CORRELATION TRANSFER THEORY: APPLICATION OF RADIATIVE TRANSFER SOLUTION METHODS TO PHOTON CORRELATION IN FLUID/PARTICLE SUSPENSIONS By NAFAA M. REGUIGUI Bachelor of Science Oklahoma State University Stillwater, Oklahoma 1987 Master of Science Oklahoma State University Stillwater, Oklahoma 1990 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfilment of the requirements for the Degree of DOCTOR OF PHILOSOPHY December, 1994 Name: Nafaa Mohamed Reguigui Date of Degree: December, 1994 Institution: Oklahoma State University Location: Stillwater, Oklahoma Title of Study: CORRELATION TRANSFER THEORY: APPLICATION OF RADIATIVE TRANSFER SOLUTION METHODS TO PHOTON CORRELATION IN FLUID/PARTICLE SUSPENSIONS Pages of Study: 235 Candidate for Degree of Doctor of Philosophy Major Field: Mechanical Engineering Scope and Method of Study: In this work, the derivation of the correlation transfer equation (CTE) governing the temporal field correlation function for multiple scattering of light through suspensions of diffusing particles is presented. It was shown here that there exists a formal similarity between the CTE and the radiative transfer equation (RTE). Several radiative transfer solution techniques (approximate and exact) were applied to obtain solutions for the field correlation function in isotropic and anisotropic one-dimensional media subjected to either natural or polarized radiation. Findings and Conclusions: Numerical solutions were presented for the field correlation function back-scattered and transmitted from plane parallel media. Overall trends of the solutions agree very well with published experimental observations. The agreement is improved when anisotropic effects and index of refraction effects are taken into consideration. It is shown that CTE is in excellent agreement with Diffusive Wave Spectroscopy and the single scattering theory. Results for the polarized CTE follow trends of published experimental results. However, the P1 approximation for the polarized CTE fails at low optical thicknesses (L≤1). ADVISER'S APPROVAL ABSTRACT In this study, the derivation of the correlation transfer equation (CTE) governing the temporal field correlation function for multiple scattering of light through suspensions of diffusing particles is presented. It is shown that there exists a formal similarity between CTE and the radiative transfer equation (RTE). Several radiative transfer solution techniques (approximate and exact) are applied to obtain solutions for the field correlation function in isotropic and anisotropic one-dimensional media. In particular, the CTE for isotropic scattering is written and solved using five different methods: an exponential kernel approximation for pre-averaged CTE which yields closed form solutions; an exact numerical solution for pre-averaged CTE based on Chandrasekhar's X- and Y-functions; a numerical solution based on the Legendre expansion of the single scattering term (g1) appearing in the CTE; a diffusion approximation which yields closed form solutions; and the P1 approximation for the polarized correlation which also yields a closed form solution. Other methods that are discussed in this work include the spherical harmonics solution method for the scalar CTE and the generalized spherical harmonics solution method for the polarized CTE. Numerical results (in graphical form) are presented for the correlation in both the forward and backward directions for a finite medium and for backscattering in the case of a semi-infinite medium. The different solution methods when compared to each other, tend to agree for a very short delay time and/or high optical thickness. The Legendre expansion of the single scattering function (gl) seems to yield the most accurate results, especially when using eight terms in the Legendre expansion of gl. Also, a comparison of the CTE behavior in both the single scattering regime and the diffusion limit to the available theories in both of these limits is successfully presented. Finally, effects of the optical thickness, scattering angle, index of refraction, an depolarization on the correlation function that have been investigated are presented and discussed. It has been shown that polarization effects can be neglected for high optical thicknesses (L≥20), but need to be considered for any realistic characterization of the suspension for lower optical thicknesses (L≤5). CORRELATION TRANSFER THEORY: APPLICATION OF RADIATIVE TRANSFER SOLUTION METHODS TO PHOTON CORRELATION IN FLUID/PARTICLE SUSPENSIONS Thesis Approved: PREFACE In this study, the derivation of the correlation transfer equation (CTE) governing the