Shock Wave Science and Technology Reference Library, Vol. 5: Non-Shock Initiation of Explosives

Shock Wave Science and Technology Reference Library

The new Springer collection, Shock Wave Science and Technology Reference Library, conceived in the style of the famous Handbuch der Physik has as its principal motivation to assemble authoritative, state-of-the-art, archival reference articles by leading scientists and engineers in the field of shock wave research and its applications. A numbered and bounded collection, this reference library will consist of specifically commissioned volumes with internationally renowned experts as editors and contributing authors. Each volume consists of a small collection of extensive, topical and independent surveys and reviews. Typical articles start at an elementary level that is accessible to non-specialists and beginners. The main part of the articles deals with the most recent advances in the field with focus on experiment, instrumentation, theory, and modeling. Finally, prospects and opportunities for new developments are examined. Last but not least, the authors offer expert advice and cautions that are valuable for both the novice and the well-seasoned specialist.

Shock Wave Science and Technology Reference Library

Collection Editors

Hans Gr¨onig Hans Gr¨onig is Professor emeritus at the Shock Wave Laboratory of RWTH Aachen University, Germany. He obtained his Dr. rer. nat. degree in Mechanical Engineering and then worked as postdoctoral fellow at GALCIT, Pasadena, for one year. For more than 50 years he has been engaged in many aspects of mainly experimental shock wave research including hypersonics, gaseous and dust detonations. For about 10 years he was Editor-in-Chief of the journal Shock Waves.

Yasuyuki Horie Professor Yasuyuki (Yuki) Horie is internationally recognized for his contributions in high-pressure shock compression of solids and energetic materials modeling. He is a co-chief editor of the Springer series on Shock Wave and High Pressure Phenomena and the Shock Wave Science and Technology Reference Library, and a Liaison editor of the journal Shock Waves. He is a Fellow of the American Physical Society, and Secretary of the International Institute of Shock Wave Research. His current interests include fundamental understanding of (a) the impact sensitivity of energetic solids and its relation to microstructure attributes such as particle size distribution and interface morphology, and (b) heterogeneous and nonequilibrium effects in shock compression of solids at the mesoscale.

Kazuyoshi Takayama Professor Kazuyoshi Takayama obtained his doctoral degree from Tohoku University in 1970 and was then appointed lecturer at the Institute of High Speed Mechanics, Tohoku University, promoted to associate professor in 1975 and to professor in 1986. He was appointed director of the Shock Wave Research Center at the Institute of High Speed Mechanics in 1988. The Institute of High Speed Mechanics was restructured as the Institute of Fluid Science in 1989. He retired in 2004 and became emeritus professor of Tohoku University. In 1990 he launched Shock Waves, an international journal, taking on the role of managing editor and in 2002 became editorin-chief. He was elected president of the Japan Society for Aeronautical and Space Sciences for one year in 2000 and was chairman of the Japanese Society of Shock Wave Research in 2000. He was appointed president of the International Shock Wave Institute in 2005. His research interests range from fundamental shock wave studies to the interdisciplinary application of shock wave research.

B.W. Asay (Ed.)

Shock Wave Science and Technology Reference Library, Vol. 5

Non-Shock Initiation of Explosives

With 298 Figures and 31 Tables

123

Blaine W. Asay Los Alamos National Laboratory Explosives Applications and Special Projects Los Alamos NM 87545 USA Email: [email protected]

Blaine W. Asay Blaine W. Asay is a Research Scientist and Group Leader in the Explosive Applications and Special Projects group at Los Alamos National Laboratory. His research interests include hypervelocity jet initiation, DDT, novel energetic materials, thermobarics, abnormal initiation, and thermal response. He has published over 50 papers in the refereed literature and presented numerous papers at international conferences, and also holds several patents. Dr. Asay received the Ph.D. in Chemical Engineering for his work in combustion from Brigham Young University in 1982 and worked at Kodak Research Laboratories before joining LANL in 1984.

ISBN 978-3-540-87952-7 e-ISBN 978-3-540-87953-4 DOI 10.1007/978-3-540-87953-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009933123 c Springer-Verlag Berlin Heidelberg 2010  This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar/Spain Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Los Alamos National Laboratory is an incredible place. It was conceived and born amidst the most desperate of circumstances. It attracted some of the most brilliant minds, the most innovative entrepreneurs, and the most creative tinkerers of that generation. Out of that milieu emerged physics and engineering that beforehand was either unimagined, or thought to be fantasy. One of the fields essentially invented during those years was the science of precision high explosives. Before 1942, explosives were used in munitions and commercial pursuits that demanded proper chemistry and confinement for the necessary effect, but little else. The needs and requirements of the Manhattan project were of a much more precise and specific nature. Spatial and temporal specifications were reduced from centimeters and milliseconds to micrometers and nanoseconds. New theory and computational tools were required along with a raft of new experimental techniques and novel ways of interpreting the results. Over the next 40 years, the emphasis was on higher energy in smaller packages, more precise initiation schemes, better and safer formulations, and greater accuracy in forecasting performance. Researchers from many institutions began working in the emerging and expanding field. In the midst of all of the work and progress in precision initiation and scientific study, in the early 1960s, papers began to appear detailing the first quantitative studies of the transition from deflagration to detonation (DDT), first in cast, then in pressed explosives, and finally in propellants. Clearly this phenomenon had been observed before, and was undoubtedly the cause of many accidents, but with the improved diagnostic techniques (e.g., imageintensified cameras, rapid oscilloscopes, etc.) workers were finally able to begin to probe this very difficult area of reactive material behavior. These formed the initial studies of what has come to be known as nonshock initiation. This behavior and the reasons it is so difficult to understand are explored in Chap. 1. If an explosive is going to react violently, an ignition event must occur at some location within the volume. That ignition will be at a single point, perhaps in a small portion of a crystal. Whatever its size, to propagate

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the reaction, mass, momentum, and energy must be conveyed (conducted, convected, radiated, or advected) to other locations on a timescale that promotes growth of temperature and pressure. Transport phenomena have been studied for hundreds of years, but in the context of nonshock initiation, we felt that a complete presentation was required to place the common terms into context. This forms the substance of Chap. 2 by Perry. Explosives are metastable molecules – they are always reacting, but the rate is extremely temperature sensitive. Depending on the material and temperature, the reaction rate can double with an increase of only a few degrees centigrade. The physics and chemistry governing this behavior are extremely complex. For example, the decomposition of HMX has been described with over 500 elementary kinetic steps, and it is known that this is not a complete treatment. Trying to use that kind of description to understand the behavior of a reactive system is impossible, even with the latest state-of-the-art computational capabilities. We require a more global approach to the entire decomposition process. The ability to do that requires enormous insight, a fundamental understanding of chemistry, and the capability of reducing processes to their most simple, yet complete, level. But no further. Henson has done just that in Chap. 3. Once initiated, an explosive grain, or hot spot, can either fail, survive for a short amount of time, or progress and grow. This is the problem of criticality, and without knowing how this process occurs, it is impossible to understand or predict the outcome of an initiation event. In Chap. 4, Hill discusses in detail the important issues surrounding the topic of criticality, upon what it depends and how it affects the outcome of an event. If a hotspot survives, the reaction may then proceed through the body of material. This process is called combustion, and involves the conversion of a solid material having chemical energy into gas at much higher temperature and pressure. The rate at which this occurs is governed by the kinetics, but the process is controlled by how the mass, momentum, and energy move. Combustion waves can progress at very slow speeds (fractions of a centimeter per second) or up to supersonic speeds in cracks. Clearly, the mechanism of combustion is central to the concept of nonshock initiation and is covered by Jackson in Chap. 5. HMX is a very powerful explosive, but at atmospheric pressure and no confinement, it burns very slowly and in a laminar fashion. However, allow the pressure to increase somewhat, and provide surface area and confinement, and that lazy laminar combustion wave can easily convert into a detonation wave with pressures of thousands of atmospheres. In Chap. 6, Parker and Rae discuss the damage mechanisms that create the surface area without which the transition to detonation cannot occur in secondary explosives. This is an area of study of relatively recent vintage and which some have been slow to acknowledge. However, our ability to directly observe reacting systems being damaged has increased substantially in the past 10 years, enabling these conclusions to be considered incontrovertible.

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Chapter 7 discusses the topic of cookoff, which comprises each of the topics discussed before. It is a general term that includes explosion as well as detonation. The chapter discusses the small and large-scale tests used to study the field of nonshock initiation and some of its important elements. The deflagration to detonation transition was alluded to in an earlier paragraph and is discussed in other portions of the text, but it needed a chapter of its own to be elucidated more fully. The earliest papers accurately described the overall behavior of explosives, but in many cases missed the quantitative response. In Chap. 8 McAfee provides a complete treatment of the DDT process incorporating all of the advances that have recently occurred, and unifies many of the observations that have been made over the past 50 years. It is possible to drop a relatively large piece of explosive from a substantial distance onto a very hard surface and have no reaction. However, take that same explosive part, and the same hard surface and add a few grains of sand, and an entirely different outcome obtains. The study of frictional initiation has a long history with many elegant experiments having been conducted. Dickson provides a well-reasoned and complete analysis of the topic in Chap. 9. Finally, in Chaps. 10 and 11, Kennedy covers the areas of initiation by shear and impact, followed by spark and laser. These topics are sometimes overlooked because they are not very well characterized, but the first three constitute primary sources of accidental nonshock initiation, and the last is a current topic of study for purposeful ignition of explosives. We hope that readers will see the organization of the text as a natural progression in the study of nonshock initiation. It is by no means an easy field of study, nor one in which all of the major questions have been worked out. Each of the authors of this volume is not only an expert in the field, but someone who has thought deeply about the topic. It took several years to formulate the concepts, reduce them to the written word, and then finally produce something worthy of publication. Each has been a great resource and an important and trusted colleague for many years. I am indebted to each one for his contribution to my understanding of the field. I also acknowledge the work of Phil Howe, without whose tireless and persistent support many of the experiments and analyses discussed in this text would not have occurred. He is an integral and crucial member of the team. I have been incredibly fortunate to have known some of the most influential men and women in the field of explosives research. At Los Alamos, I was instructed in DDT by Wayne Campbell, shock physics by Mac Walsh, detonation physics by Bill Davis and John Bdzil, detonation chemistry by Ray Engelke, propellant behavior and explosives safety by John Ramsay, shaped charge design by Bill Mautz, and the list goes on. I have also had the privilege of meeting pioneer researchers from throughout the world ranging from Kondrikov and Dremin from Russia to Price and Jacobs from White Oak. As funding and support for the kind of work we do erodes, the community is becoming ever smaller. I am grateful to have been involved during some of the high points.

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I want to thank my closest coworkers of many years, Laura Smilowitz, Bryan Henson, Steve Son, John McAfee, and Peter Dickson. The thousands of hours spent in discussion, writing, some arguing, and waiting for that elusive exotherm have been joyous. Finally, I acknowledge the inestimable influence of my four children who have each exceeded every expectation of which I could have ever conceived, and my wife and eternal companion, Patrice, who completes me. Los Alamos, NM, October 2009

Blaine W. Asay

Contents

1 Introduction Blaine W. Asay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 What Are We Able to Predict? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 What Are We Unable to Predict? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Motivation for This Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Purpose of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Final Word of Caution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Transport Phenomena for Nonshock Initiation Processes W. Lee Perry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 An Overview of Transport Theory in Explosives Problems . . . . . . . 2.2 The Origins of Transport Theory in the Field on Nonshock Initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Physical Properties Used in Transport Theory Analysis . . . . . . . . . 2.4 Transport Theory Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Selected Examples of Transport Theory Applied to Explosive Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Chemical Kinetics of Solid Thermal Explosions Bryan F. Henson and Laura B. Smilowitz . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Chemical Kinetics and Explosives Modeling . . . . . . . . . . . . 3.2.2 Types of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Modeling Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 3 5 7 9 11 11 13 14 15 15 19 22 27 30 41 42 45 45 49 49 73 76

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Specific Example: A Global Chemistry Model for HMX . . . . . . . . . 76 3.3.1 HMX Thermal Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3.2 HMX Thermal Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3.3 HMX Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.3.4 HMX Thermodynamics and Ignition . . . . . . . . . . . . . . . . . . . 103 3.3.5 Global Kinetic Models of PBX 9501 (HMX) . . . . . . . . . . . . 109 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 4 Classical Theory of Thermal Criticality Larry G. Hill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 4.1.1 Spontaneous Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.1.2 Thermal versus Physical Explosions . . . . . . . . . . . . . . . . . . . 134 4.1.3 The Three Fundamental Cookoff Safety Questions . . . . . . . 135 4.1.4 The Nature of Preignition and Post-Ignition Cookoff Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 4.1.5 Historical Development of Thermal Explosion Theory . . . . 137 4.1.6 Definition, Scope, and Modern Relevance of Classical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.1.7 Emphasis and Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.1.8 Additional Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.2 Reaction Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.2.1 Homogeneous and Heterogeneous Chemical Reactions . . . . 142 4.2.2 Spatial Averaging Over Heterogeneous Solid-Phase Reactants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.2.3 The Law of Mass Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.2.4 Arrhenius Reaction Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.2.5 Reaction Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.2.6 Reaction Rate Expressed as Mass Fraction . . . . . . . . . . . . . 147 4.2.7 Reaction Progress Function Behavior for Arbitrary Reaction Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.2.8 Zero-Order Reaction versus Negligible Depletion . . . . . . . . 149 4.2.9 Autocatalytic Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 4.2.10 Hot-Spot-Initiated Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.2.11 Reaction Progress Function for an Ideal Hot-Spot-Initiated Reaction . . . . . . . . . . . . . . . . . . . . . . . . . . 157 4.2.12 Modeling Complex Thermal Decomposition Behavior . . . . 162 4.2.13 The Arrhenius Function and its Calibration . . . . . . . . . . . . 163 4.2.14 Activation Energy as a Measure of HE Sensitivity . . . . . . . 165 4.2.15 Thermal Decomposition Parameters for PBX 9501 . . . . . . . 169

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Semenov Theory of Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.3.1 Volumetric Heat Generation and Surface Heat Loss . . . . . . 171 4.3.2 Heat Flux Balance and the Semenov Graphical Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.3.3 Semenov Criticality Condition . . . . . . . . . . . . . . . . . . . . . . . . 175 4.3.4 Explicit Solutions for the Critical Temperatures . . . . . . . . . 177 4.3.5 Semenov Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.3.6 Semenov Steady-State Solution Space . . . . . . . . . . . . . . . . . . 181 Semenov Criticality Theory with Reactant Depletion . . . . . . . . . . . . 184 4.4.1 The Path to Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.4.2 Generalized Semenov Critical State . . . . . . . . . . . . . . . . . . . . 186 4.4.3 Example: First-Order Reaction with Constant Heating Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 4.4.4 Behavior of Reaction Runaway Near the Explosion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 4.4.5 Effect of Reaction Energy Content on the Explosion Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.4.6 Sensitivity to Arrhenius Parameters . . . . . . . . . . . . . . . . . . . 192 Reactive Heat Equation and the F-K Approximation . . . . . . . . . . . 192 4.5.1 Heat Equation with Volumetric Reaction . . . . . . . . . . . . . . . 193 4.5.2 F-K Dimensionless Temperature . . . . . . . . . . . . . . . . . . . . . . . 194 Semenov Theory in the F-K Limit: The “SFK” Model . . . . . . . . . . 197 4.6.1 Heat Flux Balance, Graphical Construction, Critical Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.6.2 Criticality Expressed by the Semenov Number . . . . . . . . . . 198 4.6.3 Explicit Solutions for the Critical Temperatures . . . . . . . . . 199 Basic Frank-Kamenetskii Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.7.1 General Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.7.2 Critical Temperature and Critical Size . . . . . . . . . . . . . . . . . 201 4.7.3 Critical Condition for the Infinite Slab . . . . . . . . . . . . . . . . . 202 4.7.4 Critical Condition for the Infinite Cylinder . . . . . . . . . . . . . 204 4.7.5 Critical Condition for the Sphere . . . . . . . . . . . . . . . . . . . . . . 205 4.7.6 Critical Temperature Profiles for the Canonical Shapes . . . 207 4.7.7 Relative Critical Temperatures for the Canonical Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.7.8 F-K Critical State for General Shapes: Square Bar Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 F-K Criticality Correlations for General Shapes . . . . . . . . . . . . . . . . 213 4.8.1 Correlation for Bodies of Near-Unity Aspect Ratio . . . . . . . 214 4.8.2 Comparison of F kc Values Obtained by Various Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.8.3 Effective Dimensionality of Near-Unity-Aspect-Ratio Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

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4.8.4

Breakdown of the Shape Correlation for Elongated Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.8.5 On Extending the Shape Correlation to Elongated Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.9 F-K Theory with Generalized Boundary Conditions . . . . . . . . . . . . . 224 4.9.1 Thermal Conductance/Resistance and the Biot Number . . 224 4.9.2 Thomas’ Generalized Boundary Condition . . . . . . . . . . . . . . 225 4.9.3 Infinite Slab: Critical Condition versus Biot Number . . . . . 226 4.9.4 Infinite Cylinder: Critical Condition versus Biot Number . 228 4.9.5 Sphere: Critical Condition versus Biot Number . . . . . . . . . . 230 4.9.6 Critical State versus Biot Number for the Canonical Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 4.9.7 Quasi-Self-Similar Scaled Criticality Curves vs. Biot Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 4.10 Convergence of the Critical State to the F-K Limit . . . . . . . . . . . . . 235 4.10.1 Canonical Shape Criticality for the Full Arrhenius Rate . . 235 4.10.2 Quasi-Self-Similar Scaled Criticality Curves vs. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 4.10.3 Critical State Correction for Finite Activation Energy . . . . 237 4.11 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 5 Deflagration Phenomena in Energetic Materials: An Overview Scott I. Jackson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 5.2 Combustion and Its Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 5.3 The Conductive Deflagration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 5.3.1 A Description of the Conductive Flame Structure . . . . . . . 250 5.3.2 Factors Affecting the Burn Rate . . . . . . . . . . . . . . . . . . . . . . . 252 5.3.3 Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.3.4 Pressure Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.3.5 Quenching and Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 5.3.6 Burning Progression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 5.4 Convective Burning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 5.4.1 Gas Flow into Porosity and Isolated Voids . . . . . . . . . . . . . 264 5.4.2 Ignition of Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 5.4.3 The Influence of a Melt Layer on the Critical Pressure . . . 272 5.5 Progression of Combustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 5.5.1 Across Flat Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 5.5.2 Spreading of Convective Combustion . . . . . . . . . . . . . . . . . . . 275 5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

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6 Mechanical and Thermal Damage Gary R. Parker and Philip J. Rae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 6.2 Classification of Pertinent Morphologies . . . . . . . . . . . . . . . . . . . . . . . 294 6.2.1 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 6.2.2 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 6.3 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6.3.1 Issues of Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 6.3.2 The Representative Elementary Volume . . . . . . . . . . . . . . . . 298 6.3.3 Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 6.4 Pristine Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 6.4.1 Energetic Material Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 6.4.2 Examination of Defects in Pure Materials . . . . . . . . . . . . . . 310 6.4.3 Examination of Defects in Pressed Materials . . . . . . . . . . . . 316 6.4.4 Examination of Defects in Melt-Cast Materials . . . . . . . . . . 317 6.5 Insult and Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 6.5.1 Damage From Mechanical Insult . . . . . . . . . . . . . . . . . . . . . . 319 6.5.2 Damage From Thermal Insult . . . . . . . . . . . . . . . . . . . . . . . . . 337 6.6 The Effect of Pore Morphology on Fluid Transport in Damaged EM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 6.6.1 Fundamentals of Fluid Transport . . . . . . . . . . . . . . . . . . . . . . 352 6.6.2 Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 6.6.3 Experiments with EM Measuring Gas Transport in the Preignition Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 6.6.4 Experiments with EM Measuring Gas Transport in the Post-ignition Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 6.7 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 7 Cookoff Blaine W. Asay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 7.1 Concept of Critical Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 7.1.1 Semenov’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 7.1.2 Frank-Kamenetskii’s Approach . . . . . . . . . . . . . . . . . . . . . . . . 412 7.1.3 Critical Temperature Determination . . . . . . . . . . . . . . . . . . . 413 7.2 Visual Observation of the Cookoff Event . . . . . . . . . . . . . . . . . . . . . . 418 7.2.1 Mechanically Confined Cookoff Tests . . . . . . . . . . . . . . . . . . 418 7.2.2 Cindy Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 7.3 Small-Scale Instrumented Thermal Ignition Experiments . . . . . . . . 431 7.4 Quantification of Reaction Violence . . . . . . . . . . . . . . . . . . . . . . . . . . . 441 7.4.1 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 7.4.2 A First Law Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 7.4.3 Violence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 7.4.4 Energy and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

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7.4.5 Violence and P 2 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 7.4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 7.5 Large, Integrated Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 7.5.1 LLNL Large-Scale Cookoff Test . . . . . . . . . . . . . . . . . . . . . . . 460 7.5.2 Scaled Thermal Explosion Experiment (STEX) . . . . . . . . . 460 7.5.3 Large-Scale Annular Cookoff . . . . . . . . . . . . . . . . . . . . . . . . . . 461 7.5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 7.5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 7.5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 7.6 Final Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 8 The Deflagration-to-Detonation Transition John Michael McAfee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 8.2 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 8.2.1 Solid DDT Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 8.2.2 Solid DDT Experimental Evidence . . . . . . . . . . . . . . . . . . . . 487 8.3 Type I DDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 8.3.1 The Phenomenology of Type I DDT . . . . . . . . . . . . . . . . . . . 489 8.3.2 Experimental Evidence for Type I DDT . . . . . . . . . . . . . . . . 494 8.4 Type II DDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 8.4.1 Type II DDT Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . 508 8.4.2 Experimental Evidence for Type II DDT . . . . . . . . . . . . . . . 511 8.4.3 General Discussion of Type II DDT . . . . . . . . . . . . . . . . . . . 516 8.5 The Change Between Type I and Type II DDT . . . . . . . . . . . . . . . . 518 8.5.1 Experimental Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 8.5.2 Burn Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 8.5.3 Porosity, Permeability, and Particle Size Distribution . . . . 523 8.5.4 Particle Surface Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . 527 8.5.5 Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 8.5.6 Igniter Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 8.5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 8.6 DDT in Gases, Vapors, and Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 8.6.1 Gases and Vapors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 8.6.2 Dusts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533 9 Friction Peter Dickson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 9.1 Introduction to Frictional Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 9.2 Significant Accidents Probably Involving Frictional Processes . . . . 537 9.3 Classical Friction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 9.4 Other Factors Affecting Surface Temperatures . . . . . . . . . . . . . . . . . 540 9.5 Effects of Particulate Contaminants . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 9.6 Frictional Processes Involved in Heating and Ignition of Explosives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

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Friction Testing Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Powder Explosive Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 Pressed/Cast Explosive Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 9.9.1 Oblique Impact Testing Revisited . . . . . . . . . . . . . . . . . . . . . 545 9.9.2 LANL Friction Test Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 548 9.10 Drop-Weight Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 9.11 Analysis and Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554

10 Impact and Shear Ignition By Nonshock Mechanisms James E. Kennedy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 10.1 Effects of Shear in Material-Flow Processes . . . . . . . . . . . . . . . . . . . . 555 10.2 Drop-Weight Impact Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 10.2.1 Cavendish Laboratory Glass-Anvil . . . . . . . . . . . . . . . . . . . . . 559 10.2.2 Temperature and Flow Sensors . . . . . . . . . . . . . . . . . . . . . . . . 561 10.3 Large-Scale Crushing Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 10.3.1 IITRI Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 10.3.2 Susan Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566 10.3.3 U.S.S. Iowa Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567 10.3.4 XDT Response Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568 10.3.5 Two-Step Impacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570 10.4 Penetrating Impact of Cased Explosives . . . . . . . . . . . . . . . . . . . . . . . 571 10.5 Shear in Confined Charge Assemblies . . . . . . . . . . . . . . . . . . . . . . . . . 572 10.6 Mechanical Properties of Energetic Materials Relevant to Shear Ignition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 10.6.1 Properties of Unreacted Energetic Materials . . . . . . . . . . . . 575 10.6.2 Mechanical Property Effects on Ignition . . . . . . . . . . . . . . . . 575 10.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 11 Spark and Laser Ignition James E. Kennedy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 11.2 Spark Detonator: A Vehicle for Study of Loose Powder Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584 11.3 Initiation Behavior of Secondary Explosive Spark Detonators . . . . 586 11.4 Similarities Among EBW, Laser, and Spark Initiation of Detonation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 11.4.1 EBW Detonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588 11.4.2 Laser Detonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 11.5 Sparks as Sources for Ignition of Explosive Powders . . . . . . . . . . . . . 591 11.5.1 Charge Generation and Discharge . . . . . . . . . . . . . . . . . . . . . 591 11.5.2 Electrostatic Discharge (ESD) Sensitivity Testing of Energetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 11.5.3 Discharges from Human Body or Handling Equipment . . . 595

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11.6 Electrical Discharges Through High-Density Energetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596 11.6.1 Conclusions Concerning High-Density Energetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 11.7 Other Laser Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607

List of Contributors

Blaine W. Asay Los Alamos National Laboratory Los Alamos, NM, 87545 [email protected] Peter M. Dickson Los Alamos National Laboratory Los Alamos, NM, 87545 [email protected] Bryan F. Henson Los Alamos National Laboratory Los Alamos, NM 87545 [email protected] Larry G. Hill Los Alamos National Laboratory Los Alamos, NM, 87545 USA [email protected] Scott I. Jackson Shock and Detonation Physics Group Dynamic and Energetic Materials Division Los Alamos National Laboratory Los Alamos, NM, 87545 USA [email protected]

James E. Kennedy Hazards and Explosives Research and Education LLC 2710 Via Caballero del sur Santa Fe, NM, 87505 USA [email protected] John M. McAfee Los Alamos National Laboratory Los Alamos, NM, 87545 [email protected] Gary R. Parker Los Alamos National Laboratory Los Alamos, NM, 87545 USA [email protected] William Lee Perry Los Alamos National Laboratory Los Alamos, NM, 7545 [email protected] Philip J. Rae Los Alamos National Laboratory Los Alamos, NM, 87545 USA [email protected] Laura B. Smilowitz Los Alamos National Laboratory Los Alamos, NM, 87545 [email protected]

1 Introduction Blaine W. Asay

Accident 1 On 14 February 2005, two highly trained technicians who had helped write the standard operating procedures were transferring triethoxy-trinitrobenzene (TETNB) from a filter tray into 5 gallon buckets. TETNB is a precursor in the manufacture of triamino-trinitrobenzene (TATB), and each of these compounds is considered to be extremely insensitive explosives as they are very difficult to ignite with shock, friction, or electrostatic discharge. During the transfer, an explosion occurred, killing one of the technicians and critically injuring the other. The technicians were investigating a new approach for synthesizing TATB using a more environmentally friendly method. They had already successfully demonstrated the new technique in 23 kg batches and were scaling up to a 230 kg batch. During the transfer of the TETNB, they encountered an unusual condition, in that the material was caked hard to the bottom and the sides of the filter tray. They proceeded to remove the caked material with a metal scoop and a plastic shovel, whereupon the material exploded. Subsequent investigation revealed that a precursor to TETNB, TNPG, which is extremely impact sensitive, had segregated during the wash step and formed the hard cake. So while the workers thought that they were dealing with TETNB, they were actually forcefully removing TNPG with a metal tool. Accident 2 On 10 August 2004, a truck carrying 17,000 kg of explosives jack-knifed negotiating a tight turn while driving through a canyon. The truck skidded across the highway and started a fire on the mountainside. The fire reached the explosive load, and 3 min later, exploded, leaving a crater 9 m deep and 21 m wide, propelling concrete barriers into the river a 100 m away.

