen, 2003;Veihelmann et al., 2004; Dubovik

et al., 2006], though there is still some question as to the validity of this approximation since

real mineral dust particles may have many sharp edges, points, and internal voids (see Figure

6.1). For example, some work has suggested that the neglect of sharp edges inherent in the

assumption that particles can be treated as smooth spheroids may lead to errors that could be

appreciable in some instances [Kalashnikova & Sokolik, 2002]. Others have argued that the

errors in the scattering phase function resulting from the spheroidal particle assumption may

not be large [Kahnert & Kylling, 2004].

A number of particle shape distributions have been explored in this work. Two shape

distributions were based on analysis of electron micrographs of aerosol particles. The first,

the “SEM-Based” model, was based on images of quartz particles used in this study, and

included aspect ratios in the range AR < 3.0 consisting primarily of nearly spherical particles.

T-Matrix results for this shape distribution were very close to those predicted by Mie theory,

which is valid only for spheres, and agreed poorly with experimental phase functions and

polarizations. The “Standard” shape model is based in image analysis of micrographs of a

range of aerosol samples collected in the field [Mishchenko et al., 1997] and is a commonly

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used model to represent mineral dust aerosols. Though scattering calculations using the

“Standard” shape model agree slightly better with experiment than the “SEM-based model”,

deviation from experiment is still large for the polarimetric data, both in term of peak

position and magnitude.

In Kleiber et al. [2009], particle shape distributions for quartz were also determined

by spectral fitting of the Si-O resonance absorption band.. The shape and peak position of the

IR spectral resonances show a strong dependence on particle shape [Hudson et al., 2008a,

2008b]. This analysis gives information on the range of aspect ratios necessary for accurately

modeling the light scattering. For quartz, aspect ratios AR >> 3.0 were required to get good

agreement with experiment. These ”IR-Based” particle shape distributions that include more

extremely shape particles give much better agreement with both the experimental phase

functions and the experimental polarization data for all wavelengths investigated in this

study, 470, 550, and 660 nm.

These results suggest that using electron micrograph images of a dust sample may not

give an accurate representation of the particle shape distribution and indicates the inherent

limitations in using a 2D analysis to extract information on 3D particles. Quartz dust, studied

here, is a major component of atmospheric aerosol [Sokolik & Toon, 1999]. If these results

can be generalized to real atmospheric dust, this study also suggests that it may be possible to

develop algorithms that use correlated IR extinction and visible polarimetry data from

satellite or ground based field instruments, together with T-Matrix based simulations, to more

accurately characterize atmospheric dust composition, size, and shape distributions

Chi-square analysis suggests that the T-matrix model based on the uniform spheroid

approximation can be used with confidence to model the optical properties of highly irregular

quartz dust aerosol particles in the accumulation mode size range provided the shape

distribution is reliably modeled. However, due to convergence limitations of the T-Matrix

code, visible scattering calculations may be limited to the accumulation mode size range for

particles with high aspect ratios. The IR spectral analysis for particle shapes, on the other

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hand, can be extended to much larger particle diameters because of the longer wavelengths in

the IR because the same dimensionless size parameter, X=2r/, corresponds to a much

larger particle diameter in the infrared. This method for determining particle shapes could

then be used with other light scattering theories such as the geometric optical theory to

determine visible scattering for large particles.

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Shape Model Wavelength (nm)

“SEM-based” “Standard” “Window”

470 2.29 0.72 0.31

550 2.50 0.68 0.26

660 2.91 1.29 0.13

Table 6.1 Reduced χ2 values for comparison of the experimental phase functions with simulations based on different particle shape models, the moderate “SEM-based” and “Standard” models, and the extreme “Window” model.

Mineral Dust Wavelength gExp gSEM gStandard gWindow

470 nm 0.69 ± 0.01 0.66 0.67 0.74

550 nm 0.69 ± 0.01 0.66 0.67 0.74 Quartz

660 nm 0.72 ± 0.01 0.67 0.68 0.73

Table 6.2 Asymmetry parameter values for experimental (gExp) and T-Matrix theory assuming the SEM-Based (gSEM), Standard (gStandard), and Window (gWindow) shape model phase functions for 470, 550, and 660 nm.

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Figure 6.1 SEM image of quartz particles with best-fit ellipses determined using the ImageJ software package.

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Figure 6.2 The left panel shows the two different moderate particle shape distributions used in the experiment: (a) “Standard” shape distribution using a uniform distribution of oblate and prolate spheroids with AR ≤ 2.4; (b) “SEM-based” shape distribution as determined from Figure 6.1 using ImageJ software package. The right panel shows the corresponding comparison of the T-Matrix simulation results (dashed lines) with experimental IR resonance extinction spectrum (solid with circles).

Figure 6.3 The left panel shows the different extreme particle shape distributions as determined from IR resonance spectrum of quartz in Kleiber et al. [2009]: (a) “Unconstrained” model; (b) “Gaussian” model; (c) “Window” model. The right panel shows the corresponding comparison of the T-Matrix simulation results (dashed lines) with experimental IR resonance extinction spectrum (solid with circles).

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Figure 6.4 Comparison of visible scattering phase function (a) and polarization profiles (b) at 550 nm for the three different extreme model shape distributions shown in the left panel of Figure 6.3.

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Figure 6.5 Comparison of experimental scattering phase functions (a), ratios of experimental to theoretical phase functions (b), and polarization profiles (c) at 470 nm with T-Matrix simulations based on different particle shape models: the moderate “SEM-based”, and “Standard” models, and the extreme “IR-Based” model. Phase functions in (a) for different shape models are offset by factors of ten for clarity.

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Figure 6.6 Comparison of experimental scattering phase functions (a), ratios of experimental to theoretical phase functions (b), and polarization profiles (c) at 550 nm with T-Matrix simulations based on different particle shape models: the moderate “SEM-based”, and “Standard” models, and the extreme “IR-Based” model. Phase functions in (a) for different shape models are offset by factors of ten for clarity.

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Figure 6.7 Comparison of experimental scattering phase functions (a), ratios of experimental to theoretical phase functions (b), and polarization profiles (c) at 660 nm with T-Matrix simulations based on different particle shape models: the moderate “SEM-based”, and “Standard” models, and the extreme “IR-Based” model. Phase functions in (a) for different shape models are offset by factors of ten for clarity.

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CHAPTER 7

AUTHENTIC MINERAL DUST LIGHT SCATTERING

Using the techniques developed for single-component mineral dust aerosol, the light

scattering properties for several authentic mineral dusts were examined. Authentic samples

studied include diatomaceous earth, Saharan sand, palagonite, and Iowa loess. Not only do

these samples provide a more rigorous test for the light scattering theories, but they were also

chosen due to their importance in the Earth’s atmosphere. Diatomaceous earth (diatomite or

kieselgur) consists primarily of fossilized diatoms; a type of algae encased in a cell wall

made of silica. Recent measurements of the Bodélé Depression in northern Chad, one of the

world’s largest source regions of mineral dust aerosol, revealed that aerosol emissions were

made up of predominantly diatomaceous earth particles [Todd et al., 2007]. Saharan sand

particles make up a large fraction of atmospheric aerosol and are capable of traveling

intercontinental distances [Prospero, 1999]. The palagonite sample used in this study (JSC

Mars-1) consists of volcanic ash collected near the Pu’u Nene volcano in Hawaii. This

sample is often used as a simulant for Martian soil [Clark et al., 1990; Clancy et al., 1995;

Allen et al., 1998; Johnson & Grundy, 2001, Lann et al., 2008]. Loess includes silt blown

soils that are generally porous and contain varying amounts of quartz, mica, and clay

minerals. Significant regions of loess production include China, central Europe, Argentina,

and central United States [Pye, 1984]. Below, light scattering phase function and polarization

profiles using Mie and T-Matrix theories assuming the “Standard” shape model will be

presented. In addition, as was done in Chapter 7, attempts to determine the shape distribution

for palagonite using the IR spectral line shape will also be discussed.

Particle Size Distributions

Prior to measuring the light scattering for the authentic mineral dust, a Scanning

Mobility Particle Sizer (SMPS, Model 3034, TSI Inc.) was added to the experimental

apparatus. This instrument is able to measure particle diameters in the range of ~ 20-500 nm.

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By combining the size distribution measured with this instrument with that measured with

the Aerodynamic Particle Sizer (APS), it was possible to determine the full particle size

distribution over the range of ~ 0.02-20 µm. The details of the distribution splicing procedure

are given in Chapter 2. Briefly, size distributions are measured simultaneously with both the

APS and SMPS. A log-normal fit of the SMPS data is first generated in order to extend the

range of the of the small diameter portion of the distribution to a region of overlap with the

APS data. The APS size distribution is next converted from a function of aerodynamic

diameters to a function of mobility diameters by optimizing the overlap of the APS data with

the log-normal fit of the SMPS data by varying the aerodynamic shape factor, χ (see equation

2.5). Once the shape factor is determined, both the APS and SMPS data can be converted to a

common reference diameter, the volume equivalent diameter, and then spliced together.

For the Mie theory calculations, the spliced full size distributions were directly used

in the calculations of the scattering properties. The input to the T-Matrix code of Mishchenko

et al. [1998], however, requires the input size distribution to be specified in terms of model

fit parameters for the size distribution (i.e. log-normal mode diameter and width parameters).

For each authentic mineral dust, these parameters were determined by using a single (Iowa

loess and palagonite) or bi-modal (diatomaceous earth and Saharan sand) log-normal fit to

the full size distribution. These log-normal fit parameters are given in Table 2.1. Measured

size distributions and the log-normal fits to the distributions are shown in Figure 7.1. The use

of a model distribution rather than the actual distribution introduces some additional

uncertainty into the T-Matrix calculations, though the magnitude of this uncertainty will still

be small and is discussed below.

T-Matrix Shape Distribution

For most of the T-Matrix results presented below, the “Standard” shape model, a

uniform distribution of ARs ranging between 1.0 ≤ AR ≤ 2.4 in steps of 0.2 for both prolate

and oblate spheroids, was used in the light scattering calculations. In addition, for the

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palagonite sample, attempts were made to infer the shape distribution using a fit to either the

polarization profile or the IR spectral line shape. Though the exact shape distribution could

not be determined using this method, information on the range of ARs required for good

agreement with the scattering data was obtained. This is discussed in further detail below.

Optical Constants

The single component mineral dusts investigated previously in this work were mostly

well characterized samples with known optical constants. For the authentic mineral dust

aerosol samples, the optical constants are more uncertain. In many cases refractive indices

were only reported for these samples in one or two publications that we are aware of or over

a limited wavelength range. In addition, due to regional variations in sample composition, the

optical constants used in the theoretical calculations of the light scattering may not accurately

represent the actual indices for these particular samples. Optical constants for diatomaceous

earth and Saharan sand were obtained from Egan [1985] using the optical constants for

Saharan sand samples #4 and #2a respectively. The Saharan sand sample #4 was collected

from the Lake Chad area which has a high concentration of diatomaceous earth. Sample 2a is

from the Maradi region of Niger and was chosen for our Saharan sand sample due to its

central location in the Sahara desert.

For Iowa loess, the real part of the refractive index was obtained from Cuthbert

[1940], who made measurements of the physical and optical properties of loess soils from

western Iowa. An estimate of the imaginary part of the refractive index was obtained from

Volten et al. [2001]. In both sources, the refractive index was only given at one unspecified

wavelength which we assume to be the sodium line (589 nm). In Cuthbert’s composition

analysis of loess soil from Iowa, he found a relatively high fraction of clay particles,

primarily kaolinite, montmorillonite, and illite. For this reason, calculations of the scattering

were also performed using the kaolinite optical constants at 550 nm from Egan and

Hilgeman [1979] which are in good agreement with those measured by Cuthbert.

