multiscale modeling of reactive ni/al nanolaminates

MULTISCALE MODELING OF REACTIVE

Ni/Al NANOLAMINATES

by

Leen Alawieh

A dissertation submitted to The Johns Hopkins University in conformity with the

requirements for the degree of Doctor of Philosophy.

Baltimore, Maryland

October, 2013

c© Leen Alawieh 2013

All rights reserved

Abstract

This dissertation employs multiscale modeling for the purpose of investigating

reactions occurring in reactive Ni/Al nanolaminates. These are comprised of alter-

nating layers of Ni and Al that can react exothermically upon local ignition, even-

tually leading to the initiation of a self-propagating reaction front with speeds that

can exceed 10 m/s. A generalized thermal transport model is developed, based on

the transient multi-dimensional reduced continuum formalism introduced by Salloum

and Knio [31]. The generalized model accounts for an anisotropic thermal conduc-

tivity, that also depends on composition and temperature. A systematic analysis of

the role and ramifications that such a generalization has on the flame front structure

and dynamics is conducted, revealing that it has a dramatic impact on the ability to

successfully capture experimentally observed thermal front instabilities.

A multiscale analysis is then conducted in order to infer atomic intermixing rates

prevailing during different reaction regimes in the nanolaminates. The analysis com-

bines the results of Molecular Dynamics (MD) simulations with macroscale experi-

mental observations, and leads to the construction of a new composite atomic dif-

ii

ABSTRACT

fusivity law. Using this composite diffusivity law, a generalized reduced model is

obtained with the capability to simultaneously capture various reaction mechanisms

over a wide temperature range.

The generalized reduced model for single multilayers is then extended towards

exploring reactions occurring in layered particle networks. A further reduction of the

model is sought through identifying regimes under which spatial homogenization on

the particle level would be valid. The limiting case of a single chain of particles is

considered, and comparisons between the computational results of the heterogeneous

and the homogeneous reduced model descriptions are carried out. These reveal a

complex dependence of the reaction progress on the system properties and that simple

scaling arguments, based on particle size and rates of heat transfer, are not sufficient

for establishing a universal criterion of validity.

Advisor: Professor Omar M. Knio

Readers: Professor Timothy P. Weihs

Readers: Professor Joseph Katz

iii

Acknowledgments

First and foremost, I would like to express my sincere gratitude to my advisor,

Professor Omar M. Knio, for providing me with the opportunity to join his group,

and learn about reactive materials and numerical modeling in general. His endless

enthusiasm, stimulating discussions, guidance, patience, and support have been inte-

gral to this work. I cannot thank him enough for his kindness and all that he has

taught me over the past few years. I have been privileged to be his student.

I would also like to extend my deep appreciation to Professor Timothy P. Weihs

for his continuous insightful feedback on my work, and for providing me with the

experimental data that I needed to carry out my research. I am also thankful to

Professor Joseph Katz for serving as a member of my Ph.D. defense committee, and

as a reader for this dissertation.

For helping fund the work in this dissertation, I am grateful for the U.S. Depart-

ment of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and

Engineering Award DE-SC0002509; the Office of Naval Research Award N00014-07-1-

0740; and the Defense Threat Reduction Agency, Basic Research Award # HDTRA1-

iv

ACKNOWLEDGMENTS

11-1-0063.

I want to also thank Professors Todd C. Hufnagel, Michael L. Falk, Takeru Igusa,

Cila Herman, and Sean X. Sun for their encouragement and interesting discussions. I

am especially appreciative of Professors Hufnagel’s and Falk’s constructive input on

my work during the group meetings, and of Professor Falk’s and Mr. Rong-Guang

Xu’s invaluable help with initializing the Molecular Dynamics simulations when I first

embarked on the atomistic investigations.

Credit also goes to all the staff members in the Mechanical Engineering department

for making the annoying administrative issues much easier to deal with.

I am deeply grateful for my friends and colleagues here at Hopkins and elsewhere.

Without the fun moments, their help and support, graduate school would have been a

much less rich, memorable, and enjoyable experience. Special thanks to my childhood

and college friends back in Lebanon for all the past, and ongoing, stimulating and

heart-warming moments.