B.W. Asay (ed.), Shock Wave Science and Technology Reference Library Vol. 5: Non-Shock Initiation of Explosives, DOI 10.1007/978-3-540-87953-4 1, c Springer-Verlag Berlin Heidelberg 2010 

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Accident 3 On 16 April 1947, the French liberty ship Grandcamp was being loaded with the remainder of its cargo of ammonium nitrate fertilizer. Twenty three hundred tons were already on board. The longshoremen in the hold receiving the 45 kg pallets noticed a small wisp of smoke coming up through the bags. They estimated that the smoke originated about five layers down. Attempts to quench the fire with small quantities of water were unsuccessful. The ship’s captain did not want to use large amounts of water as it would ruin the cargo, so they closed all the hatches and turned on the steam. However, the fire continued and the local fire department was called. The pressure from the steam blew off the hatch covers and large amounts of orange smoke filled the sky. Onlookers began to gather to watch the spectacle. An hour had now passed, but the fire continued to grow. A few minutes later, an enormous explosion occurred that was heard 150 miles away, with a cloud erupting to over 600 m. Two airplanes were knocked out of the sky and ship fragments weighing several tons were found in the surrounding countryside. Hours later, another liberty ship, the High Flyer, containing sulfur and a thousand tons of ammonium nitrate was observed to be on fire. Around 1:00 AM the following morning, the High Flyer also detonated with a blast that was reported to have been even greater than when the Grandcamp was destroyed. The resulting fires burned for days until they were finally extinguished. In all, nearly 600 people perished, with another 3,500 seriously injured. Accident 4 During the development of a new warhead, a series of experiments were conducted to ensure that it would pass the spigot test. This involved dropping a weighted protrusion representing a cleat or other object present during the transfer of warheads between ships, or from ship to shore, onto a warhead. All previous warheads tested had shown a direct relationship between drop height and weapon response. That is, increasing the drop height increased the likelihood of initiation of the charge. During one such test series with the new warhead, it was shown to ignite and burn nonviolently when dropped from 10 m. When the final demonstration was scheduled, the decision was made to reduce the drop distance to ensure no reaction. When dropped from 1.75 m, the charge detonated. No injuries were reported, but significant concern over the ability of the test to predict weapon response resulted. Accident 5 While developing a new super-thermite comprised of 80 nm aluminum formulated in a tetrafluoroethylene-hexafluoropropylene-vinylidene fluoride (THV) polymer, researchers applied the nanocomposite formulation to a fiberglass e-glass substrate, baked off the acetone solvent, and then cooled the 25 cm2

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sample. The Al/THV mixture had passed all the conventional safety tests including impact, spark, and friction, showing it to be insensitive. The e-glass is inert and showed no chemical incompatibility with the Al/THV. After the sample was cooled, one individual examined the material with bare hands before giving the sample to a coworker to examine. The coworker, wearing nitrile gloves, then manipulated the sample, observing the coating uniformity and other mechanical properties. Approximately 30 s later, the material spontaneously ignited, resulting in second-degree burns to the coworker. These are five examples drawn from thousands that could have been cited. The causes of each are now well understood, but as with many accidents, the response to the stimulus could not have been predicted beforehand. We could continue for many volumes detailing explosives accidents over the past 150 years. In fact, texts have been written doing just that. The frequency of large accidents like those on the USS Forrestal (nine 1,000 lb bombs, 132 perished), Oppau Germany (4,500 tonnes of ammonium sulfate and ammonium nitrate, 500–600 perished), Hanbury England (3,500 tons of bombs, 68 perished) has been greatly reduced because of increased understanding and safer practices. However, accidents continue to occur on a daily basis. Most, if not all, are the result of a phenomenon known as nonshock initiation, the topic of this text.

1.1 Background Defining what constitutes an explosive can be difficult. The DOE Explosives Safety Manual defines an explosive as any chemical compound or mechanical mixture which is designed to function as an explosive, or chemical compound which functions through self-reaction as an explosive, and which, when subjected to heat, impact, friction, shock, or other suitable initiation stimulus, undergoes a very rapid chemical change with the evolution of large volumes of highly heated gases that exert pressures in the surrounding medium. The term applies to materials that either detonate or deflagrate. Explosives can be gas, liquid, or solid. They can be physical mixtures of fuel and oxidizers (e.g., ammonium nitrate and fuel oil) or have the fuel and oxidizer in the same molecule (e.g., RDX). Explosives can be classified as either high or low (e.g., nitroglycerin vs. black powder), primary or secondary (e.g., lead azide vs. HMX), sensitive or insensitive (e.g., CL20 vs. TATB). Their classification can change with the physical state. For example, for shipping purposes, dry HMX is considered a primary and wet HMX is a secondary explosive. PETN can be considered either sensitive or insensitive depending on the particle size. Explosives can be formulated with a binder or used neat (e.g., PBX 9501 vs. TNT), they can be cast or pressed to shape (e.g., Comp B vs. PBX 9404), they can have additives that either increase or decrease the sensitivity (e.g., microballoons vs. wax). Addition of approximately 0.1% carbon

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black can change the spark sensitivity of an explosive from extreme to zero. An explosive component that is quickly heated to 500◦ C may be totally safe and nonviolent while slowly heating the same charge to a much lower temperature might result in a detonation and much damage. Dropping an explosive component from 10 m may result in a benign reaction while dropping the same charge from 1/2 the height might result in a detonation. These facts are cited to illustrate that explosive behavior is complicated. There are a few general rules of thumb that can be quite useful, but for every rule there are a number of exceptions. Having said this, it is possible to get a fundamental and useful understanding of behavior in a relatively short period of time. That is the purpose of this text. Many people think that explosives are dangerous because they have an enormous amount of energy available. While it is true that they can contain considerable energy, when compared to well known non-detonable organic compounds, we see that the total amount is not remarkable. For example, the heat of combustion of kerosene is 43 MJ kg−1 , while HMX, an extremely powerful military explosive, has only 9.9 MJ kg−1 . The difference is attributed to the fact that the oxygen required for detonation is contained in the HMX molecule, while that used by the kerosene is taken from the air. If the oxygen required for complete combustion of the kerosene is included, its energy is reduced to approximately the same as HMX. The relatively weak explosive ammonium perchlorate (AP) has 12.1 MJ kg−1 while lead azide, the very sensitive primary explosive used in detonators, has 2.6 MJ kg−1 . So if it is not the total amount of energy that makes an explosive useful, and in some cases dangerous, what is it? The answer is the rate at which the available energy is released. Typical combustion velocities are in the order of millimeter to centimeter per second. Detonation velocities in condensed phase materials range from 2 to 9 km s−1 . So even a relatively poor explosive releases its energy orders of magnitude faster than nondetonable compounds having relatively high amounts of energy. For example, to provide a context for the numbers that will be used, the following list illustrates the rate of energy release for common processes: – – – – –

Acetylene/air flame: 100 W cm−3 Deflagrating gun propellant: 1 MW cm−3 Detonating explosive: 100 GW cm−3 US generation capacity in 2004 was 1.05 TW Average hurricane represents 1.7 TW

In commercial operations in 2005, according to the International Society of Explosives Engineers, six billion pounds of explosives and 75 million detonators were used in the US and Canada alone. This does not take into account the military explosives or the chemicals with explosive properties that are used to make nonexplosive materials. Two-thirds of these explosives are used in coal mining, and of that, more than 80% is ANFO. These are extremely large numbers. Given the large amount of explosives used, it is remarkable that we do not experience more accidents. This is due in part to the continuous

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improvement in handling and processing practices. It is not necessarily due to increased fundamental understanding of the physics and chemistry involved in explosive behavior. Explosives have been studied for nearly 1,000 years, and even longer if Greek Fire is considered to be an explosive. The Chinese described black powder in 1040 ad, and from that point forward, scientific investigations have been carried out to improve the performance, efficiency, and safety of explosives. These investigations began with attempts to purify one of the main ingredients, saltpeter, when it was recognized to have a major effect on behavior. This was followed by modification of the grain structure through corning and moderation, which led to more reproducible burning, elimination of grain disruption, and its attendant uncontrolled increase in surface area and resultant explosion. The destruction of guns and cannon by this explosion process was most likely the earliest observation of the early stages of a process we now call deflagration to detonation transition (DDT). The behavior and performance of black powder were more or less fully understood by the late seventeenth and early eighteenth centuries. Other molecules and mixtures were soon sought out because of the relatively low pressure of black powder, and its inability to effectively fragment granite in mining and tunneling operations. Nitroglycerin, nitrocellulose, picric acid, tetryl, TNT, RDX, and HMX, and many others followed as understanding of chemistry and more specialized needs developed. Many book volumes and literally thousands of research papers have been written on the history, performance, physics, chemistry, and safety of explosives. Most of the early work concentrated on gaseous detonations and combustion. Later came treatises on solid explosives. These include recent texts by Zukas and Walters, Cooper, Davis, Cherest, and others. Some of the most brilliant scientists in the world have made explosives the subject of study. Given the lengthy history and the intellectual effort expended, one would think that prediction of explosives’ response to abnormal events would be a relatively simple exercise, and that certainly another text on the topic would not be needed. What is the problem? The problem is that, even with all of the study and the thousands of years of experience, we still cannot predict with any precision, in general, what will happen to an explosive if we hit it, heat it, drag it, drop it, or do anything else outside of its design envelope.

1.2 What Are We Able to Predict? We actually do know quite a bit about HE response in certain cases. Response is predictable in cases with the following: 1. Strong, planar shocks in well-characterized materials 2. HE that is either extremely sensitive (e.g., NI3 ) or extremely insensitive (e.g., TATB) 3. Large amounts of data in specific systems with specific threats (e.g., fragment impact correlations)

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Sample Detonation Run Distance

Booster

Fig. 1.1. Illustration of wedge test configuration and analysis used for strong shock initiation

Let us address each case individually Case (1). In the case of strong planar shock waves, data from an experiment called the wedge test are used to construct a pressure vs. rundistance plot (Pop plot [1]) as shown in Fig. 1.1. In the wedge test, a planar shock wave is driven into a carefully characterized explosive and the distance required to achieve a steady detonation is recorded. Results from a series of experiments are then compiled and plotted as shown. Different explosives have different slopes and positions. Thus as shown in Fig. 1.1, explosive A will transit to a detonation in a shorter distance than explosive B when presented with a shock of the same strength. These results are functions of temperature, particle size, density, and other parameters. However, once obtained for a specific explosive, the data can be used to make accurate predictions of system response, as long as strong planar shocks are used. Case (2). There are many materials that are either so sensitive or so insensitive that their response to any stimulus is assured, or where the probability of a given event is so great that it does not matter. For example, NI3 will explode at the slightest perturbation. The mere act of touching it, or cooling a crystal, is enough to make it react. On the other hand, TATB and similar highly insensitive explosives require extremely strong shock waves to begin initiation of a detonation wave, and they have never been observed to undergo deflagration-to-detonation transition. For this case, we are able to make predictions of response based on the known extreme sensitivity or insensitivity of the compound. Case (3). In this case, a specific explosive with specific characteristics (e.g., density, particle size distribution, etc.) is tested in a specific environment to measure the response to a specific stimulus, and correlations are compiled. For example, the response of TNT to fragment impact has been measured over a range of fragment velocities and sizes. As long as one interpolates within the acquired data set, relatively precise predictions can be made. However, extrapolation outside the data is not reliable and can oftentimes lead to extremely unwelcome miscalculations. In addition, these types of correlations typically

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yield relatively imprecise predictions of response. For example, a designation of a violent response might be something that ranges from a mild explosion to a full-scale detonation. The effect of either of these outcomes on surrounding structure or personnel might be the same. However, the outcome could arise from completely different mechanisms. Thus the correlation is not useful for deriving an understanding of the behavior, but merely assessing the level of risk involved in a specific application.

1.3 What Are We Unable to Predict? Unfortunately, the cases described above do not constitute a large fraction of those in which accidents happen (e.g., Trident D5 motor detonation, USS Forrestal, Texas City, etc.). Bulk explosives by their nature are designed to be initiated by strong shock waves originating from a detonator. Accidents typically occur when an explosive is subjected to an unplanned stimulus. These stimuli are oftentimes relatively low rate (when compared to shock waves), and relatively low pressure (when compared to the pressure encountered from a detonator output). In fact, accidents involving unplanned thermal ignition occur at atmospheric pressure. In addition, the material is often damaged by the event; that is, it contains cracks or pores that are not part of the original design for the component. Because accidents do not typically occur in the restricted situations described by one of the three cases cited above, our predictive capability is negligible. The major reasons for this observation are as follows: 1. There is no corollary to the Pop Plot for weak shock or low-strain-rate processes 2. Many explosives are of intermediate sensitivity and their response to a given input depends on material and confinement properties, temperaturedependent heat transfer and kinetic parameters, and other complications 3. Sufficient data to produce correlations for even a small fraction of potential accident scenarios have not been acquired, even though many explosives have been subjected to a range of safety tests For example, if an explosive for which a basic set of kinetic data is available is heated, we can predict with fair accuracy when an event will occur, but we cannot say what the event will be (e.g., how violent), or specifically where in the explosive it will begin. Given ignition of an explosive, we cannot predict reaction rates, how the combustion wave will build (or if it will build), or if it will transition to a detonation or merely continue as a low-order deflagration wave. Under mechanical insult, if mechanical and material data are available, we can calculate deformation and heat localization, but we cannot predict how violent the response will be. The reasons that we cannot make accurate predictions are that the problems are (1) multiscale both temporally (microseconds to seconds) and

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spatially (microns to tens-of-centimeters), (2) multidimensional, (3) almost always initially subsonic with the resultant complication that boundaries and surrounding material have a large effect on the outcome, and (4) dependent on the mechanical properties of HE (these may actually dominate). In addition, (5) extremely small amounts of chemical reaction can affect behavior. For example, a chemical energy release of less than 1% of that contained in an initially nonreactive shock wave may be sufficient to build the shock to detonation. Finally, (6) if all we needed to know was whether or not a classic detonation would result, the problem would be almost tractable. However, when the energy is delivered on a relatively short time scale, even modest amounts of reaction can result in disastrous consequences. One example of the complexity of explosive behavior is illustrated in Figs. 1.2 and 1.3, which shows a Viper shaped charge penetrating the liquid explosive nitromethane. Figure 1.2 shows the structure during penetration while the jet penetration velocity is less than that required for prompt ignition. Various mach reflections are evident as well as the (relatively) inert bow shock structure. Figure 1.3 shows penetration at a slightly higher velocity, resulting in unsteady ignition. Ignition began on the lower left hand side of the shock front, failed, and then began again on the lower right side. The internal structure behind the shock front is not visible in these images because of the reacting gases. These images illustrate the immense complexity in a relatively simple liquid explosive system. The complexities and difficulties in heterogeneous systems only become more numerous.

Mach Reflections

Fig. 1.2. Shaped charge penetration of nitromethane. Penetration velocity is subcritical, and so no steady-state initiation is achieved

1 Introduction

2.99 µs

7.0 µs

10.99 µs

14.99 µs

9

Fig. 1.3. Shaped charge penetration of nitromethane. Penetration velocity is slightly higher than that in for Fig. 1.2. Ignition is unsteady

1.4 Numerical Approaches Some in the research community (particularly the funding agencies) hope that numerical simulations are currently, or soon will be, adequate to make up for the deficiency in our understanding. Although it is true that computer modeling can be used to evaluate a large number of situations and examine large portions of parameter space, it is also true that the computer codes are only as good as the physical models used to construct them. These are devised by writing the equations of conservation of mass, momentum, and energy along with the necessary equations of state and closure relations. Because of the complex nature of this set of equations, certain assumptions are employed to make the solution tractable. And because there is no analytic solution to the coupled set of conservation equations, they must be solved numerically. This requires a choice of first or higher order methods, each with its own list of positive and negative attributes, but having in common the inability to solve the equations exactly. Issues with numerical diffusion and viscosity, among many others, must be dealt with. Specifying applicable equations of state is problematic because the material state changes in nonshock processes, so the starting material is usually changed significantly during the course of the event. Thus, its new state must either be calculated (e.g., crack and surface area formation, chemical species, etc.) or assumptions about how the state evolves must be invoked and an ad hoc modification of these properties made.

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Additionally, even with a perfect model, accurate physical, material, and chemical properties are required. Information regarding thermal diffusivity, porosity, density, heat capacity, and a host of other properties must be known as a function of temperature, and to a lesser extent, as a function of pressure. These properties are rarely known to high precision, and while in some cases this does not matter, in others it is a critical deficiency. Also, chemical reaction rate constants, kinetic data, activation energies, etc. are difficult to obtain and evaluate. Because these describe the energy-producing processes, and as indicated earlier, a difference of only 1% in an energy release rate can mean the difference between a detonation and a deflagration, errors in this part of the model can have large effects on the accuracy of the prediction. Other difficulties with computer calculations are the enormous memory and computational requirements for performing high-fidelity calculations in three dimensions. Even with the continuing increase in computational power (0.1–100 Tflops from 1992 to 2006), adaptive mesh refinement (AMR), highly optimized numerical schemes, and increased visualization capabilities, we are still years away from being able to accurately compute generalized HE response in generalized geometries. This would be true even if we did possess completely accurate physical models, which we do not. For example, as computational error is linearly dependent on zone size, and the CPU time required is nonlinear, a 1,000-fold increase in computational speed yields only a tenfold decrease in error for a two-dimensional calculation, and an error reduction of only 5.6 times in three dimensions. In addition, for a generic problem of a planar detonation wave in a 200 mm3 cube with a 1 mm reaction zone, and using AMR to resolve the reaction zone, it would require 185 days of CPU time with a 1,000 processor machine [5]. Increasing the complexity by invoking multistep kinetics, unsteady reactions, material interfaces, and damage only serves to increase the time requirements. The bottom line is that we will not be able to calculate our way out of our problems any time soon. This is not to say that computational tools are not of enormous importance and utility. Even with approximate physical models, estimated material properties, and gross assumptions about material state, we can obtain useful guidance of a general nature. There are many powerful and sophisticated computer codes capable of highly resolved calculations (see, e.g., [4]). As long as the assumptions and difficulties of making such calculations are appreciated, they can be useful. The real danger is in the fact that these approaches tend to be so complicated and involved, requiring tens of thousands of lines of code in some cases, that it is likely impossible to fully understand where they may be misleading or have errors. Thus, if we need specific information about time to explosion, extent of reaction, and violence, we are still dependent on estimates based upon a combination of expert elicitation, correlations, and numerical approaches. Correlations of system response are very useful in some circumstances. However, they typically require a large number of tests. As many processes do not scale, the tests must be run at different sizes, which significantly increases

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the cost. Changing one parameter (e.g., particle size or density) may invalidate a derived relationship. For example, in one set of tests, dropping PBX 9404 vertically from 8 ft results in no reaction. If the same sample is dropped from the same height onto a plate inclined at 45◦ , a violent reaction ensues. The violence of the reaction is a strong function of whether or not loose grit is present and to a lesser extent on surface finish and thermal conductivity. For PBX 9011, the drop height required to produce a violent response at room temperature is 78 ft, while at −29◦ C it is only 4 ft. The types of tests that yield data for correlations are generally deficient, in that they typically do not isolate a single mechanism. For example, in the well-known drop weight test, the sample is undergoing multiaxial and unsteady shear, strain, and compression. This leads to an unknown material state. The only data that are produced from the experiment are whether or not a sound of sufficient magnitude was produced. This information is then compiled to find the drop height that produces this nonspecific diagnostic 50% of the time. Although quite useful for a number of applications, using data of this nature to make predictions about explosive behavior in a general sense is extremely unreliable. The same can be said for every other set of data ranging from differential scanning calorimetry, to Henkin tests, to the one-dimensional time to explosion (ODTX), to fragment impact, to small and large scale gap tests, and beyond.

1.5 Motivation for This Text These observations argue strongly for a sound understanding of the fundamentals of explosive behavior so that informed and reliable decisions can be made in the absence of specific data for a specific situation. If we understand how explosives react, what the mechanisms are for ignition, what the requirements are for propagation, and how violence is determined, then we can more accurately assess the danger of an operation or application, even if the seemingly infinite number of specific details that affect the outcome are unknown or unknowable. For example, a fundamental understanding of thermal ignition and explosives’ kinetic response and the careful application of some significant assumptions were used by Frank-Kamanetskii to formulate a simple approach enabling solutions to problems ranging from autoignition of hay bales to the design of safe shipments of thousands of tons of fertilizers in the holds of ships. However, blind application of this kind of simple approach without a fundamental understanding makes disastrous consequences inevitable. What we present in this text is the basis for that required understanding of explosives in abnormal environments.

1.6 Purpose of This Book There are many books on mining and blasting, detonation physics, explosives engineering, and explosives chemistry. And for every book on detonation,

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there are several that discuss combustion. Given that observation, a legitimate question would be, with all that has been written about explosives over many years, why do we need yet another text on the topic? While there are many books that discuss performance, shock initiation, and a host of engineering topics, none (that we found) assembled into one location a useful summary of all the relevant topics involved in nonshock initiation. For our purposes, shock initiation is defined as prompt initiation of a detonation through shock processes such as detonators, planar impact by plates, etc. Nonshock initiation is defined as arising from bulk heating, fragment impact, spark and friction, or other similar processes. Initiation is not prompt and involves a series of steps that may or may not lead to a steady detonation. Much of the early scientific experimental work (before 1940) on shock initiation was performed with gases because they are more easily studied. Parameters can be quickly varied, optical access is straightforward, and they are homogeneous. However, condensed-phase systems are much more complex with varying material properties, the possibility of damage, and surface area creation, etc. Most of the early theoretical study was performed on idealized one-dimensional systems, and much of this work concentrated not on initiation, but on propagation of steady waves. In the area of nonshock initiation, once again, serious work on DDT began with substances in the gas phase. Later, the problem of DDT in condensed phase materials was addressed at length, with an emphasis on propellants. Much of that work is published in research journals and has been nicely summarized in a few excellent review papers. Some of it appears in internal reports and classified reports and is not easily accessed. A single text that describes DDT, initiation by spark and friction, cookoff, the effects of damage, simplified kinetics, and other important and essential topics is not to be found. As illustrated earlier, it is not possible to completely describe all the physics and chemistry involved in nonshock initiation. Even if they were known, which they are not, that would be too ambitious, even for a series of volumes. This text is meant to provide an introduction to several important topical areas and a foundation in the essential areas; those necessary to understand the relevant details, evaluate situations and procedures, develop models, and to sort out the important facts from those that are not important. This latter point is very important and often overlooked. For example, when considering a scenario of a cased explosive part impacted by a fragment, or heated in a fuel fire, or dropped from some height onto a protrusion, how is the risk evaluated? Or which hazards are important to mitigate and which ones can be ignored? Many have made the mistake of thinking that if they worry about all possible scenarios and hazards, and treat each equally, they will be safer. Nothing could be farther from the truth, because unneeded complexity is just as dangerous, or more so, than not enough rigor. The key is to know what deserves your attention. Do you worry about the temperature, or the confinement, or the velocity, or the material state? And if you worry about the material state, which is more important, the initial pristine mate-

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rial or the damaged material? And before you guess the obvious, it may be important to recognize that the state of the pristine material might actually determine the importance of the damage that might occur, and in some cases, might make the damaged state irrelevant. The knowledge and ability to sort out the important facts from those that are unimportant is critical. This text presents the background and fundamental understanding required to more fully understand non-shock initiation of explosives in abnormal environments. It is not meant as a stand-alone document. Information on detonation and shock physics, explosives formulation, materials science, and other topics is required to fully round out an understanding of the behavior of explosives. References found in each of the chapters can and should be used to focus more attention on a specific topic.

1.7 Final Word of Caution Having said all of this, lest one assume that with sufficient study and acumen he will be able to completely understand this field, we offer a few interesting observations. With respect to understanding and predicting accidents, one early researcher’s view is that: Whatever method we use to predict possible accidents, we have one fundamental problem. The substances can be subjected to so many different situations, each with its own pattern of different kinds of stress, that it is impossible to imagine them all. With more understanding, we can make predictions of risks in more and more situations, but a wholly complete understanding and description of all potential risk situations will never be possible [2] Even though this statement was made in 1976, over 30 years ago, the situation and conclusion have scarcely changed. Smith [3] made the following observations regarding HE sensitivity to initiation by various means: – The sensitivity of an explosive is not a well-defined property of the material, expressible as a single number, say, but is, instead a complex pattern of behavior – Different sensitivity tests, even when intended to measure the same property, will frequently produce different orders of relative sensitivity for a given series of explosives. In other words, there is not even a unique qualitative scale of sensitivity – In most sensitivity tests, the response of an explosive varies in seemingly random fashion over some range of severity of the applied stimulus. That is, there is no sharp threshold above which the explosive will always explode, below which it will never explode

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Given the difficulties described by these authors, we will present as clearly as possible the information required to sort out the conflicting data and outline the approach to making informed decisions.

References 1. Ramsay, J.R., Popolato, A.: Analysis of shock wave and initiation data for solid explosives. In: 4th Symposium (International) on Detonation, pp 233–238. US Naval Ordnance Laboratory, White Oak, 12–15 October 1965 2. Ek, S.: Sensitivity of explosive substances a multivariate approach. In: 6th Symposium (International) on Detonation, pp 272–280. Naval Surface Weapons Center, Coronado, CA, 24–27 August 1976 3. Smith, L.C.: Los Alamos National Laboratory Explosives Orientation Course, Sensitivity and Sensitivity Tests, Los Alamos National Laboratory, LA (1987) 4. Baer, M.R.: Computational modeling of heterogeneous reactive materials at the mesoscale. AIP Conference Proceedings, pp. 27–33. Shock Compression of Condensed Matter: Conference of the American Physical Society Topical Group, Snowbird, UT, USA, 27 June–2 July 1999 5. Aslam, T.D.: Direct numerical simulation of detonation. In: Conference of the American-Physical-Society-Topical-Group on Shock Compression of Condensed Matter, Baltimore, MD, 31 July–5 August 2005. AIP Conference Proceedings, vol. 845, pt.1 and 2, p.931–935 (2006)

2 Transport Phenomena for Nonshock Initiation Processes W. Lee Perry

2.1 An Overview of Transport Theory in Explosives Problems A useful explosive material is, of course, stable under reasonable environmental conditions. It will neither release energy nor produce gas without some kind of thermal stimulus. The possible stimuli include impact, spark, friction, boundary heat, or shock. These processes raise the temperature of the explosive material, either in a localized volume, or throughout its entire volume, to a point where the exothermic, gas-producing reaction becomes self-sustaining (ignition). We refer to the events leading to self-sustaining reaction as the preignition regime, and the behavior after self-sustaining reaction commences as the post-ignition regime. We further generally characterize the post ignition regime by either laminar nonviolent deflagration; violent, convectively driven explosion; or the process may undergo a transition to detonation (DDT). Transport phenomena are inherent to the nonshock initiation process and post-ignition behavior. Specifically, transport phenomena are the movement of heat (energy), molecular species, and momentum under the influence of temperature, species concentration, pressure, and velocity gradients. The theory of these phenomena, together with the production of gas and heat from chemical reactions (kinetics and thermodynamics), in principle fully describe the processes that occur prior to ignition (preignition) and what happens after (post-ignition). In practice, the theory has only been applied to very simple situations due to the complexity of the integrated problem. The stimuli listed above all serve to raise the temperature of the explosive either within a volume small with respect to the charge (impact, spark, friction, shock) or globally (boundary heat). Each of these stimuli will be discussed in detail in other chapters of this text. As these are the initial drivers for the ignition process, we briefly introduce them here. Impact leads to localized heating when deformation becomes concentrated, where the material fails by cracking, or when the yield strength is exceeded and causes shear localization and concentration of energy along shear bands (Chap. 10). Spark energy B.W. Asay (ed.), Shock Wave Science and Technology Reference Library Vol. 5: Non-Shock Initiation of Explosives, DOI 10.1007/978-3-540-87953-4 2, c Springer-Verlag Berlin Heidelberg 2010 

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is localized along the spark channel (Chap. 11). Friction (Chap. 9) localizes heat between moving surfaces or at specific locations, where, for example, foreign matter (grit) becomes trapped between the surfaces (the latter has been implicated in accidents.) A shock wave compresses material along its leading edge and causes heat generation directly by adiabatic compression of the material itself, by collapsing void space, or by grain boundary interactions. Boundary heat, as the phrase implies, is heat applied to the outer surfaces of an explosive charge, or its container, and the most common source is fire, and ignition by this method is commonly called “cookoff” (Chap. 7). Chapter 3 discusses reaction kinetics and thermodynamics. We mention them here in the context of heat transport because, of course, exothermic chemical reaction becomes the driving force for the ignition and combustion process once one (or more) of the ignition stimuli meet a critical criterion. To a first order, the general Arrhenius expression provides an illustration of the rate of gas production as a function of temperature and some reactive state, f (X), of the explosive (e.g., reaction extent, density, concentration of reactive species, available surface area, etc.):   E ˙ · f (X). N = A · exp − RT

(2.1)

The product of enthalpy change (per mole converted) and the rate of gas production reveal the rate of heat production: Q˙ = N˙ · Δh.

