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Hybrid model for characterization of submicron particles using multiwavelength spectroscopy
h [electronic resource] /
by Alicia Garcia-Lopez.
[Tampa, Fla.] :
b University of South Florida,
Thesis (Ph.D.)--University of South Florida, 2005.
Includes bibliographical references.
Text (Electronic thesis) in PDF format.
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Document formatted into pages; contains 123 pages.
ABSTRACT: The area of particle characterization is expansive; it contains many technologies and methods of analysis. Light spectroscopy techniques yield information on the joint property distribution of particles, comprising the chemical composition, size, shape, and orientation of the particles. The objective of this dissertation is to develop a hybrid scattering-absorption model incorporating Mie and Rayleigh-Debye-Gans theory to characterize submicron particles in suspension with multiwavelength spectroscopy.Rayleigh-Debye-Gans theory (RDG) was chosen as a model to relate the particles joint property distribution to the light scattering and absorption phenomena for submicron particles. A correction model to instrument parameters of relevance was implemented to Rayleigh-Debye-Gans theory for spheres. Behavior of nonspherical particles using RDG theory was compared with Mie theory (as a reference).A multiwavelength assessment of Rayleigh-Debye-Gans theory for spheres was conducted where strict adherence to the limits could not be followed. Reported corrections to the refractive indices were implemented to RDG to try and achieve Mies spectral prediction for spheres.The results of studies conducted for RDG concluded the following. The angle of acceptance plays an important role in being able to assess and interpret spectral differences. Multiwavelength transmission spectra contains qualitative information on shape and orientation of non-spherical particles, and it should be possible to extract this information from carefully measured spectra. There is disagreement between Rayleigh-Debye-Gans and Mie theory for transmission simulations with spherical scatterers of different sizes and refractive indices.
Adviser: Arthur David Snider.
In suspension particles.
x Electrical Engineering
t USF Electronic Theses and Dissertations.
Hybrid Model for Characterization of Submicron Particles Using Multiwavelength Spectroscopy by Alicia C. Garcia-Lopez A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Arthur David Snider, Ph.D. Wilfrido Moreno, Ph.D. Kenneth Buckle, Ph.D. Stanley Deans, Ph.D. Oscar D. Crisalle, Ph.D. Date of Approval: March 29, 2005 Keywords: Particle Analysis, Spherical Partic les, Non-spherical Par ticles, In suspension Particles, Rayleigh-Debye-Gans Copyright 2005, Alicia C. Garcia-Lopez
Dedication I would like to dedicate this dissertation to my younger br other Rodrigo and my family.
Acknowledgments I would like to acknowledge Prof. L. H. Garcia-Rubio for the permission to use his ideas for this research project, the vast amount of consultation time, and his expertise in the area of particle characterization. I would also lik e to acknowledge Prof. A. D. Snider for giving me a wonderful graduate experience in pursuing my Ph.D. I have greatly enjoyed working together and your guidance and mathematics expertise have greatly enhanced my life. Finally, I would like to acknowle dge the PERC located at the University of Florida for their financial support.
Table of Contents List of Tables iii List of Figures iv List of Symbols ix Abstract xii Chapter One: Introduction and Methods 1 1.1 Introduction 1 1.2 Materials and Methods 2 1.3 Overview of Chapters 3 Chapter Two: Scattering Theories and Models 4 2.1 Background 6 2.2 General Concepts and Equations 7 2.3 Mie Theory and Model 12 2.4 Rayleigh-Debye-Gans Theory and Model 13 2.5 Hybrid Theory and Model 15 2.6 Methods Review 16 Chapter Three: Instrumentation Correction Model for Transmission 20 3.1 Instrumentation Correction Model 20 3.2 Implementation of Instrument Corrections 23 Chapter Four: Nonspherical Particles 26 4.1 Geometry and Notation for Ellipsoids 26 4.2 Ellipsoid Simulations 30 Chapter Five: Validation Study of Rayleigh-Debye-Gans Theory 34 5.1 Exploration of Theoretical Limits 34 5.2 Particle Diameter << Wavelength 35 5.3 Particle Diameter ~ Wavelength, No Absorption 38 5.4 Particle Diameter ~ Wavelength, Absorption > 0 42 5.5 Conclusion 45 Chapter Six: Corrections to the Refractive Index 46 6.1 Refractive Index 46 i
6.2 Hypochromic Effect 47 6.3 Implementation of Optical Correction for Absorption 48 6.4 Effective Refractive Index Estimation 53 6.5 Conclusion 54 Chapter Seven: New Hybrid Theory 55 7.1 Geometry and Notation 55 7.2 Internal Field 58 7.3 Dipole Scattering Approach 60 7.4 Hybrid Theory 62 7.5 Determining the Transmission 65 7.6 Scattering Intensity Ratio and Turbidity 66 Chapter Eight: Validation and Sensitivity of Hybrid Theory 67 8.1 Validation of Hybrid Theory Implementation 67 8.2 Case 1: Relative Refractive index 0nn ~1 and Absorption = 0 73 8.3 Case 2: Relative Refractive index 0nn 1 and Absorption > 0 76 8.4 Case 3: Relative Refractive index 0nn ~1 and Absorption > 0 79 8.5 Conclusions 82 Chapter Nine: Contributions and Future Work 83 9.1 Contributions 83 9.2 Recommendations and Future Work 84 References 85 Appendices 87 Appendix A: Intensity Ratio and Turbidity Model 88 Appendix B: Optical Properties 92 Appendix C: Validation for Rayleigh-Debye-Gans Theory 98 C.1 Validation of Rayleigh-Debye-Gans Theory 98 Appendix D: Estimation of Absorption Coefficient and Hypochromism Model 102 D.1 Hypochromism Model 104 About the Author End Page ii
List of Tables Table 1.1 Simulation Parameters 3 Table 4.1 Table of Form Factors 29 iii
List of Figures Figure 2.1 Diagram for Complete Particle Characterization Model 7 Figure 2.2 Diagram of Coordinate System used in the Mie and RDG Models 12 Figure 3.1 Transmission System (a) Open Detector (b)Pinhole Detector 21 Figure 3.2 Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Spheres with Pinhole Detector Setup 25 Figure 3.3 Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Spheres with Open Detector Setup 25 Figure 4.1 Light Scattering in Laboratory Frame 27 Figure 4.2 Light Scattering in Particle Frame 27 Figure 4.3 Fixed Orientations A, B, and C for an Ellipsoid 31 Figure 4.4 Calculated Transmission of Soft Body Prolate Ellipsoid =0.3 with a Volume Equivalent to a 1 m Sphere 33 Figure 4.5 Calculated Transmission of Soft Body Prolate Ellipsoid =0.8 with a Volume Equivalent to a 1 m Sphere 33 Figure 5.1 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm AgBr Spheres 36 Figure 5.2 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm AgCl Spheres 36 Figure 5.3 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm AgBr Spheres 37 iv
Figure 5.4 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm AgCl Spheres 37 Figure 5.5 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Soft Body Spheres 38 Figure 5.6 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Soft Body Spheres 39 Figure 5.7 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 5.5 m Soft Body Spheres 39 Figure 5.8 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Hemoglobin Spheres with =0 41 Figure 5.9 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Hemoglobin Spheres with =0 41 Figure 5.10 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 5.5 m Hemoglobin Spheres with =0 42 Figure 5.11 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 100 nm Hemoglobin Spheres 43 Figure 5.12 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Hemoglobin Spheres 44 Figure 5.13 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Hemoglobin Spheres 44 Figure 6.1 Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction 50 Figure 6.2 Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with 0% and 100% Hypochromocity 50 Figure 6.3 Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction to k c and an Effective n eff Calculated through Kramer-Kronig Transform 51 v
Figure 6.4 Zoom in of Figure 6.4 of Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction to k c and an Effective n Calculated through Kramer-Kronig Transform 52 Figure 7.1 Diagram of Scatterer Point and Detector Location 55 Figure 7.2 Local Unit Vectors with Respect to the Detector 56 Figure 7.3 Local Unit Vectors with Respect to the Scatterer 57 Figure 7.4 Diagram of the Volume at Height z for a Sphere 64 Figure 8.1 Coefficients c 1 ,d 1 and d 2 Versus at the Limit when k 1 =k 68 Figure 8.2 Form Factor f 1 for Hybrid Theory at k 1 =k 69 Figure 8.3 Form Factor f for Rayleigh-Debye-Gans Theory 70 Figure 8.4 Real Part of Form Factor f 1 for Hybrid Theory using Polystyrene 70 Figure 8.5 Imaginary Part of Form Factor f 1 for Hybrid Theory using Polystyrene 71 Figure 8.6 Real Part of Form Factor f 2 for Hybrid Theory using Polystyrene 71 Figure 8.7 Imaginary Part of Form Factor f 2 for Hybrid Theory using Polystyrene 72 Figure 8.8 Form Factor f for Rayleigh-Debye-Gans Theory using Polystyrene 72 Figure 8.9 Comparision of Calculated Transmission for 50 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 74 Figure 8.10 Comparision of Calculated Transmission for 100 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 74 Figure 8.11 Comparision of Calculated Transmission for 250 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 75 vi
Figure 8.12 Comparision of Calculated Transmission for 500 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 75 Figure 8.13 Comparision of Calculated Transmission for 50 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 77 Figure 8.14 Comparision of Calculated Transmission for 100 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 78 Figure 8.15 Comparision of Calculated Transmission for 250 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 78 Figure 8.16 Comparision of Calculated Transmission for 500 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 79 Figure 8.17 Comparision of Calculated Transmission for 100 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 80 Figure 8.18 Comparision of Calculated Transmission for 250 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 80 Figure 8.19 Comparision of Calculated Transmission for 500 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 81 Figure 8.20 Comparison of Calculated Transmission for 500 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 81 Figure B.1 Optical Properties for Water 92 Figure B.2 Optical Properties for Soft Body 92 Figure B.3 Optical Properties for Hemoglobin 93 Figure B.4 Optical Properties for Polystyrene 93 Figure B.5 Optical Properties of AgCl 94 Figure B.6 Optical Properties of AgBr 94 Figure B.7 Relative Refractive Index of Soft Body in Water 95 Figure B.8 Relative Refractive Index of Hemoglobin in Water 95 Figure B.9 Relative Refractive Index of Polystyrene in Water 96 vii
viii Figure B.10 Relative Refractive Index AgCl in Water 96 Figure B.11 Relative Refractive Index of AgBr in Water 97 Figure C.1 Calculated Transmissi on of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm Polystyrene Spheres 99 Figure C.2 Calculated Transmissi on of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm Polystyrene Spheres 100 Figure C.3 Calculated Transmissi on of Mie and Rayleigh-Debye-Gans for a Suspension of 100 nm Polystyrene Spheres 100 Figure C.4 Calculated Transmissi on of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Polystyrene Spheres 101 Figure D.1 Stack Arrangement of Chromophores along Chain Axis 104
List of Symbols Angle of directional cosine Angle of directional cosine Angle of directional cosine Extinction coefficient for a single chromophore Average extinction coefficient for a single chromophore m Molar extinction coefficient for a single chromophore c Correction imaginary part of the refractive index Angle between scattering and incident beam Wavelength o Wavelength in vacuo n Angle dependent function Solid angle Transmission/Turbidity n Angle dependent function Rotational Euler angle around Z axis Rotational Euler angle around Y axis Rotational Euler angle around new Z axis Size parameter ix
a Radius a n Mie coefficient b n Mie coefficient C abs Absorption cross section C ext Extinction cross section C sca Scattering cross section D Particle diameter E Extinction coefficient Average extinction coefficient E Calculated Extinction coefficient f Form factor G Cross sectional area h Hypochromicity h n Hankel functions I o Incident intensity I s Scattering intensity k Wave number k Quantity of chromophores j n Bessel functions l Pathlength of sample M w Molecular weight m Relative refractive index x
N Complex refractive index of particle N 2 Complex refractive index of medium N A Avogadros number N p Number of particles n Real part of the refractive index of particle P Probability of absorption by a photon P n Legendre polynomial Q abs Absorption efficiency factor Q ext Extinction efficiency factor Q sca Scattering efficiency factor Q ext Apparent extinction efficiency factor Q sca Apparent scattering efficiency factor R Correction factor R Path averaged correction factor r Distance between Middle of Sample and Detector S Scattering Amplitude Function s Effective Geometric Area of Chromophore V Volume of Particle vf Volume fraction of xi