Last but not least, I am profoundly indebted to my family for their unconditional

love and unwavering support. Without them, none of this would have been possible.

v

Dedication

For a better understanding of nature.

vi

Contents

Abstract ii

Acknowledgments iv

List of Tables x

List of Figures xi

1 Introduction 1

1.1 Ni/Al Nanolaminates . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Multilayer Fabrication . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Reaction Basics . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.3 Reaction Initiation and Self-Propagation . . . . . . . . . . . . 7

1.1.4 Scientific Motivations and Applications . . . . . . . . . . . . . 9

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Methodology 13

vii

CONTENTS

2.1 Multilayer configuration . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Continuum Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Model Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Effects of Thermal Diffusion 25

3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Derivation of the generalized thermal transport models . . . . . . . . 28

3.2.1 Constant κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 Concentration dependent κ . . . . . . . . . . . . . . . . . . . 30

3.2.3 Direction-dependent κ . . . . . . . . . . . . . . . . . . . . . . 30

3.2.4 Direction and temperature dependent κ . . . . . . . . . . . . . 33

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Front Properties . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.2 Axial Front Structure . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.3 Normally-Propagating Fronts . . . . . . . . . . . . . . . . . . 56

3.3.4 3D computations . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Inference of Atomic Diffusivity 66

4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Atomistic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.1 Coarse graining . . . . . . . . . . . . . . . . . . . . . . . . . . 71

viii

CONTENTS

4.2.2 Extracting D(T ) . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.3.1 MD analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4 Macroscale Information . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4.1 Low Temperature Regime . . . . . . . . . . . . . . . . . . . . 89

4.4.2 Intermediate Temperature Regime . . . . . . . . . . . . . . . . 95

4.4.3 High Temperature Regime . . . . . . . . . . . . . . . . . . . . 102

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 Reactive Multilayered Particles 116

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2 Problem Formulation and Approach . . . . . . . . . . . . . . . . . . . 127

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6 Conclusions 149

Bibliography 154

Vita 177

ix

List of Tables

x

List of Figures

2.1 2D schematic of an unreacted (1:1) Ni/Al bilayer separated by a thinNiAl premix region. The total thickness of the bilayer is λ = 2(1+γ)δ,where the Al, Ni, and premix layers have individual thicknesses of2(δ − w), 2γ(δ − w), and 4w, respectively. . . . . . . . . . . . . . . . 15

3.1 Thermal conductivity of pure Al as a function of temperature. Shownare data from Touloukian et al. [99], along with the two best fits forthe solid and liquid states of Al. The melting temperature indicatedin the plot is 933K approximately; R2 ≈ 0.9992 for the fit in the solidstate, whereas R2 ≈ 0.9998 for the one in the liquid state region. . . . 35

3.2 Thermal conductivity of pure Ni as a function of temperature. Shownare the original data reported in [94] along with the best two fits forthe data below and above the Ni Curie temperature, TC ≈ 631K.The value of κNi at TC has been obtained from [99]. R2 ≈ 0.9997 forT < TC , whereas R

2 ≈ 0.9999 for T ≥ TC . . . . . . . . . . . . . . . . 373.3 Thermal conductivity of stoichiometric NiAl as a function of tempera-

ture. Shown are the data reported by Terada et al. [97] along with thebest fit for the data. R2 ≈ 0.9955. . . . . . . . . . . . . . . . . . . . . 41

3.4 Thermal conductivity of Ni(48%)V(2%)Al(50%) as a function of tem-perature. Shown are the data reported by Terada et al. [97] along withthe best fit for the data. R2 ≈ 0.9993. . . . . . . . . . . . . . . . . . . 42

3.5 Dependence of κ on temperature, direction, and concentration. Thesolid and dashed line plots correspond to thermal conductivity of pureNi/Al multilayers along the axial (x and z) and normal (y) directions,evaluated using (3.6) and (3.7) respectively. Curves are generated fordifferent values of C, as indicated. The (N) symbol refers to the valueof κ for the constant conductivity model computed using (3.2). . . . . 43

xi

LIST OF FIGURES

3.6 Dependence of κ on temperature, direction, and concentration. Thesolid and dashed line plots correspond to thermal conductivity of NiV/Almultilayers along the axial (x and z) and normal (y) directions, eval-uated using (3.8) and (3.9) respectively. Curves are generated for dif-ferent values of C, as indicated. The (N) symbol refers to the value ofκ for the constant conductivity model, computed using (3.3). . . . . . 44

3.7 Thermal width of the front versus δ. Curves are generated for Ni/Al(solid) and NiV/Al (dashed) multilayers, using the constant, concentration-dependent, and concentration and temperature dependent κ models,as indicated. Inset shows a zoom in on the region of δ = 12− 72 nm. 48

3.8 Reaction width of the front versus δ. Curves are generated for Ni/Al(solid) and NiV/Al (dashed) multilayers, using the constant, concentration-dependent, and concentration and temperature dependent κ models,as indicated. Inset shows a zoom in on the region of δ = 12− 72 nm. 50