(2.2)

These expressions illustrate the sensitivity of the process to temperature in the exponential term and the dependence of some reactive state of the explosive. When one (or more) of the listed stimuli is applied with sufficient intensity to raise the temperature to a value where significant heat production occurs, ignition may occur. The specific ignition threshold is crossed when the rate of heat production in a volume exceeds the rate of heat removal from that volume by a heat transport process (conduction, convection, or radiation). For combustion to spread, energy must be fed back from the combustion zone to regions of unreacted material at a sufficient rate to maintain combustion. These are central concepts of ignition and propagation theory and will be discussed throughout the book. As described, ignition and reaction spread arise from the specific interplay of heat transport behavior, reaction kinetics, and the thermodynamics of the involved reactions. Heat transport theory has played a central role in determining the factors that lead to ignition. The critical conditions for ignition are determined by applying the principle of energy conservation on the stimulated volume, balancing heat production by chemical reaction and heat loss by conduction or convection. In conjunction with carefully constructed experiments, the application of heat transport theory has also provided a tool to help deduce

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or verify the specific chemical steps that occur in the preignition regime [1]. Heat transport theory is also well-developed in the post-ignition regime for low-order (nonviolent) laminar combustion of simple regressing solid or liquid surfaces where heat feeds back from the combustion zone to the surface liberating the reactive species [2, 3]. However, for cases of more critical interest where violent explosive reaction occurs, heat transport occurs in a very dynamic, multidimensional, multiphase environment. Although the basic processes and theory are fairly well understood, the complexity challenges our best computational tools and this remains a very active area of research. Species transport is the movement of specific chemical constituents throughout a system. Explosives considered in this text generally consist of a single molecular constituent. In the preignition regime, the aforementioned stimuli can cause direct decomposition to combustion products, decomposition to intermediate species, or sublimation/evaporation of the original substance. In each of these cases, species transport phenomena govern the physical movement of the products or sublimated species from a solid or liquid surface into the gas phase. The rates of these processes and any associated exothermicity are again controlled by kinetics, which of course is strongly temperature dependent, but also dependent on species distributions and/or morphology (f (X)). If the explosive is heterogeneous (e.g., HMX powder), species transport governs how the gas phase species will distribute themselves throughout pre-existing void space, or throughout the void space created by decomposition, sublimation, phase changes, etc. (mechanical or thermal damage). Recent research has concluded that the void space (pore structure) and its potential development and spatial distribution prior to ignition has a very significant effect on the post-ignition behavior (Chap. 6) [4]. The distribution of species, whether in homo- or heterogeneous system, determines, in part, f (X) in (2.1) above and therefore affects the preignition heat transport behavior. Recent work has shown this distribution of reactive species affects the temperature distribution [5]. This in turn affects the postignition spread of combustion from the ignition locus as it propagates into a larger volume at a higher temperature and in a more reactive state [6]. Further, combustion spreads more readily into regions previously saturated with combustible species. As with heat transport, the situation during violent explosive reaction becomes exceedingly complex, and to date, no significant computational work has been done, which includes rigorous species transport in the post-ignition regime. In the post-ignition regime, species transport theory has also been applied to the study of laminar flames where convection and diffusion processes carry reactive species into the combustion zone [7, 8]. Heat production, decomposition, combustion, and sublimation/evaporation all potentially lead to pressure gradients, which are the driving force for momentum transport. Momentum transport is responsible for the convective (advective) movement of both heat and species. In the pre-ignition regime, localized heating from impact, spark, or friction leads to local gas formation by decomposition or sublimation, causing a local region of elevated pressure

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relative to the bulk of the explosive. In the case of boundary heating, the evolving temperature profiles will cause a distribution of gas products and a resulting pressure distribution. In both these cases, the pressure gradients cause the transport of momentum, which carry heat and species, affecting preand post-ignition behavior as already described. The rigorous incorporation of momentum transport requires the Navier– Stokes (NS) equations, which for any practical problem require the art of Computational Fluid Dynamics (CFD). The tensorial nature of the NS equations and the presence of multiple phases add a large degree of complexity to an already exceedingly complex problem and have not been applied in the preignition regime. However, the porosity and permeability of some explosives have been determined [9, 10], and it was suggested that Darcy’s law for permeation in porous materials would provide a tractable means to incorporate momentum transport for analysis of the preignition problem [5, 11]. This approximation has provided good insight to the preignition problem and allows models to correctly predict temperature profiles for a boundary-heated system [5]. The Darcy’s law approximation requires low Reynold’s number flow, which of course is not the case for the violent post-ignition regime. In that regime, high flow rates in stationary media occurs in the initial phases; as momentum is transferred to the solid phase and the pressure builds, the solid phase disassembles and is set in motion by drag forces. In this complex multiphase flow regime the full NS system is required. These have not been incorporated into contemporary computational solutions; however, progress has been made in formally formulating the problem and examining tractable limiting cases [9]. For the most violent possible post-ignition outcome, DDT, multiphase momentum transport also governs the formation of compaction waves that build to a shock wave and initiate the shock-to-detonation transition (SDT) [12]. Although our understanding of the mechanism of DDT is fairly mature (Chap. 8) [13], only a limited amount of work has been done to apply formal momentum transport theory to the problem. Hopefully, this brief introduction has given the reader a flavor of the interconnection of the three transport modes through heat and mass production by chemical reaction and the importance of all three modes to the ignition and explosion behavior. In the rest of this chapter, we will provide an overview of the history of the application of heat transport theory to explosive systems. We will then introduce the fundamental equations of heat transport theory as an introduction to other chapters that provide a more complete development. Properties such as heat capacity, thermal conductivity, viscosity, etc. are crucial to accurate transport theory predictions, and so a section will be devoted to that topic. Finally, we will conclude the chapter with an overview of three recent problems that employ formal transport theory in a way that has enhanced our physical understanding of the ignition and combustion process.

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2.2 The Origins of Transport Theory in the Field on Nonshock Initiation In our modern world of advanced computing tools, we may lose sight of the fundamental transport processes involved in the nonshock initiation and combustion of explosives. The early work in this field, prior to the existence of modern computing tools, required very insightful analysis and therefore deserves a brief review. Explosion is merely an extreme form of flame propagation, such that early flame research provides a logical origin of transport theory development for ignition and combustion of explosives. Scientists have known for over 150 years that combustion and flame propagation involved the transport of momentum, chemical species, and heat. This is evident in the lively and engaging compilation of lectures by Michael Faraday entitled The Chemical History of a Candle, published in 1861 [14]. Early theory of flame propagation velocity was published by Ernest-Francios Mallard in 1875 [15], and further developed by both Mallard and the famous physical chemist Henry Le Chatelier [16]. The Nobel Prize winning Dutch physical chemist Jacobus van’t Hoff provided early insight in the late nineteenth century to the nonequilibrium nature of combustion by suggesting that an exothermically reacting system cannot be in equilibrium with its surroundings [17]. The early development of transport theory for combustion and ignition of energetic materials mostly belongs to the Russians. In the 1920s and 1930s, Yakov Zeldovich [18, 19] and Belyaev [20] pioneered the early application of transport theory to the combustion of gasses and solid energetic materials. In the same time period, Todes and Seminov provided the earliest recognized formal heat transport theory of the critical conditions for the initiation of thermal explosion for a boundary-heated system [21, 22]. Seminov’s unsteady treatment is widely cited and discussed today and is distinguished by its simplicity, qualitative nature, and illumination of physical features important to criticality. His analysis showed the conditions for which heat production exceeded heat removal, the condition for ignition, using simplified mathematics. To find an analytical solution, Seminov included an exponential temperaturedependent reaction term, but neglected heat transport within the explosive material, that is, the spatial derivatives, such that all resistance to heat transfer occurred at the material’s surface. However, it was known experimentally at the time that ignition generally occurred at a specific location within the explosive material, implying that an accurate analysis could not neglect internal heat transport. David Frank-Kamenetskii, a student of Seminov, took a different approach from his mentor and developed a steady-state solution that included internal heat conduction (spatial derivatives) and also included the nonlinear reaction term [23]. An essential feature of Frank-Kamenetskii’s analysis was the use of Seminov’s steady-state solution to develop a dimensionless temperature having a value near 1 at the explosion temperature and zero initially. The interplay between the works of these two researchers continued, as it was later recognized that Frank-Kamenetskii’s steady solution could

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provide key parameter information for Seminov’s unsteady solution such that a fairly clear picture of (transient) thermal explosion behavior emerged. When exactly this latter insight came about is unclear, but it is discussed in a text published by Frank-Kamenetskii in 1947 [24]. Their methods of solution and the results are especially clever and insightful considering the lack of computational resources. The work of these early researchers, embodied in the theories of Seminov and Frank-Kamenetskii, are essential for the understanding of energetic material ignition, primarily due to the clear presentation and illumination of salient features. As such, Chaps. 4 and 7 go into much greater detail of the development and predictive qualities of these theories. This early development of ignition theory was concerned with the boundary heating problem; investigation of hot-spot heating followed soon after. It was recognized in the 1940s that the size of a hot spot played a role in the critical conditions for ignition. This idea may have originated from intuition stemming from knowledge of heat transport theory, cognizance of the size effect in the Frank-Kamenetskii/Seminov boundary analyses, or more likely, both. Bowden et al. confirmed this when they observed a size effect in friction experiments, the results of which were reported in 1947 [25]. This size effect was shown theoretically in 1948 by the hot-spot analysis of Rideal and Robertson that examined impact ignition [26]. In that study, the researchers formulated the one-dimensional spherical transient and spatially dependent heat transport problem and presented an analytical solution. The analytical solution was possible because they neglected the nonlinear exothermic reaction term and imposed a temperature, rather than a heat source term. They determined critical hot-spot volume and temperature by equating the rate of heat conduction at a given temperature and spot size to heat evolved from chemical reaction over the same volume. As with the early work on boundaryheated ignition, computational tools did not yet exist to provide numerical solutions to the nonlinear equations that govern the hot-spot problem. In fact, these authors admit: “A more elegant and satisfactory treatment, however, would be to solve the general equation. . .,” referring to the unsteady 1D spherical heat equation with an Arrhenius heat generation term. Apparently, very little work was done on the hot-spot problem until the advent of numerical tools. In 1963, Merzhanov and his coworkers published the results of their numerical study that followed the dimensionless parameterization scheme of Frank-Kamenetskii [27]. They concluded that those parameters were also good indicators of criticality for the hot-spot ignition problem, although with a different functional dependency on temperature. The usefulness of the Frank-Kamenetskii parameters for finding the properties of a critical hot spot arises due to the universal physical basis for criticality. While boundary heating may be generic in nature, specific localized mechanisms cause hot-spot heating. In those cases, an additional criterion is required: localized power dissipation from the heating mechanism must also exceed the rate of local cooling such that the hot spot will reach a critical temperature. A current good review was provided by John Field et al.

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in 1992 [28]. They point out the important early contributions to impact and friction studies of Bowden et al. [25], Eirich and Tabor [29], Rideal and Robertson [26], Bowden and Yoffe [30], Afanas’ev and Bobolev [31], and Field and his coworkers [28]. Many of these studies are mechanistic in nature and some report a temperature rise under the stimulated conditions. From a transport theory perspective, it is most useful to have a rate of energy dissipation (power). Spark initiation studies provide a simpler platform to quantify energy and power from the electrical properties of the spark generator. Langevin and Bicquard published early studies of the spark initiation of primary explosives in 1934 [32]. Wyatt and his colleagues published their studies of spark initiation of primary materials in 1958 [33], and Tucker reported their research on secondary explosives in 1968 [34]. Somewhat surprisingly, these early papers, with some exceptions in spark initiation research, represent the state of the art of hot-spot ignition source research. Although mechanisms are fairly well understood, heat production in the context of transport theory has not been fully investigated and this remains an active area of research. So far, we have reviewed the foundations of ignition theory and laminar (nonviolent) combustion. It was known in the late 1950s that sufficient pressure would cause an explosive deflagrating from its outer surface to transition to a violent thermal explosion as flame penetrated into the bulk via an existing pore network or cracking induced by pressure. It was also known in this period that this phenomenon, usually called “convective burning,” was a critical step for DDT. In terms of transport phenomenon, this physical transition is reflected by the mathematical transition to theory where momentum transport analysis becomes essential for understanding the movement of heat and chemical species. The Englishmen Griffith and Groocock published their early work in 1960 [35], as did the contemporary Russian researchers Andreev and the already-mentioned pioneer Belyaev [36]. In 1962, Taylor published his work on the convective burning mechanism in HMX [37]. The evolution of this research was in part motivated by the potential for solid rocket motors, some of which were propelled by HMX and/or RDX, to explode or detonate under abnormal pressure conditions. As such, the propellant literature from the 1960s and 1970s provide the most complete theoretical transport basis for the convective burning problem and the review text of Kenneth Kuo and Martin Summerfield provides an excellent fundamental overview [38]. The process of DDT begins when the drag forces of moving gases become sufficient to set the solid phase into motion, thus amplifying the complexity of the problem. Bernecker and Price credited the aforementioned work of Griffiths and Groocock [35] in an early review in 1979 as providing the first complete description of DDT in a bed of granular explosive [39]. Perhaps it is not surprising that the theory of explosion originates coincidentally with the understanding of chemical equilibrium. The early contributions of the Russian researchers cannot be overstated, as followed by the English at Cambridge and, most recently, the researchers funded by the American Department of Defense. Over the last 20 years, researchers funded

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by the US Department of Energy have pursued this area of research in order to understand and minimize the consequences of a nonshock initiation event in a nuclear weapon system. While the early research has provided a reasonably clear picture of mechanisms and the appropriate theory, the evolution of computational tools and their ever-increasing ability to incorporate nonlinear, multiphysics, multiphase phenomena at ever-increasing spatial and temporal resolution now drives the advancement of the field. However, one cannot build good models without the basic theory and insight. That is the intent of this book. To follow the evolution of these research topics beyond their origins, the reader may perform a citation search from this section, and the texts of Frank-Kamenetskii and Zeldovich provide a complete theoretical foundation of transport theory in explosive systems. The other chapters of this book will provide the details of the most recent research.

2.3 Physical Properties Used in Transport Theory Analysis Solving the transport equations requires the physical properties including heat capacity, density, thermal conductivity, thermal diffusivity, viscosity, and species diffusion coefficient. These properties depend directly on the thermodynamic state of the system: pressure, temperature, etc. Common high performance explosives and explosive composites have well-characterized heat capacity, density, and thermal conductivity and data are available up to temperatures where decomposition reactions become significant. The reference text of Gibbs and Popalato [40] or the LLNL Explosives Handbook [41] provides these data for a wide range of explosives. For gases or liquids involved in the explosion process, viscosity and the diffusion coefficient also become factors. For tabulated data for all the properties for gases and liquids, except the diffusion coefficient, we refer the reader to general sources such as the CRC handbook [42] and the NIST Chemistry Web Book [43]. The diffusion coefficient depends on factors specific to the specific environment and tabulated data are sparse, but some data are provided in textbooks covering diffusion; See, for example, [44] and the references therein. Here we review the physical basis for each property, general value ranges, and general trends with pressure and temperature. Specific Heat. In conjunction with density and molecular weight, specific heat (on a mole basis) provides a measure of the thermal inertia of a system. That is, it is a measure of the resistance of a system to a change in energy content (temperature.) As such, specific heat affects transient heat conduction, but has no effect on steady-state heat conduction. Specific heat is mathematically defined as  c=

δq dT

 . path

(2.3)

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In order to make tractable relationships, the path is taken to be one of constant pressure or constant volume. For a constant volume process, specific heat is related to the internal energy:   du cV = . (2.4) dT V For a constant pressure process, the specific heat is related to enthalpy:   dh cp = . (2.5) dT P For ideal gases, it can be shown that cp − cv = R. And the ratio of the specific heats commonly appears in gas-phase problems: cp γ= . (2.6) cv Due to the relative incompressibility of the solid and liquid states, u = h such that cp ≈ cv . A simple relationship exists for the internal energy stored by a single degree of freedom regardless of the size or shape of the molecule: u= 

And cv =

1 RT. 2

du dT

 = v

(2.7) n R 2

(2.8)

for n degrees of freedom. For example, monatomic gas molecules translate through three dimensions and therefore have three degrees of freedom, such that 3 cv = R. (2.9) 2 The presence of rotational, vibrational, torsional, etc. modes increases the heat capacity accordingly. The free electrons in conducting solids contribute to the heat capacity at room temperature on the order of 0.03R/2. Not all modes are active at all temperatures and there are other temperature effects for more complex structures. The reader should consult a physical chemistry text or a statistical thermodynamics text for details of specific molecules and temperatures. Monatomic gasses have a specific heat exactly 3R/2 and diatomic gasses such as nitrogen, oxygen, carbon monoxide, etc., at room temperature are also near 3R/2. Liquid water at room temperature is 75.9 J mol−1 K−1 . Solid copper has a specific heat of 24.4 J mol−1 K−1 and the common organic crystalline explosive HMX has a specific heat of 325 J mol−1 K−1 at room temperature.

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Density plays a role in all three transport processes. It affects the transient solution of the heat transport equations and both the transient and steady state solution of the momentum equations. For reasonable temperatures and pressures found in the preignition regime, gas phase density may be computed from the ideal gas law equation of state. In the more nonideal post ignition regime, a more descriptive equation of state, such as the van der Waals equation of state, may be required. In extreme post-ignition conditions, with pressures and temperatures approaching detonation conditions, an equation of state such as the Abel or Becker–Kistiatowsky–Wilson equation of state may be necessary. Solid and liquid phase densities for preignition conditions are almost always measured quantities. Most liquids fall into the range of 800–1,200 kg m−3 . Most inorganic solids have crystal densities in the range of 1,500–3,000 kg m−3 . Most metals are in the range of 2,500–12,000 kg m−3 , with exceptions that exceed 13,000 kg m−3 such as mercury, uranium, plutonium, etc. Common organic crystalline explosives and their binders have densities in the range of 1,500–2,000 kg m−3 . Thermal conductivity is the proportional factor of Fourier’s Law of Heat Conduction, relating heat flux to a temperature gradient. As with density and heat capacity, thermal conductivity affects transient heat conduction. However, it also appears in the steady state solution. Gas thermal conductivity is related to the heat capacity of the particular molecule, the characteristic velocity, and the mean-free path. The kinetic theory of gases reasonably predicts the thermal conductivity of simple gases to 10 atm (or so) [45]:  κ3 T 1 , (2.10) k= 2 d π3 m where d is the molecular diameter, k is the Boltzmann’s constant, and m is the mass of the molecule. This general expression shows that pressure has a weak overall effect on thermal conductivity. Thermal conductivity of gases is on the order of 0.01–0.05 W m−1 K−1 , but can increase by a factor of 10 at combustion temperatures. In liquids, thermal conductivity is roughly an order of magnitude greater than gases because of the shorter mean free path. The kinetic theory of gases provides a starting point that leads to    Cp ∂P 2/3 2 2/3 2 , (2.11) k ≈ π d vs = π d Cv ∂ρ T where vs is the sound velocity. In most cases for liquids, the ratio of heat capacities is unity and the pressure–density relationship must be found from an appropriate equation of state. As such, liquid thermal conductivity also has a weak dependency on pressure, but no general temperature behavior. It may increase or decrease depending on the equation of state for the particular liquid. The thermal conductivity of liquids is on the order of 0.1–0.5 W m−1 K−1 .

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In solids, phonons and free electrons exchange momentum to cause heat conduction. For metallic solids, the dominant mechanism is the latter, and (2.10) above will provide an estimate and trends when evaluated for the electron gas. For non-metallic solids, phonon interaction (frequency) dominates and thermal conductivity generally increases with temperature. For non-metallic solids, conductivity ranges over 0.5–5, and metallic solids typically are on the order of hundreds of W m−1 K−1 . Common organic explosives and their binders are in the range of 0.1–0.6 W m−1 K−1 . Thermal diffusivity is the ratio of the thermal conductivity to the product of density and heat capacity: k (2.12) α= . ρc It is a measure of the efficacy of conduction of thermal energy relative to the storage of thermal energy. This may be thought of as thermal inertia or the rate of diffusion of heat. Thermal diffusivity always appears in the exponential terms of the transient solution of heat transport analyses; large values reflect short transients. The dimensionless Fourier number provides an estimate of the thermal time constant of a system when its value is near unity: Foc =

ατ ≈ 1, L2

(2.13)

where L is a characteristic length. Simple gases at STP have a thermal diffusivity near 2 × 10−6 m2 s−1 , but can increase to near 1 × 10−3 m2 s−1 at combustion temperatures. Most liquids are on the order of 1 × 10−7 m2 s−1 and solids are on the order of 1–2,000 × 10−7 m2 s−1 , with metallic solids around 2–5 × 10−5 m2 s−1 . Organic solids, including common explosives and their binders, generally fall into the range of 1–2 × 10−7 . Viscosity is the proportionality factor of Newton’s law of viscosity, relating momentum flux (shear stress) to the velocity gradient. It appears in both the transient and steady state solution of the momentum equations, and also appears in Darcy’s Law along with the permeability to relate flow to a pressure gradient. The simple kinetic theory can again be used to qualitatively understand the effect of the state variables on viscosity for gases: √ 2 mκT . (2.14) μ= 3π 3/2 d2 As with thermal conductivity, there is no pressure dependence at normal conditions. Gas viscosities are on the order of 10−5 under normal conditions and can increase by a factor of 10 at combustion temperatures. In the denser liquid phase, in order to translate, molecules must overcome an energy barrier presented by its neighbors. A kinetic theory approach therefore reveals an exponential dependence on temperature relative to the boiling temperature (Tb ):   Tb ρNa h exp 3.8 μ= , (2.15) M T

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where h is Planck’s Constant. Examples of liquid viscosity are water: 1.75 × 10−3 ; and engine oil: 3.85 N s m−2 . Kinematic viscosity is the ratio of the viscosity to density: μ ν= . (2.16) ρ Analogous to thermal diffusivity, this parameter may be thought of as the rate of diffusion of momentum. It always appears in the exponential terms of the transient solution of momentum problems; large values reflect short transients. An estimate of a system’s time constant is given by τ≈

L2 . ν

(2.17)

Simple gases at STP have a kinematic viscosity near 1–25 × 10−5 m2 s−1 and the value does not change significantly near combustion temperatures. Liquids have a wide range of values; water at STP is 1.75 × 10−6 . Mass diffusivity is the proportionality constant of Fick’s law of diffusion and relates the flux of a molecular species to the concentration gradient. It appears in the solution to both the steady and unsteady solution of the species mass transport equations. Analogous to kinematic viscosity and thermal diffusivity, the dimensionless mass transfer Fourier number provides an estimate of the time constant of a system when its value is near unity: Fom,c =

Dτ ≈ 1. L2

(2.18)

Kinetic theory tells us that translational velocity increases with temperature, leading to a more rapid redistribution of species, and collision rates increase with increasing pressure, leading to a slower redistribution. The expression for a gas mixture consisting of nearly identical molecules (e.g., isotopes) shows the dependence on pressure and temperature:  3  3/2 κ T 2 . (2.19) DAA = 3 π 3 m P d2 Diffusion gas phase coefficients for simple binary systems generally range from about 1 to 5 × 10−5 m2 s−1 . Similar to viscous momentum transport, diffusion in liquids is an “activated” process where molecules must overcome the barrier presented by nearby neighbors: 

  2/3 M Tb exp −3.8 . (2.20) ρNa h T In practice, many factors influence the process. In most cases, empirical measurements are required and (2.20) can be used cautiously to extrapolate from known data. Typically, liquid diffusion coefficients are 105 times smaller than gases. D ≈ κT

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2.4 Transport Theory Equations The equations governing transport processes have been developed ad nauseum, and there are many good texts covering each transport phenomenon. In particular, we suggest the text of Bird, Stewart, and Lightfoot as a comprehensive and detailed general text [46], and the aforementioned texts of Frank-Kamenetskii and Zeldovich for developments more specific to combustion and explosions. It is the intent of this text to provide a complete overview of nonshock initiation, and so we include a brief review of the formal development of transport theory. The development of all three transport fields begins with the fundamental concept of the conservation laws: energy, mass, and momentum are conserved within a control volume. Within this very simple framework, and through the straightforward application of differential calculus, we can develop all of the transport equations. It is always useful to begin with a very general form of the transport equations. Simplification is usually straightforward and a very general starting point allows us to see what assumptions must be made to arrive at the most tractable form for a particular situation. A balance on a differential volume equates energy stored and produced in the volume to the energy flux gradient: − ∇φ =

∂h , ∂t

(2.21)

where we have written storage and production in terms of enthalpy. For an Eulerian coordinate system (fixed reference frame) and in terms of temperature and heat capacity, we account for all fluxes (Fourier conduction and convection), storage and generation:   ∂T ρcp + (v · ∇T ) = ∇ (k∇T ) + Δhrxn r (T, X) + q, ˙ (2.22) ∂t where Δhrxn r (T, X) is the product of the enthalpy change associated with chemical reaction and the rate of that reaction. The rate, of course, generally depends exponentially on temperature and on some quantity related to the availability of reactants (X); for example, concentration of gas phase reactants or solid phase surface area (see Chap. 3.) q˙ is the heat generated by any other source. In a Lagrangian reference frame (moving with the flow), we invoke the substantial derivative and we can simplify the equation to ρcp

DT = ∇ (k∇T ) + Δhrxn r (T, X) + q. ˙ Dt

(2.23)

The Lagrangian frame might be useful in the post-ignition regime when things begin to move or for the analysis of a propagating flame. Specific enthalpy was used in this derivation. If we were to start the derivation with specific internal energy, the result would depend on the constant

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volume heat capacity and reaction internal energy, rather than the constant pressure heat capacity and reaction enthalpy. The distinction is important as the former generally relates to incompressible media such as solids, liquids, and constant density gas phase processes. The latter generally applies to compressible gas phase processes. In this latter case, one must use caution with respect to reaction energy, as enthalpy is typically reported and the relationship between enthalpy and internal energy must be accounted for. Specifically, ∂ ∂h ∂P ∂v ∂u = (h − P v) = −v −P . ∂t ∂t ∂t ∂t ∂t

(2.24)

In general, all the material and thermodynamic properties can be space-, time-, and temperature-dependent. There are three types of boundary conditions in heat transport analyses. These are the Dirlecht boundary conditions where we impose a surface temperature, Ts ; the Neumann boundary conditions where we specify the heat flux, q˙s , at the surface; and the Robin boundary conditions, which is a weighted hybrid of flux and temperature conditions. Mathematically, the Nuemann condition is (2.25) n · (−k∇T |s ) = q˙s . The Robin boundary condition can take a general form that may account for convection normal (through) the boundary, convection parallel to the boundary, or conduction normal to the boundary: n · (−k∇T|s + ρCp uTs ) = n · [ρCp u ( T∞ − Ts ] + h (T∞ − Ts ) .