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The palagonite sample used in this study (JSC Mars-1) was obtained from NASA and

consists of the <1 mm fraction of volcanic ash collected from near the Pu’u Nene volcano in

Hawaii. The real part of the refractive index was obtained from Johnson et al. [2001] who

used a linear fit to the palagonite data presented in Clark et al. [1990]. The latter study used

samples collected from Muana Kea, Hawaii. The imaginary part of the refractive index was

measured by Clancy et al. [1995] for the Muana Kea palagonite.

Error Analysis

As was done in Chapters 6 and 7, estimation of the uncertainty in the experimentally

measured light scattering was determined from the day-to-day variation in the measured

scattering. For each of the authentic dust samples, the light scattering was measured on three

separate days and averaged together. Additional systematic uncertainties due to fits of the

calibration function and the relative scaling between the parallel and perpendicular intensities

were also included in the experimental data.

For the theoretical data presented previously, the reported errors were due to

uncertainty in the small diameter region of the aerosol size distribution. To determine those

errors, multiple log-normal fits to the APS data were used to generate a range of possible

phase functions and polarization profiles. With the implementation of the SMPS, it was

possible to directly measure the small diameter particles. However, there is still some

uncertainty in the size distribution due to a day-to-day variation in the derived aerodynamic

shape factor, , which is determined during the splicing procedure. As χ varies, the

conversion between mobility and aerodynamic diameters to the common, volume equivalent,

diameters will change. The effect will be to shift the entire size distribution toward larger or

smaller diameters. In order to quantify this uncertainty, the standard deviation, σ, was

determined between the shape factors determined over multiple days for each authentic dust.

For each day of data collection, size distributions were generated using the shape factor

determined from optimizing the overlap between the APS and SMPS data, χOpt, along with

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shape distributions generated using a shape factor of χOpt + σ and a shape factor of χOpt- σ.

All three size distributions were then input into the Mie theory calculations and the standard

deviation between the scattering data for each was determined. The magnitude of the error

for the T-Matrix data was assumed to be the same as that for the Mie data. This uncertainty

was found to be much less than that arising from the uncertainty in the small diameter region

of the size distribution which, itself, is generally quite small (in some cases in the data

presented below, the error bars are contained within the width of the lines in the figures).

For the T-Matrix calculations there is an additional error due to using the model, log-

normal fits to the size distribution rather than the measured size distribution directly. These

errors were determined by generating scattering data using Mie theory for both the measured

size distribution and the log-normal fit to the distribution (see Figure 7.1). The difference

between the phase functions or polarization profiles calculated using either distribution gave

an estimate of the magnitude of the error. This uncertainty was assumed to be the same for

the T-Matrix calculations and was added in quadrature with the uncertainties due to

variations in the shape factor.

Authentic Mineral Dust Results

The light scattering results for diatomaceous earth are shown in Figure 7.2 for 550 nm

incident light. Results are presented using both Mie (dashed line) and T-Matrix theories

assuming the “Standard” shape model (dash-dot line). T-Matrix results for the phase

functions have again been offset by a factor of 10 for clarity. Reduced χ2 values, as defined in

Equation 5.1, for the phase function profiles are presented in Table 7.1. For both the T-

Matrix and the Mie theory phase functions, there is a significant over prediction in the

backwards scattering for angles greater than ~135o. This is most pronounced near 180o where

both theories predict a scattering intensity that is ~2 times higher than that which is observed

experimentally. This can be seen more clearly in Figure 7.2 (b) where the ratio of experiment

to theory has been plotted. However, the asymmetry parameters (Table 7.2) calculated using

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either the experimental or theoretical phase functions are almost identical, possibly due to

cancellations when integrating over all scattering angles (see Equation 4.4). The polarization

profile for diatomaceous earth is shown in Figure 7.2 (c). The experimental polarization

shows an exceptionally large peak near 90o which is not seen using either of the light

scattering theories. The differences between experiment and theory are very similar to that

which was seen for pure quartz mineral dust (Figure 5.3). This may not be surprising due to

the high SiO2 content of diatomaceous earth.

Results for the Iowa loess sample are given in Figures 7.3 and 7.4. Theoretical results

using the optical constants from Cuthbert [1940] are given in Figure 7.3 and those using the

optical constants for kaolinite [Egan & Hilgeman, 1979] are given in 7.4. For either set of

refractive indices, the Mie and T-Matrix calculations give very similar results for the phase

functions as can be seen clearly in Figures 7.3 and 7.4 (b), though Mie theory does give

better agreement with the measured asymmetry parameter. Using kaolinite optical constants

gives an overall better agreement with the phase functions and the polarization profiles than

those measured directly for Iowa loess. This is surprising since this sample is expected to

have contributions to the optical constants from other clay components, such as illite and

montmorillonite, as well. Possible reasons for this discrepancy will be discussed further

below.

Palagonite scattering data is shown in Figure 7.5. The T-Matrix prediction gives a

slightly better fit to the experimental phase function than the Mie, but both theory predict

scattering intensities higher than those measured for all scattering angles > 40o. As was seen

in previous samples, the T-Matrix results agree better with the experimental polarization than

the Mie, but the overall magnitude is still too low and peaks at slightly higher scattering

angles. It is possible that this is due to the limited shape distribution that was used in these

calculations. As will be shown, by going to more prolate particle shapes a better agreement

with the experimental polarization profiles can be obtained. More results for the palagonite

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sample, where different shape distributions that include particles of more extreme axial ratios

were used with the T-Matrix data, are presented below.

Phase functions and polarization profiles for Saharan sand are given in Figure 7.6.

Agreement with the experimental phase function is quite good using Mie theory and

exceptional using T-Matrix theory; theoretical predictions are within experimental error bars

for almost all scattering angles. In addition, theoretical predictions for the asymmetry

parameters are quite close to that calculated from the measured phase function. T-Matrix

theory is also able to accurately predict the polarization within estimate errors for all

scattering angles. These results are surprising since the T-Matrix data for the Saharan sand

sample agrees better with experiment than what was seen for any of the single component

mineral dust samples (see Chapter 5).

Shape-Fitting Palagonite Data

In addition to applying the “Standard” shape model to the T-Matrix calculations for

palagonite, an attempt was made to determine the shape distribution by fitting the visible

scattering data. As was seen in previous works [Mishchenko & Travis, 1994; Mishchenko et

al., 1996; Dubovik et al., 2006], the polarization profile is generally more sensitive to particle

shape than the phase function and can be used to extract particle shape information. For this

reason, a least squares fitting routine was used to fit only the polarization data using a series

of T-Matrix polarization profile basis functions calculated for spheroidal particles of varying

axial ratios. Once the shape distribution was determined from the fit, it was then used to

generate theoretical phase functions and IR spectra. This approach can be thought of as the

inverse to what was done for the quartz mineral dust in Chapter 6. There, fitting of IR

spectral line shapes determined the shape distributions which were then applied to the visible

scattering data.

Figure 7.7 (c) shows the shape distribution that was determined from fitting the

polarization profile for palagonite with 550 nm incident radiation. For these simulations, the

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shape distribution was restricted to follow a Gaussian distribution. A similar range of axial

ratios was also obtained by using a uniform (window) distribution. The resulting T-Matrix

fits to the phase functions and polarizations are shown in Figure 7.7 (a) and (b) respectively.

Not surprisingly, the agreement between the T-Matrix and experimental polarization data is

drastically improved using the shape distribution from the fitting routine. The agreement

between the phase functions is also significantly improved over that which was obtained

using the “Standard” shape model. In addition the asymmetry parameter for the T-Matrix

simulation now agrees with the experimentally determined parameter within the expected

error.

Theoretical IR spectra for palagonite were calculated using the refractive index values

from Roush et al., [1991]. A comparison between the measured (solid line) and theoretical

spectra using the shape distribution from the fit of the polarization (solid line with black

circle markers) can be seen in Figure 7.8 (a). The spectral peak of the T-Matrix simulation is

shifted to lower wavelengths by ~0.2 µm though the general peak shape is very similar. In

order to compare this approach to that which was used for quartz and to see if the shape

distribution determine above was unique, an attempt was made to determine the shape

distribution by fitting the IR spectra as well. The resulting distribution is given in Figure 7.8

(b). In order to accurately fit the palagonite IR spectra peak position, extremely (AR > 10)

oblate spheroids were required. Unfortunately, it wasn’t possible to generate visible

scattering simulations for these high axial ratios due to convergence limitations of the T-

Matrix code. A discussion of the large discrepancies between the shape distributions is given

below.

Discussion

Overall, both Mie and T-Matrix theories assuming the “Standard” shape model

performed surprisingly well at predicting the scattering phase functions and polarization

profiles. In some cases, such as for the Saharan sand and the and the Iowa loess samples, the

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agreement between experimental measurements and theory was better than that which was

seen for the single component mineral dusts, particularly the clays (see Chapter 5). This is

surprising in that the refractive index values for these authentic dusts are not available for all

wavelengths or are based on assumed mineralogy. At the time of this work, no measurements

have been made of the elemental or mineralogical composition for these specific samples.

The only improvement to the experimental apparatus that was made prior to the collection of

light scattering for the authentic samples was the implementation of the SMPS to measure the

small diameter region of the size distribution. This should result in some improvement in the

agreement between the theoretical and measured scattering. The magnitude of this

improvement is expected to be small however as the light scattering intensity generally scales

with the particle surface area (i.e. the larger diameter particle size fraction was already well

characterized using the APS).

One important point needs to be noted for the Saharan sand and the palagonite

samples. As was discussed in Chapter 2, as received, these samples were made up of particles

that were too large to be aerosolized using the atomizer. Attempts were made to aerosolize

the particles using a fluidized bed generator, but the high settling rate of these samples

prevented them from making it all the way through the aerosol flow path in the apparatus. In

order to use these samples, they were manually ground down using a mortar and pestle

followed by mechanical grinding using a Wig-L-Bug device. Scanning electron micrographs

were taken for these samples before and after grinding. As can be seen in Figure 7.10, this

did result in changes to the sample shape distributions for both the palagonite (7.10 (c) and

(d)) and the Saharan sand (7.10 (e) and (f))). The palagonite sample became more spherical

after processing whereas the large spherical particles in the Saharan sand appear to have

broken up into more irregularly shape particles. There may also be unforeseen changes in the

degree of internal/external mixing of the mineralogical components of these samples. For this

reason, readers are cautioned that the results presented above may not be truly representative

of actual Saharan sand or palagonite atmospheric aerosol.

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The fits to the diatomaceous earth sample were poor relative to the other authentic

dust samples, particularly for the polarization profiles. Since the optical constants were

available for this sample for 550 nm, the most likely reason for the poor agreement may be

particle shape effects. Scanning electron micrographs for diatomaceous earth are shown in

Figure 7.10 (a). There is a high degree of variability between the shapes of the particles for

this sample, with shape ranging from cylindrical tubes to thin plate-like structures. In

addition, many particles can be seen to have voids or “fringes” along the edges. It may

perhaps be unreasonable to approximate the diatomaceous earth samples with spheroids.

However, considering the particle shapes seen in the SEM images, it is perhaps surprising

how well the visible scattering data does agree with theoretical calculations using spheroids

within a moderate range of axial ratios (AR ≤ 2.4). It may be that orientation averaging of the

particle shapes cancel out some of the more extreme shape effects.