3.9 Average, 1D, axial flame velocity versus δ. Curves are generated forNi/Al (solid) and NiV/Al (dashed) multilayers using the constant,concentration-dependent, and concentration and temperature depen-dent κ models, as indicated. In all cases, w = 0.8 nm. . . . . . . . . . 52

3.10 Average, 1D, axial flame velocity versus σR. Curves are generated forNi/Al (solid) and NiV/Al (dashed) multilayers, using the constant,concentration-dependent, and concentration and temperature depen-dent κ models, as indicated. The same data points as in Figs. (3.8)and (3.9) are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.11 Normalized average scalar concentration, C/C0 versus the normalizedtemperature T/Tf . The scatter plot depicts results for Ni/Al multilay-ers, obtained for different values of δ using the constant, concentration-dependent, and concentration and temperature-dependent κmodels, asindicated. Note that the temperature band observed on the C = 0 axisincludes superadiabatic overshoots that are associated with transientfront propagation [45]. . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.12 Reaction width versus δ. Curves are generated for axial and normalfront propagation in Ni/Al multilayers, using the (i) constant, (ii) con-centration dependent, (iii) concentration and direction dependent, and(iv) concentration, direction and temperature dependent κ models, asindicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.13 Average front velocity versus δ. Curves are generated for axial andnormal front propagation in Ni/Al multilayers, using the (i) constant,(ii) concentration dependent, (iii) concentration and direction depen-dent, and (iv) concentration, direction and temperature dependent κmodels, as indicated. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

xii

LIST OF FIGURES

3.14 Instantaneous distributions of surface temperature. The 3D computa-tions are performed for Ni/Al multilayer with δ = 6 nm, using the tem-perature dependent κ model over a domain size of (Lx×Lz ×Ly) = (1mm ×1 mm ×10µm). Ignition was simulated by imposing a surfaceheat flux for a short duration over a small square region centered at0.1

√2 mm from the origin. . . . . . . . . . . . . . . . . . . . . . . . . 63

3.15 Instantaneous surface heat release rate profiles. The 3D computationsare performed for Ni/Al multilayer with δ = 75 nm, using the directiondependent κ model over a domain size of (Lx × Lz × Ly) = (1 mm ×1mm ×2µm). Ignition was simulated by imposing a surface heat fluxfor a short duration over a small square region centered at 0.1

√2 mm

from the origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1 Snapshot of the initial configuration of Ni (white) and Al (green) atomsin the MD system at time t = 0. The arrangement corresponds to aNi/Al bilayer of total thickness λ = 8 nm and δ = 2.34 nm. . . . . . . 70

4.2 Cumulative distribution functions (CDF) of Nickel (red) and Alu-minum (blue), computed at (a) t = 0, and (b) t = 2.2 × 104 psec.The dashed line y = x corresponds to the asymptotic limit of a com-pletely mixed system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Mixing measure versus time for an MD system with δ = 2.34 nm. Thecurves depicts the evolution of M(t) during the initial equilibrationstage, the rapid heating stage to T = 700 K, and the adiabatic stage.Inset provides an enlarged view of the late stages of the computations,during which the Ni structure collapses. . . . . . . . . . . . . . . . . . 79

4.4 Average temperature versus time for an MD system with δ = 2.34 nm.The curves depicts the evolution of T (t) during the initial equilibrationstage, the rapid heating stage to T = 700 K, and the adiabatic stage.Inset provides an enlarged view of the late stages of the computations,during which the Ni structure collapses. . . . . . . . . . . . . . . . . . 80

4.5 Inferred diffusivity versus temperature. Plotted are curves generatedfor MD systems with δ = 2.34 nm (blue) and δ = 4.78 nm (black). Thetemperature range corresponds to the adiabatic stage in Figure 4.4.The solid curves correspond to approximations obtained as best fits tothe inferred D(T ) values obtained for δ = 2.34 nm along three separatebranches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

xiii

LIST OF FIGURES

4.6 Mixing measure versus time for the MD system with δ = 2.34 nm,under homogeneous, adiabatic reaction conditions. The blue curvecorresponds to the MD data shown in Figure 4.3 during the adiabaticstage, whereas the red dashed curve corresponds to predictions usingthe reduced continuum model with the approximate D(T ) fits depictedin Figure 4.5. Note that the continuum model does not take intoaccount the initial premixing that had occurred in MD during theheating stage, and instead starts from a purely unmixed state M(t =0) = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Average potential energy of the MD system with δ = 2.34 nm ver-sus the mixing measure, M , during the adiabatic phase.The M valuescorrespond to those shown in Fig. 4.3 during the adiabatic stage. . . . 86