(2.26)

In these expressions, the subscript s indicates the boundary coordinate, ∞ indicates the condition far away from the system, and h is the heat transfer coefficient. The parameter h depends on the physical properties and movement (flow) of the medium surrounding the system. The method for computing h comes from boundary layer theory, and that discussion is beyond the scope of this text. However, simple methods to estimate h are readily available in the references provided [44, 46]. We arrive at the species transport equations by a similar route. Balancing the gradient of the mass flux (n˙ i ) with the mass density storage and production (r) of specie i in a differential volume leads to − ∇n˙ i =

∂ρi + ri , ∂t

(2.27)

and we must account for each species present. Including convection transport, diffusion according to Fick’s law and rewriting the equation in terms of the species mole fractions: − v · ∇ρi + ∇ · (ρi Dij ∇xi ) =

∂ρi + ri (T, X) , ∂t

(2.28)

where xi is the mole fraction of species i. This is a very general (and difficult to apply) form of the species mass transport equation with no restriction on

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constant density or diffusion coefficient. However, this form is only valid for binary systems, and in general the derivation of equations for more than two components with nonconstant density is difficult. If we make the assumption of constant density and diffusion coefficient, the more familiar diffusion equation results (Eulerian coordinates): ∂Ci + v · ∇Ci = Di,n ∇2 Ci + ri (T, X), ∂t

(2.29)

and in Lagrangian coordinates, DCi = Di,n ∇2 Ci + ri (T, X). Dt

(2.30)

These forms are most valid for dilute solutions at constant temperature and pressure. Of course, those conditions do not describe most of the interesting explosives problems, but there is insight to be gained by the application of simplified theory. Most notably, researchers have applied (2.29) with its heat transport analog (2.22) to describe flame structure [19]. The codependency of these equations of course arises through the heat and species generation term, which depend on temperature and the quantity X that relates to concentration. Two notable assumptions exist to decouple the equations. If the Lewis number, ρCD , (2.31) k is close to unity, the concentration and temperature fields are similar. And the assumption of zero-order chemical reaction allows decoupled solution of the heat and species transport equations. The boundary conditions for the species transport equations are analogous to those for heat transport. We can specify a Dirlecht condition at the boundary (concentration), a Nuemann condition (species flux), or the Robin hybrid condition. The velocity fields of (2.22) and (2.29), if they exist, are computed using the principle of conservation of momentum. As with heat and species transport, momentum transport occurs by convection and by molecular collisions (i.e., conduction or diffusion.) While these two mechanisms cause transport in parallel for heat and species transport, convective momentum transport by fluid motion is orthogonal to transport by molecular collisions. Consequently, the derivation leads to second order tensor expressions. The full derivation of the governing equation falls into the fluid mechanics field of study. Formal application of the full equation is rare in our field of study, but for completeness we provide what is commonly referred to as the Navier–Stokes equation:   ∂v 2 + v · ∇v , (2.32) μ∇ v − ∇P = ρ ∂t Le =

where this form requires a constant density and viscosity (μ). Note that the product of the gradient operator vector and the velocity vector and the

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Laplacian operator vector and the velocity vector are both dyadic products and the result is a tensor. The application of the Navier–Stokes equation to nonshock initiation problems requires a level of computing power not generally available. In lieu of those equations, researchers have applied Darcy’s law of permeation in porous media as an approximation for creeping flow conditions (low Reynold’s number, Re < 10): −κ ∇P, (2.33) vs = μ where κ is the permeability and vs is the superficial velocity vector. In the preignition regime, the movement of chemical species under the influence of a pressure gradient has only recently been incorporated into analysis [5], and we will review that work in a later section. Of course, conditions in the postignition regime do not meet the creeping flow requirement and we must apply the Navier–Stokes equations along with the science of Multi-phase Computational Fluid Dynamics to provide meaningful solutions. We see the further interplay of transport processes that occurs through the pressure gradient term in both the Darcy approximation and the Navier– Stokes equation. The pressure gradient must be found from the production of chemical species via coupling with the heat and species transport equations. Strictly speaking, we must simultaneously consider all three transport phenomena and the interplay originates both from chemical reaction heat and species generation and the thermodynamic linkage of pressure, temperature and density. Consideration of fully coupled transport theory remains a significant challenge, mainly due to limitations of computing (and funding) resources.

2.5 Selected Examples of Transport Theory Applied to Explosive Problems We will review three recent cases where transport theory has been applied to problems involving explosives. First, to more fully understand the fundamentals of hot spot initiation, we have chosen a model system for study that uses the internal and localized dissipation of microwave energy to generate hot spots [47]. Next we will review a boundary-heated problem that includes species and momentum transport. Zerkle has invoked Darcy’s Law for a boundary-heated problem and he showed that permeation of decomposition products through the bulk of the explosive was required to more accurately predict temperature profiles and the violence of reaction [5]. Finally, we discuss a post-ignition problem having coupled heat and species transport. The work of Ward et al. used a low activation energy kinetic expression to characterize the combustion of HMX and made assumptions about the similarity between the thermal and species diffusion coefficients [7].

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Theory of microwave-induced hot spots. Organic explosives generally interact weakly with electromagnetic energy in the microwave frequency range. Therefore, the addition of electromagnetically absorbing inclusions provides a tractable system for the study of localized heating and ignition phenomena. The work considered the connections amongst the theories of dielectric mixing, microwave absorption, heat transfer, and thermal ignition theory in order to understand a model system of idealized hot spots; namely, small high thermal conductivity spherical inclusions (<100 μm) dispersed within a homogeneous low thermal conductivity explosive. The advantage to this mode of heating compared to “normal” hot spot mechanisms is that electromagnetic excitation isolates purely thermal effects while the others rely upon mechanical stimulation of material. Here, we review the work of Perry and Glover [47], and use dimensionless numbers to illuminate dominant mechanisms and to simplify the problem [44, 46]. The most generic solution to this problem is found by writing (2.22) for both the region inside the inclusion (0 < r < R) and the region outside the inclusion (R < r < ∞). For the region inside the inclusion, we use the microwave heat generation term E2 , (2.34) q˙ = ωεi 2 where ω is the microwave angular frequency, εi is the dielectric loss (absorptivity) of the inclusion, and E is the electric field strength near the inclusion. For the region outside the inclusion, we use a zero-order form of the Arrhenius reaction term. The problem is solved by using a flux-matching boundary condition at the interface (r = R), a symmetry boundary condition at r = 0, and the initial temperature far away from the inclusion (r = ∞). Finding the time and spatially dependent temperature behavior requires a numerical solution of the coupled set of equations for r < R and r > R. However, in the case of microwave hot spot generation, the heat transport properties of a typical high thermal conductivity microwave absorbing inclusion (e.g., silicon carbide) suggest that the low thermal conductivity explosive surrounding the hot spot will limit the rate of heat transport. This is shown formally by evaluating the dimensionless Biot number: Bi =

2hR , ki

(2.35)

where h is the heat transfer coefficient and the subscript i refers to the inclusion. The heat transfer coefficient is found using the dimensionless Nusselt number: 2hR , (2.36) Nu = ke where the subscript e refers to the explosive. For a sphere in a stationary medium, the Nusselt number is exactly 2 and h=

ke . R

(2.37)

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W.L. Perry

For ke  ki and small R, Bi  1 and the transport equation for R < r < ∞ will adequately describe the behavior. Equation (2.22) in spherical coordinates for the region outside the inclusion is   ∂T 1 ∂ ρe Δhrxn 2 ∂T [ρcp ]e = ke 2 r(T, X). (2.38) r + ∂t r ∂r ∂r M Let θ=

T − T∞ r ; ξ= , Tig − T∞ R

(2.39)

and we use a Fourier number to define a dimensionless time τ=

αe t . R2

(2.40)

The dimensionless form of the equations is then 1 ∂ ∂θe = 2 ∂τ ξ ∂ξ



ξ2



  Ea ∂θe ρΔhrxn R2 + A exp − , ∂ξ M ke (Tig − T∞ ) R (θe (Tig − T∞ ) + T∞ ) (2.41)

where the expression in brackets is the dimensionless Damk¨ohler number, relating the rate of heat conduction to the rate of heat production by chemical reaction. The boundary condition is ωεi E 2 R2 dθ qR ˙ 2 ˙ = = − = Θ. (2.42) 3ke (Tig − T∞ ) 6ke (Tig − T∞ ) dξ ξ=1 Because of microwave skin depth effects, R must be less than ∼100 μm. Even with this abbreviated statement of the problem, a numerical solution is required to find the evolution of the temperature profiles around the hot spot and the time to ignition. However, we can learn salient features about the behavior of the system without the full solution. From our general understanding of hot spot ignition, we know the following physical criteria must be met for ignition to occur: 1. Microwave energy dissipation must exceed heat removal at the ignition temperature. 2. The rate of heat production via exothermic chemistry must exceed the rate of heat transport away from the inclusion at the ignition temperature. Equation (2.42) provides the relationship for condition 1. The dimensionless gradient at the inclusion surface must exceed unity for the hot spot to reach the ignition temperature. To find the ignition temperature, we use the Damk¨ ohler number in (2.41) to find when condition 2 is true and use that information to determine the ignition temperature. By definition, θ = 1 when reaction heat exceeds heat removal by conduction:   Ea ρΔhrxn R2 A exp − = 1. (2.43) M ke (Tig − T∞ ) RTig

2 Transport Phenomena for Nonshock Initiation Processes

33

Ignition Temperature [K]

800

750

700

650

600

2

4 68

10–5

2

4 68

10–4

2

10–3

Inclusion Size [m]

Fig. 2.1. The relationship between ignition temperature and hot spot inclusion size for HMX

This transcendental equation must be solved iteratively to find Tig . Figure 2.1 shows the relationship of Tig and hot spot size for values representative of the kinetic, thermochemical, and thermophysical parameters of HMX. We have learned two very important features of this system without solving the governing equations: the ignition temperature and the minimum microwave power dissipation required to reach the ignition temperature. However, we cannot deduce the heating rate or the time required to reach the ignition temperature from this approach. If we do not care about the exact nature of the spatial temperature profiles, we can further simplify the problem by ignoring the spatial dependence and performing an energy balance on the inclusion. The balance leads to a dimensionless equation describing the temperature of the inclusion (recall N u = 2 for a stagnant medium): dθ ˙ − θ) = 3(Θ dτ  where τ = and the solution is

ke t ρi Ci R2

˙ [1 − exp (−3τ  )] . θ=Θ

(2.44)

(2.45)

(2.46)

The dimensionless time to ignition is  τig

  1 1 . = − ln 1 − ˙ 3 Θ

(2.47)

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W.L. Perry

This straightforward development illuminates the temporal qualities of the ˙ < 1, a condition idensystem. Mathematically, ignition will never occur for Θ tical to the statement derived from the dimensionless boundary condition of (2.42). Here, we have used R for the characteristic dimension. This most accurately (but still qualitatively) shows the transient behavior prior to ignition ˙ that is, Θ ˙ > 5. For values of Θ ˙ closer to unity, a for larger values of Θ; characteristic length between R and 2R is more accurate. The dielectric loss (εi of (2.34)) of the inclusion depends on its electrical conductivity and the permittivity of the explosive medium. Generally, inclusions embedded in a low permittivity medium that are less than 100 μm in size and that have a conductivity in the semiconductor range (10 ohm) have 

a high dielectric loss ∼70 × 10−12 F m−1 , leading to a high rate of energy dissipation [47]. High frequency also favors a high dissipation rate (2.34). Figure 2.2 shows the transient inclusion temperature and Fig. 2.3 shows the dimensionless ignition time over a range of dimensionless microwave power ˙ While this analysis was qualitative, it illuminates the dissipation values (Θ). factors important to microwave hot spot ignition. More importantly, we may apply the analysis for any heating stimulus as long as we can satisfy the stated ˙ assumptions and find an appropriate expression for Θ. Boundary heating problem including species and momentum transport. In the early 2000s, researchers at Los Alamos conducted a series of large-scale (10 kg) PBX 9501 cookoff experiments (the Large Scale Annular Cookoff experiment: LSAC.) The annular charges (8.9 cm i.d., 15.9 cm o.d.) were held at a temperature sufficient to drive sublimation and decomposition reactions (∼180◦C) and the β − δ phase transition such that substantial thermal 1.0 0.8

θ

0.6 0.4 0.2 0.0 0.0

0.5

1.0

1.5

2.0

τ Fig. 2.2. The dimensionless transient inclusion temperature for three values of the dimensionless microwave heating term: 0.5, 1, and 5

2 Transport Phenomena for Nonshock Initiation Processes

35

1.0 0.8

τ ig

0.6 0.4 0.2 0.0 0

2

4

6

8

10

Θ Fig. 2.3. Dimensionless ignition time as a function of dimensionless microwave power dissipation. There is a threshold below which ignition will not occur

damage occurred; that is, significant porosity developed. These experiments were highly instrumented for the primary purpose of calibrating cookoff models that attempt to predict the explosion violence after ignition. The boundary conditions that lead to ignition have already been well established and documented in modern texts [48]. However, ignition theory does not tell us anything about the post-ignition explosive behavior. It was suspected, a priori, that the high degree of porosity generated during the heating phase would facilitate movement of reactive decomposition products through the charge and potentially affect both the pre- and post-ignition behavior. Indeed, it was found that the models could not accurately predict temperature profiles without including the movement of reactive species by momentum transport. Both the temperature field and distribution of reactive species affect the post ignition behavior, making the incorporation of species and momentum transport an important step towards understanding explosion violence. As modeling of the LSAC is an excellent example of the application of heat transport theory coupled with species and momentum transport, we will review the work of David Zerkle towards this end [5]. The model development began by writing the species transport equation (e.g., (2.28)) for a multiphase system: ∂ (φi ρi ) + ∇ · (φi ρi vg ) = r (T, X) , ∂t

(2.48)

where φi is the species volume fraction. Compressible gas phase processes were important to this analysis, and the heat transport equation was derived in terms of internal energy. The convective transport term of (2.22) was written as

36

W.L. Perry



  Pg ∇ · φg ρg vg ug + , ρg

(2.49)

where the subscript g refers to the gas phase. The specific internal energy is related to temperature through (2.4); pressure and temperature are related by the ideal gas equation of state. These relations lead to the following form of the heat transport equation: ρCv

∂T + ∇ · (φg ρg vg Cpg T ) = ∇ (−k∇T ) + Δurxn r (T, X) . ∂t

(2.50)

In lieu of incorporating the multiphase momentum transport equations, the velocity field was found using Darcy’s law (2.33) where vs = φg vg . The permeability for a range of thermal damage conditions was determined from the small-scale experiments of Parker et al. [49]. The overall effective density, heat capacity, and thermal conductivity were computed from (respectively):    φi ρi Cv ; and k = ρ= φi ρi ; Cv = i φi ki . (2.51) i i ρ The reaction term (r) was modeled using the Los Alamos four-step decomposition mechanism that includes the endothermic solid phase transition, the endothermic sublimation of HMX, and the exothermic gas phase reactions [50]. The coupling of momentum, species, and heat transport in this system comes about through the thermodynamic state of the system, where, in this case, the ideal gas law equation of state ties pressure, temperature, and density together and the system is driven by the reactive production (or consumption) of heat and species. The partial differential equation solver, Flex PDE [51], provided the temperature, species concentration, and velocity fields. Figure 2.4 shows the computed temperature profiles, which agree well with the experimentally measured values. Figure 2.5 shows the comparison of temperature profiles with and without including flow, just prior to ignition. The inclusion of flow indicated a much larger volume at a higher temperature and that reactive species were also spread through a large portion of the charge. These two factors mean that the material was primed for faster reaction spreading, hence more violent explosion. Finally, the reader might notice the asymmetry in the results. This comes about from variations in contact resistance imposed on the interface between the explosive and its container. This experiment is discussed more fully in Chap. 7. A laminar flame model. We have examined two preignition scenarios, and now we examine the post-ignition transport phenomena associated with laminar burning of HMX. While we are most interested in violent explosion, the physical processes that occur for laminar burning also occur during violent explosion. Most explosive combustion processes are surface phenomena: combustible gases evolve from a solid surface and burn in the gas phase at a well defined “stand-off” from that surface. Pressure controls the stand-off distance.

2 Transport Phenomena for Nonshock Initiation Processes

37

Fig. 2.4. Calculated temperature profiles just prior to ignition for the annular cookoff shot. The calculation included Darcy flow and agreed well with experimental measurements. The boundaries were held at 180◦ C and the highest temperature contour was 220◦ C. The contour intervals are approximately 2.86◦ C. The inner diameter was 8.9 cm and the outer diameter was 15.9 cm

Fig. 2.5. Comparison between temperatures computed with (upper) and without (lower) species and momentum (Darcy) transport just prior to ignition. The highest temperature contours are 220◦ C. The lowest temperature contour with Darcy flow was 198◦ C and the lowest temperature contour without flow was 191◦ C

The essential difference between laminar burning and violent explosion is that laminar burning occurs over a surface area much larger than any defects, pores, cracks, grains, etc. in that surface and the stand-off is also much larger than

38

W.L. Perry

those surface features. Violent explosive burning occurs when the pressure squeezes the flame stand-off to a dimension on the order of the aforementioned feature sizes, forcing hot combustion products into the cracks and pores, such that combustion occurs on a high surface area within the bulk of the material. Convective (momentum transport driven) processes of course play the dominant role in this regime, but the fundamental process of surface gas evolution and burning near the surface that occur during laminar combustion still occur in the violent explosion regime. Because of the relevance of the laminar analysis to the whole range of ignition response, we will review the work of Ward and Son [7]. Those authors develop the heat and species transport equations for the laminar flame and provide a straightforward and illuminating analytical solution that correctly describes experimentally observed flame structure of burning HMX, which is shown in Fig. 2.6. Their analysis consisted of two parts: development of the steady-state heat transport equation with a high activation energy decomposition reaction for the solid phase and the steady-state coupled heat and mass transport equations with very low activation energy exothermic combustion reactions for the gas phase. One-dimensional cartesian coordinates were used with the origin at the solid–gas interface for both regions. For the solid phase, (2.22) was reduced to mc  p

∂T ∂2T = ke 2 + Δhd r (T ), ∂x ∂x

(2.52)

where the subscript d refers to the decomposition reaction HMX → I,

(2.53)

where I is a representative gas phase intermediate. The kinetic expression is 1.2 1.0

Solid

Gas

θ

0.8 0.6 0.4 0.2 –1.0

Stand Off –0.5

0.0

0.5

1.0

ξ

Fig. 2.6. The quantitative laminar flame temperature structure predicted by Ward et al.

2 Transport Phenomena for Nonshock Initiation Processes

  Ed r (T ) = ρe Ad exp − . RT

39

(2.54)

The symbol m ˜ is the mass flux and was the variable in the final solution that represented the “feed” of gas phase reactants for the gas phase part of the analysis. It is also the parameter that translates into a burning rate and is thus a parameter of primary interest. A floating boundary Dirlecht boundary condition, Ts , was specified at the solid–gas interface and ambient temperature in the solid bulk (T = T0 @ x = −∞) provided the second boundary condition. Merzhanov and Dubovitskii [52] provided the original solution for the mass flux: Ad RTs2 ke pe · exp (−Ed /RTs ) . (2.55) m 2 = Ed [cp (Ts − T0 ) − Δhd /2] To develop the problem in the gas phase, the authors began with the assumption of equality between the molar heat capacity of the gas and solid phase, and that the dimensionless Lewis number was unity: Le ≡

αg = 1, D

(2.56)

where the subscript g refers to the gas phase. These assumptions are reasonably accurate and they substantially simplify the problem. Expressing the governing equations in terms of dimensionless variables further facilitates the solution; the following relationships were used: E Δhi T ; Δh∗i = ; ; E∗ = T∞ − T−∞ R (T∞ − T−∞ ) Cp (T∞ − T−∞ ) xm ˜ r cp m ˜ ξ= ; and m ˜∗ = , (2.57) kg m ˜r

θ=

where m ˜ r is a reference mass flux. These definitions lead to the following dimensionless and steady forms of (2.22) (heat) and (2.29) (species): m ∗

∂2θ ∂2X ∂θ ∂X = 2 − Dah and m = ∗ + Das , dξ ∂ξ dξ ∂ξ 2

(2.58)

where X is the mass fraction of I and Dah and Das are the heat (h) and species (s) transport Damk¨ ohler numbers. The boundary conditions were θ (0) = θs ; θ (ξ → ∞) = θf ; X (0) = Xs ;

and X (ξ → ∞) = 0.

(2.59)

The similarity of the two equations and the fact that Le = 1 leads to a relationship between the mass fraction and the dimensionless temperature: X=

θf − θ , Δh∗c

(2.60)

where the subscript c refers to the gas-phase combustion reaction. The route leading to this relationship is not entirely straightforward, and we refer the

40

W.L. Perry

reader to Zeldovich for the derivation [53]. The gas phase reaction was assumed to be of the form I + X → C + X, (2.61) and the kinetic expression was   2  PM Ec r (T, X) = Ac exp − X. R RT

(2.62)

Using (2.60), the Damk¨ ohler numbers are     E∗ Ec∗ , Dah = Da∗ (θf − θ) exp − c ; Das = Da∗ X exp − θ θf − XΔh∗c (2.63) where Da∗ =

kg Ac cp



P ·M m ˜ rR

2 .

(2.64)

They still required a numerical solution for a nonzero E ∗ ; however, an essential feature of this work was the authors’ recognition that a solution using high gas phase activation energy did not qualitatively predict experimentally observed temperatures and they investigated the solution behavior using a very small gas phase activation energy. In the limit, Ec∗ → 0 had the added benefit of allowing an analytical solution. In dimensionless variables the temperature and concentration profiles were   ξ X θ − θf = exp , (2.65) Xs θs − θf ξg where ξg the dimensionless flame stand-off distance is 2 . ξg = √ ∗2 m ˜ + 4Da∗ − m ˜∗

(2.66)

The dimensionless form of the mass flux expression (2.55) is m ˜ ∗2 =

Ad θs2 exp (−Ed /θs ) , Ed [θs − θ−∞ − Δh∗d /2]

(2.67)

and the surface temperature was θs = θ−∞ + Δh∗d +

Δh∗c . ξg m ˜∗ +1

(2.68)

The ‘−∞’ subscript in these equations refers to the far upstream (initial) condition. Unfortunately, after systematically following the route of nondimensionalization, the Lewis number assumption, determination of the surface temperature and mass flux required iteration. What the authors found, however, is that the low activation energy assumption indeed resulted in a

2 Transport Phenomena for Nonshock Initiation Processes 2.5x10−3

70 60

2.0

50 1.5

40 30

1.0

20 0.5

Flame Stand Off [m]

Mass Flux [kg/sq.m/s]

41

10 0

6

2

105

4 6

2

4 6

106

2

4

0.0

107

Pressure [Pa]

Fig. 2.7. Mass flux from solid to gas and the flame stand-off distance as a function of pressure according to Ward et al.

fairly accurate prediction of temperature profiles, mass flux, stand-off, and surface temperature. Furthermore, the theory correctly predicts the effect of pressure: m ˜ ∝ P n, (2.69) where n ≈ 0.85, as shown in Fig. 2.7, was also in good agreement with experimental observations. We have known for some time that the burning rate of solid propellants follows this type of pressure relationship, often referred to as Vielle’s law. This relationship, however, was empirically determined, whereas the relationships leading to the theoretically determined behavior depend on physical parameters and thereby provide insight to the factors that affect the overall behavior. The theory also reveals how flame stand-off depends on pressure and this is also shown in Fig. 2.7. This behavior is particularly important; as mentioned above, the flame stand-off distance relative to characteristic surface defect dimensions determines the pressure at which laminar deflagration transits to violent explosion.

2.6 Summary In this chapter, we have discussed the general framework of transport phenomena and their central role as the theory that describes ignition and combustion behavior. The development of this theory has a rich and interesting history. Undoubtedly, in all fields of study, creative and insightful individuals advance the state of the art, and our field is no exception. We have measured or we can compute (or estimate, to a reasonable degree) the physical properties of transport theory, we know the conservation-based equations of

42

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transport theory, and we can therefore compute solutions to many important problems. However, in the modern era of advanced numerical solutions, we must remain cognizant of the important physical processes that govern ignition and combustion behavior. Progress requires computational advancement while maintaining our intuitive understanding.

References 1. Tarver, C.M., Tran, T.D.: Thermal decomposition models for HMX-based plastic bonded explosives. Combust. Flame 137, 50–62 (2004) 2. Kubota, N.: Survey of rocket propellants and their combustion characteristics. In: Kuo, K.K., Summerfield, M. (eds.) Fundamentals of Solid-Propellant Combustion, Progress in Astronautics and Aeronautics, vol. 90. American Institute of Aeronautics and Astronautics, New York (1984) 3. Prasad, K., Yetter, R.A., Smooke, M.D.: An eigenvalue method for computing the burning rates of HMX propellants. Combust. Flame 115, 406 (1998) 4. Perry, W.L., Dickson, P.M., Parker, G.R., Smilowitz, L.B., Henson, B.F., Asay, B.W.: Factors affecting explosive reaction violence in PBX 9501. In: Proceedings of the 13th International Detonation Symposium (2006) 5. Zerkle, D.K., Luck, L.B.: Modeling cook-off of PBX 9501 with porous flow and contact resistance, Los Alamos National Laboratory document LA-UR-03-8077; or 21st JANNAF Propulsion Systems Hazards Subcommittee Proceedings CD# JSC-CD-24 (2003) 6. Zeldovich, Y.: Flame propagation in a substance reacting at initial temperature. Combust. Flame 39, 219–224 (1980) 7. Ward, M.J., Son, S.F., Brewster, M.Q.: Steady deflagration of HMX with simple kinetics: A gas phase chain reaction model. Combust. Flame 114, 556–568 (1998) 8. Kumar, M., Kuo, K.K.: Flame spreading and ignition transients. In: Kuo, K.K., Summerfield, M. (eds.) Fundamentals of Solid-Propellant Combustion, Progress in Astronautics and Aeronautics, vol. 90. American Institute of Aeronautics and Astronautics, New York 305 (1984) 9. Asay, B.W., Son, S.F., Bdzil, J.B.: The role of gas permeation in convective burning. Int. J. Multiphas. Flow 22(5), 923–952 (1996) 10. Parker, G.R., Peterson, P.D., Asay, B.W., Dickson, P.M., Perry, W.L., Henson, B.F., Smilowitz, L.B., Oldenborg, M.R.: Examination of morphological changes that affect gas permeation through thermally damaged explosives. Propellants, Explosives and Pyrotechnics 29, 274 (2004) 11. Zerkle, D.K., Asay, B.W., Parker, G.R., Dickson, P.M., Smilowitz, L.B., Henson, B.F.: On the permeability of thermally damaged PBX 9501. Propellants, Explosives and Pyrotechnics 32, 251 (2007) 12. Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-todetonation transition (DDT) in reactive granular materials. Int. J. Multiphas. Flow 12(6), 861–889 (1986) 13. McAfee, J.M., Asay, B.W., Bdzil, J.B.: Deflagration-to-detonation in granular HMX: ignition, kinetics and shock formation. In: Proceedings of the 10th International Detonation Symposium, 716 (1993)

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14. Faraday, M.: The Chemical History of a Candle. Cherokee Publishing Company, Atlanta, Georgia (1993) 15. Mallard, E.F.: Ann. Mine. 7, 355 (1875) 16. Mallard, E.F., Le Chatelier, H.: Combustion of explosive gas mixtures. Ann. Mine. (1883) 17. van’t Hoff, J.H.: Lectures on Theoretical and Physical Chemistry, Lehfeld, R.A. (trans.). Arnold, London (1899) 18. Zeldovich, Y.B.: On the theory of flame propagation. Zh. Fiz. Khim. 22, 27 (1928) 19. Zeldovich, Y.B., Barenblatt, G.I., Librovich, V.B., Makhviladze, G.M.: The Mathematical Theory of Combustion and Explosions (English Translation). Consultants Bureau, New York (1985); Original Russian text: Nauka Press, Moscow (1980) 20. Belyaev, A.F.: Zh. Fiz. Khim. 12, 93 (1938) 21. Todes, O.M.: Zh. Fiz. Khim. 4, 78 (1933) 22. Seminov, N.N.: Z. Phys. Chem. 48, 571.(1928) 23. Frank-Kamenetskii, D.A.: Zh. Fiz. Khim. 13, 738 (1939) 24. Frank-Kamenetskii, D.A.: Diffusion and Heat Transfer in Chemical Kinetics, Appleton, J.P. (trans.). Plenum Press, New York (1969); Original Russian text: Nauka Press, Moscow (1947) 25. Bowden, F.P., Stone, M.A., Tudor, G.K.: Proc. Roy. Soc. Lond. A 188, 329 (1947) 26. Rideal, E.K., Robertson, A.J.B.: Proc. Roy. Soc. Lond. A 195, 135 (1948) 27. Merzhavanov, A.G., Barzykin, V.V., Gontkovskaya, V.T.: A problem of local thermal explosion. Dolk. Akad. Nauk SSSR 148(2), 380 (1963) 28. Field, J.E., Bourne, N.K., Palmer, S.J.P., Wallery, S.M.: Hot-spot ignition mechanisms for explosives and propellants. In: Field, J.E., Grey, P. (eds.) Energetic Materials. The London Royal Society, London (1992) 29. Eirich, F.R., Tabor, D.: Proc. Camb. Phil. Soc. 44, 566 (1948) 30. Bowden, F., Yoffe, A.: Initiation and Growth of Explosion in Liquids and Solids. Cambridge Monographs on Physics, Cambridge University Press, Cambridge (1952) 31. Afanas’ev, G.T., Bobolev, V.K.: Initiation of Solid Explosives by Impact, Jerusalem Israel Program for Scientific Translations (1971) 32. Langevin, A., Bicquard, P.: Memor. Proud. 26, 354 (1934) 33. Wyatt, R.M., Moore, P.W., Adams, G.K., Summer, J.F.: Proc. Roy. Soc. Lond. A 246, 189 (1958) 34. Tucker, T.J.: Spark initiation requirements of a secondary explosive. Ann. New York Acad. Sci. 152, 643–653 (1968) 35. Griffith, N., Groocock, J.M.: The burning to detonation of solid explosives. J. Chem. Soc. Lond. 814, 4154–4162 (1960) 36. Andreev, K.K., Belyaev, A.F.: The Theory of Explosives. Oborongiz, Moscow (1960) 37. Taylor, J.W.: The burning of secondary explosive powders by a convective mechanism. Trans. Faraday Soc. 58, 561 (1962) 38. Kuo, K.K., Summerfield, M. (eds.): Fundamentals of Solid-Propellant Combustion. American Institute of Aeronautics and Astronautics, New York (1984) 39. Bernecker, R.R., Price, D.: Studies in the transition from deflagration to detonation in granular explosives III. Combust. Flame 22, 161–170 (1974)