temporal field correlation function for multiple scattering of light through suspensions of diffusing particles is presented. It was shown here that there exists a formal similarity between the CTE and the radiative transfer equation (RTE). Several radiative transfer solution techniques (approximate and exact) were applied to obtain solutions for the field correlation function in isotropic and anisotropic one-dimensional media subjected to either natural or polarized radiation. I am deeply grateful to my major advisor, Dr. Ronald L. Dougherty for the continuous guidance, support and encouragement he has given me throughout my graduate work. I am also indebted for comments and advice on various topics related to this work to Dr. Bruce J. Ackerson. I also wish to thank Dr. Afshin J. Ghajar and Dr. Frank W. Chambers for the trust they gave me and for their gracious help. My warm thanks go to Farhad Dorri-Nowkoorani, Ulf Nobbmann, Cho-Chun Liu, and Y. Tian for their invaluable contributions and assistance. Special thanks go to all the faculty and staff of the Mechanical and Aerospace Engineering department. Finally, and most important, I must thank my father Mohammed and my mother Meriam, whose loving encouragement and support enabled me to bring this project to completion. TABLE OF CONTENTS Chapter page I. INTRODUCTION ................................................................................. II. LITERATURE REVIEW ...................................................................... II.1. II.2. II.3. II.4. II.5. III. General Introduction ..................................................................... Radiative Transfer Solution Methods ........................................... Dynamic Light Scattering ............................................................. Multiple Scattering ....................................................................... Applications .................................................................................. FROM SINGLE TO MULTIPLE SCATTERING ................................ III.1. Single Scattering Photon Correlation Theory .............................. III.2. Multiple Scattering Theory.......................................................... III.2.a. Foldy-Twersky Integral Equation for the Average Field.. III.2.b. Integral Equation for the Field Spatial Correlation Function………………………………………………… IV DEVELOPMENT OF THE FIELD CORRELATION TRANSFER EQUATION ………………………………………………………….... IV.1. The Mutual Coherence Function for Diffusing Particles ………. IV.2. The Correlation Transfer Equation for Diffusing Particles ……. V APPROXIMATE SOLUTION METHODS …………………………... V.1. Isotropic Scattering From Plane Parallel Media ………………… V.l.a. Pre-averaging V.1.a.i. Exact Numerical Solution V.l.a.ii. Exponential Kernel Approximation …………….. V.i.b. Legendre Expansion of gl ……………………………….. V.2. Anisotropic Scattering ..................................…………………… V.2.a. Legendre Expansion …………………………………….. V.2.b. Diffusion Approximation ……………………………….. V.2.b.i. Plane Wave Incident Normal To Slab Containing Isotropic Pure Scatterers ………………………… VI POLARIZED LIGHT AND THE EQUATION OF CORRELATION TRANSFER ……………….…………………………………. VI.1. Fundamentals of Polarized Light ………………………………. VI.2. Spherical Harmonics Expansion . VI.2.a. Diffuse Intensity Vector VI.2.b. The Basic Scattering Constants VI.2.c. Azimuthally Symmetric Radiation VI.2.d. The P1 Approximation With Rayleigh Scattering VII RESULTS AND DISCUSSION VII.1. Comparison With Diffuse Wave Spectroscopy VII.2. Results With the Legendre Expansion of gl VII.2.a. Effect of Optical Thickness VII.2.b. Comparison of the CTE to the Very Thin Limit Results VII.3. Pre-averaging and Off-angle Detection VII.4. Comparison With Experimental Data; Index of Refraction and VII.5. Polarization Effects VII.6. Extensions VIII CONCLUSIONS AND RECOMMENDATIONS VIII.1. Conclusions VIII.2. Recommendations REFERENCES APPENDIX A MODIFIED BESSEL FUNCTIONS APPENDIX B EXPANSION OF THE PHASE MATRIX ELEMENTS APPENDIX C SPHERICAL AND GENERALIZED SPHERICAL HARMONIC FUNCTIONS C.I. Legendre and Associated Legendre Functions C.II. Generalized Spherical Functions APPENDIX D RECURSIVE RELATIONS FOR THE MATRICES Π im ( µ ) and Pi(µ) Matrices The Π im ( µ ) Matrices The Pi(µ) Matrices APPENDIX E FORWARD SCATTERING APPROXIMATION APPENDIX F SPHERICAL HARMONIC SOLUTIONS TO CTE F.I. Non-Polarized Radiation F.I.a. Two-Moment Expansion of Gm F.I.b. Gaussian Phase Function F.II. Polarized Radiation with Rayleigh Scattering F.II.a. Two-Moment Expansion APPENDIX G SPHERICAL HARMONICS G.I Definitions G.II Spherical Harmonics and the Transformation Matrix