Hybrid Model for Characterization of Submicron Particles Using Multiwavelength Spectroscopy Alicia C. Garcia-Lopez ABSTRACT The area of particle characterization is expansive; it contains many technologies and methods of analysis. Light spectroscopy techniques yield information on the joint property distribution of particles, comprising the chemical composition, size, shape, and orientation of the particles. The objective of this dissertation is to develop a hybrid scattering-absorption model incorporating Mie and Rayleigh-Debye-Gans theory to characterize submicron particles in suspension with multiwavelength spectroscopy. Rayleigh-Debye-Gans theory (RDG) was chosen as a model to relate the particles joint property distribution to the light scattering and absorption phenomena for submicron particles. A correction model to instrument parameters of relevance was implemented to Rayleigh-Debye-Gans theory for spheres. Behavior of nonspherical particles using RDG theory was compared with Mie theory (as a reference). A multiwavelength assessment of Rayleigh-Debye-Gans theory for spheres was conducted where strict adherence to the limits could not be followed. Reported corrections to the refractive indices were implemented to RDG to try and achieve Mies spectral prediction for spheres. xii
The results of studies conducted for RDG concluded the following. The angle of acceptance plays an important role in being able to assess and interpret spectral differences. Multiwavelength transmission spectra contains qualitative information on shape and orientation of non-spherical particles, and it should be possible to extract this information from carefully measured spectra. There is disagreement between Rayleigh-Debye-Gans and Mie theory for transmission simulations with spherical scatterers of different sizes and refractive indices. Finally, it is not possible to adequately or realistically compensate for the differences between Mie and RDG through the use of hypochromicity models and/or effective refractive indices. A hybrid model combining RDG and Mie theories was developed and tested for spheres of different sizes and refractive indices. The results of hybrid model is that it approximates Mie theory much better than Rayleigh-Debye-Gans for particle sizes smaller than the wavelength and for a broader range of optical properties in the context of multiwavelength spectroscopy. Overall, this new model is an improvement over Rayleigh-Debye-Gans theory in approximating Mie theory for submicron particles and is computationally more effective over other methods. The development of the hybrid spherical model constitutes a platform for developing nonspherical models. xiii
Chapter One Introduction and Methods 1.1 Introduction Characterization of particles entails obtaining information about size, shape, orientation and chemical composition. Particle characterization is a broad area of undertaking which encompasses many technologies, among them light spectroscopy techniques. Light spectroscopy typically involves scattering and absorption methods. Scattering measurements are performed at a single wavelength but measured as a function of the direction of observation. For absorption, the light is measured in the forward direction as a function of wavelength. Light scattering techniques typically use highly collimated sources (lasers), whereas absorption spectrophotometric techniques use broadband sources to produce multiwavelength spectra. In either case the resulting spectra can be interpreted with the theory of electromagnetic radiation, which describes interaction of light with matter (Maxwells Equations). Mie and Rayleigh-Debye-Gans theories are solutions to Maxwells Equations that relate the particles joint property distribution to the light scattering and absorption phenomena. This connection is made through the optical properties that are characteristic of the materials contained in the particle. The objective of this study is to develop a hybrid scattering-absorption model incorporating both theories to characterize submicron particles with multiwavelength spectroscopy. To accomplish this objective Mie theory and Rayleigh-Debye-Gans theory 1
are revisited and extended to account for the field alteration predicted by Mie, and for the dipole radiation mechanism employed by Rayleigh-Debye-Gans. Throughout this dissertation Mie theory and Rayleigh-Debye-Gans theories are emphasized because at this point they enable real time particle characterization for industrial and biomedical applications. The largest area of application that would profit from this study is in the biological and biomedical field in the subject of microbial and disease detection in tissue and bodily fluids. 1.2 Materials and Methods The programs for Mie theory, Rayleigh-Debye-Gans theory, instrument models and hypochromicity were developed in Matlab v6.5.1. Computations for these programs were conducted using a Dell Inspiron 4100 with 1GHz Pentium III processor and 512 MB RAM. The optical properties (refractive indices) utilized were provided by Dr. Garcia-Rubio and the SAPD laboratory through the College of Marine Science at the University of South Florida . The computer codes developed for the analysis of Rayleigh-Debye-Gans and Mie particles were tested against published values of the scattering functions [1, 14]. In testing and exploring the algorithms for Rayleigh-Debye-Gans the refractive indices selected were those of soft bodies and hemoglobin, where soft bodies are defined here as particles whose relative refractive index is close to one with no absorption component. The values of the index of refraction n+i for biological particles commonly used are soft bodies (1) and hemoglobin (1 04.145.n 6.148. n 15.001.0 ) . Polystyrene (1, 2. 25.n 82 .001.0 ), silver bromide ( 5 .36.2 n 2
6.1001.0 ) and silver chloride ( 7.22 n 0 85.0001. ) are materials found in industrial applications whose properties are used as standards for optical instruments . Water (1) was used as the suspending medium. The refractive indices as function of wavelength are reported in Appendix B. 4.13n The ranges of particle volumes were chosen between 12700 nm 3 and 87 m 3 The spherical diameter equivalents to the volume range are between 25 nm -5.5 m. The table below gives the simulation parameters used to define the suspensions for the analyses conducted in this dissertation. Table 1.1: Simulation Parameters Light Source Wavelength Particle Concentration Particle Density 200-900 nm 1E-4 g/cc 1 g/cc 1.3 Overview of Chapters This dissertation is divided into nine chapters. Chapter two presents a review of Mie and Rayleigh-Debye-Gans theory, citing the resulting formulas for the scattered field and their matrix formulations. For each theory the scattering intensity ratio that governs scattering measurements/simulations and the turbidity formula that governs transmission measurements/simulations are displayed. The latter formula contains a term proportional to the scattering cross section, which takes different forms for the two models, and a term proportional to the absorption cross section, which is the same for both models. Only simulations of transmission are reported, at multiwavelength. 3 Chapter three describes aperture correction models that account for the fact that actual transmission measurements are inevitably polluted by the presence of some near-forward scattered radiation. The simulations of this effect are for the RDG model only, for 1 m hemoglobin spheres.
Chapter four reports a study of nonspherical scatterers, simulating soft body ellipsoids. Transmission curves are compared for two eccentricities, each with three different body orientations, using the RDG model. Chapter five compares RDG theory and Mie theory for transmission with simulations for spherical scatterers of different sizes and refraction indices. Disagreement between the two theories is demonstrated. The rest of the dissertation is concerned with attempts to modify RDG to bring the transmission simulations into closer agreement with Mie (the exact solution for spheres). In chapter six, two approaches are described to increase the computed RDG turbidity to that of Mie. The first approach is to use hypochromicity as a correction to RDG to account for absorption. The second approach is based on the observation that the RDG formula for the extinction cross section is (very nearly) a simple quadratic function of n and ; therefore one can invert this function and find "effective" values of or n that will result in turbidity values calculated by RDG in agreement with those computed by Mie. Reasons for rejecting these approaches are cited. Chapter seven presents the new hybrid model, based on the rigorous Mie calculation of the internal field and the Rayleigh-Debye-Gans approach for the scattering radiation. This theory is developed in full for spheres. In chapter eight the hybrid model is tested by simulated comparisons with Mie and Rayleigh-Debye-Gans theories, employing the span of optical properties of interest. Finally, conclusions, contributions, and recommendations are covered in chapter nine. 4
Chapter Two Scattering Theories and Models This chapter describes light scattering theories and measurements of submicron particles in suspension. The first section provides information on scattering and turbidity measurements used to characterize particles. This section also provides a description of the models used to describe the observed measurement. The subsequent sections provide an outline of Mie theory and RDG theory. These theories describe the scattering phenomena observed in transmission and scattering measurements. Specifically, the scattering intensity ratio as a function of wavelength (in our case a broad wavelength range) and angle of observation, and the turbidity as a function of wavelength (again broad wavelength range) are quantified. Other more computationally intensive, techniques for solving light scattering and absorption problems are discussed. These techniques include the T-Matrix and the Purcell-Pennypacker methods. 2.1 Background There are many types of spectroscopy measurement used to characterize particles in suspension. Most interest focuses on transmission and scattering measurements. In the former the electromagnetic energy of an incident wave is measured after interaction with a particle or suspension as it leaves the system in the forward direction. In contrast, scattering measurements capture the light after interacting with a particle as it leaves the 5
system at any angle of observation. They differ because transmission measurements capture both the (forward) scattered light and the unscattered portion of the incident beam. Information concerning the properties of the scattering and absorbing particle is contained in the measured spectra, which are plots of the power intensity versus frequency or wavelength (and, direction). Through the uses of the appropriate theories and models, it is possible to obtain estimates of the size, shape, chemical composition, internal structure, and surface charge from spectroscopic measurements . A complete scheme for particle characterization must take into account various experimental conditions occurring in the lab system when spectroscopy measurements are conducted. These include the type of measurement, instrumentation setup, particle-light interaction, and other optical phenomena. Figure 2.1 illustrates how these components relate to one another. The scattering intensity ratio equation and the turbidity equation are energy balance equations that are developed from the scattering theories studied. A detailed description of the transmission measurement and analysis is provided in chapter three. Refractive indices and corrections are discussed in chapter six. The desire to characterize particulate systems for real-time continuous monitoring has led to the selection of Rayleigh-Debye-Gans (RDG) theory and Mie theory. Computation time being the restricting factor, these theories provide light scattering solutions in a suitable time. The rest of this chapter is dedicated to the description of Mie and RDG theories along with the development of the corresponding scattering intensity ratio and turbidity formulas. 6
Figure 2.1: Diagram for Complete Particle Characterization Model 2.2 General Concepts and Equations The emphasis of this section is the utilization of the scattering matrix formalism to evaluate the extinction of light predicted by the scattering intensity and turbidity equations. Throughout the recent course of light scattering history, the terms turbidity (or optical depth) and optical density have caused much confusion . Turbidity has been traditionally defined as an attenuation coefficient due to scattering (only) for the transmission of the incident beam. Herein, turbidity is described as the total attenuation observed due to scattering and absorption. The term optical density (O.D.) was originally used synonymously with absorption; the units of O.D. are absorption unit per pathlength (Au/cm). Turbidity will be described in the units of optical density. 7