An attempt was made to determine the shape distribution for palagonite using a least

squares fitting routine for the polarization profile. This shape distribution was found to

consist primarily of oblate particles with an average axial ratio of ~2.7. Though using this

distribution resulted in better agreement with the the phase function data, when it was applied

to the IR spectral calculations a significant shift in the spectral peak was seen. Only by using

a much more oblate shape distribution (average AR ~ 12.6) was good agreement obtained

between the theoretical and experimental spectra. The shape distribution determined from an

analysis of the SEM images of this sample was much more spherical than either of the two

distributions, AR ~1.7. These discrepancies call into question the reliability of using

scattering or absorption spectra to determine particle shapes for complex mixtures such as

these. It is possible that spheroidal approximations to particle shape are not sufficient for

modeling highly irregular dust particles. There is also uncertainty in the applicability of using

the Johnson et al. [2001] and Clancy et al. [1995] refractive indices for this palagonite

sample since it was processed before the scattering data was collected. This may have

changed the degree of internal mixing of the sample and thus changed the optical constants.

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The analysis and results presented here only represent the first steps towards

determining the scattering properties for for the authentic dust samples. Additional

approaches to modeling the scattering for these complex samples are currently being

explored. One approach will use the measured mineralogy of the sample along with the

known optical constants for each mineralogical component to generate averaged optical

constants, assuming internal mixing. The shape distribution will then be determined using a

fit to the IR spectral data. An alternate approach would again involve determination of the

mineral composition of the sample though separate measurements. For this method, the

sample will be assumed to be composed of externally mixed particles. The shape

distributions for each component mineral will be assumed to be the same as that determined

in the single-component measurements, such as those obtained for quartz in Chapter 6. The

optical properties could then be determined for each component and then weighted based on

the empirical mineralogy. This would serve to eliminate any fitting parameters from the

scattering calculations. For now, the results presented above can still be used to predict the

general behavior of some of these authentic mineral dust aerosol samples provided the

limitations in some of the theoretical calculations are acknowledged.

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Mineral Dust χ2 (Mie) χ2 (T-Matrix)

Diatomaceous Earth 1.4 2.4

Iowa Loess (Cuthbert [1940])

5.7 4.8

Iowa Loess (Egan & Hilgeman [1979])

1.6 1.3

Palagonite* 5.0 1.9 (0.9)

Saharan Sand 1.4 1.2

* Values given in parentheses for palagonite are for theoretical simulations using shape distributions determined from fits to the polarization profile.

Table 7.1 Reduced χ2 values for comparison of the experimental phase functions with simulations using Mie theory and using T-Matrix theory assuming a “Standard” shape distribution.

Mineral Dust gExp gMie gTM*

Diatomaceous Earth 0.73 ± 0.01 0.73 ± 0.01 0.72

Iowa Loess (Cuthbert [1940])

0.69 ± 0.01 0.62 ± 0.01 0.62

Iowa Loess (Egan & Hilgeman [1979])

0.69 ± 0.01 0.65 ± 0.01 0.66

Palagonite** 0.72 ± 0.01 0.67 ± 0.01 0.69 (0.72)

Saharan Sand 0.70 ± 0.01 0.68 ± 0.01 0.67

*Errors in asymmetry parameters generated using T-Matrix theory, due to uncertainties in the input size distributions, are expected to be comparable to those generated using Mie theory. ** Values given in parentheses for palagonite are for theoretical simulations using shape distributions determined from fits to the polarization profile.

Table 7.2 Asymmetry parameter values for experimental (gExp), Mie theory (gMie), and T-Matrix theory (gTM) phase functions for 550 nm.

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Figure 7.1 Measured size distributions obtained by splicing Aerodynamic Particle Sizer and Scanning Mobility Particle Sizer measurements (dotted line) along with log-normal fits to the size distributions (solid line) for the authentic mineral dusts used in this work.

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Figure 7.2 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for diatomaceous earth measured at 550 nm. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 7.3 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for Iowa loess measured at 550 nm. Optical constants used for the theoretical calculations were obtained from Cuthbert [1940].Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 7.4 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for Iowa loess measured at 550 nm. Kaolinite optical constants used for the calculations and were obtained from Egan & Hilgeman [1979].Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 7.5 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for palagonite measured at 550 nm. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 7.6 Normalized phase functions (a), ratios of experimental to theoretical phase functions (b), and linear polarizations (c) for Saharan sand measured at 550 nm. Mie or T-Matrix results were spliced onto the experimental phase functions for scattering angles less than 17o, the experimental phase functions were linearly extrapolated past 172o prior to normalization (black circles). Phase functions comparing experimental data to T-Matrix results in (a) have been scaled by a factor of 10 for clarity.

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Figure 7.7 Normalized phase functions (a), linear polarizations (b), and best fit shape distribution (c) for palagonite measured at 550 nm. The shape distribution was determined by optimizing the fit to the polarization.

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Figure 7.8 IR spectral data (solid line) and T-Matrix simulations using shape distribution fits to the polarization (solid line with filled circles) and to the IR spectral data (dashed line) (a), and best fit shape distribution (b) for palagonite. The shape distribution in (b) was determined by optimizing the fit to the IR spectral data.

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Figure 7.9 Particle shape distributions for palagonite (a) and Saharan sand (b). Processed samples were ground using a mortal and pestle followed by mechanical grinding using a Wig-L-Bug.

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Figure 7.10 Scanning electron micrographs of the authentic mineral dust samples. SEMs are shown for diatomaceous earth (a), Iowa loess (b), palagonite (c), palagonite post-processing (d), Saharan sand (e), and Saharan sand post-processing (f).

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CHAPTER 8

FUTURE WORK

A thorough and comprehensive effort was made to provide a complete and accurate

description of light scattering from aerosol particles throughout this work. However, there are

still a number of questions that have arisen as well as ideas for future measurements that have

yet to be completed. As is often the case in any scientific discipline, the results from one

measurement can lead to inspiration for another entire set of experiments. It was only due to

time constraints that the scope of this work was limited to what has been presented.

In this chapter, a brief overview of some of the potential work that will further our

understanding of scattering from aerosol particles will be presented. Many of the following

ideas will be slight variations on measurements and calculations that were done in this work,

but are nonetheless important for they are all applicable to real world atmospheric process.

This is far from an exhaustive list, and instead serves as a starting point to build upon the

efforts detailed in previous chapters.

Larger Mineral Dust Aerosol Particles

Our measurements were limited to light scattering from mineral dust aerosol with

maximum particle diameters on the order of ~2-3 µm. This corresponds to surface area

weighted effective diameters of ~0.2-0.5 µm. We were limited to this size range by a number

of factors. The aerosol flow was generated from a suspension of mineral dust in water using a

commercial atomizer (TSI). The maximum particle diameter was therefore limited by the

size of the droplets generated by the atomizer which have a mean diameter of ~0.3 µm. Other

aerosolization techniques were attempted, but these were not able to provide a steady flow of

significantly larger particles. This may be due in part to the long aerosol flow path (~4 m)

through the diffusion driers and drying tube in the current experimental apparatus. The long

flow path provides an opportunity for the larger particles to gravitationally settle out of the

flow before reaching the scattering zone.

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Though the aerosol particles used in these measurements are within the accumulation

size mode, and are therefore representative of a large fraction of atmospheric aerosol,

particles as large as ~2 µm can still be suspended in the atmosphere for long periods of time

and are therefore still important [Hinds, 1999]. In addition, during strong wind events, such

as those that are common during dust storms, even larger particles can be lifted into the lower

atmosphere for shorter periods of time. Since the magnitude of the light scattered from an

aerosol particle is proportional to the surface area of the particle, these larger particles can

significantly affect the atmospheric radiative balance on a regional scale. Future

measurements will therefore require a method to generate steady flows of larger aerosol

particles. One possibility is to use a mechanical means to drop the aerosol particles directly

down into the scattering region, thus removing any possible settling out of the sample. This

would also have the advantage of eliminating the need to first suspend the mineral dust in

water. This is important because suspension in water can change the physical properties of

some samples, such as certain clays that will swell when they absorb water.

In many of our studies, there were convergence errors in the simulations resulting

from well-documented convergence limitations in the T-Matrix scattering code [Mishchenko

& Travis, 1998]. This was the case, even for the more limited range of particle sizes

investigated in this work, All T-Matrix calculations were performed using the extended

precision T-Matrix Code of Mishchenko et al.. As was discussed in Chapter 5, as the axial

ratio and/or index of refraction for a particle increases, the maximum size parameter for

which the code will converge decreases. In some cases this meant truncating the input size

distribution leading to some error in the simulation results. The resulting errors in the

scattering calculations were generally small for the aerosol samples used in this work, but the

errors will be more significant when the scattering measurements involve larger particles. For

that reason, it will also be necessary to implement an alternative approach to the light

scattering calculations. One possibility is to use a combination of scattering theories, as

described in previous works [Dubovik et al., 2006; Feng et al., 2009]. In this approach, T-

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Matrix theory is still used for calculations of the smaller particles, but for cases where the T-

Matrix code fails to converge, geometric optics calculations [Yang & Liou, 1996] would be

used instead, and the results would be spliced together.

Particle Shapes Used in T-Matrix Calculations

Our simulations of the light scattering from irregularly shape mineral dust aerosol

have been limited to approximations of the particle shape using an ensemble of randomly

oriented spheroids of varying axial ratio. This uniform spheroid approximation has been used

previously in a number of works [Mishchenko et al., 1997; Nousiainen & Vermeulen, 2003;

Veihelmann et al., 2004]. In many cases, the agreement with experiment is relatively good,

provided a wide enough range of axial ratios is used in the calculations. Others have found

that the neglect of sharp edges in the particle shapes can cause significant errors in some

cases [Kalashnikova & Sokolik, 2002]. The effect of sharp edges on scattering properties is a

topic that deserves more study.

In addition to spheroids, there are two other classes of particles shapes that are

commonly used in scattering calculations and for which an exact numerical solution for the

scattering is possible using T-Matrix theory; finite cylinders and Chebyshev particles. A

number of the samples used in this study exhibit a degree of surface roughness as can be seen

in the scanning electron micrographs in Chapters 6-7. This roughness is not duplicated in

models using spheroids or cylinders. However, the use of Chebyshev particles allows for the

simulation of periodic surface roughness of a specified degree in scattering calculations. As

developed by Mugnai & Wiscombe [1980], a Chebyshev particle is a surface of revolution

with the radial component defined as follows:

))](cos(1[0 nTrr (8.1)

Above, r and θ are the usual spherical coordinates and the revolution is in the φ

direction. The deformation parameter is given by ε and Tn = cos(n θ) is the nth order

Chebyshev polynomial. The radius of the unperturbed sphere is given by r0.Using this

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method to define particle shape will result in a surface with n ridges and n valleys evenly

distributed with respect to angle and of equal amplitude, the magnitude of which depends on

ε. All particles with n ≥ 2, exhibit some degree of concavity.

Early investigations into the scattering properties of such particles found that greater

degrees surface roughness lead to a decrease in the scattering asymmetry parameter [Mugnai

& Wiscombe, 1986], i.e. an increase in the relative amount of backscatter. The results of

Mugnai & Wiscombe [1986, 1988] also found that the effects of surface roughness become

more evident with increasing size parameter. However, another study found that similar

scattering results could be obtained using smooth ellipsoidal particle and found only marginal

changes in the polarization profile [Mishchenko & Travis, 1994]. It must be noted that both

of these studies focused on deformation parameters that were small (|ε| ≤ 0.2) and for lower

order Chebyshev polynomials. In a more recent study, Rother et al. [2006] developed a

method to generalize this approach. In this case, the radius of the unperturbed sphere, r0, in

Equation 8.1 can be replaced with a radius that depends on θ.