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40. Gibbs, T.R., Popalato, A.: LASL Explosive Property Data. University of California Press, CA (1981) 41. Dobratz, B.M., Crawford, P.C.: LLNL Explosives Handbook. National Technical Information Service, US Department of Commerce, Springfield, VA (1985) 42. Lee, D.R.: CRC Handbook of Chemistry and Physics, 88th edn. CRC Press, FL (2007) 43. NIST Chemistry WebBook, http://webbook.nist.gov/chemistry/ 44. Incropera, F.P., Dewitt, D.P.: Fundamentals of Heat and Mass Transfer, 5th edn. Wiley, New York (2002) 45. Loeb, L.B.: The Kinetic Theory of Gases, 3rd edn. Dover Publications, Mineola, New York, (2004) 46. Bird, R.B., Stewart, W.E., Lightfoot, E.N.: Transport Phenomena. Wiley, New York (1960) 47. Perry, W.L., Sewell, T.D., Glover, B.B., Dattelbaum, D.M.: Electromagnetically induced localized ignition in secondary high explosives. J. Appl. Phys. 104(1), 094906-094906-6 (2008) 48. Cooper, P.W.: Explosives Engineering. Wiley-VCH, New York (1996) 49. Parker, G.R., Asay, B.W., Dickson, P.M., Henson, B.F., Smilowitz, L.B.: In: Shock Compression of Condensed Matter, American Physical Society Topical Conference, Portland, OR, 20–25 July 2003 50. Dickson, P.M.: Private Communication 51. FlexPDE 3. Copyright PDE Solutions Inc., Antioch, CA (2003) 52. Merzhanov, A.G., Dubovitskii, F.I.: Proc. USSR Acad. Sci. 129, 153–156 (1959) 53. Zeldovich, Y.B., Barenblatt, G.I., Librovich, V.B., Makhviladze, G.M.: The Mathematical Theory of Combustion and Explosions (English Translation) Consultants Bureau, New York, 98–103 (1985)

3 The Chemical Kinetics of Solid Thermal Explosions Bryan F. Henson and Laura B. Smilowitz

3.1 Introduction In this chapter, we discuss the physics and chemistry of thermal decomposition in energetic materials. We adopt the traditional approach of chemical kinetics, where the chemistry of decomposition and the resulting conversion of chemical potential into heat are captured by systems of differential equations whose variables are the concentrations of original reagent and product chemical species. Rates of transformation are simulated using temperature-dependent equations for the rate constant and the particular mechanisms of chemical transformation are reproduced in the form of the concentration equations. The energy liberated or consumed by reaction is then a function of the calculated concentration change either as a Hess’s Law calculation constrained by real chemical species and their heats of formation or more generally as a free parameter for an unconstrained, unidentified chemistry. These systems of equations then serve as thermal source (sink) terms in the nonsteady heat equation, within which other thermo-physical parameters, such as the average heat capacity or thermal diffusivity, as well as overall geometry and thermal boundary condition, must also be specified. Such models are necessarily global in nature, and given realistic computational resources cannot hope to encompass the full chemical complexity of the decomposition of a material like octahydro-1,3,5,7-tetranitro-1,3,5,7tetrazocine (HMX), an 8 member, 28 atom nitramine ring. Indeed in a paper published more than 20 years ago, Shackleford [1] noted The scientific literature is replete with decomposition studies of the cyclic nitramines but their critical reaction steps and overall chemical mechanisms are still subject to spirited scientific debate. Depending on how one counts them, as many as 8–13 chemical mechanisms have been proposed, and collectively, they eventually identify each type of covalent bond in the HMX molecule as being an important participant in the decomposition process B.W. Asay (ed.), Shock Wave Science and Technology Reference Library Vol. 5: Non-Shock Initiation of Explosives, DOI 10.1007/978-3-540-87953-4 3, c Springer-Verlag Berlin Heidelberg 2010 

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On the one hand, it is likely that the authors of these mechanisms, which were generated by sound methods of elemental chemical analysis, were entirely correct in their individual attributions of importance to each type of covalent bond in this molecule. On the other hand, it also seems unlikely that the full suite of possibilities in such a complex decomposition process will be equally important for our goal, which is the construction of models that reproduce a very specific mode of decomposition in an energetic material, that lead to thermal ignition. The fundamental idea in the theory of solid thermal explosions is a simple, albeit highly nonlinear one, and aids in the identification of the subset of the full chemical complexity of a molecule necessary here. We are interested in thermal ignition, a nonlinear process wherein chemical energy is released and temperature increased, with a rate that depends on temperature. We therefore have important criteria for selection of the decomposition chemistry relevant to this process. The reactions we are concerned with must either themselves release or consume energy sufficient to modify the temperature of the mixture, or provide reagents that participate in such reactions. Furthermore, the extremely nonlinear heating and acceleration of decomposition observed prior to a thermal explosion indicates that some reactions must possess an extreme sensitivity of reaction rate on temperature. These criteria require that not just elemental transformation from one species to another be considered but that those transformations which can be shown to ultimately generate heat and accelerate reaction be identified and incorporated into models discussed here. Historically, the earliest attempts to model this problem were based on analytical solutions to the nonsteady heat equation with necessarily very simple source terms, for example, the Semenov and Frank-Kamenetskii treatments discussed in Chaps. 2 and 7. As discussed previously, the analytical solutions generated in this work were of considerable value in elucidating the overall behavior of thermal explosions. Early systems of equations meant to simulate chemical decomposition in the computer modeling of thermal explosions were also very simple but led to important new insight into the relationship between decomposition and thermal explosion, as well as identifying avenues for the introduction of more realistic, and complex, models of decomposition [2, 3]. These simple models serve as the starting point for this chapter. After a discussion of classical kinetics and the construction of the set of differential equations that encodes a decomposition model, we review the early work and discuss the strengths and weaknesses of the general source term models and applications of such models to a number of energetic materials. We discuss the relative advantages and disadvantages of systematically increasing complexity of the source term model used, from simple homogeneous material heating to multiphase considerations, complex solid states of the material, to a final

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consideration of a fully chemical global decomposition model. As an example of such a complex model we present the current state of attempts in our group to produce a chemically accurate global decomposition model for HMX-based explosive formulations. Although we present the modeling considerations here primarily in the context of direct heating and classic thermal explosions, we do not restrict our considerations to a particular regime of temperature or heating rate. Indeed, particularly in the context of the fully chemical global models, the application of the thermal response model is in principle independent of the temperature or rate of heating, and in the limit that the relevant chemistry over the entire range of temperature is included, such models represent a comprehensive thermal response model. We have devoted a considerable amount of effort toward validating this idea as applied to HMX [4–7]. In this spirit, we do not intend to present this work as equivalent to the very complex chemical models constructed for combustion applications [8, 9], or the complex mathematical models used for the deflagration-to-detonation transition [10]. We do intend to explore where the overlap in response to such models exists, and where the solid state thermal response discussed here may compliment or expand upon these other models. Finally, the quote noted above begs a further question. Why pursue more complex models, and in particular, why pursue a more complex model of HMX? The answer is straightforward and has been reiterated throughout this book. Much more is now required of these models than ever before. We are well past a time when the calculation of a time to ignition as a function of temperature sufficed to answer questions of safety regarding explosive systems. Models of decomposition must now be coupled with models of ignition and the subsequent release of energy to fully understand thermal response. Thermal decomposition models must therefore provide more than a spatially resolved source (sink) of heat but must also provide a spatially resolved rate of mass loss to enable calculations of gas phase pressurization. They must also provide the mechanism to couple that mass loss into a model of solid phase morphology in order to address the ability of the material state to support either convective or conductive burning. There has been significant progress recently toward realizing these ambitious goals, and it is useful to note what has been required to realize it. The temperature and kinetics of the β-δ phase transition in HMX have been the subject of study for more than 40 years, yet until recently there was considerable uncertainty even in the equilibrium transition temperature. To fully resolve these difficulties, a means of resolving the relative fraction of δand β-HMX within the solid phase during transformation was required. The discovery of the extreme nonlinear optical susceptibility of HMX and other energetic molecules provided such a probe in the form of the laser spectroscopic technique second harmonic generation (SHG) [11–13]. With this new probe, the separation of nucleation and growth components of the transformation was

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enabled, and the detailed temperature dependence of the transformation was determined. It was immediately recognized that this temperature dependence was consistent with a process governed by melting, albeit at temperatures more than 100 K below the melting point [14], and this led to an entirely new model of solid phase transformation [14–19]. This is not only a new model of HMX transformation or even of energetic material transformation, but an altogether new and general model of solid phase transformation that has already found application to other fields of solid state physics [20]. Further work has revealed a similar thermodynamic constraint in the intermediate steps of the decomposition and ignition of HMX-based materials. The ignition rate was shown to be composed of rates of both sublimation and vaporization, again even though no direct evidence of equilibrium liquid HMX was apparent [4, 7]. Investigations into the complex physical chemistry of liquid phases below the triple point, including the quasiliquid interfacial phase and premelting phenomena, resulted in another general breakthrough in a broader field of physical chemistry, with a new theory of the quasiliquid phase based on the interfacial liquid activity that has resolved decades old questions regarding the extent of the phase as a function of temperature [21]. Work on the final nonlinear acceleration and ignition mechanisms has led to new laser techniques to control and synchronize thermal ignition for the first time [22, 23], and using these techniques, the first radiographic images of the evolution of density within the solid during ignition and the transition to burning have recently been obtained [24, 25]. We will present a suite of global chemistry models for HMX-based materials, which are the results of this progress. Clearly, the significance and scientific breadth of the progress required to produce these models is a testament to the difficulty of these problems. The chapter is organized as follows. We initially discuss the various modeling strategies traditionally applied to the energetic thermal decomposition problem. These span the spectrum of generic source terms in the nonsteady heat equation discussed previously to attempt to seriously constrain the modeling of response by the application of realistic global models of known chemical steps in the decomposition. These strategies are then discussed in the pragmatic context of modeling objectives, where the computational and conceptual cost of more sophisticated models are justified by increased fidelity and additional observables generated as output. We then discuss a specific global modeling effort applied to the HMX plastic-bonded explosive PBX 9501, with particular attention first to the state of understanding of the endoand exothermic decomposition chemistry of HMX and how this understanding constrains the construction of rate constants and kinetic order in the thermal source (sink) model. We then discuss the current state of the resulting global models. We finish with a discussion of the output of these models and how they inform the final, post-ignition component of the problem.

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3.2 General Considerations 3.2.1 Chemical Kinetics and Explosives Modeling The canon of chemical kinetics established over the last two centuries is of course the subject of a voluminous literature and a number of classic texts. In this brief discussion we will review some central ideas that will repeatedly appear as we construct the models of thermal decomposition that are the topic here. In particular, a review of the derivations of the chemical rate constant, the temperature dependence of the chemical rate, and the relationship between kinetic rate and thermodynamics will be of value in Sect. 3.3 as we construct rate constants and kinetic equations based on assumptions of mechanism. While there are a number of classic texts on chemical kinetics, we rely heavily upon [26] and references therein for the particular derivations reviewed here. We refer here, as is traditional, to the rate constant k. The tradition stems from the reality that early kinetic measurements were almost exclusively performed under isothermal conditions, where the temperature dependent rate constant is actually constant, enabling analysis of data in the pre-computer era. In problems of thermal decomposition and ignition, of course some experiments are not isothermal. Also in some instances it is preferable to conduct calorimetric measurements of even very fundamental components of the overall chemistry under nonisothermal (typically linear heating) conditions. We will not change the nomenclature here and will refer to k as the classic rate constant, even in nonisothermal conditions, with the hope that this caveat is sufficient to avoid confusion. Temperature Dependence: The Arrhenius Rate Constant The classic Arrhenius equation for the temperature dependence of the rate constant is used throughout the chemical kinetics literature, and particularly as the generic form for the temperature dependence of the rate of heat generation by decomposition in the nonsteady heat equation discussed in Chaps. 2 and 4. The empirical form of the equation is given by   −ΔE , (3.1) k = A exp RT where k is the reaction rate constant, A is a prefactor (integration constant) whose units are determined by the order of the reaction, E is an activation energy, R is the ideal gas constant, and T is the absolute temperature. It is noteworthy here that the original justifications of this equation were derived from consideration of the temperature-dependence of the equilibrium constant. Applying the idea that chemical equilibrium represents a dynamic balance between opposing rates rather than a static condition, van’t Hoff derived (3.1) as the form of the rate constant governing both forward and reverse reactions in equilibrium (k1 and k−1 , with energies E1 and E−1 , in modern

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nomenclature) [27]. He then showed that the two energies must be such that the difference E1 –E−1 = ΔE, the internal energy difference between the product and reagent species. It was Arrhenius who first proposed that ΔE in (3.1) represented the energy difference between reagent species and an activated form of the reagents in a state intermediate between reagents and products [28]. Subsequent work using similar considerations of chemical equilibrium and utilizing early versions of what would become the Gibbs free energy yielded similar equations [29]   −ΔGa k = νa exp , (3.2) RT where νa is again an integration constant that determines the units of the rate constant and ΔGa is now a free energy of activation, which can be decomposed into an activated entropy and enthalpy. Although providing conceptual models of reaction still favored today, there was no theory in classical physics or chemistry with which to deduce the form of the prefactor A, the rate constant at infinite temperature. Conventional transition state theory (TST) provides a theoretical framework for a rigorous understanding of the rate constant. Although the subject of much work and again a voluminous literature [30], the original formulation of TST in its simplest form will suffice here [31, 32]. TST is built upon Arrhenius’ idea of an activated complex, consisting of the combined reagent species and somehow representing a transitional structure on the way to products, and utilizes van’t Hoff’s equilibrium to construct the formula for the rate constant. Four original assumptions concerning the dynamics of transition through the activated complex are required: (1) passage through the transition state is irreversible, (2) the internal energy distribution of the reagent species is at thermal equilibrium, (3) the degree of freedom representing the motion of the complex through the transition (e.g., translational, vibrational) may be separated from the other degrees of freedom of the reagent species, and (4) quantum mechanical tunneling is not involved in passage through the transition state. The derivation of the rate constant begins with a set of equalities derived from the aforementioned assumptions. We review a derivation of the rate constant for bimolecular reaction, as it is of the most utility in the problems discussed in this chapter. The starting point is the expression of equilibrium between the reagent species and the activated complex [26], expressed as a function of concentrations (where A + B form the activated complex Xa ), the classic equilibrium constant, the free energy difference between the two states, and the expression of the equilibrium constant as a function of the molecular partition functions of all species and their internal energy difference [33]     −ΔEa −ΔGa [Xa ] [qa ] = Ka = exp exp = . (3.3) [A][B] RT [qA ] [qB ] RT The next step requires specification of the degree of freedom assumed to capture the motion through the transition state. Classic derivations exist based

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on the assumption of both translational [31, 32] and loose vibrational [34] degrees of freedom. Both derivations yield equivalent results. We will review the derivation assuming that a passage through the transition state proceeds via a loose vibrational state of the activated complex. We start with (3.3) and note that the molecular partition functions, qi , are product functions of the translational, rotational, and vibrational degrees of freedom of each species. We make the assumption that one of the degrees of vibrational freedom associated with the activated species represents the coordinate describing the passage through the transition state. The function qa is then a product of one loose degree of freedom and 3 (nA + nB ) − 7 remaining vibrational degrees of freedom (typically assumed to be the uncoupled product states of the two original reagent species), where ni are the number of atoms of each species. The term loose is specifically used to indicate that this degree of freedom ultimately leads to separation of the activated complex into stable product states, and therefore is likely a very low frequency vibration. This assumption allows one to use the molecular partition function for simple harmonic vibration in the zero frequency limit, qv = kB T /hν, where kB and h are Boltzmann’s and Plank’s constants, respectively, and ν is the vibrational frequency. Separating out the single vibrational degree of freedom and assuming qa = qa qv , where qa is the product molecular partition function of the remaining degrees of freedom in the complex yields   −ΔEa [Xa ] kB T [qa ] = exp , (3.4) [A][B] hν [qA ] [qB ] RT which can be rearranged to give ν [Xa ] = [A][B]

kB T [qa ] exp h [qA ] [qB ]



−ΔEa RT

 .

(3.5)

Finally, ν [Xa ] is the classic rate of reaction, the frequency of passage through the transition state multiplied by the concentration of activated species. The concentration dependence [A][B] is that expected of a second order bimolecular reaction. Equation (3.5) is then the classic second order kinetic equation, with rate constant   −ΔEa kB T [qa ] exp k= . (3.6) h [qA ] [qB ] RT Equation (3.6) is a very useful starting point for the application of the Arrhenius equation to problems related to explosives and explosive modeling. The first and most fundamentally satisfying feature is an elucidation of the prefactor in (3.1) and (3.2), particularly the universal frequency factor kB T /h, which establishes the molecular time scale for reaction and enables calculation of activation enthalpy and entropy if temperature-dependent rate data exist. Also satisfying is the identification of the mechanism by which higher order rate constants derive their units. The assumption of equilibrium between reagent and activated species establishes the concentration dependence through the equilibrium constant.

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The assumption of thermal equilibrium is of course of significant practical importance. The exponential temperature dependence of the rate constant ultimately derives from the equilibrium distribution of energy amongst the mechanical degrees of freedom of the reagent species. This is a molecular characterization of equilibrium and should not be confused with the overall thermal equilibrium of the bulk sample. This assumption of molecular equilibrium distinguishes the use of this rate constant from other approaches such as mechano-chemistry or electronically excited chemical pathways initiated by, for instance, a laser. The assumption of the equilibriuim distribution of internal energy, typically achieved on the order of nanoseconds or less, is an excellent one when considering most reactive processes. It can fail in explosive applications, however, when considering processes fast on this time scale, such as applications to detonation or the transition to detonation. However, even in this case, for HMX, we have argued in the past that the assumption of molecular equilibrium appears to be justified [6]. The dependence of the rate constant on pressure can be shown by expansion of the activation enthalpy as an internal energy and work term due to isotropic pressurization of an activation volume, ΔVa , assumed to be the volume necessary to form the activated state     T ΔSa − (ΔEa + P ΔVa ) T ΔSa − ΔHa kB T kB T exp exp k= = . h RT h RT (3.7) It is important to properly define the pressure in (3.7), and such definition requires assumptions concerning the mechanism it is intended to reflect. For instance, in solid state processes P is commonly taken to be the isotropic, quasistatic pressure in the sample, and the volume of activation reflects the perturbation to the solid density required to affect the transformation. More sophisticated understanding of the effects of mechanical perturbation to the chemistry might be incorporated, however, by expanding the pressure dependence as a function of the stress tensor of the material. This enables sophisticated considerations of the effects of crystal orientation and nonuniformities in pressurization to be included. Of course, if the mechanism under consideration is a gas phase reaction, then the volume of activation is difficult to interpret, as P and pressure effects are more commonly incorporated in the concentration dependence of the reaction. For models that include both gas and solid phase processes, it is of paramount importance to distinguish between solid stress and gas phase pressurization, which are only equivalent in a truly quasistatic environment such as a fluid supported DAC experiment. The expansion of (3.1) into more fundamental components, (3.6) or (3.7), is not always necessary in the construction of models. It is nonetheless motivated by the additional information carried by the more complex equations, particularly the connection to thermodynamics and thermodynamic phase. For instance, many fundamental kinetic processes in solid decomposition may proceed with rates activated by energies, which are equivalent to changes

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in thermodynamic state, such as when sublimation or vaporization rates are constructed from the latent heats of transformation to the gas phase. In particular for HMX, it was the observation that the rate of growth in the βδ polymorphic phase transformation was thermally activated by an energy equal to the heat of fusion that led to the discovery of a new general class of solid phase transformation [15], and the construction of a very comprehensive specific model of the β-δ transformation in HMX [12, 14], which is the first important step in normal decomposition. This will be discussed further in sections “The Crystalline Phases of HMX and Rates of Transformation” and “The β-δ Transition.” As a brief example, we construct a rate expression for sublimation, which will help to illustrate this increased information content, provide a specific rate expression for use in the next section, and will be used as a step in the global chemistry model of HMX decomposition presented in Sect. 3.3. The process of sublimation is extremely important in solid explosive modeling. Sublimation is a process known relatively well, in particular when compared with covalent bond breaking in large solid explosive molecules, and can be sufficiently endothermic to impact energy balance during decomposition, particularly as the temperature increases to ignition. The rate of sublimation is one of the two competing rates that determine the equilibrium vapor pressure, the other being the rate of collision with the surface by gas phase molecular species. The two rates balance at equilibrium, but are not necessarily equal, as every collision from the gas phase may not lead to incorporation into the solid. The fraction representing this probability of incorporation is the accommodation coefficient [35]. For our purposes, as in some of the earliest applications of TST [34], we consider this coefficient to be equal to one. Experiments on fundamental aspects of sublimation continue and are remarkably recent, with one study in particular demonstrating that molecules leave the surface with a Maxwell–Boltzman (MB) distribution in the velocity reflecting the surface temperature, and resolving the accommodation coefficient as a function of the internal quantum states of the molecule [36]. The collision frequency with a surface follows from the MB distribution of velocities for particles of mass m in the gas phase at temperature T . Briefly, the number of molecules, n, in a volume element dV in the gas phase is given by [37] (3.8) n = ρ dV = ρr2 dr sin θ dθdϕ, where ρ is the gas phase density, r is a ray emanating from the origin of a coordinate system on the surface to that volume element, and the spherical polar volume element is defined using the polar angle θ and azimuthal angle in the surface plane using ϕ. The fraction of the number of molecules in this volume element with velocities between v and v +dv is given by an integration over the MB distribution, f (v)dv. n = f (v) dvρr2 dr sin θ dθdϕ,

(3.9)

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where the MB distribution is    m 2/3 −mv 2 exp f (v) = 4π v2 . 2πkT 2kT

(3.10)

Assuming a radially symmetric flux of molecules out of the volume element, we calculate how many will intersect an area A on the surface, centered at the origin of the spherical polar coordinate system. This fraction of the total number of molecules in the volume element is given by the ratio of the projected cross section of the area A, viewed at an angle θ from the volume element dV , over the total solid angle at a distance r from A, or n = ρf (v) dvr2 dr sin θ dθdϕ

A cos θ . 4πr2

(3.11)

The flux of such collisions is calculated by integrating r over velocities sufficiently close to the surface to reach it in 1 s, or integrating over r from zero to v (r = vt, where t = 1 s) as ∂n =ρ ∂A∂t



∞



v

dr [f (v) dv] v=0

r=0

1 4π





π/2

2π

sin θ cos θ dθ θ=0

dϕ.

(3.12)

ϕ=0

The final integration over the MB distribution multiplied by v then gives the mean velocity as  ρ ∞ ρ ∂n = vf (v) dv = v, (3.13) ∂A∂t 4 0 4 where

 v=

8kT πm

This yields ∂n ρ = ∂A∂t 4



1/2 .

8kT πm

(3.14)

1/2 (3.15)

and using the ideal gas equation ρ = P/kT leads to P ∂n . = √ ∂A∂t 2πmkT

(3.16)

This is the classic Hertz–Knudsen equation for the rate of collision with the surface from the gas phase per unit surface area. A TST rate equation for an equivalent reaction whereby a gas phase molecule becomes incorporated into the solid can be written, beginning with (3.3), as (for a single reagent species)   −ΔEa [Xaarea ] [q area ] exp = a , (3.17) ρgas [qgas ] RT

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where the variables have the same assignments as before. The concentration and molecular partition function for the activated state are in units of molecules per unit area and for the reagent gas phase in molecules per unit volume. As before, a frequency of vibration along a reaction coordinate is extracted from the molecular partition function of the activated state and rearranged to give   −ΔEa kB T [qaarea ]  area exp ν [Xa ] = ρgas . (3.18) h [qgas ] RT At this point a model of the reaction must be provided. If we wish to recover (3.16), a suitable model involves partition functions that describe two- and three-dimensional (2D, 3D) translation for the activated and reagent states, respectively, and a mechanism of incorporation into the solid with no activation barrier, ΔEa = 0. This yields

ν

[Xaarea ]

(2πmkB T ) kB T h2 = ρgas . h (2πmkB T )3/2 h3

(3.19)

Again using ρ = P/kT leads to ν [Xaarea ] =

P ∂n = √ . ∂A∂t 2πmkB T

(3.20)

The assumption of a 2D translational partition function for the activated state evokes a mobile surface phase as a precursor to incorporation of a molecule that has struck the surface. Finally, and equivalently, we construct a TST rate constant for crystalline solid sublimation using   −ΔEa [q area ] [Xaarea ] exp = a , (3.21) ρsolid [qsolid ] RT where we again assume a two-dimensional activated state and a threedimensional solid reagent state. For a crystalline solid, which we have carefully defined here, ρsolid is fixed and independent of the amount of solid. If one is dealing with a non-crystalline solid solution or amorphous solid, for which ρsolid is not fixed and is better represented by a true concentration [csolid ], then (3.21) takes the form of a true first order TST rate constant with concentration dependence as   −ΔEasolution ] kB T [qaarea  exp ν [Xaarea ] = [csolid ] , (3.22) h [qsolution ] RT where [csolid ] denotes a non-crystalline phase of varying concentration. Two important considerations of the reagent phase are important here. A condensed phase at equilibrium with the vapor is at the same chemical potential

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as the vapor. This is the definition of equilibrium. If the solid is a crystalline solid, then, neglecting surface and grain boundary effects, the chemical potential of the solid is constant. No matter how much of the crystalline solid is present, the vapor pressure above it is constant, as is the sublimation rate. The non-crystalline phases of varying concentration are also phases of varying vapor pressure and consequently varying sublimation rate. Proceeding with the assumption of constant crystalline solid density, ρsolid = n/V , as well as writing the dependence on dimension in the solid state molecular partition function leads to n area   [qa ] −ΔEa area V Xa = qsolid exp . (3.23) RT V We define the complete solid phase partition function as q=qsolid exp (ΔEa /kT ) in molecular units (substitute k for R), and substitute into (3.23), [qaarea ] q . n We use a series of equalities from Hill [33],   q μ = exp − , n kB T   μ − μo , P = exp kB T Xaarea =

and for an ideal gas

 (2πmkB T )3/2 μo = ln kB T − kB T h3 

or exp

μo kB T

 =

h3 1 . kB T (2πmkB T )3/2

(3.24)

(3.25) (3.26)

(3.27)

(3.28)

Substituting (3.24) into (3.25) gives Xaarea =

[q area ] a . μ exp − kB T

Rearranging (3.26) and substituting into (3.29) gives   μo Xaarea = [qaarea ] exp − P. kB T

(3.29)

(3.30)

Substituting (3.28) into (3.30) gives Xaarea = [qaarea ]

h3 1 P. kT (2πmkB T )3/2

(3.31)

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Making the same assumption of a delocalized 2D activated state at the interface modeled with a 2D translational partition function and extracting the vibrational partition function of the reaction coordinate yields [qaarea ]= 

kB T 2πmkB T . hν h2

(3.32)

Substituting (3.32) into (3.31) gives Xaarea =

h3 kB T 2πmkB T 1 P, 2 hν h kB T (2πmkB T )3/2

which reduces to

(3.33)

P ∂n = √ . (3.34) ∂A∂t 2πmkB T We see that once again the assumption of a mobile 2D activated state at the surface is necessary to reconcile the three different derivations of the sublimation flux. Such a precursor could play a significant role in a decomposition mechanism, and we will discuss this possibility in the mechanism of HMX decomposition in Sect. 3.3. The detailed evaluation of these rate expressions for the general process of solid sublimation leads to several important questions concerning realistic chemical modeling of solid thermal decomposition. The identification of a third phase, delocalized on the surface, could have serious ramifications to the mechanism. Does the molecule in this phase exhibit different reactivity than in the gaseous or solid phase, perhaps more resembling the liquid phase? Is there direct decomposition from this phase, especially at elevated temperatures or as the melting point is approached? Does the phase persist long enough, or lead to branches in the decomposition pathway giving rise to different products, such that the latent heat associated with the conversion of a molecule in the crystalline lattice to one in the delocalized surface state needs to be expressed? A framework to discuss the impact of these decisions is illustrated in Fig. 3.1 using the heat of formation and the latent heats for the main thermodynamic phases of HMX. On the left of the figure, the energy levels of the β and δ solid phases, the liquid and the vapor phase are shown plotted on the left ordinate relative to the heat of formation. On the right of the figure an energy diagram of a model trajectory from the δ solid phase to the vapor phase is shown as either a direct path or one composed of two steps and explicitly including the liquid. These trajectories are plotted on the right hand ordinate as the latent heats of each phase, assuming the gas phase potential of zero. Both ways of plotting the energy are equivalent, and differ only in the relative zero of energy. The right hand side is often encountered when dealing with the thermodynamics of a material, and the latent heats of the different phases are more apparent and easier to track. The left hand side is necessary when decomposition and the rearrangement of atoms from reagents to their products are calculated, as the relative zero on the left hand side is the thermodynamic standard where the elements in their pure state are taken to have ν [Xaarea ] =

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Fig. 3.1. Energy levels in the HMX system plotted as a function of a generalized reaction progress from β to δ to liquid to vapor. On the left ordinate, the energies are plotted relative to the heat of formation, and on the right as latents heats of transformation relative to a gas phase energy of zero

zero heat of formation. From the trajectory for sublimation, the assumption that the latent heat of sublimation serves as the energy of activation is intuitive, as is the assumption that condensation proceeds without a barrier. Neither is of course strictly or always true; however, the sense of the approximation is clear. The ramifications of separating out a trajectory that proceeds through the liquid (or delocalized surface) state is also made more clear. If, on the one hand, there is no role for the liquid other than as a transient state in between the solid and the gas phase, then one can consider the overall process as the product of two dependent probabilities, that of transition from solid to liquid, and then from liquid to gas. In this case, the two rates can be compared as ksub = kmelt kvap . Taking the product of the exponentials it is clear that the activation energy is the same. The comparison of the entropy for the overall process as compared to the component processes is more interesting, but we will not pursue that further here. If for some reason the component processes cannot be considered dependent, for instance if the time spent in the liquid state is too long by some measure, then an expression of rates based on the sum of the component rates would clearly not be equivalent to the single overall step.