The amplitude scattering matrix is used to relate the incident and scattered fields. iizrkissEESSSSikreEE||1432)(|| 2.1 where are the asymptotic incident and scattering fields parallel and perpendicular to the scattering plane; r is the distance from the scattering center to the detector, z is distance along the axis of propagation of the incoming wave, and k is the propagation constant or wave number in the medium surrounding the particle; ssiiEEEE,,,|||| oonk 2 where o is the wavelength in vacuo and n o is the refractive index of the medium. Van de Hulst  describes in detail various assumptions made for simplifying the scattering functions with regard to rotation and symmetry of particles. For a spherical particle S 3 and S 4 are equal to zero. S 1 and S 2 are complex amplitude scattering elements; they depend on the indices of refraction, particle size, and the scattering direction. S 1 and S 2 are given in the Mie model by formidable series expansions involving Bessel, Neumann, and Legendre functions. The expressions for S 1 and S 2 predicted by Mie and RDG theory are provided in subsequent sections. If the detector is not situated in the forward direction, it is illuminated only by light that is scattered by the particle; its construction shields it from the incident beam. The term scattering measurement refers to this configuration. This scattered intensity I s is given by 8
2,12||,222,2||,2Re121Re21Re21iisssssESESrkEEEI 2.2 where is the permittivity and is the permeability. If the incoming light is unpolarized, 22,2||,21iiEE oE so ooosI r k SSErkSSESSrkI2221222222122221222221Re21212Re121 2.3 I s /I o is known as the scattering intensity ratio. On the other hand, if the detector is aligned with the incoming beam, it measures the forward scattered wave together with the transmitted incident wave. The analysis of such a transmission measurement is most easily conducted by accounting for the energy loss suffered by the original incident beam. There are two loss mechanisms which attenuate the incident beam, scattering and absorption. The power scattered out of the beam by a particle is evaluated by the integrating the scattered intensity over an enclosing sphere at infinity (in spherical coordinates). ddrrIpowerscattereds0202sin,,, 2.4 9
It is convenient to define a scattering cross section C sca as the area over which one would integrate the incoming intensity I o to balance the scattered power; thus powerscatteredCIscao 2.5 or ddrIrICosscasin,,,2020 2.6 The scattering efficiency Q sca is the ratio of the C sca to the actual cross section G that the particle presents to the incoming beam: GCQscasca 2.7 The power absorbed by the particle will be discussed in detail in chapter five; it too can be expressed using an equivalent area C abs : powerabsorbedCIabso 2.8 According to Van de Hulst , the absorption cross section for a particle of volume V and relative refractive index of refraction oninm is given (in both theories) by 21Im322mmkVCabs 2.9 The absorption efficiency is expressed analogously: GCQabsabs 2.10 Thus one can mathematically characterized the power loss in the incident light as if an area (C sca + C abs ) were blocked out of the incident beam by each particle. The extinction cross section and efficiency then, are given by 10
absscaextCCC GCextext Q 2.11 If the particle density in the medium is N p then in an infinitesimal length dz there are N p dz particles per unit area; each effectively depletes the beam power by C ext I. Therefore the attenuation of the beam intensity I(z) as a function of pathlength z is governed by )()()(zICdzNzIdzzIextp 2.12 or )(zICNdzdIextpz 2.13 The solution to the differential equation is zCNextpeoIzI)()( 2.14 The optical theorem  states that the extinction cross section can be expressed in terms of the elements of the scattering amplitude matrix in the forward direction 0Re40Re42212 S k S k Cext 2.15 The terms turbidity, or optical density are used to characterize the total attenuation in a sample of length l. lGQNlCNextpextp 2.16 In practice, a detector placed in the forward direction will have a finite aperture, and thus capture some of the radiation scattered at small angles; corrections for this effect are discussed in chapter three. 11
2.3 Mie Theory and Model The exact solution to the boundary value problem for light scattering by a sphere is generally referred to as Mie Theory . Mie theory assumes that the spherical scattering object is composed of a homogeneous, possibly absorbing isotropic and optically linear material irradiated by an infinitely extending plane wave. Figure 2.2: Diagram of Coordinate System used in the Mie and RDG Models In figure 2.2 (x,y,z) refer to Cartesian coordinates and (,, r ) refer to spherical coordinates. The amplitude scattering matrix elements S 1 and S 2 are expressed explicitly as 1211coscos112coscos112nnnnnnnnnnabnnnSbannnS 2.17 12
where n (cos) and n (cos) are defined in terms of the associated Legendre polynomial ; 1nP coscoscossin1cos11nnnnPddP 2.18 Note that there is no dependence in the Mie model. To calculate the scattering intensity ratio using Mie theory, the equations in 2.17 for the amplitude scattering matrix elements can be substituted into equation 2.2 The amplitude scattering matrix elements for Mie theory at 0 are given by: 1211221000nnnbanSSS 2.19 The turbidity formula for Mie theory is calculated by substituting 2.l9, into equations 2.15 and 2.16. Through the direct calculation of the turbidity and scattering intensity ratio equations, Mie theory has been shown to be an effective tool for determining particle size distributions of nonspherical shapes, internal structures, and optical properties . 2.4 Rayleigh-Debye-Gans Theory and Model The basis of the theory for Rayleigh-Debye-Gans scattering is Rayleigh scattering. Rayleigh presented an approximate theory for particles of any shape and size having a relative refractive index near unity, Debye and Gans later added refinements. Kerker  states that the fundamental approximation in the Rayleigh-Debye-Gans approach is that the phase shift, the change of the phase of a light ray that passes through the sphere is 13
negligible. A restriction is therefore put upon the particle size, the wavelength, and the refractive index. For a particle of radius a the restriction is 11mka 2. The physical assumption of Rayleigh-Debye-Gans scattering is that each infinitesimal volume element of the particle gives rise to Rayleigh scattering and does so independently of the other volume elements. The waves scattered in a given direction by these elements interfere due to the different positions of the volume elements in space. For spherical and non-spherical particles, a form factor dVeVfi1 2.20 is introduced that averages the phase difference throughout the volumes V of the (spherical and nonspherical) particle; the scattering amplitude elements S 1 and S 2 then take the form of fVmmikS21432231 2.21 cos21432232fVmmikS 2.22 were Kerker lists form factors from various shapes, of which some are quoted in this dissertation. In subsequent sections the specific derivation of the form of f is given, as well as a discussion for utilizing f to deduce orientation. A detailed derivation of the scattering intensity ratio equation and the turbidity equation for RDG is developed in Appendix A. The scattering intensity ratio is easily obtained by substituting equations 2.20 and 2.21 for the amplitude scattering elements into equation 2.2. For unpolarized light, 2.3 becomes 043SS 14
222222224cos12132k9,fVmmrIIos 2.23 The scattering cross section C sca is expressed from 2.6 as 02222224sincos121169dfmmVkCsca 2.24 The general expression for the absorption cross section is, from equation 2.9, 21Im322mmkVCabs 2.25 The extinction cross section is the sum of these two (equation 2.24 and 2.25): 21ImV3sincos1211692202222224mmkdfmmVkCext 2.26 The expression for the turbidity for a monodispersed system can then be explicitly expressed by the following equation. 21Im3sincos1211692202222224mmkVdfmmVkNp 2.27 2.5 Hybrid Theory and Model The new scattering model proposed in this dissertation is a hybrid combination of Mie theory and Rayleigh-Debye-Gans theory. Chapter six is dedicated to the description and development of the hybrid theory and model. 15
2.6 Methods Review In sections 2.3 and 2.4 the models presented for characterizing particles have been Mie theory for spherical particles and Rayleigh-Debye-Gans theory for arbitrary shaped particles. To summarize, Mie theory is an exact mathematical solution to Maxwells Equations for light scattered by spheres. It has been used extensively to characterize non-spherical particles approximately for a broad range of sizes and optical properties. Although Mie theory provides good estimates of the size and chemical composition it does not posses the ability to estimate the actual shape and orientation of non-spherical particles. Rayleigh-Debye-Gans theory provides information on shape and orientation for spherical and nonspherical particles; however, its applicability has limitations with regards to size and optical properties of systems. For the case of nonspherical particles, many exact and approximate methods have been developed. Singham and Bohren discuss advantages and disadvantages of several [6, 7]. Two methods considered significant to this investigation are the T-Matrix and Purcell-Pennypacker method. The T-Matrix is the linear transformation connecting the coefficients of the eigenfunctions in the scattered field and those for the incident field. It is through the linearity of Maxwells equations and the boundary condition satisfied by the electromagnetic field that these coefficients are linearly related. Bohren states that in principle the coefficients of the T-matrix are obtainable by integration; however, computational difficulties arise if the particles are highly absorbing or their shapes extreme. Recent progress has been made for computing the T-matrix as discussed by Mishchenko et. al . Mishchenko provides a review and description of several 16
numerical techniques developed for single and aggregated particles. The scope of this dissertation is on single particles and as such a brief overview follows. The standard approach for computing the T-matrix for single scatterers is based on the extended boundary condition method developed by Waterman for homogenous particles. Nearly all numerical results computed by the T-matrix relates to bodies of revolution. The first computational advance of the T-matrix is using nonspherical particles of fixed orientation using the extended boundary condition method. This method proved to be faster than the conventional separation of variables method for spheroids and discrete dipole approximation integral equation formulation. The disadvantage to using the extended boundary condition method is its poor numerical stability for particles with very large real and/or imaginary parts of the refractive index, large size compared to the wavelength, and/or large extreme geometries. Mishchenko discusses the iterative extended boundary condition as another approach for overcoming the problem of numerical instability in computing the T-matrix for highly elongated spheroids. The disadvantage of this technique is that the numerical stability comes at the expense of considerable increase in computation code, complexity, and CPU time. The last computational approach discussed by Mishchenko is that of using extended precision instead of double precision floating point variables. This technique provides an increase in size parameter for spheroids and better accuracy. The use of the extended precision variable requires only a negligibly small additional memory and its approach is simple with little additional programming. The disadvantage to the extended precision is the CPU time. 17
In the Purcell-Pennypacker method a particle is approximated by a lattice of dipoles, each small compared with the wavelength but still large enough to contain many atoms. Each dipole is excited by the incident field and by the fields of all the other dipoles. Two methods are used to solve for these equations; either iteration or matrix inversion. The iterative method is slow for large parameters, dipoles and values of the refractive index. The matrix inversion method, although useful for calculating scattering by a particle in more than one orientation and for orientational averaging over an ensemble of particles, is limited to small number of dipoles. An embedding method using a scattering-orders approach was developed by Singham and Bohren . The scattering-orders perturbation method is an extension of the series for the perturbation method which looks upon a nonspherical particle as a sphere, the boundary of which is distorted or perturbed by different amounts at different points . In the case of Singham and Bohren, the scattering-order perturbation method is used to formulate the coupled dipole method. This method uses the dipolar interactions as infinite series in scattering-orders. Two interactions exist, those between dipoles within the sphere and all others. The scattering fields resulting in the dipole interaction within the sphere are described by Mie theory or any other equivalent theory. It was demonstrated that this method worked and shortened computation time. However, it was limited by the scattering-order series, which can diverge. The greater the refractive index, the smaller the particle size for which it diverges. 18
19 The endeavor to characterize biopartic ulates in the context of engineering application requires identifying the problem, finding solution s, analyzing, designing and testing of the solution all taking place in real-time. Mie, Rayleigh-Debye-Gans, and the new hybrid theory provide a more palatable m eans to characterize bi ological systems as compared to those aforementioned. The rest of this dissertation describes how these theories are utilized and explored.