))](cos(1)[( nTRr (8.1)

The functional form of R(θ) can then be set to take the form of an ellipsoid. In

addition to implementing this more general particle shape, Rother et al. made calculations for

much higher order Chebyshev polynomials (n ≤ 45) than were done previously. Their work

confirmed earlier calculations which found increased backscattering for roughened particles,

but also found an increased dependence of the light scattering on surface roughness for

particles with a large imaginary part of the refractive index. However, these studies were

limited to single size parameters and the effects of averaging over the entire size distribution

are yet unknown.

Including surface roughness into the T-Matrix calculations increases the complexity

of the particle shape, requiring two parameters for the Chebyshev particles rather than a

single a single parameter for the spheroidal particle shape. In addition, limitations on the size

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parameter for which convergence is obtained in T-Matrix calculations become more

pronounced as ε or n is increased. Nonetheless, these model shapes may offer a more

representative model for authentic particle shapes.

Particle Coatings

Many of the samples examined in this work consisted of well characterized single

component mineral dusts. Authentic mineral dusts present in the atmosphere are far from this

simple. Dust particles often consist of agglomerations of different minerals. In addition, due

to long atmospheric residence times for particles within the accumulation size mode, there is

a high potential for chemical reactions with trace gas species, and for surface deposition of

various compounds from the atmosphere onto the dust particles. These reaction and coating

effects depend on particle surface area, exposure time, dust surface chemistry, and

environmental conditions [Clarke et al., 2004]. Dust particle coatings can include sulfates,

nitrates, organics, and fine soot particles [Buseck & Posfai, 1999; Gao & Anderson, 2001;

Lee et al., 2002; Conant et. al., 2003]. As these coatings form on the surface of the dust

particle, they can also change the physical shape of the aerosol [Khalizov et al., 2009].

Though atmospheric dust is normally found to by hydrophobic [Li-Jones et al., 1998;

Kaaden et al., 2009], recent work analyzing single mineral dusts particles using TEM, has

found a large fraction of particles collected within regions with a high degree of pollution

become coated with a layer of nitrate which enhances the hygroscopic properties of the

mineral dust [Li & Shao, 2009]. This results in greater uptake of water on the surface of the

particle leading to larger, more spherical particles.

It is sometimes possible to treat a coated particle as a series of concentric spheres in

scattering calculations [Aden & Kerker, 1951; Toon & Ackerman, 1981]. Theoretical

calculations by Bauer et al. [2007] found that scattering properties of shell-core models are

not significantly affected unless the shell thickness was at least 20% of the core radius. These

effects could be neglected in many radiative forcing calculations since sulfate and nitrate

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coating layers are not expected to be this thick on atmospheric aerosol. In the study by Bauer,

moderate indices of refractions were used for the core material, ~1.5-1.6. Another study

measured the extinction efficiency of a core material with a high degree of absorption coated

by a non-absorbing shell [Abo Riziq et al., 2008]. They found discrepancies between their

measured results and calculations using the concentric sphere shell-core model that increased

with increasing shell thickness. It should also be noted that spherical cores were used in both

of these studies. Measurements using an irregularly shaped core may provide better insight

into coating effects on the radiative properties of actual atmospheric aerosol. With a spherical

core, the particle shape will not change with an increase in core thickness as might be

expected with an irregular core. A possible set of measurements would involve applying a

nitrate coating to a mineral dust particle through vapor deposition or via heterogeneous

processing then monitoring changes in the scattering properties as the relative humidity is

increased. As more water is coated on the surface of the particle, the overall shape would

become more spherical; changing the scattering properties.

Expanding the Light Scattering Database

Finally, one of the most straightforward ways of expanding upon the previous work

described here and elsewhere is to add to the database of experimental measurements and

theoretical calculations of light scattering for a wider range of aerosol particles and

wavelengths. Soil composition and thus aerosol source composition can vary dramatically

from one region to another [Petrov, 1976; Sokolik & Toon, 1999; Reid et al., 2003; Lafon et

al., 2006]. As was shown in this work, particle shape, size, and optical properties all play an

important role in the scattering properties of mineral dust aerosol. For this reason, the aerosol

community would benefit from more extensive light scattering studies of authentic mineral

dusts such as volcanic ash and various sands and soils from regions throughout the world.

The range of wavelengths that were investigated in this work were chosen to

correspond to wavelength bands used by remote sensing satellites such as MODIS and MISR

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[Diner et al., 1998; Ichoku et al., 2004; Bruegge et al., 2007]. However, the wavelengths

employed in our work only cover the visible bands, which are used for over-land

measurements. In over-ocean measurements, wavelengths out to ~2 µm are also used for

aerosol atmospheric loading measurements by MODIS. In addition, calculations of the

radiative forcing due to mineral dust aerosol must take into account scattering and absorption

at many wavelengths, not just those within the visible spectrum. For some mineral dusts,

such as quartz or calcite, the wavelength dependence on the scattering as shown in Chapter 5

was seen to be small. For others, such as hematite or goethite, the refractive index and

therefore the scattering is highly dependent on the wavelength. It is therefore important to

expand these scattering measurements out to the IR region and into the near UV.

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APPENDIX

Included below are relevant sections of code used in the data analysis and modeling

throughout this work. All code has been written using the Matlab computing language.

Comments, text following the “%” symbol, have been included throughout the codes to

enhance readability. Example codes for determining the calibration curve (see Chapter 2),

fitting APS size distributions and splicing APS data to SMPS size distributions (see Chapter

2), and modeling the experimental apparatus (see Chapter 3) are giving in Appendices A.1-

A.3 along with relevant sub-functions.

A.1 Determining the System Calibration and Angle Mapping

Functions

The following code is used to calculate the system calibration function using

experimentally measured light scattering for polystyrene latex spheres (PSL). The code uses

a look-up table to select appropriate theoretical phase function and polarization profiles,

calculated using Mie theory, to compare to. After using a rolling average to reduce signal

noise, the relative amplitude between the scattering measured with perpendicular polarized

incident light, I , and that measured with perpendicular polarized incident light, //I , is

corrected by forcing the polarization to go to zero near 0o and 180o (see A.1.1 Polarization

Correction). Next, the angle mapping function is determined by optimizing the agreement

between the experimental and Mie calculated polarization profiles (see A.1.2 Angle

Mapping), which are independent of the system calibration. Finally, the calibration function

is generated by taking the ratio of the Mie theory phase function to the experimental phase

function. In order to remove noise and features in the calibration function that result due

resolution limits in the experimental apparatus, a final optimization routine (see A.1.3

Calibration Curve Fitting) is used to generate a fit to the calibration curve. The input for the

code is two nx1 arrays (where n is the number of data points) representing the integrated

signal intensity for perpendicular and parallel polarizations ( I and //I ). For a detailed

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discussion of the methods used in calculating calibration and angle mapping functions, see

Chapter 2.

%Set parameters for PSL used in calibration. These will be used to find the %correct Mie scattering data in a look-up table. Diameter = 771; %Particle diameter of PSL (in nm) LaserWavelength = 550; %Incident laser wavelength (in nm) %Fitting options FitPolar = 1; %Set to 1 to set the polarization to zero at the ends FitAngleMapping = 1; %Set to 1 to optimize angle mapping function FitCalibrationCurve = 1; %Set to 1 to fit the calibration curve BaselineCorrect = 1; %Set to 1 to do baseline correction on data BaselinePoints = 20; %Number of points for baseline correction NumPoints_Smooth = 5; %# of points used in data smoothing (odd integer) %Angle range (in radians) over which to optimize angle mapping AngleLowAngleMap = 0.3; AngleHighAngleMap = 3.04; %Angle range (in radians) over which to optimize calibration curve fitting AngleLowCaliFit = 0.12; AngleHighCaliFit = 3.11; %Load all necessary Mie data for PSL %Use a look-up table to find PSL data for the wavelength (LaserWavelength) %and diameter (Diameter) set above. MieDataLocation = ['C:\Program Files\MATLAB\R2006b\work\CalibrationCode'... '\MieData\PSL\' num2str(LaserWavelength) 'nmLaser\' ... num2str(Diameter) 'nmDiameter\']; MieData = load([MieDataLocation 'MieData_PSL_' num2str(Diameter) '.txt']); MieRadians = MieData(:,1); MiePara = MieData(:,2); MiePerp = MieData(:,3); AngleRadians = load([MieDataLocation 'AngleRadians.txt']); %Initialize all global variables %Globals for "PolarizationFit.m" global gPerpAvg gParaAvg gIndexLow gIndexHigh %Globals for "AngleFit.m" global gMieRadiansInterp gPolarAvg gMiePolar gPixels... gAngleLowAngleMap gAngleHighAngleMap %Globals for "CalibrationCurveFit.m" global gCalibrationRaw gAngleRadiansMod gIndexLowCaliFit gIndexHighCaliFit %Select and load parallel experimental data if ~exist('PathOfFile','var'), PathOfFile = ''; end FileDialogHeader = 'Choose a parallel experimental data file'; [Path,Name1,Ext,PathOfFile] = DialogBox(PathOfFile,FileDialogHeader,0); ParaRaw = load([Path Name1 Ext]); %Select and load perpendicular experimental data FileDialogHeader = 'Choose a perpendicular experimental data file'; [Path,Name2,Ext,PathOfFile] = DialogBox(PathOfFile,FileDialogHeader,0); PerpRaw = load([Path Name2 Ext]); Pixels = 1:1:length(PerpRaw); %Array defining the pixels on detector %Generate experimental phase function and polarization profiles

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PhaseRaw = PerpRaw + ParaRaw; PolarRaw = (PerpRaw - ParaRaw)./(PerpRaw + ParaRaw); %Interpolate Mie data so it contains the same number of data points as %PerpRaw and ParaRaw. MieRadiansInterp will still be linearly increasing AngleInterpStep = (MieRadians(end)-MieRadians(1))/(length(AngleRadians)-1); MieRadiansInterp = MieRadians(1):AngleInterpStep:MieRadians(end); MieRadiansInterp = MieRadiansInterp'; MieParaInterp = interp1(MieRadians,MiePara,MieRadiansInterp); MiePerpInterp = interp1(MieRadians,MiePerp,MieRadiansInterp); MiePhaseInterp = MiePerpInterp + MieParaInterp; MiePolarInterp = (MiePerpInterp - MieParaInterp)./ ... (MiePerpInterp + MieParaInterp); %Signal Averaging. Do a rolling average on the PerpRaw and ParaRaw data to %reduce noise. The number of points used in this rolling average is %selected be the user (must be an odd number of points) AdjPoints = (NumPoints_Smooth - 1)/2; %Adjacent points %Preallocate memory for PerpAvg and ParaAvg ParaAvg = zeros(length(ParaRaw),1); PerpAvg = zeros(length(PerpRaw),1); for x = 1:length(ParaRaw); if x > AdjPoints && x <= length(ParaRaw) - AdjPoints; ParaAvg(x) = mean(ParaRaw(x-AdjPoints: x+AdjPoints)); PerpAvg(x) = mean(PerpRaw(x-AdjPoints: x+AdjPoints)); else ParaAvg(x) = ParaRaw(x); PerpAvg(x) = PerpRaw(x); end end %Do a baseline correction if BaselineCorrect == 1; %Baseline correction uses the "tail" end of the data since the light %scattering intensity is lower than in the forward direction, thus %cutting down on the chance of light "bleeding" over into this area ParaAvg = ParaAvg - mean(ParaAvg(end-BaselinePoints:end)); PerpAvg = PerpAvg - mean(PerpAvg(end-BaselinePoints:end)); end %Generate experimental phase function and polarization profiles using the %"smoothed" data. PhaseAvg = PerpAvg + ParaAvg; PolarAvg = (PerpAvg - ParaAvg)./(PerpAvg + ParaAvg); %The polarization data should go to zero at both 0 degrees and 180 degrees. %Since the data is very noisy near these angles I will check near 5 degrees %and 175 degrees. The code will check if the polarization is close to zero %here within some tolerance. If it is not, it will multiply PerpAvg by some %constant then recalculate PolarAvg. It will continue doing this until the %data is within tolerance. if FitPolar == 1; %Find the index of both 5 and 175 degrees in the AngleRadians data AngleLow_PolarFit = 5*(pi/180); %Convert to radians AngleHigh_PolarFit = 175*(pi/180); %Convert to radians AngleStep = abs(AngleRadians(20) - AngleRadians(19)); IndexLow = find(AngleRadians >= (AngleLow_PolarFit - 2*AngleStep) & ... AngleRadians <= (AngleLow_PolarFit + 2*AngleStep),1,'first'); IndexHigh = find(AngleRadians >= (AngleHigh_PolarFit - 2*AngleStep)... & AngleRadians <= (AngleHigh_PolarFit + 2*AngleStep),1,'last');