4.8 Temperature evolution with time for a homogeneous reaction regimein a NiV/Al multilayer with δ = 56 nm. The blue curve correspondsto experimental observations by Fritz [19], while the red dashed curvecorresponds to predictions (truncated at T = 800 K) using the reducedcontinuum model with optimized pre-exponent and activation energyvalues, D0 = 2.08×10−7m2/s and Ea = 92.586 kJ/mol. Inset providesa zoom into the optimized region following the heating stage. . . . . . 92

4.9 Comparison between the D(T ) correlation obtained by Fritz [19] withD0 = 5.58× 10−9m2/s and Ea = 78.9 kJ/mol (red dashed curve), andthose obtained using the reduced continuum model with optimizedpre-exponent and activation energy values, D0 = 2.08×10−7m2/s andEa = 92.586 kJ/mol (solid black curve) and D0 = 5.176 × 10−8m2/sand Ea = 88.796 kJ/mol (green dashed curve) when assuming either alinear or a quadratic Q(C), respectively. . . . . . . . . . . . . . . . . 93

4.10 Inferred diffusivity, D, versus temperature, T . The estimates rely onthe experimental measurements of [106] for a nanocalorimeter incor-porating a Ni/Al bilayer with δ = 15 nm and a variant of the thermalmodel developed in [107]. Inset shows that the inferred D(T ) data inthe temperature range of interest does not exhibit an Arrhenius rela-tionship when plotted as ln(D) versus 1/T , and that rather the naturallogarithm of a quadratic fit, shown by the red solid curve, would bemore appropriate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.11 Combined extracted D(T ) values (on a semi-log scale) in the low andintermediate temperature regimes. The blue data points were plottedusing the Arrhenius diffusivity parameters optimized in Figure 4.8,while the red circles were plotted using the quadratic fit shown in theinset of Figure 4.10. Inset provides a zoom near the overlap regionbetween the D(T ) predictions (on a linear scale) using the outcomes ofthe low temperature regime optimization and that of the intermediatetemperature regime inference. . . . . . . . . . . . . . . . . . . . . . . 101

xiv

LIST OF FIGURES

4.12 Average axial self-propagating flame velocities as a function of δ on thebottom axis and λ on the top axis. The blue dots and error bars corre-spond to experimental observations of Knepper et al. [29], whereas theopen circles and red dots correspond to predictions using the reducedcontinuum model with optimized pre-exponent and activation energyvalues, D0 = 2.56× 10−6m2/s and Ea = 102.1910 kJ/mol in the hightemperature range, concurrently with the optimized and inferredD val-ues reported in Figures 4.8 and 4.10 at the lower temperatures. Theopen circles were obtained using a mesh size of ∆x = 0.5 µm, whereasthe red dots were obtained using a coarser mesh of size ∆x = 1 µm.Inset shows the variation of the finer-mesh velocity predictions whentaking smaller δ increments around the region where a velocity plateauis observed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.13 Temperature profiles along the foil length (direction of front propaga-tion) at the time instant t = 97.445 µs for δ values around the regionwhere there is a shoulder (plateau) in the reduced model velocity pre-dictions reported in Fig. 4.12. Also shown for comparison are thetemperature profiles at the time instants t = 55.645 µs for δ = 3 nmand δ = 5.3 nm, and t = 79.945 µs for δ = 9.1 nm. . . . . . . . . . . 108

4.14 Average axial self-propagating flame velocities as a function of δ onthe bottom axis and λ on the top axis. The blue dots correspondto experimental observations by Knepper [29], while the open circlesand red dots correspond to predictions using the reduced continuummodel. The open circles correspond to the open circle data pointsshown in Fig. 4.12 obtained using a premix width w = 0.8 nm. Thered data points were obtained for a premix width w = 0.91 nm with are-optimized pre-exponent and activation energy values, D0 = 1.91 ×10−6m2/s and Ea = 97.103 kJ/mol in the high temperature range,concurrently with the optimized and inferred D values reported inFigures 4.8 and 4.10 at the lower temperatures. The arrows highlightthe points where a velocity plateau is exhibited in both cases, and thered line provides a guide for the eye. . . . . . . . . . . . . . . . . . . 110