Chapter Three Instrumentation Correction Model for Transmission The research presented in this document utilizes many models to represent those conditions present when taking experimental measurements. As stated in the previous chapter and illustrated in figure 2.1, light scattering, instrumentation and optical formulas are integrated to make up a complete model for experimental conditions observed when taking turbidity measurements. The previous chapter described the light scattering equations for the intensity ratio and transmission spectroscopy measurements. This chapter is dedicated to discussing the instrument formulas available for simulating transmission measurements and implementation of the formulas to RDG theory. 3.1 Instrumentation Correction Model Deepak et al.  defines the expression forward scattering for a single scattering phenomenon as scattered radiation reaching the detector after being scattered only once by a scatterer situated within the path of the direct radiation. Ideally, the conditions for turbidity measurements entail a non-divergent beam illuminating a homogeneous medium and a detector directly, opposite the light source, possessing an acceptance angle close to zero (to capture light solely in the forward direction). The turbidity equation presented in the previous chapter does not account for the instrument setup, therefore an instrument model or correction to the turbidity is introduced in this chapter. On the basis 20
of the scattering theories presented and the instrument setup used for transmission measurements, the correction factor provided by Deepak et al. was chosen. The correction factor is dependent upon the geometry of the transmission measurement through the angle subtended at the scatterer by the detector window (figure 3.1). Therefore the instrument design has to be taken into account. For completeness the two designs provided by Deepak et al. were implemented : the open detector and pinhole detector. The geometries are illustrated in figure 3.1 (a) and (b). Figure 3.1: Transmission System (a) Open Detector (b)Pinhole Detector Deepak et al. states that the true optical depth from transmission measurements cannot be obtained directly. Deepak et al. notes that the finite aperture of the detector, in a transmission measurement, picks up some additional light that has been scattered into the aperture. As seen in chapter two the turbidity formula (equation 2.16) 21
extpabsscapabsscaplQNQQlGNCClN 3.1 accounts for light that has been scattered out of, or absorbed from, the incident beam. Recall that the power scattered by a particle equals (equation 2.4) oscasICddrrIpowerscattered0202sin,,, 3.2 However, if the finite aperture captures scattered light within a cone of half-angle in the forward direction, then the beam depletion due to scattering only ddrrIs202sin,,, 3.3 giving an effective scattering cross section of ddrIrICCosscasca202sin,,, 3.4 The apparent extinction cross section of the particle can be written ddrIrICCCCCosabsscaabsscaext202sin,,, 3.5 or PddrIrICCCosextextext202sin,,,11 3.6 In the open detector system (figure 3.1a) a radiation source is placed at the focal length of the transmitter lens L 1 which transmits a parallel beam through the medium of thickness l, which is in turn measured by an open detector of radius R 2 The path averaged correction factor P for particles on axis is defined as 22
22csc,tandRP 3.7 where the angle for the scatters satisfies 12tanLR The pinhole detector system (Figure 3.1b) consists of a radiation source placed at the focal point of a lens L 1 A second lens L 2, with focal length f, focuses the light through an aperture of radius r in the focal plane and onto the detector. The path averaged correction factor for the pinhole detector is simply P PP 3.8 The result of equation 3.2 stems from the fact that is determined by fr tan which stays constant. The correction factor corresponding to the design set up can be introduced into the turbidity equation, equation 2.11, by multiplying through by P : PmQalNextp),()(2 3.9 3.2 Implementation of Instrument Corrections The corrections for scattering developed by Deepak  have been evaluated for RDG theory for hemoglobin spheres. Hemoglobin spheres are hypothetical spheres whose refractive indices are those of hemoglobin. The refractive indices of hemoglobin were used to test these effects for strong scattering and absorption relative refractive indices ( 1.0001.0,2.1 1 oonnn ). The acceptance angle for spectrometer detectors are typically less than or equal to 2 degrees. The acceptance angles studied for both detector 23
configurations were 1, 2, and 5 degrees for a spherical particle of 1 m. The parameters used to conduct the simulations are provided in table 1.1. Figures 3.2 and 3.3 show the simulated transmissions using Rayleigh-Debye-Gans theory with and without acceptance angle corrections for each detector configuration. The (uncorrected) Mie theory simulation is included as a reference. The pinhole detector setup (figure 3.2) shows smaller subtle changes in the corrected spectra for all angles (RDG 1 RDG 2 RDG 5 ) compared to uncorrected (RDG) spectra. The open detector setup (figure 3.3) shows dramatic changes in the spectra as the aperture angle increases. It is evident that the angle of acceptance plays an important role in being able to assess and interpret spectral differences. Bohren has made similar statements concerning the angle of acceptance though here calculations have been provided. Having demonstrated the methodology for correcting for finite detector apertures in transmission measurements, we next turn to a study of the effects of particle shape and orientation. Aperture corrections are omitted in the simulations reported in the remainder of the dissertation, except where noted. 24
200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength (nm)Optical Density RDGRDG1RDG2RDG5 Figure 3.2: Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Spheres with Pinhole Detector Setup 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength (nm)Optical Density RDGRDG1RDG2RDG5 Figure 3.3: Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Spheres with Open Detector Setup 25
Chapter Four Nonspherical Particles Form factors have been developed for many shapes such as cylinders, disks, rods, spheres, and ellipsoids. We are interested in biological systems, whose particles are best represented by ellipsoids. 4.1 Geometry and Notation for Ellipsoids It is convenient to carry out the present analysis in two non-standard, left handed coordinate systems, the laboratory and particle frame. The laboratory frame establishes the position and orientation of the particle relative to the source and detector. The particle frame exploits the convenience of particle coordinates in describing the scattering. Therefore scattering calculations are in the particle frame and related back to the laboratory frame. Unit vectors along the x L y L and z L axes in the lab frame are denoted by i, and LLj, Lk ; and similarly for the particle frame. 26
Figure 4.1: Light Scattering in Laboratory Frame Figure 4.2: Light Scattering in Particle Frame The behavior of light scattered from the particle is dependent upon its size, shape, orientation and chemical composition. Figure 4.1 illustrates the incident beam approaching the particle in the z L -direction in the laboratory frame. The light scattered from the particle can radiate in all directions. The line in the scattering plane that bisects the angle between the incident and scattered beam is called the bisectrix , denoted 27
by BIS in figures 4.1 and 4.2. The plane through the bisectrix ( BIS ) and perpendicular to the plane of the scattering ( SCA ) will be called the bisectrix plane. Equation 4.1 describes the directions of incident light, scattered light and the bisectrix from the laboratory frame. LLLLLkiINSCABISkiSCAkIN 1cossincossin 4.1 The description can also be expressed in the particle frame. Figure 4.2 shows the angles used to relate the bisectrix to the particle axes. Calculation of the scattered light intensity at different angles in the laboratory frame can be reconstructed through its relation to the particle frame. pppLLLkjiEkji 4.2 where i p =|1 0 0|, j p =|0 1 0|, k p =|0 0 1| in the particle frame, the Euler angles , are used to relate the frames using the zyz Euler rotation sequence, and E is given by cossinsinsincossinsincoscossincossincossinsincoscoscossinsincoscoscossinsinsincoscoscosE 4.3 Using this relationship, the directional cosines of angles can be found in the laboratory frame to mathematically describe the bisectrix (equation 4.1). LLLkBISBISjBISBISiBISBIScos,cos,cos 4.4 28
In the case of an ellipsoid with a fixed orientation, we can see the use of the directional cosines in the form factor computed by Kerker: 22222232232coscoscos29cbaAhAuuuJf 4.5 where A is a vector described by the directional cosines and a, b, c are the semiaxes for ellipsoid. The variable h equals 2sin4h The following is a short table of various form factors, pertinent to this work, taken from Kerker . The equations presented in this section were programmed; simulations conducted are reported in the next section. Table 4.1: Table of Form Factors Shapes Definition of variables Form factor f 2 () Sphere Radius=a u=ha Concentric Sphere with spherical shell Inner radius=a Outer radius=b u=ha v=hb Inner refractive index=m1 Inner refractive index=m2 Ellipsoid of Revolution Semi axes are a,b, is angle between figure axis & bisectrix u=h(a 2 cos 2 b 2 sin 2 Ellipsoid Semi axes are a,b,c u=hA A 2 =a 2 cos 2 b 2 cos 2 c 2 cos 2 are the directional cosines of the bisectrix 32232)(9uuJ 232333212323)(12)(29uuJbammmvvJ 32232)(9uuJ 32232)(9uuJ 29
4.2 Ellipsoid Simulations For the simulation of the effects of shape and orientation on the multiwavelength transmission spectra, prolate ellipsoids with refractive indices of soft particles have been selected. Ellipsoids have been used to model the scattering behavior of a large variety of biological systems such as microorganisms and red blood cells, and offer the possibility of exploring geometrical extremes between spheres and needle-like particles. The semimajor and semiminor axes for prolate ellipsoid were determined for a volume equivalent to 1 m diameter sphere. For absorption it is apparent from equation 2.9 that the pathlength for the particle is not relevant since the absorbed power is proportional to the volume. Nevertheless, scattering is proportional to the cross sectional area and therefore orientation. For a prolate ellipsoid with length a along its semimajor axis and lengths b=c (
Orientation A Orientation B Orientation C Figure 4.3: Fixed Orientations A, B, and C for an Ellipsoid For each of these orientations the projected cross sectional area in the laboratory xy plane and average pathlength along the laboratory z axis were calculated. These projected values were used to compute the transmission (using the RDG form factors as presented earlier). The algorithms for Rayleigh-Debye-Gans calculations for non-spherical particles with volumes equivalent to that of a 1 m diameter sphere, were tested against Mie calculations for the sphere, using the relative refractive indices in the range where the RDG assumptions are met; for these case studies the refractive indices of soft 31
bodies were used. The multiwavelength turbidity spectra were calculated for soft body ellipsoids for two eccentricity values and three fixed orientations. The multiwavelength turbidity spectra for the soft body ellipsoids are plotted along side the calculated turbidity using Mie theory. Figure 4.4 shows the effect of the particle orientation (A, B, C) for a prolate ellipsoid with an eccentricity of 0.3. Figure 4.5 shows the turbidity for an ellipsoid with an eccentricity of 0.8. (An eccentricity of 1 for an ellipsoid results in a sphere; thus the predicted spectra for RDG should be the same as for Mie theory.) Comparison of ellipsoids with varying eccentricities (figures 4.4 and 4.5) shows that particle orientation and shape have a significant effect on the features and amplitude of the multiwavelength turbidity, in the RDG model. Multiwavelength transmission spectra contain quantitative and qualitative information on shape and orientation of non-spherical particles, and it should be possible to extract this information from carefully measured spectra. This conclusion is in agreement with the results reported in Buehler . 32
200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Wavelength (nm)Optical Density RDGARDGBRDGCMie Figure 4.4: Calculated Transmission of Soft Body Prolate Ellipsoid =0.3 with a Volume Equivalent to a 1 m Sphere 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 Wavelength (nm)Optical Density RDGARDGBRDGCMie Figure 4.5: Calculated Transmission of Soft Body Prolate Ellipsoid =0.8 with a Volume Equivalent to a 1 m Sphere 33
Chapter Five Validation Study of Rayleigh-Debye-Gans Theory This chapter scrutinizes Rayleigh-Debye-Gans theory by simulating a broad range of refractive indices and particle sizes, probing the limitations imposed by the assumptions and approximations implicit in the theory. 5.1 Exploration of Theoretical Limits The limits of Rayleigh-Debye-Gans model were discussed in chapter two. To summarize: the relative refractive index m must be close to one and the size of the particle must be much smaller than 1m There exists a trade off for the limits of RDG theory; first, if m is close to one and no absorption is present then the size of the particle can be the same order of magnitude as the wavelength. Conversely, if absorption is present and m is greater than one, the particle size must be smaller than the wavelength. Although these assumptions are invoked in the derivation of the theory, scope of the present research and the complexity of its models calls for a reevaluation of these restrictions for multiwavelength measurements where strict adherence to the limits cannot be followed. Three approaches were taken to explore, through simulation, the constraints of this theory for spheres. First, the sizes of the spherical particles were kept small compared to the wavelengths, but the wavelength-dependent relative refractive index was allowed to significantly exceed one typical of actual materials. 34
