Diffusion-Wave Fields

Transcription

1 Diffusion-Wave Fields

2 Springer Science+Business Media, LLC

3 Andreas Mandelis Diffusion-Wave Fields Mathematical Methods and Green Functions With 174 Illustrations, Springer

4 Andreas Mandelis Department of Mechanical and Industrial Engineering Photothermal and Optoelectronic Diagnostics Laboratory University of Toronto 5 Kings College Road Toronto, Ontario M5S 1 A4 Canada Library of Congress Cataloging-in-Publication Data Mandelis, Andreas. Diffusion-wave fields: mathematical methods and Green functions 1 Andreas Mandelis. p. cm. Includes bibliographical references and index. ISBN ISBN (ebook) / Diffusion-Mathematics. 2. Heat equation. 3. Green functions. I. Title. QC185.M362oo '75'OI51-dc Printed on acid-free paper Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in Softcover reprint ofthe hardcover 1st edition 2001 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Timothy Taylor; manufacturing supervised by Jacqui Ashri. Reproduced from author's supplied camera copy SPIN

5 To the memory of my parents, Alexandros and Eleonora Mandelis

6 Preface "IX KalVtl 0' t1ke1 v6~ <j>epouoa <j>tlo~a'ta., A.e~(J) 1tp6<; aioep', ei n 0"; 'too' em' ctko<;" Euripides, Iphigeneia in Taurica, The purpose of this treatise is to develop a consistent and unifying mathematical framework for treating a diverse array of phenomena comprising what is now called diffusion-wave fields. For some time, it has become apparent that several, largely independently developed, diffusion-related periodic phenomena may become unified under the global mathematical label of diffusion-wave fields. At this point in its evolution, in terms of sheer levels of activity, the rapidly growing science of diffusion waves mainly hinges on the distinct categories of thermal waves, charge-carrier-density waves and diffuse-photon-density waves. Some other emerging important areas, such as modulated eddy currents, neutron waves, harmonic mass-transport diffusion waves and the (still controversial existence of) diffusive viscosity waves are straightforward variants of one of the three basic diffusion-wave fields and has not been treated separately. They can be developed through straightforward extrapolation of the wave-field mathematics presented in this book. The ever-increasing applications of diffusion-wave fields to very broad ranges of experimental situations have confronted researchers with the task of quantification, who, thus, have to use a number of mathematical tools ranging from elementary to sophisticated. In recent years, several monographs have appeared with reviews of specific mathematical formalisms relevant to one or more specialized experimental configurations. Among those, the monograph Photoacoustics and Photoacoustic Spectroscopy by Allan Rosencwaig (Chemical Analysis Vol. 57, Wiley, New York, 1980) and Stephen Bialkowski's monograph in the same series under the title Photothermal Spectroscopy Methodsfor Chemical Analysis (Chemical Analysis Vol. 134, Wiley, New York, 1996) offer detailed examples of theoretical formulations in the service of specialized experimental methodologies and configurations. Of special importance is the monograph by Darryl Almond and Pravin Patel, Photothermal Science and Techniques (Chapman & Hall, London, 1996). This small book presents a variety of fundamental theoretical and experimental aspects of the thermal-wave field. It is an excellent introductory text to the mathematics of thermal-wave fields with emphasis on interferometric formalism as a straight extension of propagating wavefields. Other multi-authored books with a strong mathematical content also lean toward the requirements of specialized experimental situations and can be found largely in the photothermal literature. This is natural, as the mathematical treatments of thermal-wave fields are historically the earliest ones to appear, dating back to 1861 in the classic papers by A. J. Angstrom (Ann. Physik, Leipzig 114, 513) on the periodically varying

7 V1Il Preface temperature of a semi-infinite rod; and by Lord Kelvin (Trans. Roy. Soc. Edin. 22, 405) on the thermal oscillations of the surface of the Earth. Among the multiauthored books with a mathematical slant, the volume edited by Jeffrey A. Sell, Photothermal Investigations in Solids and Liquids (Academic, New York, 1989) is an excellent effort to match theoretical formalism and experimental results pertinent, again, to the thermal-wave field. The volume edited by Ashley 1. Welch and Martin 1. C. Van Gernert, Optical-Thermal Response of Laser Irradiated Tissue (Plenum, New York, 1995), contains much mathematical background on which the formalism of the diffuse-photon-density-wave field can be built. This, of course, is solidly founded on A. Ishimaru's seminal treatise Wave Propagation and Scattering in Random Media, (Academic, New York, 1978). The series Progress in Photothermal and Photoacoustic Science and Technology (PPPST), originally edited by myself and recently co-edited with Peter Hess, Institute of Physical Chemistry, University of Heidelberg, and published by the Society of Photo-optical Instrumentation Engineering (SPIE), is yet another venue for matching specialized theoretical treatments to experimental research. Experimentally motivated mathematical formalisms of the charge-carrier diffusion-wave field have appeared in the book Photoacoustic and Thermal-Wave Phenomena in Semiconductors (North-Holland, New York, 1987), edited by myself, and in Volume IV (Semiconductors and Electronic Materials, SPIE Press, 2000) in the PPPST book series. As the diffusion-wave field matures, it has become apparent that there is a need to set the underlying mathematical science on a proper generalized and easy-to-follow foundation independent of experimental configuration, which can be used as a springboard for further theoretical investigations and experimental interpretations. This treatise is the outcome of the realization that a suitable pedagogical tool is currently unavailable, although several textbooks have given exposure to one aspect or another of diffusion-wave mathematics besides the above-mentioned research-level books. The most notable among them include the classic treatises by H. S. Carslaw and 1. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, Oxford, 2nd Ed., 1959), and Methods of Theoretical Physics by P. M. Morse and H. Feshbach (McGraw-Hill, New York, 1953). Significant mathematical insights in the complex nature of thermal-wave fields and one-dimensional case studies have been described in the textbook by Vedat S. Arpaci, Conduction Heat Trasfer (Addison-Wesley, Reading, MA, 1966). Unfortunately, the purely discipline-based (and widely disparate) literature does not adequately address the common mathematical issues, the geometries, and the emerging convergence of several scientific disciplines which can be globally categorized under the unifying concept of diffusion waves. Furthermore, to date, there exists no other work dealing with a wide range of analytical aspects of diffusion-wave science in consistent mathematical formulations. Based on the belief that the maturing of diffusion-wave science and technology is inevitably giving rise to new generations of students and researchers who require pedagogically presented, rigorous, and relevant education in the mathematical foundations of this field, I have developed the material for this treatise. The level is suitable for advanced undergraduate and

