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Application of Fractional Calculus to Polarization Dynamics in Solid Dielectric Materials by Gary William Bohannan a Dissertatio Application of fractional calculus to polarization dynamics in solid dielectric materials by Gary William Bohannan A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Montana State University © Copyright by Gary William Bohannan (2000) Abstract: The fractional calculus has been suggested as an appropriate mathematical tool for the description of a wide variety of physical, chemical and biological processes, in particular non-Debye relaxation in dielectrics. While there have been substantial improvements in our understanding of the physical interpretation of the results of fractional differential equations, there is still much unknown. There is, as yet, not even complete agreement over the notation to be used, let alone the underlying definition of the fractional derivative. This dissertation explores the hypothesis that there must be an intimate relation between the ac and dc response in any given material represented by a dynamical equation of motion. That at least one of the forms of the fractional calculus does, in fact, provide a useful and practical tool for describing some of the dynamics of immediate interest is demonstrated. The process investigated is that of polarization dynamics in condensed matter dielectric materials. The dielectric material studied in detail is poly(vinylidine fluoride) (PVDF) which finds use in piezoelectric sensor and actuator applications. A dynamical model, based on the fractional calculus, is shown to reproduce both the ac and dc responses to an applied electric field. Many of the results are directly applicable a wide range of other dynamics including mechanical relaxation processes. APPLICATION OF FRACTIONAL CALCULUS TO POLARIZATION DYNAMICS IN SOLID DIELECTRIC MATERIALS by Gary William Bohannan A dissertation submitted in partial fulfillment of the requirements for the degree of . Doctor of Philosophy m Physics MONTANA STATE UNIVERSITY Bozeman, Montana November 2000 11 APPROVAL of a dissertation submitted by Gary William Bohannan This dissertation has been read by each member of the dissertation committee and has been found to be satisfactory regarding content, English usage, format, citations, bibliographic style, and consistency, and is ready for submission to the College of Graduate Studies. George F. Tuthill Approved for the Department of Physics John C. Hermanson I l - I f ' PD (Signature) Date Approved for the College of Graduate Studies Bruce R. McLeod 'SZ Ssts <? ip r (Signature) Date iii STATEMENT OF PERMISSION TO USE In presenting this dissertation in partial fulfillment of the requirements .for a doctoral degree at Montana State University, I agree that the Library shall make it available to borrowers under rules of the .Library,- I further agree that copying of this dissertation is allowable only for scholarly purposes, consistent with “fair use” as prescribed in the U. S, Copyright Law. Requests for extensive copying or reproduction of this dissertation should be referred to Bell & Howell Information and Learning, 300 North Zeeb Road, Ann Arbor, Michigan 48106, to whom I have granted “the exclusive right to reproduce and distribute my dissertation in and from microform along with the nonexclusive right to reproduce and distribute my abstract in any format in whole or in part.” Signature Date > r j z o o & ACKNOWLEDGEMENTS With great appreciation for the assistance from fellow graduate and undergradu­ ate students who participated in the development and execution of the experiments crucial to this work: Darin Arbogast, Michael Chase, Jil Hallenberg, Michelle Lind- mier, Erica McKenzie, and Delphine Medicine Horse. ' ■ I am especially grateful to Recep Avci and Sara MaCcagnano for allowing me to participate in their study of anomalous diffusion. This work was supported by a Montana Space Grant Fellowship, NASA. EPSCoR Grant NCC5-240, and NSF Grant DMR-9805272. V TABLE OF CONTENTS LIST OF TABLES................................................................................................... viii LIST OF FIGURES......................................................................................... ;....... ix 1. INTRODUCTION ............................................................................ ................ I 2. PROBLEM STATEMENT AND HYPOTHESIS................. ................... ..., 7 Example Application............................. 8 The Perm ittivity................................ 9 Permittivity of Dilute Dipolar Liquids ............................................................ 11 Permittivity of Solid Dielectrics......................................................................... 12 Equivalent Circuits............................. ........................................................ 15 Other Proposed Symmetric and Asymmetric Permittivity Functions........... 17 Low-Frequency, Long-Time Anomalies......... ................................................... 20 Polarization versus Conductivity................... 22 From Whence the Power-law?......................................................................... 23 Hypothesis................................................................ 25 3. THE MODEL.......................................... ,......................................................... 27 Polarization and Conduction as Statistical Quantities.................................. 27 A Statistical Digression....................................................................................... 28 Polarization.................................................. 30 Conductivity.....................................................................:................ .................. 31 Competition and Mixing........................................................................ 32 Distributions of Relaxation Times..................................................... 33 A Plethora of Models?........................................................................................ 34 4. THE FRACTIONAL CALCULUS ................................................................. 35 The Definition of the Fractional Derivative..................................................... 36 Causality and Time Inversion............................................................................ 41 Fractional Derivative of a Sinusoid ............................................................ 41 Fractional Relaxation ........................................................................ 42 5. DERIVATION OF THE DIFFERENTIAL EQUATION ........................... 46 The Permittivity Equation................... 47 Applying the Fractional Calculus to Non-exponential Processes ................... 48 Development of a Differential Equation ........................................................... 49 A Fractional Form of the Constitutive Equation ............................................ 50 vi A Second Fractional Form of the Constitutive Equation. .. 54 An Alternative View of the Models................................... 58 Assumption Regarding Initial Conditions........................ 59 Disclaimer ............... ............ ................................................ 59 6 . COMPUTATIONAL METHOD . .................................... 61 Development of a Numerical Integration Scheme.... ........ 63 Gfiinwald Algorithm.............................. ............................ : 64 Direct Integration................................................................. 66 Example ac Reconstruction.......................................... 68 Spectrum of Response and Noise Masking.................. 69 Example Pulsed dc Reconstruction.............................. .. 71 Stepwise Integration ............................................................ 75 7. EXPERIMENTAL METHODS AND RESULTS.............. 80 ' Material Studied .......................................................... 80 Equipment............... .......................................... ...... ............ 81 Sample Holder for Varying Temperature. .. __ _...___ .. 82 Faraday Cage for Low Frequency Measurements__ . 82 ac Measurements............................................................ 83 Transient Response Measurement ........................... 83 Data Analysis ............ .......................................................... 85 8 . OTHER APPLICATIONS AND FUTURE QUESTIONS . 98 Extending the Dynamic Range............................................ 98 NMR Relaxation Times ............ .................................. ..... 99 Domain Wall Relaxation and Self-Pinning........................ 99 Where is the Charge? ..... — .....................:....................... 100 Application to Mechanical Systems................................... 101 PIxDim Control Systems ...'...............:................................. 102 Boundaries and Barriers...................................................... 104 Many Body Dynamics .......................................................... 104 Relation to Generalized Thermodynamics......................... 105 Fractional versus Fractal............................. ...................... 105 Quantum Mechanical Treatment......................................... 106 1/f Noise in Resistors .................. ......................................... , 107 9. SUMMARY AND SIGNIFICANCE.................................. 108 Lesson Learned ............................ ..................... ................. 108 Conclusion............ ................................................................
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