Second, the relative refractive index was kept close to one while the absorption was held at zero, and particle sizes comparable to the wavelengths were considered. Third, the contribution of absorption in the relative refractive index, kept close to 1, was investigated for particle sizes comparable to the wavelengths. The following subsections describe in more detail the parameters used and the conclusions and observations drawn. 5.2 Particle Diameter << Wavelength The first of the sensitivity studies conducted tested the limits of Rayleigh-Debye-Gans for relative refractive indices greater than one and the absorption greater than zero, while keeping small sized spherical particles, compared to the wavelengths (200 nm-900 nm). The multiwavelength transmission spectra were calculated for Mie and Rayleigh-Debye-Gans using spheres of silver bromide ( 4.21. 1 onn 0 85.00001. ) and spheres of silver chloride ( 4.21. 1 onn 6.00001.0 ). The spherical diameter sizes chosen were 25 nm and 50 nm. Particle concentration, particle density, and wavelength range were kept constant Figures 5.1, 5.2 show that Rayleigh-Debye-Gans gives an adequate approximation to Mie for particle sizes much smaller than the wavelength. Figures 5.3, 5.4 reveals that for slightly larger particles Rayleigh-Debye-Gans no longer closely follows Mie Theory. Notice that in the spectral region where absorption is small (300-900 nm) both theories coincide even though n/n o >1. However, where strong absorption is present, the theories rapidly diverge (see the optical properties of AgBr and AgCl reported in Appendix B), clearly suggesting that absorption plays an important role in the disparity between the theories. 35
200 300 400 500 600 700 800 900 0 5 10 15 20 25 30 35 40 45 Wavelength (nm)Optical Density RDGMie Figure 5.1: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm AgBr Spheres 200 300 400 500 600 700 800 900 0 5 10 15 20 25 Wavelength (nm)Optical Density RDGMie Figure 5.2: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm AgCl Spheres 36
200 300 400 500 600 700 800 900 0 10 20 30 40 50 60 Wavelength (nm)Optical Density RDGMie Figure 5.3: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm AgBr Spheres 200 300 400 500 600 700 800 900 0 5 10 15 20 25 Wavelength (nm)Optical Density RDGMie Figure 5.4: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm AgCl Spheres 37
5.3 Particle Diameter ~ Wavelength, No Absorption The restriction of Rayleigh-Debye-Gans theory with respect to size was tested through the calculation of transmission spectra for nonabsorbing spherical particles with relative refractive index close to one. The refractive indices chosen were soft bodies ( 04.1 n ) and hemoglobin ( 2.101. 1 onn ). Only the real part of the refractive index was used for hemoglobin. Particle diameters used were 500 nm, 1 m, and 5.5 m. Figures 5.5 and 5.6 show that Rayleigh-Debye-Gans theory approximates Mie theory for 500nm and 1m. Figure 5.7 shows that for a particle size of 5.5 m Rayleigh-Debye-Gans and Mie no longer coincide. The combination of zero absorption and refractive index ratio close to 1 increases considerably the particle size ranges for which RDG is applicable; this is in agreement with the results reported in Kerker . 200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Wavelength (nm)Optical Density RDGMie Figure 5.5: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Soft Body Spheres 38
200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Wavelength (nm)Optical Density RDGMie Figure 5.6: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Soft Body Spheres 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 Wavelength (nm)Optical Density RDGMie Figure 5.7: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 5.5 m Soft Body Spheres 39
The multiwavelength transmission calculations conducted with only the real part of the refractive index of hemoglobin show that for 500 nm diameter particles (figure 5.8), the theories follow one another closely in spectral shape but there are quantifiable differences in amplitude. If the turbidity is used for analysis the spectral differences between the two theories would result in considerable variation in the estimate of particle size and concentration. With increasing of the particle diameter to 1 m (figure 5.9), the spectral shape for Mie theory relative to Rayleigh-Debye-Gans flattens considerably at the shorter wavelengths. Figure 5.10 shows a semi-logarithmic turbidity plot of 5.5 m particles to show the differences in shape and amplitude for the two theories. The effect of a relative refractive index greater than one with no absorption results in a limited particle size range for RDG theory, in contrast to particles with a refractive index close to one with no absorption. 40
200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 Wavelength (nm)Optical Density RDGMie Figure 5.8: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Hemoglobin Spheres with =0 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 Wavelength (nm)Optical Density RDGMie Figure 5.9: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Hemoglobin Spheres with =0 41
200 300 400 500 600 700 800 900 10-1 100 101 102 Wavelength (nm)Optical Density RDGMie Figure 5.10: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 5.5 m Hemoglobin Spheres with =0 5.4 Particle Diameter ~ Wavelength, Absorption > 0 The limits of validity Rayleigh-Debye-Gans theory with a relative refractive index close to one and an absorption value greater than zero were tested through the calculation of the transmission spectra for spherical particles whose sizes are comparable to those of the multiwavelength range. The refractive indices of whole hemoglobin ( 2.101. 1 onn 1.0001.0 ), meaning the real and imaginary part of the complex refractive index were considered. The particle diameter sizes used were: 100 nm, 500 nm, and 1 m. Figure 5.11 shows that Rayleigh-Debye-Gans and Mie closely follow one another for a 100 nm sphere. As the particle size was increased to 500 nm and 1 m the calculated turbidity from Rayleigh-Debye-Gans slowly deviates from Mie (see figures 5.12 and 42
5.13). As the size increases, the features of the spectra calculated with Mie theory flatten. This observed difference appears to be caused by absorption cross section C abs which is proportional to the volume in case for RDG theory, but not in the case of Mie theory. 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Wavelength (nm)Optical Density RDGMie Figure 5.11: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 100 nm Hemoglobin Spheres 43
200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 Wavelength (nm)Optical Density RDGMie Figure 5.12: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Hemoglobin Spheres 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength (nm)Optical Density RDGMie Figure 5.13: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Hemoglobin Spheres 44
45 5.5 Conclusion There is disagreement between Rayleigh-Debye-Gans and Mie theory for transmission simulations with spherical scatterers of different sizes and refractive indices. The disagreement is most severe when absorption is present. The rest of this dissertation will be concerned with attempts to modify RDG theory to bring the transmission simulations into closer agreement with Mie.
Chapter Six Corrections to the Refractive Index This chapter is dedicated to examining the effect of the complex index of refraction on the transmission characteristics of a particle. Schemes for adjusting the complex refractive index to bring RDG predictions into agreement with Mie theory are presented. 6.1 Refractive Index The complex refractive index is given by inN 6.1 where n and are non negative values ,n is the refractive index (real), is the absorption coefficient (imaginary). The scattering of light is due to differences in refractive indices between the medium and the particle. The refractive index of the particle (N 1 ) relative to the suspending medium (N 0 ) is, 001101ininNNm 6.2 referred to as the relative refractive index, as presented in chapter two The real and imaginary parts of the complex refractive index expressed as function of frequency, are related through the integral Kramers-Kronig relations. 46
02221)(dkPn 6.3 022,2)(dnP 6.4 here is the angular frequency measured and P is the principal value of the integral . In principle, if either n() or () is known or can be measured, the other can be calculated directly through equations 6.3 and 6.4. Measurements over the complete range of frequency (0 to ) are required when applying this transform. 6.2 Hypochromic Effect The observable light scattering phenomena depends on the instrumentation configuration and optical properties of the material. The optical properties (real and imaginary parts of the refractive index) are intrinsic properties of matter. It is known that the optical properties depend on the state of aggregation . However, under certain conditions (i.e. infinite dilution) the optical properties are additive and independent of concentration. The presence of absorbing groups (chromophors) in high concentration within particles gives rise to a concentration dependence of the observed optical phenomena. This phenomenon generally results in a decrease of the imaginary component of the refractive index relative to its value in solution (hypochromicity). Hypochromicity is a phenomenon in which an individual molecule, containing several chromophores, has a certain absorptivity at a given wavelength that is less than the sum of the absorptivities of the individual chromophores at that same wavelength. When analyzing particulate systems, the state of aggregation of chromophores within the 47
particles may result in hypochromic effects that bias the estimation of their concentration. For this reason, hypochromism was used to determine a correction factor for the imaginary part of the refractive index. The most recent models for hypochromicity are those developed by Veshkin [11,12,and 13] and take into consideration the molecular structure and the number of chromophoric groups per unit volume of particle. The procedure of Veshkin was extended to the multiwavelength scenario and implemented; details are given in Appendix D. The differences in behavior observed between the spectra calculated with Mie and with RDG theory suggests, in agreement with the work of Latimer , that it may be possible to compensate RDG theory through the use of effective optical properties estimated from particles of known shape and composition. 6.3 Implementation of Optical Correction for Absorption Rayleigh-Debye-Gans is limited to small changes in refractive index n() close to one and small values of absorption (). Rayleigh-Debye-Gans theory assumes each dipole absorbs and scatters independently and only considers the interference of the scattering wave. As a result, the angular scattering intensity is shape and orientation dependent, whereas the absorption cross-section is independent of the particle shape (equation 2.9); in other words, the total absorption is only dependent on the particle volume (the total number of chromophoric groups in the particle). When compared with Mie the latter causes a large discrepancy where the absorption efficiencies calculated with Mie theory always smaller (hypochromic) than the values calculated with RDG for large absorption coefficients (i.e., Hemoglobin, DNA). This apparent hypochromicity suggests that the 48
theoretical models developed to account for hypochromic or screening effects may be able to bring RDG into a better agreement with Mie. To explore the potential application of Veshkins correction the volume fraction of chromophoric groups (v f in equations 6.126.21) is treated as an adjustable parameter. Two cases are considered: v f =0 corresponds to 100% hypochromicity which translates to the corrected c () being equal to zero; and v f =1 corresponds to using the value of () directly. Spherical hemoglobin particles with a diameter of 1 m were where Veshkins correction was applied only to (). The volume fraction values used in this study were 0.15, 0.20, 0.33, and 0.50. The molecular diameter of hemoglobin is 68 with the cross sectional area of 20 and a molecular weight of 16100 . The orientation value q was set to one, meaning the molecules are randomly oriented . The results of the hypochromicity corrections implemented in RDG theory are shown in figures 6.1 and 6.2. Figure 6.1 shows the spectra calculated with RDG and Mie without any corrections for hypochromicity, together with the spectra calculated with RDG and several levels of hypochromicity (i.e., volume fractions). Notice that, although intermediate levels of hypochromicity result in improved RDG-calculated spectra, Veshkins model is not very effective in reducing the differences between Mie and RDG theories. This point is demonstrated more dramatically when 100% hypochromicity is considered. Figure 6.2 shows the extreme cases of 0% and 100% hypochromicity applied to both theories. 49
200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength (nm)Optical Density Mie-No correctionRDG-No correctionc15c20c33c50 Figure 6.1: Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength (nm)Transmission Mie=100%RDG=100%Mie=0%RDG=0% Figure 6.2: Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with 0% and 100% Hypochromicity 50
The use of Veshkins model for the correction of the absorption coefficient brings about the problem of the inconsistency in terms of the Kramers-Kronig transforms since, after the correction, equations 6.3 and 6.4 will no longer hold. To demonstrate this inconsistency, an effective value of n eff () was calculated through the Kramer-Kronig transform after () was corrected, using Veshkins model. All the conditions were kept the same for calculating the transmission as previously in this section. Figure 6.3 shows the results of calculating the transmission for Rayleigh-Debye-Gans with an effective n eff and a corrected c using Veshkins model compared, to uncorrected values of and n for RDG and Mie theory. 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength (nm)Optical Density RDG-No correctionMie-No correctionneff&15neff&20neff&33neff&50 Figure 6.3: Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction to k c and an Effective n eff Calculated through Kramers-Kronig Transform 51