8 Preface IX graduate students, as well as for working researchers. Emphasis is given in detailed and easy-to-follow derivations of diffusion-wave fields. The Greenfunction approach was chosen due to its power, elegance, and generality. The material has been divided into an Introduction followed by 10 chapters. The historical precedence and relative maturity of the thermal-wave field over other diffusion waves is the reason that the first eight chapters are associated with the mathematical foundations of thermal wave physics. Chapters 1 through 8 alternate between derivations of Green functions and their use in calculating thermal-wave fields for specific case studies in the three most popular coordinate systems: rectangular (Cartesian), cylindrical, and spherical. Results from those chapters are then used in a rather compact mathematical scheme to develop the material for Chapters 9 and 10, while allowing for the unique physical characteristics of non-thermal diffusion waves. Chapter 9 describes chargecarrier-density wave fields as a special case of the transport equations in physical electronics and derives the pertinent Green functions. Chapter 10 develops a similar mathematical formalism for diffuse photon-density-wave fields, a relatively new area with promising biomedical impact. Much of the material is original and is intended to be used in an analytical course for developing the necessary mathematical skills needed for springboarding into further theoretical and experimental studies in diffusion waves. Problem sets at the end of each chapter partially complement the material in the main text. They range in difficulty between simple applications and extensions of the presented material and research-paper level. They should be used for further absorption of the material. The background required for a graduate course on diffusion-wavefield analysis based on this material is mostly limited to a first course in ordinary and partial differential equations with some familiarity of the technique of separation of variables and complex analysis. Special theorems appear whenever possible, to facilitate the adaptation of results of low dimensionality or complexity to more complicated, higher-dimensional analysis. I am especially grateful to my students, Yan Chen, Lena Nicolaides, Chris Feng and Elmer Ting. Their contributions to a number of aspects in the adaptation of the original manuscript material into a book format were seminal, ranging from verifications of the theory to assembling plots and figures. Without their help, this book would not have been possible. I also wish to express my sincere gratitude to my wife Nancy and daughters Alexandra and Nicole for their patience, understanding and support throughout the long process of bringing the book from a concept into reality. Toronto, February 2001 Andreas Mandelis

9 Contents Introduction I.1 The Mathematical Nature of Diffusion-Wave Fields The Fourier and Laplace Transformation Approaches The Green Function Method. Advantages, Properties, and Mathematical Preliminaries Uniqueness Theorems of Diffusion-Wave Field Functions Problems References Chapter 1. Green Functions of One-Dimensional Thermal-Wave Fields Introduction Fundamental Green Function Solutions. The Infinite Space Green Function Green Functions for the Semi-Infinite Solid Green Functions for a Medium of Thickness L Improper Green Functions for Domains with Interfaces Green Functions for Composite Solids I Green Functions for Composite Solids II Green Functions for Composite Solids III Green Functions for Composite Solids with a Spatially Impulsive TW Source Below the Uppermost Layer Problems References Chapter 2. Thermal-Wave Fields in One Dimension Problems References

10 Contents Xl Chapter 3. Green Functions in Three- and Two-Dimensional Cartesian Thermal-Wave Fields Laterally Infinite Domains Laterally Finite Domains Two-dimensional Green Functions in Cartesian Coordinates Three-Dimensional Green Functions of Structures with Edges and Comers Problems References Chapter 4. Cartesian Thermal-Wave Fields in Three and Two Dimensions Problems References Chapter 5. Green Functions of Thermal-Wave Fields in Cylindrical Coordinates Laterally Infinite Domains Cylindrical Geometries with Finite Radii Cylindrical Sector and Wedge Geometries with Finite Radii Problems References Chapter 6. Thermal-Wave Fields in Cylindrical Coordinates Thermal-Wave Fields in Laterally Infinite Domains Thermal-Wave Fields in Cylindrical Geometries with Finite Radii Thermal-Wave Fields in Laterally Infinite Domains with Arbitrary Source Distributions Thermal-Wave Fields in Cylindrical Wedges and Edges Problems References Chapter 7. Green Functions of Thermal-Wave Fields in Spherical Coordinates Problems

11 Xli Contents References Chapter 8. Thermal-Wave Fields in Spherical Coordinates Point Sources, Spherically Symmetric Source Distributions, and Azimuthally Symmetric Sources Spherically Symmetric Sources and Hollow Spheres Spherical Cones Problems References Chapter 9. Carrier-Density-Wave Fields in Electronic Solids/Semiconductors Carrier Transport Equations One-Dimensional Cartesian Geometries Three-Dimensional Cartesian Geometries Three-Dimensional Cylindrical Geometries Composite Electronic Solids Problems References Chapter 10. Diffuse Photon Density Wave Fields in Turbid Media and Tissue Problems References Appendix: Special Mathematical Functions of Diffusion-Wave Fields Subject Index