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The construction of a sublimation rate constant from (3.21), assuming a solid reagent and 2D activated state, also illuminates the particular nature of the crystalline solid as a reagent phase of constant chemical potential. The difference between such a state and one where the chemical potential varies as a function of concentration or morphology, such as a solid solution or amorphous phase, respectively, is apparent from (3.22). In states of variable chemical potential, the loss of reagent to reaction decreases the concentration in a way that can naturally be accommodated by an assumed concentration equation. This is not so for the crystalline solid decomposing by sublimation and vapor phase reaction, for instance. In this later case the flux of molecules to the vapor phase evolves not as the concentration, which is constant, but as the shape or morphology of the solid and the surface area presented to the vapor. This is a fundamentally different type of kinetic order and involves different assumptions. We will discuss this in more detail in the next section on concentration dependence. It is also useful to be familiar with actual rates of the component processes in a model, particularly when they have been directly measured. For instance, again using HMX as an example and the measured vapor pressure as a function of temperature of δ phase HMX in (3.34), we obtain the sublimation flux at 473 K as 9.14 × 1020 molecules m−2 s−1 . This is the areal flux through a surface cross section of 1 m2 in 1 s. Put another way, a square meter of HMX surface exchanges about half a gram per second with the vapor at equilibrium. Given an approximate molecular diameter of 6.5 A, this also corresponds to almost 400 layers a second. These rates are quite important to a number of quantitative considerations, like rates of Ostwald ripening and the evolution of solid morphology over time. They are also important to a number of more qualitative considerations, such as the suitability of the idea of a surface as a dividing plane between the solid and vapor phase. On the time scale of molecular vibration or an elastic collision, say 50 fs, this idea is quite robust. On the other hand, how does this flux affect the diffusion of energy or mass along the surface, which occurs on millisecond timescales? Or more interestingly, how is the idea of an activated molecular state in sublimation, the lifetime of which we have not specified, impacted? A useful expression for comparison to data involves conversion of the flux into a linear regression rate using the molar volume Vm , Avagadro’s number No , and the equation ∂L Vm ∂n = , (3.35) ∂t No ∂A∂t which yields a linear rate of 214 nm s−1 at 473 K. At this rate the sublimation of a typical 100 μm diameter spherical crystal of HMX would take 234 s (three and a half minutes, faster if one includes the change in solid chemical potential as a function of the radius of curvature for diameter less than ∼100 nm through the Kelvin effect). Of course this rate would be difficult to achieve, even under high vacuum, as the high flux from the surface at this temperature

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would likely constitute a hydrodynamic flow condition, rather than a Knudsen regime where ballistic flight from the surface essentially removes material from the surface without return [35]. It is a useful approximation however, and distillation of HMX from hot surfaces in the lab is actually readily achievable on this time scale. A second approximation involves an expression where the specific area per unit mass or volume can be used to generate a volumetric rate. This is useful for decomposition models that are implemented without sophisticated solid morphology models. We have shown that the specific area per unit mass for HMX particles smaller than approximately 100 μm can be modeled as solid oblate cylinders, very nearly spherical [38]. For simplicity, the ratio of the surface area to volume for a sphere is A/V = 6/d, where d is the diameter. For the same 100 μm diameter particle this is ∂A/∂V = 6 × 104 m2 m−3 . Multiplying this surface to volume ratio by the surface flux and normalizing the volumetric rate to the constant molar volume of δ HMX yields the first order rate constant Vm ∂A ∂n . (3.36) k= No ∂V ∂A∂t A calculation of the rate constant at 473 K yields 1.40 × 10−2 s−1 , or a half life for sublimation, independent of the total mass of 100 μm particles, of 50 s. This approximation is of the same order of magnitude, although faster. This volumetric approximation is very simple, however, as the surface to volume ratio will typically change as a function of regression. This will be discussed in more detail in the next section on concentration dependence. Another tempting volumetric approximation involves (3.22) and the assumption that the crystalline solid is a solid solution of concentration equal to the solid density. We assume that the partition functions can be combined with the activation energy to yield the activation free energy of the solid as   T ΔSa − ΔHa kB T exp k= . (3.37) h RT If we make the further assumption that the activation free energy is equal to the normal solid free energy change on sublimation, as we do in the derivation of (3.34), ΔGa = ΔGsolid , then the rate constant at 473 K is equal to 1.3 × 107 s−1 . This is far too large to be reasonable and indicates the danger of simply applying variations of (3.2) because one can make a simple association of the mechanism with some known physical process. The details of the mechanism, particularly the dimensionality indicated by the mechanism, are very important. On the other hand, it is possible to make useful approximations in this way, and we will discuss one such volumetric approximation to a sublimation rate for HMX in the section “Comparison of Kinetic and Thermodynamic Rates.” Briefly, using an independent method for approximating the activation entropy for the process, we use (3.37) to describe the rate of sublimation of

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HMX and obtain a rate constant equal to 2.3 × 10−5 s−1 . This is considerably smaller than the first order rate of (3.36), and implies half times for sublimation of hours instead of seconds, but this is expected for this very simple zero order approximation. The last point concerning sublimation is a reminder that unless a loss mechanism for the gas phase species exists, the high rates of sublimation and collision with the surface do not lead to any net transfer of mass from the solid. However, the loss of vapor to leaks or, more likely, to gas phase reactions are loss mechanisms, and if vapor is lost at rates comparable to those discussed above for sublimation, then the overall rate of reaction may be sublimation limited. Finally, the temperature sensitivity of a rate constant is determined by the magnitude of the activation energy in relation to kb T (or RT). This is another consequence of the assumption of internal thermal equilibrium in the reagent molecules. It is useful to examine the simple temperature dependence of a few real rate constants, and this is done in Fig. 3.2. These constants have been taken from the global decomposition chemistry mechanism used to model HMX decomposition and will be described in detail in Sect. 3.3. The general point is the concave down shape of the rates as a function of temperature for most of the rates shown. In the limit of infinite temperature e−1/T goes to

Fig. 3.2. Rate constants plotted as a function of temperature to illustrate the asymptotic approach to the limiting rate at infinite temperature. These rate constants are taken from the global model of HMX decomposition to be discussed in Sect. 3

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1. As the temperature increases in relation to RT the rate approaches this limit, the shape of the curve being determined by Ea , and the limiting rate by A in (3.1), or kB T /h exp (ΔSa /R) in (3.7). Rates k2a and k2b exhibit this behavior. The situation is more complex for highly coupled rate expressions, such as k1 of Fig. 3.2, which represents the overall rate of the β-δ phase transition in HMX. The rate decreases from above and below near the equilibrium transition temperature of 432 K. This is due to the typical observation that the thermodynamic driving force for an equilibrium transformation increases as the difference in free energies of the two phases increases. This difference goes to zero at the equilibrium transition temperature and exactly at that temperature is zero. The mechanism from which the rates shown in Fig. 3.2 are taken reproduces this behavior while accurately reflecting the actual kinetic mechanism of transformation. Rates k3a and k3b are examples of radical reactions that do not typically exhibit a large activation energy, and proceed at very fast rates independent of temperature. We will discuss this in more detail in the next section and in section “The Crystalline Phases of HMX and Rates of Transformation.” Concentration Dependence: Kinetic Order The overall rate of a process is also determined by the dependence on concentration expressed by the mechanism. Kinetic order is a classic term used to describe this concentration dependence as a power law, order being the value of the exponent, as in j  ∂[P ] =k [Ci ]oi (3.38) ∂t i=1 for a reaction forming P and involving j reagents of concentration [Ci ]. A particular reaction described by (3.38) is said to be of oi th order in Ci , and the overall order is the sum of the oi . The particular form of (3.38) is often suggested by the mechanism of the reaction, particularly for reactions that proceed in well mixed, homogeneous phases such as liquids or gases, as in the product of concentrations in the binary reaction of (3.5) in section “Temperature Dependence: The Arrhenius Rate Constant,” where the product can be thought of as proportional to a collision probability. In chemical systems that are not homogeneous or well mixed, such as the solid phase reactions typical of the solid explosives under study in this book, the use of (3.38), or even the idea of a dependence on concentration, can be difficult. In such cases, it is of course more correct to use a model describing morphology as well as amount, such that sublimation rate, for instance, may be calculated as a function of the actual surface area available in the sample. There are many reasons, however, that a simplified concentration model might be required instead. Typically the complexity of a full model of morphology is neither obvious from available data nor allowed within the computational capability of a full thermal response simulation.

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Table 3.1. Kinetics of sublimation with various mechanistic approximations Order

Rate equation ∂L Linear = kL ∂t

Rate constant ∂L Vm ∂n = ∂t No ∂A∂t

Units

Integrated rate law

m s−1

c = c0  3 kL t × 1− R

Zero

∂c = −k0 ∂t

k0 =

1 ∂A ∂n No ∂V ∂A∂t

mol m−3 s−1

c = c0 − k0 t

First

∂c = −k1 c ∂t

k1 =

Vm ∂A ∂n No ∂V ∂A∂t

s−1

c = c0 e−k1 t

Second

∂c = −k2 c2 ∂t

k2 =

Vm2 ∂A ∂n No ∂V ∂A∂t

m3 mol−1 s−1

1 1 − = −k2 t c0 c

Mixed

∂c V 2 ∂A ∂n = −kM (c0 − c) kM = m m3 mol−1 s−1 c = c0 ∂t No ∂V ∂A∂t   f (1−e−c0 kM t ) ×(f c0 + c) × f +e−c0 kM t

We use the rate of sublimation again to illustrate the use of (3.38), the approximation of a solid phase process using an effective concentration, and to quantitatively compare the resulting rates of loss of the solid. One exact and four approximate schemes for calculating the rate of sublimation of a crystalline solid are shown in Table 3.1. The linear scheme is essentially an exact solution obtained by applying a linear rate based on the Hertz–Knudsen equation, calculated previously as (3.35), to the regression of a sphere of radius R. The rate equation is zero order in the concentration and one-dimensional, in units of meters per second (m s−1 ). The integrated rate equation is cubic in the time, however, and the loss of solid can be expressed as a scaled concentration, c/c0 , going from 1 to 0 and representing the regression to completion of an idealized ensemble of spherical crystals. The zero, first, and second order schemes make use of (3.38), with j = 1 and o1 = 0, 1, or 2, respectively. We approximate a concentration-based rate constant by applying a surface area to volume ratio to the 2D Hertz–Knudsen solution, as discussed earlier. The mixed order scheme is more interesting. In the zero, first, and second order schemes, we have made the approximations necessary to accommodate the 2D nature of the sublimation mechanism in the rate constant only, and used a concentration variable as it would normally be used. Mixed order rate laws using only a single component are used very often to approximate a more complex morphology model using the form of the concentration equation as well. The form of the mixed order rate equation is very general and has been shown to reproduce the behavior of a number of known reaction mechanisms. In this application it is simply a differential equation of the general form

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dx/dt = x(1–x) and could represent, for instance, a nucleation and coalescence type of process. Sublimation in a complex, pressed composite may proceed in this fashion. This equation is often referred to as the Prout–Tompkins equation after the authors of the first application to a kinetic system [39]. By inspection it is apparent that the rate will rise from zero at x = 0 or 1, peak at x = 0.5, and fall again to zero at x = 1 or 0. The integrated rate law is a sigmoidal (step) function going from 0 to 1 where the sharpness of the step is determined by the rate constant. It is also apparent that for a boundary condition reflecting a pure phase, such that x = 0 or 1, there will never be any reaction progress, that is, dx/dt will always be zero, x always 1. Mixed rate equations such as this always require an amount of material to begin progress. This is the reason for the explicit initial concentration dependence, c0 , in the rate equation shown in Table 3.1. This initial concentration of material has been discussed by many authors when using such a mixed order rate equation. For applications involving pure chemical reactivity between stable phases of material, this is straightforward and an equilibrium or pre-existing concentration of product is often not a conceptual difficulty. When using such mixed rate laws to simulate a physical transformation between phases of material, however, the issue of initial concentration is more complex. At temperatures above or below the equilibrium, boundary temperature between the phases one or the other phase is only metastable and therefore cannot simply be considered a pre-existing concentration. Nucleation or some other means to convert the metastable phase must be included in the rate law as well. We have constructed a model of the β-δ phase transition based on such a system of equations and have considered the initial concentration a stable pre-existing concentration [11], a variable concentration based on additional first order components to the overall rate law [14], and as a concentration derived from an independent nucleation rate law [18, 19]. Several important and interesting points with respect to concentration dependence arise in such an application. In addition to initial concentration, the rate equation must reproduce thermodynamic reversibility and the slowing of the transformation rate as the equilibrium boundary temperature is approached. We will discuss this in more detail in the context of the β-δ transformation in HMX in section “The Crystalline Phases of HMX and Rates of Transformation.” We show calculations of reaction progress based on each of the integrated rate laws of Table 3.1 in Fig. 3.3a. In the calculations we use the measured surface to volume ratio of pristine β-HMX based PBX 9501 [38]. The measured specific area was 0.45 m2 g−1 , which we convert to a surface area to volume ratio of 8.39 × 105 m−1 . We have divided the initial concentration from each rate law and plotted reaction progress from 1 to 0. The assumed temperature was 200◦ C, chosen for convenience. The zero order equation provides the fastest overall rate, the linear regression the slowest. Despite the effect of reaction order on the shape of the curves and half times, when plotted as a function of temperature over a large temperature range, this variance is small compared to the temperature dependence, as illustrated in Fig. 3.3b. The mechanisms

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in Fig. 3.3b are ordered, from the fastest to the slowest, as zero, first, second, mixed, and linear. This is an important point and we will return to it in section “Measurements of Rate Constants.” Finally, many other aspects of a thermal explosion kinetic model are dependent on the evolution of concentration produced by the various rate laws. In particular, the evolution of temperature is sensitively dependent on the average heat capacity and thermal diffusivity of the solid or mixture during reaction. These averages must be taken from the values of the individual components and some weighting or mixture rule. This dependence couples with the rate of heat generation to define the nonlinear sensitivity in these models. Classic Kinetic Schemes The rate equations and integrated laws listed in Table 3.1 represent the evolution of one species. The decomposition of a complex (or simple for that matter) molecular species is necessarily modeled by involving subsequent intermediate and product species that represent the entire decomposition. Each of the subsequent species that forms and then reacts requires a rate equation to govern its evolution. The resulting kinetic scheme represents the conceptual and mathematical representation of the chemical decomposition mechanism discussed in the introduction. There are several excellent compilations of simple kinetic schemes, their integration, and mathematical behavior [26, 41, 42]. These compilations are very useful for the development of intuition regarding the typical behavior of a class of kinetic equations, such as the different reaction progress curves shown in Table 3.1 and Fig. 3.3 above. The integral behavior of a coupled series of reactions, each described by some version of (3.38) can be quite complex. In addition, there are a number of classic, named equations that have been applied to particular problems. One such equation is the Avrami–Erofeev equation, which has been shown to reproduce some aspects of a nucleation driven process [43–48]. Another is the Prout–Tompkins equation discussed earlier [39]. We discussed the general form of this equation in terms of replicating a nucleation and growth type behavior. Prout and Tompkins first analyzed the solutions to this equation in the context of the termination of chains in a solid state chain reaction. This illustrates briefly the danger in using such popular, named equations. The equations themselves are quite general and can be shown to reproduce a number of different physical phenomena. They are very useful; however, they carried significantly more importance in the precomputer era when the analysis of kinetic data was very difficult and required many assumptions and approximations to generate a solution for a given kinetic scheme. It is possible now not only to generate sophisticated analytical solutions to even relatively complex kinetic schemes, but also to numerically solve any general scheme. The kinetic model applied to a given system now may be driven by hypotheses concerning the physics and chemistry, and not by the ease or possibility of solving the resulting system of equations. And regardless of the scheme or method of solution, the fit to data by any one of these equations should never be considered sufficient to attribute mechanism.

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Fig. 3.3. Rates of sublimation using the five mechanisms listed in Table 3.1. The mechanisms are labeled in (a). In (a), the progress is shown as a function of time for the complete sublimation of an ensemble of 20 μm HMX particles. In (b), the temperature dependence of the half-time of sublimation for each mechanism is also shown. The mechanisms in (b) are ordered, from the fastest to the slowest, as zero, first, second, mixed, and linear. The free energy of sublimation used was that reported by Lyman et al. [40] and discussed in section “The Crystalline Phases of HMX and Rates of Transformation”

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Summed Component Models We will briefly review a few classic kinetic mechanisms that are used routinely in explosives modeling. The first class of mechanisms we call summed component models and are typified by a sequence of reaction steps or several reactions involving competition for a single reagent species. The summed component models are characterized by regions of rate as a function of temperature, which are dominated by a single step in the mechanism. This is due to the strong dependence of the rate constants on temperature, which results in dramatically different rate dependence on temperature if the activation free energies differ for the series of steps. All mechanisms currently in the literature on energetic material decomposition can be characterized as summed component models. Processes that compete for a single reagent species are common and provide a particularly transparent example of the effects of temperature on the rates of decomposition in a summed component model. A competitive scheme is typically written as j  ∂c ki c, =− ∂t i=1

(3.39)

where the ki represent a set of first order processes of decomposition, all of which proceed with a rate proportional to the concentration of the same reagent species. When j = 1 the system is a particularly simple first order rate equation. The half life of a first order rate law is t1/2 = ln(2)/k. Likewise, for the sum of competitive processes in (3.39), it is clear that the sum of rate constants ki is effectively a single global first order rate constant. Furthermore, it follows that the half life of the competitive system is simply t1/2 =

ln(2) , j  ki

(3.40)

i=1

which will be dominated by the smallest term in the sum, the term changing as a function of temperature according to the various activation energies associated with each rate. A second common scheme applied to complex reaction systems involves a series of sequential steps, coupled by the successive intermediate concentrations as aA → bB → cC → dD, where a moles of A react to form b moles of B and so on to the formation of D. For a, b, c, and d = 1 the scheme describing such a mechanism involves three coupled first order differential equations

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∂A ∂t ∂B ∂t ∂C ∂t ∂D ∂t

= −ka A,

(3.41)

= ka A − kb B,

(3.42)

= kb B − kc C,

(3.43)

= kc C.

(3.44)

An analytical solution to this system is available and is presented in a number of kinetics and physical chemistry textbooks. For example, the concentration of D with time goes as   D=A0 −e−(ka +kb +kc )t fa (ka ) e(ka +kb )t +fb (kb ) e(kb +kc )t − fc (kc ) e(ka +kc )t , (3.45) where A0 is the initial concentration of A and the fi are simple functions of the rate constants ki . The rates appear as sums in exponential functions. The overall rate will therefore again be dominated by individual rates (processes) as a function of temperature if the activation energies for the reactions are significantly different. Significance here is defined by the ability to experimentally resolve the different slopes as a function of temperature. Three examples of the concentration as a function of time for different relative rates are shown in Fig. 3.4. The concentrations are scaled to the initial concentration of A, all other concentrations being zero at t = 0. In Fig. 3.4a, the full sequence of reactions is apparent for ka = kb = kc = 0.005 s−1. In Fig. 3.4b, subsequent reaction rates kb and kc = 0.2 s−1 , 40 times faster than the initial rate, and the two intermediate concentrations are only just apparent on this linear scaling. In Fig. 3.4c, the subsequent rates kb and kc = 5.0 × 10−5 s−1 , 100 times slower than the initial rate, and the two final concentrations are only just apparent on this linear scaling. This example illustrates that even such simple kinetic schemes can generate very complex behavior, and that the experimental work necessary to constrain a complete kinetic scheme is considerable. For instance, if only the loss of reagent A is monitored, then very little is learned about the mechanism, and the three examples are indistinguishable. In Fig. 3.4b if all possible species are observable, one would still conclude that A → D, and the role of B and C in the mechanism would remain unknown. The most significant characteristic of the summed component models, however, remains the dominance of different individual rate constants as a function of temperature. This has been realized for quite some time, and was pointed out as early as 1949 by Eyring et al. in a classic monograph on detonation [49]. The authors cite the example of ignition data from sodium azide [50], which illustrates the dominance of two rate constants over different regimes of temperature, and issue a caution against using low temperature decomposition data to model detonation phenomena.

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Fig. 3.4. Progress as a function of time for a simulated chemical mechanism involving four sequential reactions. By modifying the magnitudes of the rate constants, the system can be shown to exhibit measurable concentrations of either all species (a), only the original reagent and final product (b), or only the original and first intermediate species (c). This illustrates the complexity of observing and determining even a relatively simple mechanism

All models of thermal decomposition and particularly ignition of which we are aware of are the summed component type. This is in part due to the relative lack of data on elemental, solid phase processes in the decomposition of most solid energetic materials. It is also a function of the need for simple models that may be incorporated into larger scale simulations. Of course there are, however, any number of more complex kinetic schemes that can be used to model chemical reactions that are coupled in more complex ways. Coupled Two State Models Another important family of systems involves what we call reversible two-state systems of pure order, which can be written as a family of rate equations as ∂x = k1 (c0 − x)n − k−1 xn , ∂t

(3.46)

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where c0 is an initial concentration of reagent, x is a progress variable denoting the product concentration, the reaction is a reversible two state system and is mass conserved, that is, c0 = x + (c0 –x). The forward-going rate is given by k1 and the reverse rate by k−1 and the reaction is of nth order in the concentration of each state. For n = 1, (3.46) describes the simplest first order equilibrium between two species. The solution of (3.46) for n = 1 is given by k1 (1 − exp (− (k1 + k−1 ) t)) . (3.47) x= k1 + k−1 There are a number of important characteristics of this simple equilibrium system. The system will relax to an equilibrium state where the ratio of the concentrations of the final to the initial state is the ratio of rate constants, k1 /k−1 . This is again a summed component behavior, as can be seen most clearly by solving (3.47) for the half time of the reaction    1 1 k−1 ln t=− . (3.48) 1− k1 + k−1 2 k1 Again we see the summed rate constant that characterizes the separated dependence on temperature. Also note that the reaction time goes to infinity for k1 = k−1 , and that t is only defined for k1 > k−1 . There is therefore a temperature where the direction of flow in the reaction switches from one state to the other. All of these attributes lend themselves to using this scheme to model some aspects of phase transition behavior, although this scheme is not sufficient by itself to model a first order phase transition, as it predicts an equilibrium concentration of the two states. One must go to higher, mixed order so as to mathematically reproduce the first order thermodynamic requirement that one phase exist at a time. These are the considerations that led to the construction of one of the nucleation and growth models of the β-δ phase transition that will be discussed in section “The Crystalline Phases of HMX and Rates of Transformation” below [12, 14]. A very different behavior is observed from the system of (3.46) if n is set to 2. The solution in the progress variable is given by        k1 k1 k−1 tanh arctan h − k1 k−1 + c0 k1 k−1 t . (3.49) x= k1 + k−1 Equation (3.49) is more complex than the simple first order case, but essentially produces a sigmoidal shape in x(t), with an induction time determined by the hyperbolic arctangent and a sharpness in t determined by the square root of the product of rates. There is also a singularity for k1 = k−1 . Most important is the reaction half time, given by      k1 1 k1 + k−1 1  arctan h − arctan h . (3.50) t=−  k−1 2 k1 k−1 c0 k1 k−1

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Now the half time is a function of the product of rate constants, specifically a single new rate constant, which is the exponential average of the two originals. This is a first example of a coupled kinetic scheme, where the individual processes are coupled to the extent that the overall rate is a function of both and are inseparable as a function of temperature. In addition, the particular form of the coupling, the inverse product of the two rate constants, will prove very important to the modeling of HMX ignition in section “Comparison of Kinetic and Thermodynamic Rates.” Another example can be constructed as a strongly coupled sequence of steps that are cyclically coupled, as in ∂y ∂x = k1 y and = −k−1 x, (3.51) ∂t ∂t where x and y represent progress variables in the mechanism. Such systems of equations yield cyclic solutions, the period of which is also a product of the square root of the two rate constants. The overall rate behavior of such systems is not characterized by any single rate constant but rather some function of the constants. Such systems are not currently used in explosive modeling, but some possible advantages to such a formulation will be discussed in section “Comparison of Kinetic and Thermodynamic Rates.” Measurements of Rate Constants The measurement of rate constants and the elucidation of mechanism are closely coupled, and understanding how to extricate one without complete knowledge of the other is the art of the experimental or theoretical chemical kineticist. While much of the work necessary to understand solid thermal explosions ultimately involves the kinetics of gaseous intermediate species, which are relatively easy to observe in isolation and whose interactions constitute a well posed mixing problem, the real difficulties lie in the solid phase chemistry of the problem. Solid phase chemical kinetics have lagged decades, perhaps many decades, behind our understanding of gas phase kinetics precisely because of the difficulties alluded to previously and in sections “Temperature Dependence: The Arrhenius Rate Constant” and “Concentration Dependence: Kinetic Order” involving the complex relationship between state of mixing and the system free energy. We do not pretend to offer a solution to this general difficulty. We will instead offer a number of specific points that are critical in extracting rate constants from data in the literature or by experiment, and finish with a specific example of our work deducing a sub-mechanism of HMX decomposition, the β-δ phase transition. Specific Considerations in the Determination of Rate Constants 1. The null observation of a proposed intermediate species should always be regarded provisionally, and possibly due simply to insufficient sensitivity in detection. This is the straightforward lesson of Fig. 3.4, but the literature is replete with attributions of mechanism based on the lack of observation.