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%Set fitting options and initial values Options = optimset('MaxFunEvals',1000,'MaxIter',300,'Tolfun', ... 0.00000001,'Tolx',0.00000001); val = optimget(Options,'MaxFunEvals'); %Define global variables for "PolarizationFit.m" gIndexLow = IndexLow; gIndexHigh = IndexHigh; gPerpAvg = PerpAvg; gParaAvg = ParaAvg; PerpAmpInitial = 1; %Guess for amplitude of PerpAvg coefficient InitialValues = PerpAmpInitial; %Optimize amplitude of PerpAvg. Uses the Matlab minimization routine %"fminsearch" to minimize the sub-function "PolarizationFit.m" Results = fminsearch('PolarizationFit',InitialValues,Options); PerpAmp = Results(1); %Scale perpendicular data PerpAvg = PerpAmp*PerpAvg; PolarAvgMod = (PerpAvg - ParaAvg)./(PerpAvg + ParaAvg); else PolarAvgMod = PolarAvg; end %Generate the angle mapping function. This will map pixel position to %scattering angle for the experimental data if FitAngleMapping == 1; %Set fitting options and initial values Options = optimset('MaxFunEvals',1500,'MaxIter',400,'Tolfun', ... 0.00000001,'Tolx',0.00000001); val = optimget(Options,'MaxFunEvals'); %Define global variables gMieRadiansInterp = MieRadiansInterp; gMiePolar = MiePolarInterp; gPolarAvg = PolarAvgMod; gPixels = Pixels; gAngleLowAngleMap = AngleLowAngleMap; gAngleHighAngleMap = AngleHighAngleMap; %Set initial values. These are based on previous fitting results. A_Initial = 1.7; B_Initial = 1.57; C_Initial = 0.0013; D_Initial = 4.946; InitialValues = [A_Initial B_Initial C_Initial D_Initial]; %Find angle calibration by minimizing the sub-function "AngleFit" using %the Matlab minimization function "fminsearch". Results = fminsearch('AngleFit',InitialValues,Options); A = Results(1); B = Results(2); C = Results(3); D = Results(4); AngleRadiansMod = B + A*sin(C*Pixels + D); %Interpolate the Mie data onto the new axis MiePolarInterpMod = interp1(MieRadiansInterp,MiePolarInterp,... AngleRadiansMod); MiePhaseInterpMod = interp1(MieRadiansInterp,MiePhaseInterp,... AngleRadiansMod); MiePolarInterpMod = MiePolarInterpMod'; MiePhaseInterpMod = MiePhaseInterpMod';

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%Find the indices of the user selected ranges in AngleRadiansMod AngleStep = abs(AngleRadiansMod(1020) - AngleRadiansMod(1019)); IndexLow = find(AngleRadiansMod >= ... (gAngleLowAngleMap-2*AngleStep) & AngleRadiansMod <= ... (gAngleLowAngleMap+2*AngleStep),1,'first'); IndexMid = find(AngleRadiansMod >= (1.5-2*AngleStep) & ... AngleRadiansMod <= (1.5+2*AngleStep),1,'first'); IndexHigh = find(AngleRadiansMod >= ... (gAngleHighAngleMap-2*AngleStep) & AngleRadiansMod <= ... (gAngleHighAngleMap+2*AngleStep),1,'last'); else AngleRadiansMod = AngleRadians'; PhaseAvgMod = PhaseAvg; %#ok<NASGU> end %Scale the measured phase function to match up with Mie data. Do this %by scaling the first peak in the Mie data and the observed data to the %same amplitude. PhaseMax = max(PhaseAvg(IndexLow:IndexMid)); MiePhaseMax = max(MiePhaseInterpMod(IndexLow:IndexMid)); PhaseAmp = MiePhaseMax/PhaseMax; PhaseAvgMod = PhaseAmp*PhaseAvg; %Generate a calibration curve defined as the ratio of the phase function %calculated using Mie theory to the experimental phase function CalibrationRaw = MiePhaseInterpMod./PhaseAvgMod; CalibrationRaw = CalibrationRaw'; %Fit the calibration curve if FitCalibrationCurve == 1; %Fitting will be done between AngleLowCaliFit and AngleHighCaliFit %radians. First find the index of these points in AngleRadiansMod. AngleStep = abs(AngleRadiansMod(20) - AngleRadiansMod(19)); IndexLowCaliFit = find(AngleRadiansMod >= ... (AngleLowCaliFit - 2*AngleStep) & AngleRadiansMod <= ... (AngleLowCaliFit + 2*AngleStep),1,'Last'); IndexHighCaliFit = find(AngleRadiansMod >= ... (AngleHighCaliFit - 2*AngleStep) & AngleRadiansMod <= ... (AngleHighCaliFit + 2*AngleStep),1,'First'); %Set fitting options and initial values Options = optimset('MaxFunEvals',3000,'MaxIter',1000,'Tolfun',... 0.00000001,'Tolx',0.00000001,'LargeScale','off'); val = optimget(Options,'MaxFunEvals'); %Define global variables gCalibrationRaw = CalibrationRaw; gAngleRadiansMod = AngleRadiansMod; gIndexLowCaliFit = IndexLowCaliFit; gIndexHighCaliFit = IndexHighCaliFit; %Since we are fitting a function with 10 variables, we may need to %manually adjust the fitting ranges for the following variables. The %variables will be used to fit the following function: %CalibrationFit = Base + A*exp(-B*(AngleRadiansMod - C)) + ... % D*exp(-E*(AngleRadiansMod-F).^2) + G*exp(H*(AngleRadiansMod-I)); %Set initial values A_Initial = 5; B_Initial = 10; C_Initial = 0.3; D_Initial = 4.6; E_Initial = 4; F_Initial = 1.5; G_Initial = 1; H_Initial = 70;

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I_Initial = 3; Base_Initial = 1; %Set lower bounds on variables A_Lower = 0.25; B_Lower = 1; C_Lower = -2.0; D_Lower = 0.0; E_Lower = 3; F_Lower = 1.3; G_Lower = 0.00; H_Lower = 0; I_Lower = 2.0; BaseLower = 0; %Set upper bounds on variables A_Upper = 250; B_Upper = 50; C_Upper = 0.6; D_Upper = 0.5; E_Upper = 10; F_Upper = 1.5; G_Upper = 10; H_Upper = 75; I_Upper = 5.5; BaseUpper = 1.1; InitialValues = [A_Initial B_Initial C_Initial D_Initial E_Initial ... F_Initial G_Initial H_Initial I_Initial Base_Initial]; LowerBounds = [A_Lower B_Lower C_Lower D_Lower E_Lower ... F_Lower G_Lower H_Lower I_Lower BaseLower]; UpperBounds = [A_Upper B_Upper C_Upper D_Upper E_Upper ... F_Upper G_Upper H_Upper I_Upper BaseUpper]; %Fit the claibration curve with by minimizing the sub-function %"CalibrationCurveFit" using the predefined Matlab minimization %function "fmincon" Results = fmincon('CalibrationCurveFit',InitialValues ... ,[],[],[],[],LowerBounds,UpperBounds,[],Options); A = Results(1); B = Results(2); C = Results(3); D = Results(4); E = Results(5); F = Results(6); G = Results(7); H = Results(8); I = Results(9); Base = Results(10); CalibrationFit = Base + A*exp(-B*(AngleRadiansMod - C)) + ... D*exp(-E*(AngleRadiansMod - F).^2) + G*exp(H*(AngleRadiansMod - I)); Calibration = CalibrationFit; else Calibration = CalibrationRaw; end

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A.1.1 Polarization Correction (Sub-function)

This code is used with Calibration_PSL code to find the optimum value of the

averaged perpendicular data so that the magnitude of the polarization data goes to zero near

both zero and 180 degrees. This corrects for variability in laser output power and particle

number density.

function Chi = PolarizationFit(Input) global gPerpAvg gParaAvg gIndexLow gIndexHigh PerpAmp = Input(1); %Coeficient to multiply the PerpAvg data PolarAvgMod = (PerpAmp*gPerpAvg - gParaAvg)./(PerpAmp*gPerpAvg + gParaAvg); Baseline = zeros(size(PolarAvgMod)); %Array of zeros %The fitting will only occur for angles near 5 and 175 degrees. Subtracting %off the array of zeros isn't necessary, but it serves to remind the user %that we're trying to fit the data to zero in these regions Range = 10; ResidualLow = PolarAvgMod(gIndexLow-Range:gIndexLow+Range)... - Baseline(gIndexLow-Range:gIndexLow+Range); ResidualHigh = PolarAvgMod(gIndexHigh-Range:gIndexHigh+Range)... - Baseline(gIndexHigh-Range:gIndexHigh+Range); Chi = sum((ResidualLow).^2) + sum((ResidualHigh).^2);

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A.1.2 Angle Mapping (Sub-function)

This code will be used with Calibration_PSL code to optimize the angle mapping

function. The experimental polarization, which is originally a function of pixel position, is

mapped to a function of scattering angle. The experimental data is then iteratively compared

to the Mie polarization profile to optimize the angle mapping.

function Chi = AngleFit(Input) global gMieRadiansInterp gPolarAvg gMiePolar gPixels gAngleLowAngleMap ... gAngleHighAngleMap A = Input(1); B = Input(2); C = Input(3); D = Input(4); %Generate the angle mapping function AngleRadiansMod = B + A*sin(C*gPixels + D); %Find the indices of the user selected angle ranges in AngleRadiansMod AngleStep = abs(max(AngleRadiansMod) - min(AngleRadiansMod))/ ... length(AngleRadiansMod); IndexLow = find(AngleRadiansMod >= ... (gAngleLowAngleMap - 2*AngleStep) & AngleRadiansMod <= ... (gAngleLowAngleMap + 2*AngleStep),1,'first'); IndexHigh = find(AngleRadiansMod >= ... (gAngleHighAngleMap - 2*AngleStep) & AngleRadiansMod <= ... (gAngleHighAngleMap + 2*AngleStep),1,'last'); %Interpolate the Mie data onto the new axis MiePolarMod = interp1(gMieRadiansInterp,gMiePolar,AngleRadiansMod); MiePolarMod = MiePolarMod'; %The residual will be defined between the lower and upper angle ranges %chosen by the user. Residual = MiePolarMod - gPolarAvg; N = length(Residual(IndexLow:IndexHigh)); Chi = sum((Residual(IndexLow:IndexHigh)).^2)/N;