Using the effective n calculated from corrected c one would expect the transmission by RDG more closely the transmission calculated by Mie; however, the contrary is observed. A close look at the transmission values of figure 6.3 can be seen in figure 6.4. At different volume fractions of the chromophore relating to k c and n eff values, there are distinct differences in the shape of the spectra. The differences in the spectra are rooted in determining the values n from a revised using the Kramers-Kronig transform. 200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Wavelength (nm)Optical Density neff&c15neff&c20neff&c33neff&c50 Figure 6.4: Zoom in of Figure 6.4 of Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction to k c and an Effective n Calculated through Kramers-Kronig Transform An alternate approach for bringing together Mie and RDG theories is the mathematical adjustment of the refractive indices at each wavelength. This is explored in the next section. 52
6.4 Effective Refractive Index Estimation The concept behind calculating effective refractive indices is that given the absorption efficiency Q abs determined by Mie theory there is a set of n() and () values that would allow Rayleigh-Debye-Gans to predict some extinction efficiency Q ext to coincide with the extinction efficiency calculated by Mie. The absorption efficiency for either Rayleigh-Debye-Gans or Mie theory was expressed in chapter two as: kamkammkaQabs38132Im421Im422 6.5 The scattering efficiency Q sca in RDG is expressed (equation 2.24): dfmmkaQscasincos1212022224 6.6 For simplification introduce which isindependent of n and for 022sincos1df 1m 2242419432132nkainkaQsca 6.7 Therefore, the extinction efficiency can then be expressed as the sum of equations 6.5 and 6.7 kakankankakaQext3894194194382424224 6.8 If we assume Q ext and n are known, then we can solve for explicitly using the quadratic formula. 53
The same type of algebra manipulation can be done to solve for the refractive index n if Q ext and are known. okoextnqQn11,1 6.9 onoextnqQqqq1,1222124 6.10 Equations 6.9 and 6.10 are written in terms of relative values. Note that for equations 6.9 and 6.10 to give real effective values for n 1 and 1 ; Q ext no Q ext,ko q 1 and q 2 must always be positive. The artificiality of this mathematical juggling is clear; changing only or n, and not the other, would lead to a violation of the Kramer-Kronig relations. Therefore the implementation of this adjustment is rejected. 6.5 Conclusion The effect of changes in the refractive indices has been explored as a means to extend the range of application of RDG and to bring it into better agreement with Mie theory for larger particles and for particles containing strong chromophoric groups. It was concluded that, it is not possible to adequately or realistically compensate for the differences between Mie and RDG through the use of hypochromicity models and/or effective refractive indices. Therefore, a different type of approach is required. The following chapter discusses exploiting the internal field calculation of Mie theory as a vehicle to improve the Rayleigh-Debye-Gans and Mie conflict in the presence of absorption. 54
Chapter Seven New Hybrid Theory The hybrid theory developed in this chapter uses the Mie solution to compute the internal field of a sphere, and the Rayleigh-Debye-Gans approach to solve for the scattering fields. The induced Rayleigh-Debye-Gans dipole moment is computed from the internal Mie field, rather than the incoming field. From this we solve for the scattering fields in terms of the parallel and perpendicular components of the incoming light. Form factors were generated through the scattering amplitude functions. The following is the mathematical development of the hybridized theory for a spherical particle. 7.1 Geometry and Notation Figure 7.1: Diagram of Scatterer Point and Detector Location 55
For describing the electric field scattered by a particle in the laboratory system there are two objects of interest, the detector and scatterer. Figure 7.1 illustrates the detector located at r with spherical coordinates (r,,) or Cartesian coordinates (x,y,z). Points within the scatterer are identified by R with coordinates (R,,) or (X,Y,Z). The curvilinear unit vectors attached to the detector in figure 7.2 can be expressed in rectangular coordinates through the following equations: cossinsincossinrzryrx 7.1 yxzyxzyxreeeeeeeeeee cossinsinsincoscoscoscossinsincossin 7.2 Figure 7.2: Local Unit Vectors with Respect to the Detector 56
Figure 7.3 similarly depicts the unit vectors attached to a point inside the scatterer. The transformation equations for these vectors are identical to equations 7.1 and 7.2, with corresponding subscripts and angles. Figure 7.3: Local Unit Vectors with Respect to the Scatterer As indicated, the incident wave moves in the z-direction and is presumed to be plane-polarized in the x-direction. It impinges upon the particle and is scattered. The scattered wave is detected at some angle and measured from the direction of propagation of the incident wave; see figure 7.1. The following section provides a mathematical description of the fields induced by the particle. As will be seen, the scattering dynamics are best described using the vectors e eeR ,, ; the scattered radiation is best described by eeer ,, Therefore the transformation equations play an important role in unifying the description. 57
7.2 Internal Field The incoming field for light illuminating a spherical particle, propagating in the z-direction, and polarized in the x-direction, is described in generic Cartesian coordinates as xtiikoieeeEtE ,,, 7.3 where k is the wave number in the medium. The time factor will be omitted in the following. tie The resulting field inside the sphere is given by Mie theory as 11111112nnEnnOnoNidMcnnnERE 7.4 where M and are the solutions to the vector wave equation in terms of Bessel functions and spherical harmonics . We truncate the series above in the following manner N 2122111111652323aONdiNidMcEREEEOo 7.5 where a is the radius of the spherical particle and is the wavelength. Bohren and Huffman provide the general expressions for the terms M and N as series themselves, which we also truncate as: 211111cossin3cos3RkOeRkeRkRMO 7.6 2111sin32coscos32sincos32RkOeeeRNRE 7.7 58
21121112cossin531cos2cos53cossincos56RkOeRkeRkeRkRNRE 7.8 where k 1 is the wave number inside the sphere. The coefficients for c n and d n are calculated through mkamkajahkakahmkajkakajkahkakahkajcnnnnnnnnnk1111111 7.9 mkamkajkahkakahmkajmkakajkamhkakahamjdnnnnnnnnn11121111k 7.10 where is the permeability of the sphere and is presumed to equal the permeability of the medium, and k is the wave number in the medium. The primes denote differentiation with respect to ka. The expressions of 7.6, 7.7 and 7.8 are translated to rectangular coordinates as zxOeXkeZkM331111 7.11 xEeN 3211 7.12 zxEeXkeZkN 11125353 7.13 resulting in 21111111122RkOeXikcdeZikcddEREzxo 7.14 This expression can be written in exponential form to the same order of accuracy; since 21xOxex 7.15 59
we implement the following, ,212212111112121111111111RkOedZikcddRkOeXikcdZikdcdXikcd 7.16 resulting in the following approximation for the Mie field inside the sphere. zXcdkixZdcdkioeeeedERE 12211211121 7.17 Note that in the limit as if we have ,, then kk1 11d 11c 12d xikZoeeERE the incoming field value. In chapter eight we demonstrate by computer studies that, indeed, c 1 d 1 and d 2 all equal 1 when k 1 =k. This is consistent; if the dielectric properties of the scatterer match those of the medium the incoming field is unaltered. 7.3 Dipole Scattering Approach Electromagnetic theory states that a dipole located at R of intensity tieRp radiates in the far field according to the following : RpeeeikRrikeERrRrtiRriks43 7.18 where is the scattered electric field radiated by the dipole and is the permittivity or dielectric constant of the medium. It also states that a small dielectric sphere of radius placed in a uniform static electric field sE E generates a dipole moment. The induced dipole moment is proportional to the field and is given by 60
dVEmmEmmp2133421322322 7.19 where is the permittivity, m is the relative refractive index, and dV is the volume of the scatterer. Rayleigh scattering assumes that an oscillating, nonuniform field tieRE generates a dipole moment in a spherical volume given by the same expression in equation 7.19 and that the dipole re-radiates according to equation 7.18. Following the RDG approach, we assume that each infinitesimal volume within the scatterer (not the entire scatterer itself) behaves in this fashion. By substituting equation 7.19 into 7.18 we obtain the following expression for the incremental electric field radiated by the infinitesimal dipole located at R : dVREmmeeikRrikeEdRrRrRriks2134223 7.20 For rR Rayleigh approximates rRr1 1 rRree and rreRikrieRrikRrikeeeek When substituting these approximations into equation 7.20 the following expression is obtained. REeedVeemmrkEdrreRikikrsr2143222 7.21 61
7.4 Hybrid Theory The difference between Rayleigh-Debye-Gans theory and the hybrid theory presented herein is as follows: RDG assumes that the local field RE generating the infinitesimal dipole in equation 7.21 is given by the incoming field, whereas the hybrid theory takes the internal field that Mie theory gives for the sphere as the field inducing the dipole moment. By using the internal Mie field, we are taking some account of the effect of the surrounding dipole field alterations to the incoming field (such as attenuation, which is highlighted in chapter five as a major shortcoming in RDG theory). The validity of either approach presumes that the incoming electric field is roughly uniform over the sphere, so that the radius a of the sphere must be a small fraction of the wavelength (a<< ). If we substitute the expression for the internal electric field, equation 7.17, into equation 7.21 an explicit formula for the scattered electric field is obtained. ozZdcdikxZdcdikrreRikikrsEeeeedeedVeemmrkEdr121431111112121222 7.22 In order to evaluate xrreee and zrreee one has to use the identities in equations 7.1,2. The following expression is a result of the conversion and mathematical manipulation, with the identifications zeRZ and xeRX eeeeeddVeemmrkEEdxrreRdcdikeRdcdikeRikikrossin1coscossin21431111112121222 7.23 62
As in Rayleigh-Debye-Gans theory, we sum (integrate) this over the total scatterer volume. We introduce f 1 and f 2 as form factors for the sphere: efefefVmmrkeEEdEikrosssinsincoscos2143121222 7.24 where dVedVdVeedVfrzrzekedcdkRieRieRdcdik1121112121k21111, 7.25 dVeVdVeVdVeeVfrrxrxekRiekecdkRieRikeRcdik1111,222121121 7.26 Note that if since (as noted above) kk1 1211 ddc f 1 reduces to eVZeeRik1 dVR the form factor f in the RDG theory. Furthermore observe that the factor f 2 which does not appear in the RDG theory, goes to zero when kk1 The problem now becomes how to calculate the integrals in equations 7.25 and 7.26. They all have the form dVeSRi with constant S Consider a local coordinate system in the sphere with its z axis aligned with S If we look at figure 7.4, the element of volume at height z is '''22dzyxheightbasedV 7.27 63
Figure 7.4: Diagram of the Volume at Height z for a Sphere However and z runs from -a to a; therefore, 2222'''azyx aazzSieRSiSRidzzaedVedVez'22'''' 7.28 We can use Maple to perform these integrals. The results are 331121AieiAaeeiAaedVfiAaiAaiAaiAa 7.29 3333221CieiCaeeiCaeBieiBaeeiBaeVfiCaiCaiCaiCaiBaiBaiBaiBa 7.30 kkCcdkkcdkkBdcdkkdcdkkA212121221212121212212sincos4cos14 7.31 64
Observe from figure 7.2 that and are the detector angles and that e is in the scattering plane while e is perpendicular. Therefore parallel and perpendicular components of the scattered field equation 7.24, are expressed in terms of the form factors as sincoscos214321222||,ffVmmrkeEEikros 7.32 sin21431222,fVmmrkeEEikros 7.33 The scattering intensity is given by 7.34 2|2||,Re21sssEEI 7.5 Scattering Amplitude Matrix Formulation for the Hybrid Model Note that in this new model the scattered field can still be expressed using a scattering matrix in the manner of Van de Hulst, Bohren and Huffman, and Kerker. To do so, the incoming field must be expressed in terms of its components parallel and perpendicular to the scattering plane. In spherical coordinates the incoming field is given by eeeeEErzioisincoscoscossink 7.35 Here e is perpendicular to the scattering plane while the unit vector ree sincos lies in the plane. As a result the incoming field can be written as: sincos,||,ikzoiikzoieEEeEE 7.36 65
After some manipulation the scattered field (equations 7.32, 7.33) can be related to the incident field in a scattering matrix format: iissEEfffVmmrkEE,||,121222,||,00cossincos2143 7.37 7.6 Scattering Intensity Ratio and Turbidity The scattering intensity ratio is expressed using equation 7.24. 21221222222422sinsincoscos21329fffVmmrkEEIIosos 7.38 It can be written in terms of the scattering amplitude matrix equation 7.37; however this is not recommended due to the singularity 1cos The formula for turbidity in the hybrid model is derived by calculating C sca from insertion of equation 7.38 into equation 2.6 from chapter two; the absorption cross section C abs remains as in equation 2.9: 21Im322mmkVCabs 7.39 the turbidity is finally determined by (equation 2.16) extpabsscaplGQNCClN 7.40 66
Chapter Eight Validation and Sensitivity of Hybrid Theory The previous chapter gave a detailed mathematical description of the hybrid model. This chapter is dedicated to performance evaluation of the model in comparison to those of Rayleigh-Debye-Gans and of Mie through a series of transmission simulation studies. First the hybrid model was tested with the propagation constant of the medium equal to that of the particle to verify correct programming implementation. Second the validity of the hybrid theory using various particle sizes was tested for relative refractive indices close to one. The third study tests the hybrid theorys effectiveness by introducing absorption through the imaginary part of the refractive index. The last study investigates the behavior of the hybrid theory for refractive indices exceeding the conditions required for Rayleigh-Debye-Gans theory, that is, having strong scattering and absorption components, for various diameter sizes. The other parameters (concentration, pathlength, etc.) required to calculate the transmission were kept constant ,as listed in table 5.1. 8.1 Validation of Hybrid Theory Implementation As we indicated in chapter six, when the internal Mie field approaches the incoming field; c kk1 1 d 1 and d 2 all approach 1. The calculations displayed by figure 8.1 confirm that 1211 ddc when kk 1 67