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2. The determination of rate is not the same as the determination of a rate constant. The determination of a rate constant benefits exponentially from the breadth in temperature over which it can be measured. Because a linear regression is performed to deduce energy and entropy of activation, error in the activation energy is exponential in the temperature. Brill has published a wonderful illustration of this in the form of the kinetic compensation effect [51]. Breadth in temperature also renders a decoupling of rate constant from mechanism, as in the illustration of Fig. 3.3a and 3.3b. Despite the factor of two difference in the rate of progress for identical sublimation rates as a function of mechanism illustrated in Fig. 3.3a, one could fix that rate with good precision if the progress at some fixed point (usually x = 0.5) could be measured over the 250◦ C range of Fig. 3.3b. 3. Particularly in work toward ignition models, calorimetry remains the gold standard observable, in that it has an historic use as an observable in chemical kinetics and directly reflects the observable of interest in nonlinear ignition problems. As discussed below, however, as an integral observable it has limitations. 4. Although calorimetry is the gold standard observable, the use of calorimetry is replete with difficulties in interpretation. Probably the most frustrating of such difficulties are debates in the literature regarding the more correct type of experiment, isothermal or nonisothermal, or worse yet, which type of thermal trajectory leads to the more correct model. Of course, unless the rates of heating considered are sufficiently fast to effect molecular equilibrium (and therefore the use of the canonical thermal rate constant as discussed in section “Temperature Dependence: The Arrhenius Rate Constant”) only a model that can accurately predict both types of experiment is correct. We have demonstrated in both the phase transformation models [16, 18] and ignition models [52] that indeed both such types of experiments can be modeled using the same mechanism. This is a standard to which any serious mechanism should be held. A very good recent discussion of this and the general problem of the extraction of rates and mechanisms from calorimetry is provided by Vyazovkin and Wight [53, 54]. Specific Example of Work on the β-δ Phase Transformation Kinetic studies of the phase transformation were conducted using a second harmonic generation (SHG) probe. SHG provides a very sensitive probe that distinguishes the noncentrosymmetric δ phase from the centrosymmetric β phase, allowing a detailed study of the kinetics of the transformation by measuring the conversion efficiency of an input laser wavelength ω into its second harmonic at 2 ω [11, 14]. The SHG probe gives an integral measure of the δ mass fraction. For comparison, differential scanning calorimetry (DSC) studies have been performed, which measure the latent heat during transformation as a progress variable for the reaction. DSC is sensitive to the energy of the phase change, rather than to either the absolute product or reactant mass fractions.

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SHG, being an integral measure of the progress between the phases, is very sensitive and can distinguish the very early onset of the phase transition. It is very difficult, however, to perform an absolute calibration for SHG as the measurement of intensity at 2ω is sensitive to uncontrolled optical properties of the sample through the collection efficiency. DSC is less sensitive, but can be calibrated to provide an absolute rather than a relative progress variable. The kinetics observed by SHG and DSC are in agreement. Another observable often used is Raman spectroscopy with which both the loss of a reactant or the rise of a product species may be monitored. Some discrepancies have been reported between the phase transition kinetics observed by SHG, DSC, and Raman spectroscopy. This is due to the volume of material accessed in a heterogeneous material and how well the total average mass fraction may be infered. DSC studies measure the enthalpy of the transition summed over the entire sample. SHG studies probe an area equal to the size of the laser spot used to generate SHG signal, typically on the order of 0.5–1 cm diameter by several millimeters thick. Raman, on the contrary, probes a sample volume of a focused spot size with a penetration depth determined by the laser wavelength. The heterogeneous nature of the phase transition with multiple nucleation sites and growth from these sites causes the Raman studies to see the kinetics of the growth front propagating across the probe volume [12, 14, 55]. This explains why they appear with a delayed onset and steeper rise time compared to the volumetric nucleation and growth kinetics observed by SHG and DSC [56]. 3.2.2 Types of Models Homogeneous Source Term Models The most common type of models encountered in the literature are those characterized as homogeneous source term models. They are characterized by the use of Arrhenius rate constants and simple, nonchemically specific progress variables to model the dependence on concentration. These terms are then used to determine the rate of heat generation in the nonsteady heat equation. Such models treat only energy transfer and are therefore limited in applicability to calculations where mass transfer are important, such as the observed dependence of time to ignition on the loss of gas from a sample. They must be fit to data, and are therefore interpolative models, rather than extrapolative. The advantages to such an approach include ease of parameterization and computational simplicity. The literature contains many such models for the decomposition of energetic material. Zinn and Mader [2] reported some of the earliest computer simulations of energetic materials using numerical solution of the nonsteady heat equation and Zinn and Rogers [3] presented a reduced single term model for a

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number of materials that captured the time to ignition over the temperature range. The family of One Dimensional Time to Explosion (ODTX) models provide an example where the evolution of homogenous source term models can be followed [57–61]. Dickson et al. reported a four step model that uses a combination of identified chemical steps (the β-δ phase transition) and intermediate species to capture not only the time to ignition, but also the distribution of internal temperatures during the thermal trajectory. In this way, it captures the ignition location as well as time [62]. Zucker et al. have put forth a series of models of this type, including melt steps for explosives such as RDX and PETN which melt prior to thermal ignition to successfully capture ignition times and locations [63]. The strengths of the homogenous source term models is their ease of use. However, neglecting spatially heterogenous conditions such as mass transport limits the applicability of the model. In particular, while condensed phase energy transport may be expected to scale with sample size, mass transport or gaseous transport of sensible heat will depend on the complex morphology of the sample, perhaps determined by the spatially resolved extent of reaction, and therefore must not be expected to be invariant to sample size. Multiphase Models A level of complexity above the homogeneous source term models are multiphase models that distinguish between solid or liquid, and gas. Such models enable the treatment of mass transfer as well as energy transfer and have proven necessary in some instances. These models are significantly more computationally demanding than the homogeneous source term models, requiring finite element meshing of the problem. However, the ability to include mass transport allows the capture of behaviors where mass transport are evident [64]. The Los Alamos Large Scale Annular Cookoff (LSAC) experiment modeling is an example of a case where homogenous models were unable to capture thermal decomposition behavior without the inclusion of gas transport by Darcy flow through the thermally damaged PBX 9501 [65]. Although potentially far superior in fidelity, the computational cost of such models may be prohibitive in large-scale simulation. Complex Material Models The multiphase models used to capture internal temperature distributions in the LSAC experiment on PBX 9501 were parameterized using porosities comparable to those measured for thermally damaged samples [66–68]. To model the evolution of this porosity, more sophisticated consideration of the material is necessary. Such models enable a number of effects known to perturb explosive response to be addressed, and this is the topic of Chap. 6.

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For our purposes here, we note some of the ramifications of morphology on the chemical kinetics. For instance, how are solid and void created/partitioned? What is the best representation of kinetic order of a given process, for example, the kinetic order of a sublimation reaction as discussed in sections “Temperature Dependence: The Arrhenius Rate Constant” and “Concentration Dependence: Kinetic Order.” What are the relative effects of the transport of sensible heat vs. chemical potential. The evolution of these morphological changes can significantly impact the kinetic order or simulated concentration dependence of particular component reactions. The sublimation example described earlier is such a morphologysensitive reaction. The evolving state of the material can have a large impact on the behavior of the material to energy and mass transport through changes in porosity, permeability, on diffusivity, as has been discussed. While clearly preferable, only a few serious modeling efforts have weighted such attention to morphology [69–71]. It is possible that clever modeling strategies, particularly with respect to the simulation of the extensive dependencies like concentration or areal density, may allow inclusion of these effects in model of sufficient simplicity to find application. It is likely that as global chemical modeling is pursued and reasonably accurate mass loss mechanisms are applied, these will enable the calculated loss of mass to generate morphological change. It is certainly clear that as ignition models are used to produce the initial conditions for internal burning in explosions, the state of the material and the ability to support various modes of burning will be required. Global Chemical Models The most sophisticated models are based on a global approximation of the real chemical and physical transformations known to proceed in an explosive. These models incorporate known steps in the decomposition process, which release or transform chemical potential to generate heat. This typically involves a considerable reduction in the chemistry that must be considered compared to the full decomposition mechanism for the molecule. The advantages of such models are that they enable the calculation of many parameters of interest such as mass loss and pressurization. It is also possible to indirectly measure rates and dependence on temperature for component steps in the process. The difficulties of such a modeling strategy, however, include the fact that significant additional constraints on the model are implicit. These constraints arise from the fact that the model is constructed around the transformation of real chemical species, so a host of properties such as reaction enthalpy, thermophysical parameters, and in many cases rate constants, are fixed and cannot (or should not) be modified upon inclusion into the model. For these models, concentration variables are real chemical species rather than unspecified intermediate states. The models require independent measurements of component chemical and physical processes. The use of real species implies constraints

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on the model such as the use of independently measured reaction enthalpies and thermophysical parameters such as thermal conductivity. These independently constrained parameters limit the tunability of the model. The benefit of such constraint, however, is the ability to predict behaviors outside the directly measured space based on the understanding afforded by the model. 3.2.3 Modeling Objectives The level of fidelity required by the model determines which of the modeling schemes described above is necessary. If the only requirement of the model is a prediction of time to ignition at a given temperature, a single term homogenous model will suffice. If, however, in addition to reaction time, the location of the ignition volume is needed, then a multistep homogenous model will be needed, which captures more complex material behavior such as phase transformation. These phase transformations are responsible for the observation of “memory effects” in explosives such as HMX, which once converted to the delta phase, remains more sensitive than its initial beta phase despite being taken back to room temperature. If the model is required to predict the reaction violence of a system response to a thermal condition, more complex models yet are needed. Assumptions such as internal ignition locations and conditions that maximize the volume of heated explosive will contribute to the final reaction violence. Specific information needed to predict the final outcome or violence of a response are pressurization and pressurization rate to determine if an assembly will fail before reaction propagates. In addition, information on how pressure drives internal combustion is needed. This requires knowledge of the material state and pressurization rates at the time when the material transits from thermal decomposition to burning.

3.3 Specific Example: A Global Chemistry Model for HMX We have pursued global chemistry models of HMX decomposition and ignition for the reasons discussed earlier, the inclusion of greater model complexity in order to produce answers to more complex questions with greater reliability. Specifically we will ultimately require the following, in addition to the time to explosion: 1. A spatial map of HMX morphology, that is, regions of β and δ HMX at ignition 2. The rate of generation of gaseous products and the ultimate number of moles of gas generated per HMX molecule at the temperatures generated at ignition 3. Using the temperature trajectory of the simulation, and the spatially resolved conversion of condensed HMX to gaseous products, calculate a

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morphological description of the remaining solid, specifically containing a description of porosity and tortuosity (the length scale and complexity of pathways for gaseous diffusion) 4. Using the morphological model, a conversion of rates of gas generation and ultimate conversion into rates of pressurization and pressures generated by ignition 5. Finally, using the morphological model of tortuosity and pore length scale in concert with a model of pressurization describe whether the calculated ignition will transit to convective burning Such complexity of course leads to many more degrees of freedom in what are typically already grossly underconstrained ignition models. It is for this reason that additional constraints arise from the use of known, albeit simplified, chemical mechanisms in the decomposition mechanism. We will finish this section with our latest examples of such models for HMX. We are not yet ready for (5) and a full safety envelope calculation; however, we are well into (3), and as the rest of this book indicates, we now possess many of the pieces of the puzzle to quantitatively address (5) in the near future. 3.3.1 HMX Thermal Decomposition The literature on the reactions and decomposition pathways of the HMX molecule is considerable, and as the introductory quote by Shackelford [1], indicates often contradictory. The first measurements of the rate of decomposition were conducted by Robertson [72] shortly after the discovery of HMX as a by product of the RDX Bachman synthesis. Work on HMX proceeded over the next several decades, largely as studies incidental to more serious work on the RDX molecule. In 1977, Shaw and Walker compiled various kinetic measurements for what was typically described as HMX gas phase decomposition [73]. The compilation is unfortunately weighted heavily to reports that are more or less difficult to obtain and private communications. The theme of the paper is indicative of the efforts of researchers confronted with the problem of HMX decomposition and energetic response. Shaw and Walker identify eight different possible decomposition pathways, summarized as either identified conceptually or indicated by the measurements of a number of groups up to that time. The rate constants of four of these pathways were approximated using the measured rates (most notably dimethylnitramine) and thermodynamics of small model molecules and it was concluded that N–NO2 covalent bond scission is the rate controlling step in the gas phase decomposition. In the early 1980s, considerable work appeared by Brill et al. [74–82] on the kinetics of transformation among the family of polymorphs of crystalline HMX. In the course of the work, Brill arrived at a conclusion, which we will discuss below as critical in understanding the energetic response of HMX; that the lattice energy of the solid is critical in determining the rate of decomposition of HMX [74]. In the mid 1980s, Shackelford et al. [1] used kinetic isotope

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effects on calorimetrically measured rates of decomposition in the condensed phase to attribute different rate limiting processes depending on the phase of material. In particular, Shackelford et al. identify C–H bond breaking as rate limiting in the solid phase, C–N breaking in the melt, and reiterate Brill’s observation that the lattice, or more particularly its disassembly during decomposition, is a component of the rate limiting process in decomposition. At about the same time much of the compiled Soviet literature on HMX decomposition was recapitulated, and new measurements presented by Maksimov et al. [83]. In this work, the complexities of thermodynamic phase are again encountered, with special kinetic models incorporating mixed phases formulated to explain measurements [83]. Of particular importance here is the observation that there is a characteristic acceleration in the decomposition rate as the melting point is approached. This phenomena was observed to be relatively general as early as 1967 [84, 85] and has since proven to be quite robust [86], although there is no specific mention of HMX. Throughout the 1980s and 1990s, Brill’s group produced a very significant body of work, culminating in the high heating rate temperature jump techniques with which the dominant product distributions were identified and their rates measured [87? –96]. Beginning in the 1990s, Behrens [97, 98] and then Behrens and Belusu [99] published a significant body of work on the decomposition of HMX based upon gravimetric and very sensitive mass spectrometric techniques. With these techniques, many of the pathways of condensed phase decomposition were identified, as well as very important characteristics of the solid morphology of the HMX single crystal during decomposition and their influence on decomposition rates and products. The compilation of data continues to the present, important examples including the extremely low temperature manometric studies of solid HMX decomposition by German et al. [100] and the isoconversional calorimetric techniques applied by Vyazovkin and Wight to HMX [54]. The principle and very significant result of that work was the model-independent determination that HMX condensed phase decomposition is not chemically autocatalytic, as has often been supposed, but rather acceleratory in rate due to morphological effects of the kind previously observed by Behrens and Belusu [99]. We consider that this observation that a single activation energy controls decomposition throughout the reaction progress and over a temperature range of approximately 100 K is also very important and we will discuss our interpretation of the implications of this for the mechanism of ignition below. This brief compilation of the history of HMX decomposition is of course woefully incomplete. In particular we will not include with any justice the body of classic calorimetric work, begun by Rogers [101] and Hall [102] in the seventies, which is extensive and quite valuable. Combined with additional work discussed below, it does however give a flavor for the major bodies of work that have guided our thinking in the HMX problem and so serve as background to the final work of this chapter, which will be to illustrate the current state of our modeling of thermal ignition.

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Rates of HMX Decomposition A compilation of measured rates of HMX decomposition taken from the literature is shown in Fig. 3.5a. The data are either plotted as reported individual measurements or calculated using the reported rate constants and shown over the temperature range reported. The data are also plotted as the inverse rate, or characteristic time of reaction, to facilitate comparison with times to ignition that will be discussed in the next section. The data are plotted against the inverse temperature in order to linearize the exponential behavior expected of a single Arrhenius rate constant. The legend identifies the studies by first author and gives the thermodynamic phase of HMX reported or the solvent used in solution studies. Data from Maksimov et al. The data from Maksimov et al. [83] includes data presented for the first time in that publication, as well as a review of the contemporary Soviet literature. We cite these studies by their reference number in the Maksimov paper, which may be relatively easily obtained in English translation [83], rather than the more difficult original Russian publications, only one of which is an archival journal publication. The authors attribute pure HMX decomposition to gas phase processes in all but one study, that of the solid. In Maksimov et al. [83], a theory is presented of partitioning between gas phase and solid phase components of an experiment with the overall rate observed being a weighted combination of rates from the different phases. It is then supposed that this rationalizes the several order of magnitude discrepencies apparent in the data of Fig. 3.5a. We assume here, and for all the data reported, that the phase of HMX controlling the decomposition is very uncertain. This is not meant to be a criticism of any of the work presented, which together constitute a valuable suite of data for analysis, only a testament to the difficulty of rigorously attributing the phase in a solid-state semi-volatile chemical system below the melting point, or of understanding the importance of extremely rarified solid phases, such as would be found on the surfaces of an experiment where the amount of material used is such that a true bulk condensed phase should not precipitate. We finally mention that in Maksimov [83] it is noted that some of the prior data reported there merits less confidence. For example, the data of Maksimov (Ref. 3) is considered “provisional” (English translation). Solution Studies Data in Fig. 3.5a in grey denotes studies of decomposition conducted on molecular HMX in solution. Data obtained in acetonitrile by FTIR measurement of HMX concentration with time are reported as the single grey square [104], in acetone by chromatographic detection of product species by the grey line [103], and in m-nitrophenol by manometric detection of products by the grey dashed line, Maksimov (ref 5) [83]. In the case of the solution studies, it is

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Fig. 3.5. Compilation of measured rates of decomposition for HMX, plotted as the inverse of the measured rate constant, or the characteristic time of reaction, as a function of the inverse temperature. We have plotted individual points or a calculation based on the reported rate constants and the range of temperature over which the measurements were made. The entire suite of data are plotted in Fig. 3.5a. Measurements reported from the solid phase are plotted as the open diamonds [100], bold solid black line [83], black dashed line [103], and open triangles [1]. Measurements reported from the liquid or mixed phases are plotted as open squares and circles [1] and black dot-dashed line [72]. Measurements reported from the vapor phase are plotted as the solid black lines, and the dashed black line [83]. Measurements reported from solution are plotted as the grey square [104], the grey solid line [103] and the grey dashed line [83]. One calculated result is plotted as the long black dashed line [105]. A subset of the data, discussed in the text, is plotted in Fig. 3.5b, along with a solid black curve added as a guide to the eye

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clear that phase is not an issue, but presumably the solvent environment still presents a profound perturbation to the decomposition rate, exhibited by the four decade variability, and expected due to the strong effect of solvent on precipitate solid phase [106]. In addition to the single measurement of rate at ambient pressure, Naud and Bower [104] determine the pressure dependence of the rate and extract a volume of activation, and Naud and Bower [104] and Oxley [103] both attribute the rate limiting step in their studies to N–NO2 bond scission. Select Sub-Set of the Data The remaining data of Fig. 3.5a for neat HMX decomposition in the solid phase were obtained by Oxley et al. using chromatographic detection of products [103] and are plotted as the short dashed line, by Maksimov (ref 5) [83] using manometric detection of products and plotted as the bold solid black line, and the low temperature solid phase by German et al. [100] also using manometric detection of products and plotted as diamonds. The liquid phase data of Robertson [72] obtained using manometric detection of products are plotted as the single dashed black line. The data from Shackelford et al. [1] were obtained calorimetrically and are reported as representing a heating trajectory through the solid (triangles), mixed liquid/solid (squares), and liquid phases (circles). We also plot the rate constant calculated by Zhang and Troung using ab initio quantum mechanics as the dashed line in Fig. 3.5a [105]. In an attempt to rationalize the trends and differences in these measured rates, we select a subset of the data and plot them again in Fig. 3.5b. In Fig. 3.5b we have removed the solution studies, the “provisional” reference (ref 3) and also (ref 4) in Maksimov [83], and the calculated rates [105] and liquid rates of Robertson [72]. We further plot a guide to the eye through the remaining data set. What remains in Fig. 3.5b looks remarkably similar to the acceleration of rates observed for a number of other energetic materials, commented upon by Maksimov [84, 85] as early as 1961, and recently reviewed [86]. A fairly linear, single Arrhenius slope is observed at low temperatures, until some nonlinearity is exhibited in the German et al. [100] data set at extremely low temperature. This accelerates in a fairly smooth way to an asymptotic form at the HMX melting point, ∼280◦ C. The assignment of phase in the reported measurements is consistent except for the measurement of Maksimov [83], reported as gas phase measurements, which seem more consistent with a condensed phase measurement that approaches the melting point. The data rely heavily for this appearance on the calorimetric measurements by Shackelford et al. [1]. These measurements (reported for normal and deuterated HMX, only the normal HMX are shown here) were deliberately obtained with great temperature control very near the melting point. Extremely high sensitivity to temperature was observed within each phase, with rates changing by factors of two or higher over a three degree range. This is difficult to explain without the asymptotic acceleration that has been observed with many other molecules.

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This is a significant reinterpretation of the data, and at odds with some of the conclusions reported by the authors. It does seem to be the most consistent interpretation of the entire data set however, and provides a rational attribution of solid, liquid, and gas phase decomposition for HMX. Rates of Product Formation T-Jump/FTIR Spectroscopy While decomposition products have been measured since the earliest experiments of Robertson [72], the quantitative measurement of appearance rates and gaseous product distributions from the decomposition of HMX was significantly enhanced by the evolution of fast heating experiments, in conjuction with fast FTIR measurement of gas phase concentration. These experiments began at modest rates of heating, (2 K s−1 ) to temperatures near the melting point of HMX and were used to characterize the IR spectra of the melt phase and identify decomposition products in the melt [94]. Ignition times were subsequently measured [90] and as rates of heating increased to 600 K s−1 , the two well known families of gas phase product species were identified [91, 95, 107], and their rates of appearance measured [92]. Other groups also pursued these types of measurements, with general agreement [108, 109]. Significant discussion of the correct distribution of product species has followed this work, with particular emphasis on the surface of burning HMX [88, 89]. There is now general agreement that two families of products appear during HMX decomposition from 200 to nearly 1,000◦ C and rates of heating from a few degrees to 1,500 K s−1 . We call the first channel the 2a channel and at least three different specific product distributions have been proposed. δ − HMX → 4HCN + 4HONO [93]

ΔH2a = 149 kJ mol−1

(R2a ) −1

δ − HMX → 4HCN + 4NO2 + 4H [93] ΔH2a = 1,460 kJ mol (R2a ) δ − HMX → 4HCN + 2NO2 + 2NO + 2H2 O [110]ΔH2a = 219 kJ mol−1 (R2a) We calculate the heat of reaction, ΔH, for each stoichiometry from the heats of formation tabulated in the NIST Chemistry webbook (positive denoting an endothermic process). While there is disagreement on the product distribution, there is fair agreement between the value of the rate constant, k2a , that governs the appearance of these species [89, 109]. Conversely, there is general agreement on the simple product distribution that describes the second appearance channel, which we call channel 2b, δ−HMX → 4CH2 O+4N2 O [89, 110]

ΔH2b = −220 kJ mol−1

(R2b)

and disagreement on the parameters of the rate constant, k2b [89, 109].

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Fig. 3.6. Measurements of the concentration of two representative gas phase product species in the decomposition of HMX, plotted as the concentration of N2 O over NO2 as a function of the inverse temperature [89]. The solid black line is a fit to an Arrhenius rate constant, with parameters listed in Table 3.2 Table 3.2. Parameters of the fit of (3.1) to the data of Fig. 3.6

 ΔHp kJ mol−1  −1 ΔSp J mol K−1

Value

Standard deviation

117.66 72.073

10.4 6.2

The canonical data set regarding these relative rates was obtained by Brill et al. [92] and is expressed as the measured ratio of the two branches, captured by the measurement of the two representative concentrations [N2 O]/[NO2 ]. The data set, taken from [89] is shown in Fig. 3.6, plotted as the concentration ratio as a function of the inverse temperature. Plotted in such a way, linearity implies a functional form similar to the Arrhenius rate constant (3.7), and while such linearity does not imply a single elementary rate constant here, we perform a regression analysis of these data nonetheless and list the parameters in Table 3.2. This set of data and this linear regression parameterization will figure prominently in our attribution of mechanism in subsequent sections.

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Simultaneous Thermogravimetric Modulated Beam Mass Spectrometry (STMBMS) At the same time as Brill and Thynell were pursuing the high heating rate experiments, Behrens [97, 98] and Behrens and Belusu [99] were perfecting the high precision thermogravimetric and mass spectrometric techniques that have come to be called the STMBMS family of techniques. These experiments, designed to achieve electron impact ionization of species for mass spectrometric detection with the minimum possible perturbation [98], significantly clarified the view of product formation at low temperatures and isothermal conditions. While observing a suite of product species in agreement with those being observed at high rates, these experiments also enabled the observation of larger molecular weight, less stable intermediate species as well [97]. The most compelling results of this work have been the focus on the condensed phase and a number of complex mechanisms that have been deduced in the inductive, acceleratory, and late stages of decomposition, particularly the length scales and morphology of the porosity that results from the decomposition [99]. Rates of Reaction for Some Gas Phase Products In the calculation of ignition and thermal response based upon rigorous approximations of known chemistry, some important gas phase reactions among the decomposition products of HMX must be included. We incorporate two approximations to chemical oxidation systems for CH2 O and HCN by NO2 that were worked out in fairly complete detail by the groups of Lin and Melius in the early 1990s. We refer to these two systems by their categorization within detailed chemical combustion mechanisms for the family of energetic nitramine molecules [8, 9, 110]. We refer to the CH2 O + NO2 system therefore as dark zone chemistry, after this reference in these models and because these reactions proceed at the emergence from the solid surface and are not luminous nor sufficiently hot to easily detect by optical emission. We refer to a highly simplified HCN + NO2 approximation as bright zone chemistry after the importance of CN radical formation in the luminous, hot, so-called bright zone of the combustion. With these simple approximations, we hope ultimately to couple the decomposition and ignition models of solid HMX formulations with simple boundary conditions for the burning that follows ignition within a solid explosion. Formaldehyde Oxidation (Combustion Dark Zone Chemistry) It was recognized quite early that the gas phase reaction of CH2 O with NO2 would provide a significant source of exothermicity to the overall decomposition [92] and should play a significant role in gas phase ignition and unconfined burning in particular. The reaction had been studied as early as the 1940s by

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decomposition and ignition studies in the gas phase [111, 112]. As with the two solid phase decomposition pathways, several proposals were put forward regarding the stoichiometry of this reaction. The issue has been settled for all practical purposes by a series of experimental and modeling studies undertaken by Melius and Lin over a temperature range spanning some 1,200 K [113]. These results, in combination with the earlier work of Pollard [111, 112], Feifer and Holmes [114], and He [115] indicate the following overall reaction system. CH2 O + NO2 → HONO + HCO ΔH3 = 49.6 kJ mol−1 NO2 + HCO → CO2 + NO + H ΔH3a = −162 kJ mol−1 NO2 + HCO → HONO + CO

ΔH3b = −264 kJ mol−1

(R3 ) (R3a) (R3b)

The reaction enthalpies have again been determined using the heats of formation of all species taken from the NIST Chemistry Webbook. This system has been shown to be valid from 400 to 1,600 K and in particular is very insensitive at low temperature to a number of radical reactions that presumably proceed concurrently [113]. The reaction has actually been shown to be reversible, with a measured inverse rate several orders of magnitude larger than forward going [116]; however, in the presence of NO2 R(3a) and R(3b) are several orders of magnitude faster still and lead to the completion of this system of reactions. The rate constants determined for the system are plotted in Fig. 3.7 [113]. Data on R(3) is also plotted, with the data of Pollard and Wyatt [112] plotted as open squares, Feifer [114] as the black dashed line, He et al. [115] as the open circles, and Lin [113] as the black filled circles. The reverse rate constant is also plotted as a dashed black line [116]. The radical rate constants have been published both as zero activation energy temperature independent constants; R(3a) as the solid bold black line and R(3b) as the solid bold grey line [115] and with a modest temperature dependence of black and grey bold dashed lines [113]. If one assumes the temperature independent form of the radical rate constants and ignores the reverse reaction, one can sum the system into one overall reaction, limited in rate by R(3) and giving rise to both product branches, weighted by the relative value of the rate constants k3a and k3b , where Fa = k3a / (k3a + k3b ) and fb = k3b / (k3a + k3b )

(3.52)

The summed stoichiometry and heat of reaction are then CH2 O+2NO2 → HONO+fa (CO + NO + H)+fb (HONO + CO2 )

(R3)

and with overall thermochemistry given by a competitive ratio of R(3a) and R(3b) as (3.53) ΔH3summed = ΔH3 + fa ΔH3a + fb ΔH3b . We use this single term as the initial gas phase chemistry in models of thermal ignition of HMX to be presented at the end of this chapter.