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A.1.3 Calibration Curve Fitting (Sub-function)

This code is used with Calibration_PSL code to fit the calibration curve with a

predefined function. This serves to reduce noise and remove some features in the calibration

curve that are a result of limits on the angular resolution of the apparatus. This is a 10

variable fit, so results determined below will not necessarily be unique.

function Chi = CalibrationCurveFit(Input) global gCalibrationRaw gAngleRadiansMod gIndexLowCaliFit gIndexHighCaliFit A = Input(1); B = Input(2); C = Input(3); D = Input(4); E = Input(5); F = Input(6); G = Input(7); H = Input(8); I = Input(9); Base = Input(10); CalibrationFit = Base + A*exp(-B*(gAngleRadiansMod - C)) + ... D*exp(-E*(gAngleRadiansMod - F).^2) + G*exp(H*(gAngleRadiansMod - I)); Residual = CalibrationFit(gIndexLowCaliFit:gIndexHighCaliFit)... - gCalibrationRaw(gIndexLowCaliFit:gIndexHighCaliFit); Chi = (sum((Residual).^2))/length(Residual); %Chi-Squared

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A.2 Splicing APS and SMPS Data

Currently, there is no commercially available sizing instrument that is able to measure

the full particle size distribution of mineral dust aerosols. However, using the procedures

outline in Khlystov et al. [2004], it is possible to combine size distributions measured using a

Scanning Mobility Particle Sizer (SMPS) with the distribution measured using an

Aerodynamic Particle Sizer (APS). Since each instrument relies on different measurement

principles (measuring either mobility or aerodynamic diameter) , it is first necessary to

convert the size distributions to a common x-axis before splicing the size distributions

together. If the shape factor is known for the aerosol being measured, this is trivial (see

equations 2.4 and 2.5). If not, an optimization of the overlap of the two size distributions over

a specified region can be used, as is done in the following code. The APS instrument used in

our experimental apparatus is able to measure particle diameters in the range ~0.5-20 µm.

The SMPS is able to measure distributions in the range 0.02-0.5 µm. Since there is no region

of overlap between the two instruments, it is necessary to first generate a log-normal fit (see

A.2.1 Fitting SMPS Data with a Log-normal Distribution) of the SMPS data (which can later

be extrapolated to larger diameters) to perform the overlap with the APS data (see A.2.2

Optimizing Overlap Between APS and SMPS Data). The inputs required for the code are two

size distribution data sets, one for the APS data and one for the SMPS. More information on

this procedure for combining sizing information can be found in Chapter 2.

ParticleDensity = 3.8; %Density of particle (g/cm^3) SMPS_Fit_Low = 20; %Lower mobility diameter (nm) for fit of SMPS data APS_LowerIndex = 8; %starting data point in APS to use in fits BiModal_SMPS_Fit = 1; %set to 1 to do a bi-modal fit of SMPS data SpliceInterpPoints = 88; %# of points for interpolation of spliced data %Initialize global variables global gMobDiameter_SMPS_Extended gLogNormalFit_SMPS gAeroDiameter_APS... gdNdlogDP_APS gParticleDensity gAPS_LowerIndex X_Data Y_Data ... gBiModal_SMPS_Fit if ~exist('PathOfFile','var'), PathOfFile = ''; end %Create default path %Choose an APS file FileName_Compare_APS = {}; NameArray_Compare_APS = {};

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FileDialogHeader = 'Choose a APS size distribution file'; [Path,Name,Ext,PathOfFile] = DialogBox(PathOfFile,FileDialogHeader,0); FileNames_Compare_APS(1) = {[Path Name Ext]}; NameArray_Compare_APS(1) = {Name}; %Choose a SMPS data file FileName_Compare_SMPS = {}; NameArray_Compare_SMPS = {}; FileDialogHeader = 'Choose a SMPS size distribution file'; [Path,Name,Ext,PathOfFile] = DialogBox(PathOfFile,FileDialogHeader,0); FileNames_Compare_SMPS(1) = {[Path Name Ext]}; NameArray_Compare_SMPS(1) = {Name}; %Data files should be mx2 arrays. The fist column should contain the %diameters (aerodynamic for APS, mobility for SMPS). The second column %should contain the size distribution (dN/dlog(Dp)) %Load data SMPS_Data = load(char(FileNames_Compare_SMPS(1))); APS_Data = load(char(FileNames_Compare_APS(1))); AeroDiameter_APS = APS_Data(:,1); dNdlogDP_APS = APS_Data(:,2); MobDiameter_SMPS = SMPS_Data(:,1); dNdlogDP_SMPS = SMPS_Data(:,2); %First, generate a lognormal fit of the SMPS data, this will be used to %overlap the APS and SMPS data since there are no diameters that overlap %for the instruments used in the experimental apparatus. Options = optimset('MaxFunEvals',1500,'MaxIter',300,'Tolfun',0.00000001,... 'Tolx',0.00000001); val = optimget(Options,'MaxFunEvals'); %Set initial values for the lognormal parameters SigmaInitial = 1.5; AmplitudeInitial = 0.8*max(dNdlogDP_SMPS); MeanInitial = 300; %Set lower bounds on values SigmaLower = 0.2; AmplitudeLower = 0.1*max(dNdlogDP_SMPS); MeanLower = 150; %Set upper bounds on values SigmaUpper = 3.0; AmplitudeUpper = 1.2*max(dNdlogDP_SMPS); MeanUpper = 450; InitialValues = [SigmaInitial AmplitudeInitial MeanInitial]; LowerBounds = [SigmaLower AmplitudeLower MeanLower]; UpperBounds = [SigmaUpper AmplitudeUpper MeanUpper]; %Fit the SMPS data with two log-normals if desired if BiModal_SMPS_Fit == 1; %set initial values for the second log-normal SigmaInitial_2 = 1.5; AmplitudeInitial_2 = 0.8*max(dNdlogDP_SMPS); MeanInitial_2 = 30; %set lower bounds for second log-normal SigmaLower_2 = 0.2; AmplitudeLower_2 = 0.1*max(dNdlogDP_SMPS); MeanLower_2 = 10; %set upper bounds for second lognormal SigmaUpper_2 = 2.5;

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AmplitudeUpper_2 = 1.2*max(dNdlogDP_SMPS); MeanUpper_2 = 75; InitialValues = [InitialValues SigmaInitial_2 AmplitudeInitial_2 ... MeanInitial_2]; LowerBounds = [LowerBounds SigmaLower_2 AmplitudeLower_2 MeanLower_2]; UpperBounds = [UpperBounds SigmaUpper_2 AmplitudeUpper_2 MeanUpper_2]; end %Find index of lower fitting point fot the SMPS data Tolerance = 2; SMPS_Fit_Low_Index = find(MobDiameter_SMPS >= (SMPS_Fit_Low-2*Tolerance)... & MobDiameter_SMPS <= (SMPS_Fit_Low+2*Tolerance),1,'first'); %Set variable names used for the optimization X_Data = MobDiameter_SMPS(SMPS_Fit_Low_Index:end); Y_Data = dNdlogDP_SMPS(SMPS_Fit_Low_Index:end); gBiModal_SMPS_Fit = BiModal_SMPS_Fit; %Use an optimization to fit the SMPS data with a log-normal distribution Results = fmincon('Fit_SMPS_Data',InitialValues ... ,[],[],[],[],LowerBounds,UpperBounds,[],Options); %Allocate variables for the resulting lognormal fit Sigma_SMPS = Results(1); Amplitude_SMPS = Results(2); Mean_SMPS = Results(3); if BiModal_SMPS_Fit == 1; Sigma_SMPS_2 = Results(4); Amplitude_SMPS_2 = Results(5); Mean_SMPS_2 = Results(6); end %Generate an extended range for mobility diameters to be used to generate %fits over the entire measured range of APS and SMPS instruments. Step = 2000/500; MobDiameter_SMPS_Extended = 0:Step:2000; %0 to 20 um %Expand the lognormal fit to the SMPS data over the extended range of %diameters defined above. DataFit_SMPS = Amplitude_SMPS*exp((-(log(MobDiameter_SMPS_Extended/... Mean_SMPS)).^2)/(2*log(Sigma_SMPS)^2)); if BiModal_SMPS_Fit == 1; DataFit_SMPS = DataFit_SMPS + Amplitude_SMPS_2*... exp((-(log(MobDiameter_SMPS_Extended./Mean_SMPS_2)).^2)/... (2*log(Sigma_SMPS_2)^2)); end %The shapefactor can be determined by optimizing the overlap of the APS %data with the lognormal fit to the SMPS data. This is done by converting %the APS data from aerodynamic diameter to mobility diameter. The %conversion depends on the shapefactor. By varying the shapefactor in %the optimization the APS data will shift back and forth along the x-axis %until the overlap is optimized. %Set global variables gMobDiameter_SMPS_Extended = MobDiameter_SMPS_Extended; gLogNormalFit_SMPS = DataFit_SMPS; gAeroDiameter_APS = AeroDiameter_APS; gdNdlogDP_APS = dNdlogDP_APS; gParticleDensity = ParticleDensity; gAPS_LowerIndex = APS_LowerIndex;

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%Set the range for ShapeFactor in the optimization ShapeFactorInitial = 1.0; %Initial value ShapeFactorLow = 0.7; %Lower bound ShapeFactorHigh = 2.0; %Upper bound Results = fmincon('OptimizeShapeFactor',ShapeFactorInitial ... ,[],[],[],[],ShapeFactorLow,ShapeFactorHigh,[],Options); ShapeFactor = Results(1); MobDiameter_APS = AeroDiameter_APS*ShapeFactor*... ((ShapeFactor/ParticleDensity)^0.5); %Splice together the raw SMPS data with the shifted APS data. We don't %want to use the first 7 points of the APS data since those bins have lower %efficiency for our instrument. In addition, the APS and SMPS data don't %line up exactly so there might be a "kink" in the data when splicing. The %first thing we need to do is find the point where the SMPS and APS data %intersect by using a linear fit of both data sets in the region of %overlap. We can then use SMPS data before this point and APS data after %this point. %Do a linear fit of the APS data from APS_LowerIndex to APS_LowerIndex+3 Coefficients_APS = polyfit(MobDiameter_APS... (APS_LowerIndex:APS_LowerIndex+3)... ,dNdlogDP_APS(APS_LowerIndex:APS_LowerIndex+3),1); %Do a linear fit of the last 3 points of the SMPS data Coefficients_SMPS = polyfit(MobDiameter_SMPS(end-4:end), ... dNdlogDP_SMPS(end-4:end),1); %Find the x intersect of the above two lines X_Intersect = (Coefficients_SMPS(2)-Coefficients_APS(2))/... (Coefficients_APS(1)-Coefficients_SMPS(1)); %Find data point on SMPS before this point SpliceIndex_SMPS = find(MobDiameter_SMPS <= X_Intersect,1,'last'); %Find the data point on the APS after the intersect point SpliceIndex_APS = find(MobDiameter_APS >= X_Intersect,1,'first'); %Splice togehter dNdlogDp data and mobility diameter data FullDistribution = [dNdlogDP_SMPS(1:SpliceIndex_SMPS); ... dNdlogDP_APS(SpliceIndex_APS:end)]; FullMobDiameter = [MobDiameter_SMPS(1:SpliceIndex_SMPS); ... MobDiameter_APS(SpliceIndex_APS:end)]; %Convert the mobility diameter to volume equivalent diameter Full_VE_Diameter = FullMobDiameter./ShapeFactor; %Now interpolate APS data to an axis that is evenly spaced in base 10 %log-space. Each decade on the x-axis should have BinsPerDecade points. %The 40.5 is an arbitrary number chosen to fit an appropriate range. Bins = 1:1:SpliceInterpPoints; BinsPerDecade = 32; Full_VE_Diameter_Interp = 10.^((40.5+Bins)/BinsPerDecade); FullDistribution_Interp = interp1(Full_VE_Diameter,FullDistribution,... Full_VE_Diameter_Interp);