200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength (nm)Coefficients c1d1d2 Figure 8.1: Coefficients c 1 d 1 and d 2 Versus at the Limit when k 1 =k Another effect of taking k 1 equal to k is that the form factor f 1 of the hybrid model for spheres reduces to the form factor f of Rayleigh-Debye-Gans for spheres (and f 2 goes to zero). Figures 8.2 and 8.3 are 3-D images of these form factors graphed by wavelength and azimuthal angle of observation (with 0 ). Figure 8.2 shows f 1 of the hybrid model. The form factor f of Rayleigh-Debye-Gans is shown in figure 8.3. As an example of the case where k 1 (particle) is different from k (medium), the refractive indices of polystyrene ( 6.001.01 ) in water ( 0 ) were used to illustrate the behavior of the form factors for the hybrid model compared to that of Rayleigh-Debye-Gans. Figures 8.4, 8.5, 8.6, and 8.7 are the real and imaginary parts of the form factors f 1 and f 2 for the hybrid model. Figure 8.8 is the form factor for Rayleigh68
Debye-Gans. Note that the hybrid models factors contain real and imaginary parts even if 0 (nonabsorbing). Therefore our working program for the hybrid theory has been validated. The next sections will reveal the superiority of the hybrid theory in approximating the exact (Mie) solution, for different particle diameters and relative refractive indices. Figure 8.2: Form Factor f 1 for Hybrid Theory at k 1 =k 69
Figure 8.3: Form Factor f for Rayleigh-Debye-Gans Theory Figure 8.4: Real Part of Form Factor f 1 for Hybrid Theory using Polystyrene 70
Figure 8.5: Imaginary Part of Form Factor f 1 for Hybrid Theory using Polystyrene Figure 8.6: Real Part of Form Factor f 2 for Hybrid Theory using Polystyrene 71
Figure 8.7: Imaginary Part of Form Factor f 2 for Hybrid Theory using Polystyrene Figure 8.8: Form Factor f for Rayleigh-Debye-Gans Theory using Polystyrene 72
8.2 Case 1: Relative Refractive index 0nn ~1 and Absorption = 0 The validity of the hybrid theory was tested against Rayleigh-Debye-Gans and Mie theory using the relative refractive indices of soft bodies ( 04.1 onn ) to calculate the transmission. The spherical diameter sizes used were 50, 100, 250, and 500 nm. The results of this study are shown in transmission spectral plots provided in figures 8.9, 8.10, 8.11 and 8.12. Figures 8.9 (50 nm) and 8.10 (100 nm) show that the hybrid theory for very small particles at the shorter wavelengths is a much better approximation to Mie theory than is RDG theory. At wavelengths much larger than the particle size, the hybrid spectrum is still superior to Rayleigh-Debye-Gans. In figure 8.11 the hybrid model for 250 nm particles closely estimates Mie theory above 300nm wavelength and outperforms RDG even down to 200 nm wavelength (which is shorter than the diameter). A significant change in the calculated transmission spectra is observed in figure 8.12, where the diameter size is 500 nm. Here the hybrid spectrum no longer resembles that of Mie theory or RDG at wavelengths shorter than half the diameter. Nonetheless, for larger wavelengths the hybrid model again is a better approximation to the exact Mie theory than Rayleigh-Debye-Gans. The inset of figure 8.12 is a zoom in of the spectra between 500 nm to 900 nm wavelength showing the hybrid model approximating better than RDG at the longer wavelengths. In summary, the hybrid theory is seen to be a vastly improved model for estimating the transmission for nonabsorbing soft particles whose diameter is smaller than the wavelength. The following section will explore the effect of including absorption in the hybrid model. 73
200 300 400 500 600 700 800 900 0 0.005 0.01 0.015 0.02 0.025 0.03 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie Figure 8.9: Comparison of Calculated Transmission for 50 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 200 300 400 500 600 700 800 900 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie Figure 8.10: Comparison of Calculated Transmission for 100 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 74
200 300 400 500 600 700 800 900 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie Figure 8.11: Comparison of Calculated Transmission for 250 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie 0.08 Figure 8.12: Comparison of Calculated Transmission for 500 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 500 550 600 650 700 750 800 850 900 0.07 0.06 0.05 0.04 0.03 0.02 0.01 75
8.3 Case 2: Relative Refractive index 0nn 1 and Absorption > 0 The previous section showed that the hybrid model provided an improved approximation to Mie theory for nonabsorbing particles and whose relative refractive index is greater than or equal to one. In this section the contribution of absorption is included in the refractive index n, while the relative refractive 0nn was kept close to one. The optical properties of polystyrene ( 6.001.0,5.11. 1 onn ) were chosen in keeping with the requirements aforementioned. The diameter sizes selected for the transmission calculations were, again, 50, 100, 250, and 500 nm. Figures 8.13 and 8.14 are plotted on a semi log scale to enhance the features of the spectra. For particles diameters of 50 and 100 nm, figures 8.13 and 8.14, the calculated transmission by the hybrid model continues to approximate Mie theory closer than Rayleigh-Debye-Gans. As the particle diameter is increased to 250 nm and 500nm, figure 8.15 and 8.16, interestingly the hybrid spectra qualitatively mimics the reduced transmission features displayed by Mie theory at wavelengths shorter than the diameter. At wavelengths comparable to or larger than the diameter, the hybrid model remains the better estimate to Mie theory. The insets of figures 8.15 and 8.16 show the spectra where the particle diameter is that of the wavelength and emphasize how well the hybrid model behaves compared to RDG in approximating Mie theory. Evidently from the graphs, the differences between the incoming field and the Mie field are quite significant when absorption is present. The results demonstrate that the hybrid model provides an improved approximation over Rayleigh-Debye-Gans theory for absorbing scatterers whose relative refractive index is close to one, over a very large 76
range of wavelengths. The next section studies the behavior of the hybrid model for absorbing scatterers whose relative refractive index is approximately 1. 200 300 400 500 600 700 800 900 10-2 10-1 100 101 Wavelength (nm)Optica Density HybridRayleigh-Debye-GansMie 8.13: Comparison of Calculated Transmission for 50 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 77
200 300 400 500 600 700 800 900 10-3 10-2 10-1 100 101 102 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie 8.14: Comparison of Calculated Transmission for 100 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 200 300 400 500 600 700 800 900 0 10 20 30 40 50 60 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie 8 8.15: Comparison of Calculated Transmission for 250 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 300 400 500 600 700 800 900 7 6 5 4 3 2 1 0 78
79 200 300 400 500 600 700 800 900 0 10 20 30 40 50 60 70 80 90 100 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie 1 0.5 0 1.5 800 850 900 500 550 600 650 700 750 8.16: Comparison of Calculated Transmission for 500 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 8.4 Case 3: Relative Refractive index 0nn ~1 and Absorption > 0 Now we turn to more general scatterers, with large relative indices of refraction and nonzero absorption. Hemoglobin ( 15.001.0,2.1 1 onn ) is both a strong scatterer and strong absorber and thus a good test study. The diameter sizes used to calculate the transmission were 50, 100, 250, and 500 nm. Figures 8.17, 8.18, 8.19, and 8.20 show that at 50, 100 and 250 nm diameters the hybrid theory spectra approximates Mie theory better than does Rayleigh-Debye-Gans at all wavelengths, and at the diameter size of 500 nm (figure 8.20), the hybrid model provides a better estimate to Mie theory than Rayleigh-Debye-Gans for wavelengths larger than 300nm.
200 300 400 500 600 700 800 900 0 0.5 1 1.5 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie 8.17: Comparison of Calculated Transmission for 50 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie 8.18: Comparison of Calculated Transmission for 100 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 80
200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie 8.19: Comparison of Calculated Transmission for 250 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 Wavelength (nm)Optical Density HybridRayleigh-Debye-GansMie 8.20: Comparison of Calculated Transmission for 500 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 81
82 8.5 Conclusions The hybrid model for submicron spheres has been shown to approximate Mie theory much better than Rayleigh-Debye-Gans for part icle sizes smaller than the wavelength. For a wide range of relative refractive indice s the improvement is particularly marked for absorbing materials. For the cases were absorption is introduced, the curves displayed that attenuation becomes more significant for the larger particles, and the hybrid theory is superior in accommodating attenuation. The results of the simulations conducted in this chapter demonstrate that the st rategy of using the Mie internal field, rather than the incoming field, to energize the RDG dipoles reaps very significant benefits. One important benefit from the hybrid model wa s the computation time for calculating these spectra which was rapid.
Chapter Nine Contributions and Future Work 9.1 Contributions Mie theory is an exact solution to the wave equation for spherical scatterers . This rigorous solution is limited to spheres and although it provides a good estimate for some characteristics of nonspherical particles, Mie theory cannot provide information on shape and orientation. Rayleigh-Debye-Gans theory is an approximation to Mie theory and provides, through form factors, information about nonspherical particles. The principal limitation of Rayleigh-Debye-Gans theory is that the complex relative refractive index must be close to one. Other theories exist for determining the light scattering behavior of nonspherical submicron particles, such as the T-matrix and Purcell-Pennypacker methods. In terms of real-time applications, these methods are more computationally intensive and therefore time consuming both in code generation and computer time. The methods available are time consuming to the extent of making them impractical for engineering applications such as real time particle characterization. The hybrid model presented here provides another tool for the analysis of submicron particles for real-time computations at multiwavelength. The key to the superior performance of the hybrid theory is the incorporation of the Mie field, rather the incoming field, to generate the scattering produced by a particle. This hybrid model has demonstrated vastly improved accuracy and applicability for a broader range of optical 83
properties than that of Rayleigh-Debye-Gans for multiwavelength particle characterization applications. 9.2 Recommendations and Future Work Like most original work, the hybrid theory can be improved for the spherical model. As the theory stands, the truncation at the first term of the series for the internal field can be extended to include second order terms. These second order terms will influence the series for particles whose size is comparable with the wavelength. In other words, at the shorter wavelengths the hybrid model does not exactly match Mie theory or Rayleigh-Debye-Gans, but by extending the series we can include terms that will validate the model where a/1. The hybrid model has been worked out for spherical particles; a proposed method of extending this model to other shapes such as ellipsoids could proceed by assuming that the internal field of the ellipsoid can be described by mapping the Mie internal field for a volume equivalent, or perhaps a circumscribing, sphere evaluated, as presented in chapter six, from the induced dipole moment using the postulated field. To account for the shape of the particle, the form factor dVeSRi needs to be evaluated for the ellipsoid shape, as demonstrated with a sphere in chapter six. These factors can be determined directly from table 4.1 by reinterpreting the constant vector S Although the mathematics may appear complex for the form factors, they are relatively straightforward though time consuming. 84
References  Bohren, Craig F. and Huffman, Donald R.; Absorption and Scattering of Light by Small Particles, Wiley Science Paper Series, 1998.  Van de Hulst, H.C.; Light Scattering by Small Particles, Dover Publications, Inc, New York, 1957.  Garcia-Rubio, L.H.; Averages from Turbidity Measurements, ACS Symposium Series, vol 332, 11, 161-178, 1987.  Garcia-Lopez, A.C.; Investigation into the Transition between Single and Multiple Scattering for Colloidal Dispersions, 2001.  Kerker, Milton; The Scattering of Light and other Electromagnetic Radiation, Academic Press, New York, 1969.  Singham Shermila B. and Bohren Craig F., Hybrid Method in Light Scattering by an Arbitrary Particle, Applied Optics, vol 28, No. 3, 1989.  Bohren Craig F, Backscattering by Nonspherical Particles: A Review of Methods and Suggested New Approaches, Journal of Geophysical Research, vol 96, No. D3, 1991.  Mishchenko Michael I., Travis Larry D., and Mackowski Daniel W., TMatrix Computations of Light Scattering by Nonspherical Particles: A Review, J. Quant. Spectrosc. Radiat., vol 55, No. 5, 1996.  Deepak Adarsh, Box Michael A.; Forward Corrections for Optical Extinction Measurements in Aerosol Media: Monodispersions, Applied Optics, vol 12, No.18, 1978.  Maron, Samuel H. and Prutton, Carl F.; Principles of Physical Chemistry, The Macmillan Company, New York.  Veshkin, N.L; Screening Hypochromism of Biological Macromolecules and Suspensions, Journal of Photochemistry and Photobiology, B: Biology, vol 3, 625-630, 1989. 85