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Fig. 3.7. Compilation of data for the oxidation of formaldehyde by nitrous oxide. The rate constant for R(3) is plotted as the solid black line [113]. The data of Pollard and Wyatt [112] are plotted as open squares, of Feifer [114] as the black dashed line, of He et al. [115] as the open circles, and of Lin [113] as the black filled circles. The reverse rate constant is also plotted as a dashed black line [116]. The radical rate constants have been published both as zero activation energy temperature independent constants; R(3a) as the solid bold black line and R(3b) as the solid bold grey line [115] and with a modest temperature dependence black and grey bold dashed lines [113]

Hydrogen Cyanide Oxidation (Combustion Bright Zone Chemistry) We also include a system of reactions, again determined by Lin and Melius, for HCN oxidation based upon the initial isomerization of HCN [117, 118] HCN → HNC, NO2 + HNC → HNCO + NO,

(R4) (R4a)

NO2 + HNCO → HNNO CO2 .

(R4b)

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Fig. 3.8. Calculation of the rate constants for the isomerization of HCN to HNC and subsequent oxidation of HNC by nitrous oxide [118]

The reaction enthalpies must be calculated with respect to the HNC and products of HNC oxidation and are given by Melius [118]. The system is extremely exothermic overall. The rate constants are plotted again as the inverse characteristic temperature in Fig. 3.8 [118]. The system is clearly relevant only at temperatures higher than the dark zone chemistry, and is simply limited by the isomerization reaction R(4). We again use a very simplified approximation of the system, assuming excess NO2 , and write down the single reaction HCN → HNC

(R4)

whose overall enthalpy is quite high. These reactions are not particularly important to ignition calculations in the low temperature ignition regime of HMX, but are expected to be necessary as temperature increases, the approximation of excess NO2 becomes more accurate, and as will be discussed further below, as the highly exothermic dark zone system is less favored by higher temperature branching ratios.

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Fig. 3.9. Time to ignition data for two HMX-based formulations. The data are plotted both linearly (Fig. 3.9a), as the ignition data vs. the inverse temperature, and logarithmically (Fig. 3.9b). Data on PBX 9501 from the ODTX [61] experiment are plotted as open circles, the SITI experiment as solid triangles [119], and from the LARE experiments as solid squares. Data on LX-10 from the ODTX [59] experiment are plotted as open squares and the SITI experiment as solid inverted triangles [119]. The solid black lines in both figures are the calculated using a single Arrhenius rate constant to be discussed below

Summary The gas phase chemical systems discussed here are quite widely used and accepted, and from the standpoint of a global chemical model are the most robust in our models, followed perhaps by the polymorphic phase transformations. We do not routinely include the bright zone chemistry in our models at this time; however, we have shown that such inclusion leads to multiphase ignition processes. We have recently obtained data that may support such phenomena, and if so, this bright zone chemistry may become important for very high fidelity modeling of ignition phenomena. 3.3.2 HMX Thermal Ignition The literature on the ignition behavior of HMX and HMX-based plastic bonded explosive formulations is again considerable. We begin with the confined thermal explosion experiments and early computer modeling first by Zinn and Mader [2] and then by Zinn and Rogers [3]. The work concentrates on the computer solution of the nonsteady heat equation using a thermal

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source term whose rate is exponential in the inverse temperature, thus simulating a simple one reaction chemical decomposition. Using this simulated chemistry generates a particular curve for the time to explosion as a function of boundary temperature for simple geometries. This particular curve may furthermore be expressed in a reduced variable form where the boundary temperature is normalized to the Frank–Kamenetskii critical temperature, Tm , and the time to explosion is normalized to a term containing a dimensional parameter and the material thermal diffusivity (Chap. 4). Zinn and Rogers [3] show that a number of explosives that undergo thermal runaway and explosion in the melt phase obey this simple one term model. They also show that HMX in particular does not, noting that “The behavior of HMX is believed to be essentially different from the rest of the explosives studied, and will be discussed separately.” Thus as early as 1962 it was becoming apparent that the high melting explosive HMX presents a more complex chemical decomposition mechanism than would be admitted by a simple one term source model. More complex computer models were soon developed. Following the lead of Zinn and Rogers [3], Catalano et al. [57] showed that accurate calculation of induction time in HMX-based explosives required more than a single chemical source term. This work marked the introduction of the ODTX. This experiment was subsequently used to generate a number of data sets on many explosives and explosive formulations and has proven to be one of the more valuable experiments and suites of data in the literature. Tarver et al. [60], again using ODTX data, showed that three terms, including an initial endothermic step, were sufficient to accurately reproduce HMX induction times over the range of temperature from 450 to 550 K and McGuire and Tarver [58] subsequently identified these source terms with three possible first order reactions, the rates of which were described by three activation energies and prefactors. The chemistry attributed to these terms included an early endothermic ring opening step, semi-volatile intermediate chemistry and subsequent gas phase reactions [59]. Following the work of Brill [74] and Henson et al. [14], where the initial endothermic step in the decomposition was attributed to the β-δ polymorphic phase transition in HMX, the model has recently been expanded [61] to include the use of binder decomposition models, and early steps in the decomposition have been relabeled to indicate that the β-δ transformation is the initial step, although the comprehensive models that followed on the work of Henson et al. [12–19] have not been incorporated [120]. Next in order of heating rate and ultimate temperature was the T-jump FTIR experiments first of Brill et al. [90] and then of Thynell et al. [108, 109]. Although predominantly experiments were designed to probe distributions and rates of appearance of gas phase products, ignition behavior was observed. Through extensive modeling and measurement, the temperature of the volume of material probed was determined as well. The second type of higher rate ignition experiments conducted since this time have been those related to laser ignition of a surface, primarily in the context of laser sustained combustion. While most of these studies parameterized rate or induction

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time in the laser energy applied, and are thus difficult to interpret without a direct measurement of temperature, some studies were conducted with explicitly modeled temperatures. In particular, the laser ignition study of Lengelle provides a temperature parameterized ignition study in this range [121]. Important for the interpretation of both of these types of experiments is the fact that ignition was generated fast compared to gas phase diffusion rates from the samples, effectively confining ignition to the solid. Throughout the 1990s, work at Los Alamos, associated initially with experiments on mechanical deformation of PBX 9501 by speckle interferometry [122], began to produce ignition times as a function of measured temperatures at higher rates of heating. The first such measurements were conducted with one-color InSb or InGaAs radiometry to measure surface temperatures. These experiments were based on carefully determined IR emmissivities from the constituents of PBX 9501, and rigorous determination of experimental error in the inferred temperature [123]. Using these techniques, ignition times were measured for PBX 9501 subjected to shear impact and high rate frictional heating [5], as well as high rate material flow [7], where very interesting effects of crystalline phase were observed [124]. The boundary temperature applied in these experiments was at times isothermal and at others linear, depending on whether the volume of material was overdriven to ignition during motion or not [7]. The ultimate heating rate experiments considered at this time were actually a reconsideration of an experiment conducted in the 1980s by Tarver and Von Holle, where a run to detonation experiment was imaged radiographically and material temperatures were recorded on the way to ignition [6, 125]. The interpretations of these data at the time centered about the radiographic resolution and hot spot size and the observation of bulk heating. We instead derived (not for the first time) a method of analysis using the rate of heating and observed ignition temperature to determine an effective single Arrhenius rate constant controlling the time to ignition in those experiments and showed that the rate constant determined was consistent with rate constants previously determined at slower rates of heating [6]. Menikoff has since applied this thermal rate constant successfully to the calculation of steady detonation properties of PBX 9501 [126, 127]. In more recent experiments, fast rates of heating resulting from laser irradiation of the central volume of a PBX 9501 during thermal runaway to ignition have been made and used to synchronize thermal explosions to a high energy radiographic source to obtain direct radiography of density evolution during burning within a solid explosion for the first time [23–25]. We have calibrated fast thermocouple measurements of these internal temperatures [22] and shown that observed induction times and temperatures are also consisted with the known rates of thermal decomposition and ignition in HMX [128]. Taken together, the suite of data represents a fairly complete interrogation of HMX ignition behavior over the full temperature range of energetic response, from the minimum critical temperature, Tm , to detonation.

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Small Scale Confined Thermal Explosion Data One Dimensional Time to Explosion (ODTX) The ODTX test was developed at Lawrence Livermore National Laboratory and has served as a benchmark test of energetic response for some time [57, 58, 60]. The experimental configuration utilizes a spherical sample, hence the one-dimensional character of the boundary condition and subsequent analysis, and a relatively high pressure applied to the mechanical confinement. In recent papers, the current status of the experiment, as well as the best data provided for LX-10 [59] and PBX 9501 [61] are reported. ODTX is also a common acronym used to identify the family of thermal decomposition models designed to model these data. Los Alamos Radial Explosion (LARE) The Los Alamos National Laboratory version of a one-dimensional thermal explosion test has been a series of experiments based on solid cylindrical geometry. The earliest versions were designed around a 2:1 length to diameter ratio sufficient to achieve a one-dimensional condition at the midplane. These experiments were designed for internal instrumentation and have been used to measure internal temperature distributions as well as times to explosion. While early versions of the test suffered from a lack of strong confinement and the leaking of gaseous products, later versions have been demonstrated to be sealed. The initial 2:1 aspect ratio experiment was 1 in diameter and was used to construct early versions of thermal decomposition models of PBX 9501 [62]. Later 2:1 versions of 1/2 diameter were used to probe the effects of boundary temperature on reaction violence subsequent to ignition and to field internal radiometry by optical fiber and fast thermocouple measurements of temperature during ignition [22]. Subsequently, laser synchronization was achieved via optical fiber and a laser induced temperature jump at the central volume where thermal runaway occurs [23]. Ultimately this synchronization was successfully used to conduct the first proton radiography of a thermal explosion [25]. The current experimental configuration is a 1:1 aspect ratio 1 diameter version, which enables better radiographic contrast and resolution imaging across the diameter, but defeats the original one-dimensional character of the experiment, offering better agreement with simulation when modeled one-dimensionally as a sphere. Sandia Instrumented Thermal Ignition (SITI) The SITI test is based on the Los Alamos Radial Explosion test. It is a well confined test with internal temperature measurements. Some experiments have been attempted with gaseous pressure measurements. The test has been applied to a broad suite of explosive formulations with good results [119].

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Low Temperature Thermal Ignition Data Set The suite of thermal ignition data for PBX 9501 and LX-10 provided by these techniques is shown in Fig. 3.9. In Fig 3.9a the data are plotted linearly as the time to ignition as a function of the inverse temperature and in Fig 3.9b they are plotted logarithmically. The data from each experiment are labeled in the figure legend with spherical data as open points and cylindrical as the solid. Data on PBX 9501 from the ODTX [61] experiment are plotted as open circles, the SITI experiment as solid triangles [119], and from the LARE experiments as solid squares. Data on LX-10 from the ODTX [59] experiment are plotted as open squares and the SITI experiment as solid inverted triangles [119]. The solid black lines in both figures are calculated using a single Arrhenius rate constant to be discussed below. Note the simple linear relationship in Fig. 3.9b expected for a single rate constant, but inconsistent with the thermal ignition curve of a single rate model presented in Zinn and Rogers [3]. High Heating Rate, Inertially Confined Thermal Explosion Data Temperature Jump Experiments by Hot Wire and Laser The data from Brill and Brush [90] and Lengelle et al. [121] represent a valuable suite of ignition data that are of intermediate rate between the direct heating experiments discussed above and the high rate mechanical impact experiments to follow. The work of both of these groups concentrated on understanding the temperature induced in the material and the subsequent ignition response. In the hot wire experiments of Brill and Brush [90], the extraction of temperature by heating the platinum wire involved considerable modeling support. Laser-induced ignition in the experiments of Lengelle et al. [121] was observed to occur within the solid, below the surface, and temperatures were again constructed with modeling support using the applied laser flux. In both these experimental series, we consider the ignition to be inertially confined, and so comparable to the other series discussed here as will be explained below. Taken together, these experiments represent rates of heating from 300 to 1,500 K s−1 and ultimate temperatures between 500 and 700 K. Los Alamos Mechanical Impact Experiments The mechanical impact experiments were all conducted using light gas guns and projectiles impacting at a few hundreds of m/s. All experiments were confined in steel fixtures and ignition volumes were observed both visibly and by radiometry through sapphire windows. PBX 9501 subject to shear by impacting with a sharp edged plunger at approximately 200 m s−1 yielded temperatures on the order of 900 K over 20 μs and ignition in slightly less than 100 μs [5]. Fast frictional heating by SiC particles at the explosive/sapphire interface, again impacting at approximately 200 m s−1 yielded somewhat lower temperatures with a more linear trajectory and ignition in slightly over 200 μs [5].

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Temperatures in both of these experimental series were obtained by InSb radiometry [123]. Higher velocity impact directly onto the sapphire window was utilized to induce ignition during material flow in PBX 9501 [7]. Velocities on the order of 350 m s−1 yielded temperatures above 1,300 K over approximately 1 μs and led to overdriven ignition during heating in β HMX PBX 9501 and relatively isothermal ignition after a few microseconds in δ HMX PBX 9501 [7]. Temperatures in both of these experiments were again obtained by InSb radiometry. In the most recent series of such experiments, we have also generated very high rate heating in PBX 9501 by application of a 1,064 nm, 100 μs laser pulse to the center of a heated, cylindrical sample. In these experiments, both overdriven and relatively isothermal ignition behavior is observed depending on the rate of heating. In these experiments, we observe heating to 900–1,100 K and ignition after approximately 100 μs. Taken together, these experiments probe ignition over rates of heating from 107 to 109 K s−1 and ultimate temperatures of between 900 and 1,500 K. Inclusion of the run-to-detonation experiments of Tarver and Von Holle do not enhance the range probed, but do introduce yet another means of generating temperature in the material [125]. The Confined, Non-Prepressurized Ignition Data Set A compilation of the suite of ignition data is plotted as the ignition time as a function of the inverse temperature in Fig. 3.10. The ignition time is either determined directly from isothermal like boundary conditions in the experiment or we take the inverse of the heating rate in two nonisothermal experiments as a characteristic time indicating ignition, as discussed later. The data were obtained from experiments on thermal explosion [59, 61] (filled and open circles), fast direct heating [90] (crosses), laser ignition [121] (open inverted triangles), friction and shear ignition [5] (filled squares), fast compression and flow [7] (filled triangles), shock to detonation [125] (open triangles), and laser-induced ignition within a thermal explosion [23] (filled inverted triangles). The data set is compiled subject to the requirement that the experiments be either confined or that ignition proceeds fast compared to material or thermal diffusion. All experiments in this compilation are confined, with the exception of the fast direct heating experiments [90, 121]. In these experiments, laser power and platinum wire current were used such that ignition was observed sufficiently fast that the diffusion of gas phase species away from the heated solid was negligible and the chemical composition of the solid volume element was conserved. Also, all experiments in this compilation were not previously pressurized, although as discussed in a previous publication the pressurization is certainly a component of the higher rate impact experiments [6]. The plotted temperature in each case is either the boundary temperature applied to the experiment, as is the case for the two thermal explosion data sets, or the actual measured temperature of the volume of material observed

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Fig. 3.10. A compilation of ignition data plotted as the ignition time as a function of the inverse temperature. The data were obtained from experiments on thermal explosion [59, 61] (filled and open circles), fast direct heating [90] (crosses), laser ignition [121] (open inverted triangles), friction and shear ignition [5] (filled squares), fast compression and flow [7] (filled triangles), shock to detonation [133] (open triangles), and laser induced ignition within a thermal explosion [23] (filled inverted triangles)

to ignite, as is the case in the laser (external and internal), shear, friction, and compression and flow (on delta PBX 9501) experiments. In these experiments, the temperature is observed to approximate a fast initial heating followed by a relatively constant temperature and are therefore good approximations to an isothermal boundary condition on the material persisting for a long time compared to the final self heating that generates ignition. In the two sets of data representing the fastest rates of heating, the shock initiation [125], and compression and flow (beta PBX 9501) experiments [7], a nonisothermal boundary condition is observed. In these two cases the data are plotted as the inverse rate of heating as a function of the inverse of the temperature at ignition. The analysis that leads to this method of plotting, which accommodates a heating profile where ignition is generated at the temperature where rates of self heating exceed the applied heating rate, is a traditional one and we have presented it previously [6].

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Table 3.3. Parameters of the fit of (3.7) to the data of Fig. 3.10

 ΔEign kJ mol−1  ΔSign J mol−1 K−1

Value

Standard deviation

148.54 243.71

1.0 1.8

The HMX formulations presented in this compilation are primarily PBX 9501, although one data set on thermal explosion in LX-10 [59] and one on shock initiation in PBX 9404 are also used [125], as are laser ignition data on neat HMX [121]. The inclusion of data from formulations other than PBX 9501 was necessitated by a desire to provide the most comprehensive coverage in temperature. This may of course introduce complications due to the different binders, but we do not believe this to be the case in the data that we have included here. The LX-10 data is sufficiently close in behavior to PBX 9501 in the thermal explosion regime and PBX 9404 is also sufficiently close in the detonation regime. The principal first observation of the data of Fig. 3.10 is of course the linearity of the data set. We have replotted the data of Fig. 3.10 as the natural log of the ignition time (s) as a function of inverse temperature (K) and performed a linear regression analysis. In the regression we did not weight the times using the error bars plotted in Fig. 3.10 as such error was not established in most of the experiments used here. The resulting parameterization of ΔSign and ΔEign using (3.7) are shown in Table 3.3. Also shown are the standard deviations in each parameter resulting from the linear regression. The calculated ignition time as a function of temperature using (3.7) and the parameters of Table 3.2 is shown as the solid line in Figs. 3.9 and 3.10. Summary The data included in the suite of Fig. 3.10 represent essentially the full range of heating and ultimate temperature applicable to the thermal response of HMX. There is already a considerable conceptual product that arises from the consideration of this HMX thermal ignition data from the point of view of real, global chemical mechanism. These in particular arise due to the linearity of the data set of Fig. 3.10. We list these as follows: 1. The extrapolation of a thermal Arrhenius rate law into the detonation regime suggests the same purely thermal model of chemistry in the detonation reaction zone [6]. If temperature precedes very high pressures in the leading edge of the shock, due to dissipation upon plastic flow or from radiation pre-heating, then thermal chemistry is fast enough to go to completion within the reaction zone defined by the sonic plane. 2. The experiments compiled in Fig. 3.10 represent very many different mechanisms of heating, from direct heating to laser and mechanical impact. The analysis of all these experiments using the single thermal rate

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constant is strong evidence that thermal response in HMX depends solely on the temperature, not on the means of heating [6, 7]. 3. The decomposition remains multiterm and categorized essentially as Tarver did from the outset [60]. We believe there is no single term that exhibits that rate constant in an elemental way. If this is true then a multiterm mechanism cannot be of the summed component type discussed in section “Temperature Dependence: The Arrhenius Rate Constant” [4, 52]. This will be discussed further below. 4. Finally, we note that Menikoff [126, 127] has recently modeled steady detonation in PBX 9501 using a single term kinetic rate published by us previously [4] and equivalent to that shown here in Table 3.3. 3.3.3 HMX Thermodynamics The Crystalline Phases of HMX and Rates of Transformation The solid phase structures and thermodynamics of HMX have been the subject of study for more than 40 years. However, significant deficiencies in the understanding of the crystalline phases have persisted throughout that time, characterized by such questions as the absence of a transition to α-HMX at ambient temperatures [106], or simply the uncertainty in the transition temperature of one of the more common transformations between the β and δ phases [14]. Although the role of the endothermic phase transformation in thermal decomposition and ignition behavior was the subject of early discussion [74], direct verification has come rather late as well [11, 62]. The structures of three stable polymorphs and a fourth hydrate of HMX have been determined under ambient conditions. The typical polymorph observed under ambient conditions is the β form [129]. The lattice structure has been determined to be monoclinic of P21 /c space group by X-ray [130] and neutron [131] diffraction. The molecular conformation involves two NO2 groups above and two below the ring plane. The remaining phases have been generated either by heating or by specific precipitation steps in the synthetic process [106]. The δ phase is the high temperature polymorph most often observed under low pressure heating. The δ lattice structure is of space group P61 (P65 ), again determined by X-ray diffraction [132]. The molecular conformation of δ involves all NO2 groups on one side of the ring plane. There is a 6.7% volume expansion on transition from the β to the δ phase [133]. The remaining polymorph, α, and hydrate, γ, are not considered here, but α HMX is of interest because of the question of stability mentioned above and as a phase sometimes observed upon reversion from δ during cooling. In addition, two more phases have been postulated recently under high pressure conditions [134]. The β-δ phase transition thus involves both a change in lattice symmetry, from a centrosymmetric to a noncentrosymmetric structure, and a change in the molecular conformation. The transition is of the first order

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thermodynamically, with a measured latent heat. Reports of the transition temperature at moderate rates of heating vary by some 50 K, from 140 to 190 K. Brill et al. used the Raman and Fourier transform infrared spectroscopy (FTIR) [76] of the β and δ phases to measure both the kinetics of transformation [74, 79] and the equilibrium pressure–temperature phase diagram [78, 82]. Effects of crystal size were also noted in that work [78, 82]. With respect to crystal morphology, many groups have observed significant cracking of individual crystals upon transition from β to δ, due to the significant molar volume increase [78]. Rates of Transformation The rate of transformation is thermally activated, as shown both in the original work of Brill et al. [74, 79] on HMX crystals and recent work in this laboratory on PBX 9501 at faster rates of heating induced by laser illumination [11]. In the early kinetic work [74, 79, 81], the transition kinetics were assumed first order, and Arrhenius parameters for the rate constant determined. In that work, parameters were determined based on limited extents of transition, as noted by the authors. Subsequent preliminary measurements, based on second harmonic generation [11], showed the full transformation in an ensemble of loose crystals from the β to δ phase. Based on these observations an irreversible second order kinetic rate law was proposed, yielding Arrhenius parameters for the rate constant in fair agreement with those proposed by Brill and Karpowicz [74, 79, 81]. The first step in the thermal decomposition kinetics of HMX is the β-δ transformation. The transition is endothermic with a transition enthalpy of 9.9 kJ mol−1 [14, 74] and a volume increase of 6.7% on converting from β to δ phase [74, 133]. The energy difference for the individual HMX molecules is small (∼30 kJ mol−1 ) [74, 130]. However, the reconstructive nature of the transition means that it is kinetically limited with a high activation energy. The activation barrier for nucleation has been measured [12, 74, 79, 81] and the barrier to growth has been shown to be equal to the heat of fusion [12]. We have published a kinetic model for the β-δ phase transition based upon the data shown in Fig. 3.11. It is a reversible nucleation and growth mechanism that has been described in several previous publications [12–20]. The nucleation is believed to be controlled by defects and leads to significant differences in the observed temperature of transformation, as found in the literature. We have proposed a number of models of nucleation, based on the observation that binder components in HMX formulations such as PBX 9501 may act as solvents at elevated temperatures and tend to reduce the apparent transformation temperature [13, 19]. The rate of growth of the nucleated phase has been shown to be controlled by an activation energy equal to the heat of fusion [12], giving rise to a model for growth occurring by a virtual melt mechanism [15–17]. The virtual melt mechanism explains the fact of melt mediated kinetics for a transition occurring approximately 120 K below the

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Fig. 3.11. Measurements of the β-δ phase transition observed by SHG. The data are plotted as the approximate reaction half-time as a function of the inverse temperature. Data are shown for conversion (open circles) and reversion (open squares) induced by direct heating [12, 14], as well as conversion induced by laser heating (open triangle) [11] and mechanical impact (inverted triangle) [5]. We also show data from Karpowicz et al., which seem to reflect what we model as the nucleation rates [81]. The solid line is a calculation using our reversible model of first order nucleation and second order growth, and the dashed line is a calculation using only the reversible nucleation kinetics [12, 14]

equilibrium melting point. The mechanism is based upon a state of tension induced orthogonal to the direction of propagation of the incoherent phase front in these molecular, reconstructive transformations. This state of tension lowers the effective melting point and enables the formation of a transient liquid layer, which immediately crystallizes to form the new (equilibrium) phase.

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The virtual melt mechanism helps explain many seemingly inconsistent observations regarding the β-δ transformation. In addition to β and δ polymorphs, HMX has a stable α phase, which has the same noncentrosymmetric symmetry as the δ phase, but a volume and thermodynamic stability regime between that of β and δ. It has a 3.8% volume expansion above β, and is thermodynamically stable in the temperature range above 378 K, yet, it is not attainable by direct heating from the β phase [106]. On heating to 432 K, β HMX converts directly to δ. The virtual melt mechanism accounts for the “hidden” α phase. The lattice strains that are induced by the smaller activation volume β-α transformation are not sufficiently high to depress the melt temperature far enough to activate the virtual melt mechanism. We have detailed a number of other observations and inconstancies in the understanding of the HMX polymorph system that are rationalized by this mechanism [16, 17]. Reversibility and Rates of Reversion Reversion from the δ phase to the β phase has been studied using the second harmonic generation (SHG) and Raman spectroscopic probes [12]. Data on reversion obtained using SHG are also shown in Fig. 3.11. DSC as a differential probe is not sensitive enough to capture the enthalpy of reversion due to the extremely slow nature of the transition. While the forward kinetics can be driven to occur on convenient laboratory time scales of seconds to hours (determined by temperature), the reverse transition occurs only on the time scale of hours. It is slowed kinetically due to the fact that the reverse transition is thermodynamically driven only at temperatures below the 432 K δ phase equilibrium temperature, with the highest rates observed at approximately 398 K with reversion to β phase taking tens of minutes at this temperature. SHG is used to follow the kinetics of reversion and Raman spectroscopy is used to confirm the presence of β phase and absence of δ and α phases. Reversion to β phase is found to occur in PBX 9501. For HMX crystals, reversion is observed only when the sample has not been completely transformed to δ phase. The nitroplasticizer is believed to nucleate reversion in PBX 9501. For HMX, the β phase must remain to seed nucleation of the δ to β transition in order for it to occur. Thermally driven transformation from β to α phase has not been observed. It has been observed to occur from δ phase, however, although the details of the kinetics are not well known. The δ has been observed to persist for several weeks, and the transition from δ to α phase has been observed over hours to days at ambient temperature and pressure. Mechanical Damage Conversion of β PBX 9501 to δ PBX 9501 is found to induce significant mechanical damage in the samples that results from the crystallization step in the VM mechanism [15, 19]. Observations include optical micrographs [12, 15] and post-transformation analysis by scanning electron microscopy (SEM) [99]. Thermal expansion and the β-δ transition are predicted to cause an

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expansion of ∼9% on heating from room temperature to 180◦C [133]. Proton radiographic images confirm the filling of void space in cased samples. Allowing extra volume above this allows the sample to expand further during the heating and phase transition with the extra volume incorporated throughout the sample as macroscopic cracks and void volume. Parker et al. [67] performed imaging on unconfined samples heated to 180◦ C and found macroscopic crack formation. The orientation of these cracks was conditioned by the level of confinement of the sample. Radially confined samples with open ends to allow for axial expansion showed cracks oriented parallel to the sample faces. Such features of thermal and mechanical damage are discussed at length in Chap. 4. The β-δ transformation is important in the thermal decomposition and ignition process both due to thermal damage induced throughout the sample, as well as due to the increased sensitivity of the δ phase compared to the β phase. Certain tests such as impact and drop height show that while β HMX is a secondary HE, the sensitivity of δ HMX is comparable to PETN [133]. The ability to revert PBX 9501 from δ back to β phase on cooling without annealing out mechanical damage to the sample caused by the phase transformation has enabled studies to separate out sensitivity due to mechanical damage from sensitivity due to the polymorphic phase of the HMX. A series of impact experiments performed with a single stage gas gun on β PBX 9501, δ PBX 9501, and PBX 9501 reverted from δ back to β phase show an increased sensitivity of the δ PBX 9501 and a decreased sensitivity of the reverted β PBX 9501 [124]. This indicates an increased sensitivity in these experiments due to polymorph phase rather than damage state that is difficult to rationalize simply from the endothermicity of the transformation and may involve the kinetics of transformation in an important way. This is currently an open question. An area of controversy regarding the β-δ phase transition is the ability to retard the formation of δ phase by the application of pressure. As the phase transition involves a volume expansion, the application of pressure could retard the kinetics of the transition according to the volume of activation. However, studies by Renlund et al. [136] using their constant volume hot cell show that the availability of internal volume to accommodate the volume increase of the phase transition is such that even for experiments with fixed volume, the phase transition occurs. They found in this study that the pressure applied to the w