171

A.2.1 Fitting SMPS Data with a Log-normal Distribution (Sub-

function)

This sub-function is used with the APS_SMPS_Splice code. It is used in an

optimization to find a lognormal fit (single or bi-modal) to the size distribution measured by

the SMPS.

function Chi = Fit_SMPS_Data(Input) global X_Data Y_Data gBiModal_SMPS_Fit %Set variables for lognormal fit Sigma = Input(1); Amplitude = Input(2); Mean = Input(3); if gBiModal_SMPS_Fit == 1; Sigma_2 = Input(4); Amplitude_2 = Input(5); Mean_2 = Input(6); end if gBiModal_SMPS_Fit == 0; %Do a lognormal fit of the data DataFit = Amplitude*exp((-(log(X_Data/Mean)).^2)/(2*log(Sigma)^2)); elseif gBiModal_SMPS_Fit == 1; %Use two lognormals to fit the data DataFit = Amplitude*exp((-(log(X_Data/Mean)).^2)/(2*log(Sigma)^2))+ ... Amplitude_2*exp((-(log(X_Data/Mean_2)).^2)/(2*log(Sigma_2)^2)); end Residual = DataFit - Y_Data; Chi = sum((Residual).^2)/length(Residual);

172

A.2.2 Optimizing Overlap Between APS and SMPS Data (Sub-

function)

function Chi = OptimizeShapeFactor(Input) global gMobDiameter_SMPS_Extended gLogNormalFit_SMPS gAeroDiameter_APS... gdNdlogDP_APS gParticleDensity gAPS_LowerIndex ShapeFactor = Input(1); %Convert from aerodynamic diameter to mobility diameter MobDiameter_APS = gAeroDiameter_APS*ShapeFactor*... ((ShapeFactor/gParticleDensity)^0.5); %The first 7 data points in the APS data have reduced efficiency so we'll %want to through them out in the optimization. For now, determine the value %of the 8th point to be used later. MobDiameter_APS_LowerLimit = MobDiameter_APS(gAPS_LowerIndex); %Interpolate the APS data onto the same axis as the SMPS data dNdlogDP_APS_Interp = interp1(MobDiameter_APS,gdNdlogDP_APS,... gMobDiameter_SMPS_Extended); %Now that we've done the interpolation, there will be points in the APS %data that register as NaN since we interpolated out past the range that %APS data extended. Now we need to cut that data out. Tolerance = 2; Low_Index = find(gMobDiameter_SMPS_Extended >= ... (MobDiameter_APS_LowerLimit - 2*Tolerance) & ... gMobDiameter_SMPS_Extended <= ... (MobDiameter_APS_LowerLimit + 2*Tolerance),1,'last'); %The first 8 bins of the APS data have reduced efficiency from the other %bins so we'll cut them out for the optimization Residual = dNdlogDP_APS_Interp(Low_Index:end) - ... gLogNormalFit_SMPS(Low_Index:end); Chi = sum((Residual).^2)/length(Residual);

173

A.3 Modeling the Experimental Apparatus

The following code simulates the experimental apparatus by tracking a series of

scattered rays originating near one focus, f1, of a partial ellipse. Other optical elements,

including an aperture located at f2 and a detector, are approximated as lines with the rays will

collide. Briefly, the code generates a random distribution of particles near f1. Each of these

particles uniformly scatters rays throughout scattering angles 0o-180o. Each ray is

individually tracked until it reaches the detector, collides with the aperture, or goes outside of

the mirror boundary. For each successful detector “hit”, the initial scattering angle, the

particle position, and the position of the detector collision are logged for later analysis. A

full discussion of this model along with simulation results are presented in Chapter 3. Sub-

functions necessary for this code to function are included in Appendices A.3.1-A.3.3.

%Mirror Properties MirrorEnable = 1; %Set to 1 to check if rays travel off the mirror AngleMin = -0.1903*pi; %Angle where elliptical mirror starts, (-0.1903*pi) AngleMax = 0.1903*pi; %Angle where elliptical mirror ends, (0.1903*pi) SemiMajor = 30; %Semimajor axis (30 cm) SemiMinor = 17.5; %Semiminor axis, SemiMinor < SemiMajor, (17.5 cm) NumReflect = 10; %Number of reflections permitted before loop is canceled %Detector Properties DetectorEnable = 1; %Set to 1 to enable the detector NumPixels = 2184; %Number of pixels in detector,(2184 pixels) PixelWidth = 6.8*10^-4; %Width of 1 pixel (6.8 um = 6.8*10^-4 cm) DetectorWidth = 2*NumPixels*PixelWidth; %Width of the detector (cm) DetectorOffset_x = -2.5; %Offset of detector from F2 in x direction DetectorOffset_y = -0.0; %Offset of detector from F2 in y direction TiltDetector = 0; %Set to 1 to rotate the detector DetectorAngle = pi/2 - 0.4; %Clockwise rotation of the detector (radians) %Aperture properties ApertureEnable = 1; %Set to 1 to enable the aperture ApertureWidth = 10; %Width of the boundary of the aperture (3 cm) OpeningWidth = 0.05; %Size of opening of the aperture (cm) ApertureOffset_x = 0; %Offset of aperture from F2 in x direction ApertureOffset_y = 0.0; %Offset of aperture from F2 in y direction %Extended area scattering properties NumRays = 180; %Number of randomly emitted rays (50) EmitterOffset_x = 0.0; %Offset of scattering area from F1 in x direction EmitterOffset_y = 0.0; %Offset of scattering area from F1 in y direction EnableExtendedArea = 1; %Set to 1 to enable extended scattering areas NumParticles = 100; %Number of particles distributed within scattering area DistributionShape = 2; %Set to 1 for a rectangle, 2 for a circle DistributionLength = 0.15; %Length of particle distribution along y (cm) DistributionWidth = 0.1; %Length of particle distribution along x (cm)

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DistributionRadius = 0.47; %Radius of distribution from center point (cm) TiltExtendedArea = 0; %Set to 1 to tilt the extended area ExtendedAreaAngle = 0.1; %Angle of rotation of scattering area (radians) %Initialize global variables global StopAction Step X Ray Slope DetectionEvents DetectionEventsAll ... EmissionAngleLog EmissionAngleLogAll DetectorCollisionLog ... DetectorCollisionLogAll ParticleEmissionLogAll x4 y4 ParticleIndex %Determine location of foci. F1 is located at (c,0), F2 is located at %(-c,0) (in Cartesian coordinates). c is the center of the ellipse c = sqrt(SemiMajor^2 - SemiMinor^2); F1(1) = c; F1(2) = 0; F2(1) = -c; F2(2) = 0; %Find the location of the emitter relative to F1 clear EmitterPosition EmitterPosition(1,1) = F1(1) + EmitterOffset_x; EmitterPosition(1,2) = F1(2) + EmitterOffset_y; %Define the mirror boundary MirrorAngleRange = AngleMin:0.01:AngleMax; xMirror = SemiMajor*cos(MirrorAngleRange); yMirror = SemiMinor*sin(MirrorAngleRange); MirrorEdge_x_1 = SemiMajor*cos(AngleMin); MirrorEdge_x_2 = SemiMajor*cos(AngleMax); %Define the detector boundary, located near F2 DetectorPosition_x = F2(1) + DetectorOffset_x; %Center of detector location Step = DetectorWidth/10; yRangeDetector = (-(DetectorWidth/2)+DetectorOffset_y):Step:... ((DetectorWidth/2)+DetectorOffset_y); if TiltDetector == 1; %Now the detector will have some tilt so we can just define it as some %line with a slope but with definite limits on the x and y range DetectorSlope = tan(DetectorAngle); xMaxDetector = (DetectorWidth*cos(DetectorAngle))/2 + DetectorPosition_x; xMinDetector = -(DetectorWidth*cos(DetectorAngle))/2 + DetectorPosition_x; Step = (DetectorWidth*cos(DetectorAngle))/10; xRangeDetector = xMinDetector:Step:xMaxDetector; b_Detector = -DetectorSlope*DetectorPosition_x + DetectorOffset_y; yRangeDetector = DetectorSlope*xRangeDetector + b_Detector; else DetectorSlope = 0; end %Define aperture boundary PinHolePosition_x = F2(1) + ApertureOffset_x; %Center of aperture Step = (ApertureWidth-OpeningWidth)/10; yRangePinHoleLower = (-(ApertureWidth/2)+ApertureOffset_y):Step:... ((-OpeningWidth/2)+ApertureOffset_y); yRangePinHoleUpper = ((OpeningWidth/2)+ApertureOffset_y):Step:... ((ApertureWidth/2)+ApertureOffset_y); yRangePinHole = [yRangePinHoleLower yRangePinHoleUpper]; %Define location of scattering particles, the first particle will be %located at the scattering center as defined by emitter offset. The %remaining particles will be randomly distributed around this point. if EnableExtendedArea == 1; for z = 2:NumParticles; %Particle 1 has been assigned to the center if DistributionShape == 1; %Rectangle distribution xPosition = EmitterPosition(1,1) + ... (rand(1)-0.5)*DistributionWidth; yPosition = EmitterPosition(1,2) + ...

175

(rand(1)-0.5)*DistributionLength; elseif DistributionShape == 2; %Circular distribution Radius = rand(1)*DistributionRadius; Angle = (rand(1)-0.5)*2*pi; xPosition = EmitterPosition(1,1) + Radius*cos(Angle); yPosition = EmitterPosition(1,2) + Radius*sin(Angle); end EmitterPosition(z,1) = xPosition; EmitterPosition(z,2) = yPosition; end %Rotate the distribution if desired if TiltExtendedArea == 1; %Define the rotation matrix RotationMatrix = [cos(ExtendedAreaAngle) -sin(ExtendedAreaAngle);... sin(ExtendedAreaAngle) cos(ExtendedAreaAngle)]; %Center the distribution at x=0, y=0 We'll shift this back after %rotating all of the points EmitterPosition(:,1) = EmitterPosition(:,1) - EmitterPosition(1,1); EmitterPosition(:,2) = EmitterPosition(:,2) - EmitterPosition(1,2); %Transpose the EmitterPosition for easier calculations EmitterPosition = EmitterPosition'; for z = 1:NumParticles; EmitterPosition(:,z) = RotationMatrix*EmitterPosition(:,z); end EmitterPosition = EmitterPosition'; %Shift the distribution back near the focal point EmitterPosition(:,1) = EmitterPosition(:,1) + F1(1) + EmitterOffset_x; EmitterPosition(:,2) = EmitterPosition(:,2) + F1(2) + EmitterOffset_y; end else NumParticles = 1; %If extended area is not enables, only loop once end Theta = -pi:0.01:pi; %Angle used for parametric equations, measured from %the x-axis at center of the ellipse x = SemiMajor*cos(Theta); y = SemiMinor*sin(Theta); %Preallocate memory to log all the scattering events, collisions, %scattering areas, etc. The number of points in the following arrays will %most likely be longer than is required to store all the data as some of %the rays will miss the mirror or hit the aperture. Therefore we must %later clear out extra elements in the array at the end. %Use this value to keep track of the number of collisions with the %detector. Only includes rays emitted from the center point. DetectionEvents = 0; DetectionEventsAll = 0;%Same as above, but all points in scattering area. MemoryArraySize = NumParticles*NumRays; %Maximum number of events %Array of all emission angles that lead to collisions with the %detector. Only includes rays emitted from the center point. EmissionAngleLog = zeros(NumRays,1); %Same as EmissionAngleLog, but for all points in scattering area EmissionAngleLogAll = zeros(MemoryArraySize,1); %Array of locations (in y direction)of the collisions with the detector, %Only includes rays emitted from center point. DetectorCollisionLog = zeros(NumRays,1);