 Veshkin, N.L; Screening Hypochromism of Molecular Aggregates and Biopolymers, Journal of Biological Physics, vol 25, 339-354, 1999.  Veshkin, N.L; Screening Hypochromism of Chromophores in Macromolecular Biostructures, Biophysics, vol 44, 1, 41-51, 1999.  Wiscombe, W.J.; Mie Scattering Calculations: Advances in Technique and Fast, Vector-Speed Computer Codes, NCAR/TN-140 + STR. National Center for Atmospheric Research, Boulder Colorado, 1979.  Latimer, Barber P.; Scattering by Ellipsoids of Revolution; A Comparison of Theoretical Methods, Journal of Colloid Interface Science, vol 63, 310-316, 1977.  Narayanan, Smita; Aggregation and Structural Changes in Biological Systems: An Ultraviolet Visible Spectroscopic Approach for Analysis of Blood Cell Aggregation and Protein Conformation, 1999.  Garcia-Rubio, L.H; Private Communication.  Buehler, Christopher S.; Measurement of Orientation Distribution of Spheriod Particles by Light Scattering, 1991. 86
Appendix A: Intensity Ratio and Turbidity Model Derivation of the intensity equation for RDG and non-polarized light: Beginning with the equation of light taken from Van de Hulst  2221||21)(,rkIiiIIIo A.1 where 222211,,SiSi A.2 The scattering functions are described as function of the polarizability cosk),(k),(3231iSiS A.3 were k is the propagation constant, m 2k and m is the wavelength in the material. The polarizability for particles with a refractive index close to 1 and homogeneous with no approximation for the complex refractive index is VmmdVmm214321432222 A.5 To account for the interference effects for each dipole, the phase factor is included to scattering function, equation (3) cosk),(k),(3231iieiSeiS A.6 The scattering function can be explicitly written in terms of the polarizablity by replacing equation A.5 into A.6 88
Appendix A (Continued) cos2143k),(2143k),(22322231mmVeiSmmVeiSii A.7 Let dVeVRi1),( this is the form factor; which is replace it in equation A.7 cos),(2143k),(),(2143k),(22322231RmmViSRmmViS A.8 Now that the scattering functions are defined in terms of shape, replace equation A.8 into equation A.2. cos),(2116k9),(),(2116k9),(22222262222222226211RmmVSiRmmVSi A.9 Take the equation above and replace it in equation A.1 oIRmmrVIII222222224||cos1),(2132k9, A.10 Equation A.10 is the intensity ratio model which mathematical describes the light scattered by an arbitrary particle in terms of a non-polarized light source. Per this derivation, it can be seen that the wavelength dependence of these functions. The transmission equation as described by Kerker is mathematically described as dDDfQGNextp)(0 A.11 89
Appendix A (Continued) For a monodispersed system f(D)dD is represented by a delta function, therefore its integral equal to one. The scattering efficiency factor is function of the absorption and scattering of the particle and is described as extQ absscaextQQQ A.12 The scattering efficiency factors are defined in terms of the particle scattering cross sectional C sca and its cross sectional area G. G CQscasca A.13 The general equation for the scattering cross section  ddrIICoscasin2 A.14 The intensity ratio in the equation above can be replaced with equation A.10 and evaluated as 02222224200222222242002222222224sin)cos1(),(21169sin)cos1(),(21329sin)cos1(),(21329dRmmVkddRmmVkddrRmmrVkCsca A.15 Substituting the C sca into equation A.13 results in 90
Appendix A (Continued) 91 02222224sin)cos1(),(21169dRmmGVkQsca A.16 The absorption efficiency factor is defined similarly to that of the scattering efficiency factor, equation A.13, except in terms of absorption. The absorption cross section C abs is defined as 21Imk322mmVCabs A.17 The absorption efficiency factor can then be expressed as 21Imk322mmGVQabs A.18 The transmission equation A.11 can be explicitly expressed in terms of the scattering and absorption components of the efficiency factor. 21Imk3sin)cos1(),(2116k92202222224mmGVdRmmGVGNp A.19 Equation A.19 is the general form of transmission equation for all form factors. Since the nomenclature of Kerker is being used, |R(| 2 =P(and can be replaced. 21Imk3sin)cos1(2116k92202222224mmGVdPmmGVGNp A.20
Appendix B: Optical Properties 92 200 300 400 500 600 700 800 900 1.32 1.34 1.36 1.38 1.4 1.42 Wavelength (nm)Refractive Indices n-refractive index Figure B.1: Optical Properties for Water 200 300 400 500 600 700 800 900 0 0.1 0.2 0.3 0.4 0.5 Wavelength (nm)Refractive Indices (n-1)-refractive index-absorption Figure B.2: Optical Properties for Soft Body
Appendix B (Continued) 93 200 300 400 500 600 700 800 900 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Optical Properties for HemoglobinWavelength (nm)Refractive Indices (n-1)-refractive index-absorption Figure B.3: Optical Properties for Hemoglobin 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Wavelength (nm)Refractive Indices (n-1)-refractive index-absorption Figure B.4: Optical Properties for Polystyrene
Appendix B (Continued) 94 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Wavelength (nm)Relative Refractive Indices (n-1)-refractive index-absorption Figure B.5: Optical Properties of AgCl 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 Wavelength (nm)Relative Refractive Indices (n-1)-refractive index-absorption Figure B.6: Optical Properties of AgBr
Appendix B (Continued) 95 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength (nm)Relative Refractive Indices n/no-relative refractive index Figure B.7: Relative Refractive Index of Soft Body in Water 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 Wavelength (nm)Relative Refractive Indices n/no-relative refractive index/no-relative absorption Figure B.8: Relative Refractive Index of Hemoglobin in Water
Appendix B (Continued) 96 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Wavelength (nm)Relative Refractive Indices n/no-relative refractive index/no-relative absorption Figure B.9: Relative Refractive Index of Polystyrene in Water 200 300 400 500 600 700 800 900 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wavelength (nm)Relative Refractive Indices n/no-relative refractive index/no-relative absorption Figure B.10: Relative Refractive Index AgCl in Water
Appendix B (Continued) 97 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 Wavelength (nm)Relative Refractive Indices n/no-relative refractive index/no-relative absorption Figure B.11: Relative Refractive Index of AgBr in Water
Appendix C: Validation for Rayleigh-Debye-Gans Theory 98 A validation study for the Rayleigh-D ebye-Gans theory was conducted, here RDG was programmed and tested against Mie th eory. The range of particle diameters was chosen between 25 nm -500 nm. The si mulation parameters used to define the suspensions for the transmission calculations are: light source wavelength 200-900nm particle concentration 1E-4 g/cc, particle density 1g/cc. C.1 Validation of Rayleigh-Debye-Gans Theory The limits of applicability of RDG theory ar e established by the approximations made in its derivation (each volume element behaves as a Rayleigh scatterer and there is no phase shift through the particle) which require that 10 n n and d, where d is the diameter of the particle. Because our analysis is for the Uv-vis-NIR (Near infrared) spectrum (190-900 nm), it is impossible to simultane ously satisfy both conditions at every wavelength. Prior to any de tailed study it is important to validate the software implemented for Rayleigh-Debye-Gans theory as a function of the wavelength, and to explore the behavior of th e calculated spectra relative to a reference material. Under the conditions of a pplicability of RDG, both Mie theory and RDG theory should yield the same results. Polystyren e is a reference material with known wavelength-dependent optical pr operties that is used in th e manufacturing of spherical particles used as standards for particle an alysis. Comparison of the turbidity spectra calculated with RGD and Mie for polystyre ne particles suspended in water should provide a good indication of when the spectra deviate, and the theories no longer agree to an acceptable level.
Appendix C (Continued) The diameter sizes chosen were 25 nm, 50 nm, 100 nm, and 500 nm. The calculated spectra have been plotted on a semilog axis to better illustrate the differences between theories. Figures C.1 and C.2 demonstrate the expected close approximation of Rayleigh-Debye-Gans to Mie theory for small sized particles. Figures C.3 and C.4 show divergences of Rayleigh-Debye-Gans approximation from Mie as the particle size increases while maintaining the optical properties within the limits of the theory. Appendix B shows the optical property requirements of RDG are met as functions of wavelength for polystyrene. These results demonstrate that the programs developed for Rayleigh-Debye-Gans theory yield the expected values when compared with Mie theory and that the software developed can be reliably used for the simulations reported herein. 200 300 400 500 600 700 800 900 10-3 10-2 10-1 100 101 Optical Density RDGMie Figure C.1: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm Polystyrene Spheres 99
Appendix C (Continued) 200 300 400 500 600 700 800 900 10-2 10-1 100 101 Optical Density RDGMie Figure C.2: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm Polystyrene Spheres 200 300 400 500 600 700 800 900 10-3 10-2 10-1 100 101 102 Optical Density RDGMie Figure C.3: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 100 nm Polystyrene Spheres 100
Appendix C (Continued) 101 200 300 400 500 600 700 800 900 10-1 100 101 102 Optical Density RDGMie Figure C.4: Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Polystyrene Spheres
Appendix D: Estimation of Absorption Coefficient and Hypochromism Model The absorption coefficient is the imaginary value of the refractive index. A mathematical derivation given by Maron , on the absorption of radiation as a function of thickness and concentration of absorbing material, describes how the absorption coefficient can be obtained. The decrease in intensity of incident light of any wavelength passing through an absorbing substance is given by Lamberts law. The law states that the rate of decrease of intensity with thickness of absorbing material is proportional to the intensity of the light at point l, txIdldI D.1 where I t is the intensity at thickness l and is the absorption coefficient characteristic of the medium. The original intensity I o, is given at l=0 and the intensity at any point l can be found from the equation above. Thus we can obtain lIIdldldIotlllIIttoln0 D.2 loteII D.3 Accordingly, ln I t falls off linearly and I t exponentially, with the distance l the light travels through the absorbing medium. In the case of absorbing solutes the decrease in intensity with l is proportional only to I t and the concentration of the solution C. CIdldItt D.4 102
Appendix D (Continued) where is the molar absorption coefficient which is a proportionality constant determined by the nature of the absorbing solute and the wavelength used. Integrating the equation above using the same limits as those in equation D.2 results in the following expression: ClIICdlIdIotlllIItttoln0 D.5 CloteII D.6 Equation D.5 and D.6 are the expression of Beers law for absorption of light by solutions. These two laws can be combined to form Beer-Lambert law which says the absorbance is directly proportional to the pathlength and the concentration. This law is stated as equation D.5, however the absorbance and the complex refractive index are coupled through the absorptivity 4 D.7 Through a transmission measurement and the use of equations D.5 and D.7 the absorption coefficient can be calculated. This only holds for homogeneous solutions, otherwise scattering has to be considered. 103
Appendix D (Continued) D.1 Hypochromism Model Hypochromism was quantified using the model developed by Vekshin [11,12]. Vekshins model describes screening of chromophores when stacked along the molecular chain axis, see figure below. Figure D.1: Stack Arrangement of Chromophores along Chain Axis Experimentally the hypochromism value h at a given wavelength is defined by: %100 h D.8 where is the extinction coefficient for the situation of single chromophore in units of cmM1 and is the average extinction coefficient to account per 1 chromophore. From the screening model [12,13] ksEqqksE3.2113.2 [=] 2 D.9 This equation predicts the hypochromic extinction coefficient in a solution of stack chromophores (cluster) if the values E, s, q, and k are known. E is in units of molecular 104
Appendix D (Continued) extinction coefficients (/molecule) which is function of wavelength, E is the average extinction coefficient, s is the effective geometric area of a chromophore ( 2 ), q is the orientation factor, and k is not to be confused with the wave number but rather is the quantity of chromophores. Transforming Vekshins model from units cmM 1to results in rewriting the above equation to kAmEsEqNqks4022.63.2113.2 D.10 where N A is Avogadros number. Equation D.10 can now be used to correct the imaginary component of the refractive index by first calculating the extinction coefficient using equation D.7. The extinction coefficient is then transformed to molar units VMwm D.11 where M w is the molecular weight of the particle and V is the unit volume transformation of 1000 cm 3 /L. The number of chromophores k, is solved through the volume fraction or concentration of the sample dvkf D.12 where v f is the volume fraction, is the wavelength, d is the diameter of the sample. The probability of absorption of a photon by a molecule P can be presented as sEqP3.2 D.13 105
Appendix D (Continued) 106 NaEVEm2)81(* D.14 where E is the calculated value from equation D.9. Vekshins screening equation can therefore be written in the following form kAmpqEkEVsN11812 D.15 where the extinction coefficient for one chromophore is calculated by wmMV D.16 From this, the hypochromicity can be calculated using equation D.8. The corrected imaginary part of the refractive index c can be solved for using the following equation 4c D.17
About the Author Alicia C. Garcia-Lopez was born in Ha milton, Ontario Canada. She came to the United States in 1984 and attended the local sch ools in Tampa, Florida. Alicia attended University of South Florida and received he r bachelors in Chemical Engineering in 1998 and her masters in Chemical Engineering in 2001. Alicias keen inte rest in the science and mathematics combined with interest in problem solving and the desire to learn new subjects lead her to pursue a PhD in Elect rical Engineering. Her other interests and hobbies enjoyed, besides pursuing a PhD, are learning foreign languages, hiking, camping, cycling, climbing, running and traveling.