176

%Same as DetectorCollisionLog, but for all points in scattering area. DetectorCollisionLogAll = zeros(MemoryArraySize,1); %Keeps track of which particle emitted the ray that struck the detector %so we can later index EmitterPosition to trace out the paths. ParticleEmissionLogAll = zeros(MemoryArraySize,1);% %Create an array of evenly spaced angles that covers the entire scattering %angle range for each particle. EmittedAngle_Step = pi/(NumRays-1); EmittedAngle_Array = -pi/2:EmittedAngle_Step:pi/2; %Loop through particles and scattered rays for ParticleIndex = 1:NumParticles; for z = 1:NumRays; EmittedAngle = EmittedAngle_Array(z); Slope = tan(EmittedAngle); %Construct the line the ray will travel from the scattering area x0 = EmitterPosition(ParticleIndex,1); y0 = EmitterPosition(ParticleIndex,2); %Set StopAction = 0. This will tell the program that the ray has not %intersected with the detector, has not gone off the mirror, and has %not intersected with the aperture StopAction = 0; %Determine where ray will intersect with the ellipse Note, there will %be two solutions, we must pick the correct one a1 = Slope*(Slope*x0 - y0); a2 = (SemiMinor^2) + 2*Slope*x0*y0 - (Slope^2)*(x0^2) - (y0^2); a3 = SemiMinor*sqrt((a2/(SemiMajor^2)) + (Slope^2)); a4 = (SemiMajor^2)/(SemiMinor^2 + (Slope^2)*(SemiMajor^2)); %Choose which point is the correct solution based on the angle that %the ray was emitted. if EmittedAngle > -pi/2 && EmittedAngle < pi/2 x1 = a4*(a1 + a3); else x1 = a4*(a1 - a3); end y1 = Slope*(x1 - x0) + y0; Step = (x1 - x0)/20; X = x0:Step:x1; Ray = Slope*X - Slope*x0 + y0; %Check for intersection with aperture boundary if ApertureEnable == 1; ApertureCollisionCheck(x0,y0,PinHolePosition_x,... yRangePinHoleUpper,yRangePinHoleLower); end %Check if ray intersects with the detector. if DetectorEnable == 1 && StopAction == 0; DetectorCollisionCheck(x0,y0,TiltDetector,DetectorSlope,... yRangeDetector,DetectorPosition_x,DetectorOffset_y,... EmittedAngle); end %Check if ray goes off the defined mirror boundary. if MirrorEnable == 1 && StopAction == 0; MirrorCollisionCheck(x1,y1,xMirror); end

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%Redefine variables x2 = x1; y2 = y1; x1 = x0; y1 = y0; %Reflect the ray around until a collision or NumReflect times Reflect = 1; while StopAction == 0; %Now (x1,y1) are the points of the previous emission, (x2,y2) are %the points of the current intersection, (x3,y3) is a point on the %normal line at (x2,y2), (x4,y4) is the next intersection. The %angle of reflection at the mirror can be found geometrically %Find the normal line at x2,y2. First find the angle on the ellipse %corresponding to the intersection. Note that this is the polar %angle. It is related to the angle Theta by the relationship %PolarAngle = atan((b/a)*tan(Theta)) PolarAngle = atan2(y2,x2); if x2 >= 0; ThetaPrime = atan((SemiMajor/SemiMinor)*tan(PolarAngle)); else ThetaPrime = pi + atan((SemiMajor/SemiMinor)*tan(PolarAngle)); end %Find the tangent at some point on the ellipse. TangentSlope = (-SemiMinor/SemiMajor)*cot(ThetaPrime); NormSlope = -1/TangentSlope; x3 = 3*x2; XX = (x2-0.35*SemiMajor):0.05:(x2+0.35*SemiMajor); y3 = NormSlope*(x3 - x2) + y2; %A point on the normal line NormLine = NormSlope*(XX - x2) + y2; TangentLine = TangentSlope*(XX - x2) + y2; %Determine the reflected angle ReflectAngle1 = atan2((x3 - x2),(y3 - y2)); ReflectAngle = -EmittedAngle - 2*ReflectAngle1; %The ReflectAngle will not be changed by the addition of factors of %2*pi. In order to keep with the same angle conventions as above, I %will add or subtract factors of 2*pi until reflect angle is %withing the range (-pi,pi) if ReflectAngle < -pi; while ReflectAngle < -pi; ReflectAngle = ReflectAngle + 2*pi; end end if ReflectAngle > pi; while ReflectAngle > pi; ReflectAngle = ReflectAngle - 2*pi; end end Slope = tan(ReflectAngle); a1 = Slope*(Slope*x2 - y2); a2 = (SemiMinor^2) + 2*Slope*x2*y2 - (Slope^2)*(x2^2) - (y2^2); a3 = SemiMinor*sqrt((a2/(SemiMajor^2)) + (Slope^2)); a4 = (SemiMajor^2)/(SemiMinor^2 + (Slope^2)*(SemiMajor^2)); %Choose which point is the correct solution based on the angle %that the ray was emitted. if ReflectAngle > -pi/2 && ReflectAngle < pi/2 x4 = a4*(a1 + a3);

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else x4 = a4*(a1 - a3); end y4 = Slope*(x4 - x2) + y2; Step = (x4 - x2)/20; X = x2:Step:x4; Ray = Slope*(X - x2) + y2; %Check for intersection with aperture boundary if ApertureEnable == 1; ApertureCollisionCheck(x2,y2,PinHolePosition_x,... yRangePinHoleUpper,yRangePinHoleLower); end %Check if ray intersects with the detector. If it does, stop and %go on to the next scattered ray if DetectorEnable == 1 && StopAction == 0; DetectorCollisionCheck(x2,y2,TiltDetector,DetectorSlope ... ,yRangeDetector,DetectorPosition_x,DetectorOffset_y,... EmittedAngle); end %Check if ray goes off the defined mirror boundary. if MirrorEnable == 1 && StopAction == 0; MirrorCollisionCheck(x4,y1,xMirror); end %Update all variables for next reflection Reflect = Reflect + 1; %Stop loop if the number of allowed reflections has been reached if Reflect == NumReflect; StopAction = 1; end x1 = x2; y1 = y2; x2 = x4; y2 = y4; EmittedAngle = ReflectAngle; end end end

179

A.3.1 Determining Aperture Collisions (Sub-function)

function ApertureCollisionCheck(x0,y0,PinHolePosition_x,... yRangePinHoleUpper,yRangePinHoleLower) %Initialize global variables global StopAction Step X Ray Slope x4 y4 %Determine where the ray will intersect the line defined by %the aperture. Intercept_y = Slope*(PinHolePosition_x - x0) + y0; %Determine if this is within the range of the aperture's boundaries if Intercept_y >= min(yRangePinHoleUpper) && ... Intercept_y <= max(yRangePinHoleUpper) %Check if the ray is traveling towards or away from the aperture if PinHolePosition_x <= max(X) ... && PinHolePosition_x >= min(X) StopAction = 1; %Tell the program to go to next ray %Redefine ray so it stops at the aperture Step = (PinHolePosition_x - x0)/20; X = x0:Step:PinHolePosition_x; Ray = Slope*X - Slope*x0 + y0; x4 = PinHolePosition_x; y4 = Intercept_y; end elseif Intercept_y >= min(yRangePinHoleLower) && ... Intercept_y <= max(yRangePinHoleLower) %Check if the ray is traveling towards or away from the aperture if PinHolePosition_x <= max(X) ... && PinHolePosition_x >= min(X) StopAction = 1; %tell the program to go to next ray %Redefine ray so it stops at the aperture Step = (PinHolePosition_x - x0)/20; X = x0:Step:PinHolePosition_x; Ray = Slope*X - Slope*x0 + y0; x4 = PinHolePosition_x; y4 = Intercept_y; end end

180

A.3.2 Determining Detector Collisions (Sub-function)

function DetectorCollisionCheck(x0,y0,TiltDetector,DetectorSlope ... ,yRangeDetector,DetectorPosition_x,DetectorOffset_y,EmittedAngle) %Initialize global variables global StopAction Step X Ray Slope DetectionEvents DetectionEventsAll ... EmissionAngleLog EmissionAngleLogAll DetectorCollisionLog ... DetectorCollisionLogAll ParticleEmissionLogAll x4 y4 ParticleIndex %Determine where the ray will intersect the detector Intercept_y = Slope*(DetectorPosition_x - x0) + y0; if TiltDetector == 1; b_Detector = -DetectorSlope*DetectorPosition_x + DetectorOffset_y; b_Ray = -Slope*x0 + y0; Intercept_x = (b_Detector-b_Ray)/(Slope-DetectorSlope); Intercept_y = Slope*Intercept_x + b_Ray ; %ImpactPoint = [Intercept_x,Intercept_y]; %point of impact end %Determine if this is within the range of the detector if Intercept_y >= min(yRangeDetector) && ... Intercept_y <= max(yRangeDetector) %Check if the ray is traveling towards or away from the detector if DetectorPosition_x <= max(X) && DetectorPosition_x >= min(X) if TiltDetector == 1; %Now map this y-intercept to the corresponding position on the %detector if it was properly oriented DetectorInterceptDistance = ((Intercept_y-DetectorOffset_y)^2 + ... (Intercept_x-DetectorPosition_x)^2)^0.5; if Intercept_y >= DetectorOffset_y; Intercept_y = DetectorInterceptDistance + DetectorOffset_y; else Intercept_y = -DetectorInterceptDistance + DetectorOffset_y; end end %Keep a log of all collisions with the detector if ParticleIndex == 1; DetectionEvents = DetectionEvents + 1; DetectionEventsAll = DetectionEventsAll + 1; EmissionAngleLog(DetectionEvents) = EmittedAngle; EmissionAngleLogAll(DetectionEventsAll) = EmittedAngle; DetectorCollisionLog(DetectionEvents) = Intercept_y; DetectorCollisionLogAll(DetectionEventsAll) = Intercept_y; ParticleEmissionLogAll(DetectionEventsAll) = ParticleIndex; else DetectionEventsAll = DetectionEventsAll + 1; DetectorCollisionLogAll(DetectionEventsAll) = Intercept_y; EmissionAngleLogAll(DetectionEventsAll) = EmittedAngle; ParticleEmissionLogAll(DetectionEventsAll) = ParticleIndex; end StopAction = 1; %Tell the program to go to next ray %Redefine the ray so it stops at the detector Step = (DetectorPosition_x - x0)/20; X = x0:Step:DetectorPosition_x; Ray = Slope*X - Slope*x0 + y0; x4 = DetectorPosition_x; y4 = Intercept_y; end end

181

A.3.3 Determining Mirror Collisions (Sub-function)

function MirrorCollisionCheck(x1,y1,xMirror) global StopAction %Check if the ray goes off the mirror if y1 >= 0 %Check top half of the mirror range if x1 < xMirror(1) StopAction = 1; %Tell the program to go to next ray end else %Check lower half of the mirror range if x1 < xMirror(end) StopAction = 1; %Tell the program to go to next ray end end

182

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