4.15 Final composite atomic diffusivity, D, values as a function of temper-ature combining results reported in Figures 4.8 – 4.12. Also shownfor comparison are the D(T ) values inferred from the MD simulationsreported in Figure 4.5 for δ = 2.34 nm, and the original global Ar-rhenius correlation obtained in [42] with D0 = 2.18 × 10−6m2/s andEa = 137 kJ/mol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

xv

LIST OF FIGURES

5.1 Normalized peak values of the instantaneous maximum and minimumtemperature differences in a given particle, Smax, as a function of ther-mal contact resistance Rc. Plotted are curves corresponding to differ-ent values of particle size, L, and half-layer thickness, δ. Solid lineswith solid dots correspond to δ = 50 nm, while dashed lines with opencircles correspond to δ = 250 nm. . . . . . . . . . . . . . . . . . . . . 136

5.2 Normalized time average of the instantaneous maximum and minimumtemperature differences in a given particle, < S(t) >, as a function ofthermal contact resistance Rc. Plotted are curves corresponding todifferent values of particle size, L, and half-layer thickness, δ. Solidlines with solid dots correspond to δ = 50 nm, while dashed lines withopen circles correspond to δ = 250 nm. . . . . . . . . . . . . . . . . 137

5.3 Relative time error in the reaction progress between the meshed andhomogeneous computations as a function of thermal contact resistanceRc. Plotted are curves corresponding to different values of particle size,L, and half-layer thickness, δ. Solid lines with solid dots correspondto δ = 50 nm, while dashed lines with open circles correspond toδ = 250 nm. Inset shows the same data points plotted on a log-log scale.141

5.4 Relative time error in the reaction progress between the meshed andhomogeneous computations as a function of Smax. Shown are curvescorresponding to different values of L, Rc, and δ. Solid lines with soliddots correspond to δ = 50 nm, while dashed lines with open circlescorrespond to δ = 250 nm. Both sets of data points correspond tothose shown in Figs. (5.1) and (5.3). . . . . . . . . . . . . . . . . . . 142

5.5 Relative time error in the reaction progress between the meshed andhomogeneous computations as a function of < S(t) >. Shown arecurves corresponding to different values of L, Rc, and δ. Solid lineswith solid dots correspond to δ = 50 nm, while dashed lines with opencircles correspond to δ = 250 nm. Both sets of data points correspondto those shown in Figs. (5.2) and (5.3). . . . . . . . . . . . . . . . . . 143

5.6 Relative time error, terror, as a function of the non-dimensional ratio ofthe particle’s internal thermal resistance, L/κ, to its thermal contactresistance, Rc. Shown are curves corresponding to different values ofL, Rc, and δ. Solid lines with solid dots correspond to δ = 50 nm,while dashed lines with open circles correspond to δ = 250 nm. Insetprovides a close-up into the region of small error and high Rc values. 145

xvi

LIST OF FIGURES

5.7 3D plot of terror as a function of L/κRc and the non-dimensional ratioof particle size to thermal front width, L/σT . Shown are curves cor-responding to different values of L, Rc, and δ. Solid lines with soliddots correspond to δ = 50 nm, while dashed lines with open circlescorrespond to δ = 250 nm. σT = 100 µm for δ = 250 nm and 20 µmfor δ = 50 nm. The terror 2D slice helps highlight the points at whichthe curves cross the 10% error threshold. . . . . . . . . . . . . . . . . 146

xvii

Chapter 1

Introduction

Most of the matter in nature is not in a state of equilibrium, but is rather under-

going constant change either through physical processes, chemical reactions, or both.

Systems that are seemingly in a stable equilibrium, could be driven out of this stag-

nant state after applying a sufficiently large perturbation as witnessed for example

by extreme occurrences such as avalanches, sinkholes, and fractures, or everyday oc-

currences such as ice formation, photosynthesis, and lightning. Energetic materials,

which are the focus of this dissertation, are intimately related to another instance of

such events, namely combustion.

Combustion typically involves the reaction, after a certain degree of initial heating,

of hydrocarbons (or simply solid carbon) and oxygen to form carbon dioxide, water

vapor, and other possible byproducts, along with the release of a certain amount

of heat. This phenomenon has been observed since the very early ages, however

1

CHAPTER 1. INTRODUCTION

it was not until the late 17th century when Lavoisier helped lay down its scientific

foundations [1]. This was followed by a wave of investigations in the 18th and 19th

centuries into the nature of combustion and its applications, some of which include

the discovery of detonation reactions, the invention of the combustion engine, and

the development of explosive, ballistic, and propulsion technologies [1, 2].

Energetic materials are classified as chemical compounds, typically containing

carbon, hydrogen, nitrogen, and sometimes oxygen, that can react rapidly when trig-

gered, with the release of large amounts of energy and oftentimes force [3]. They

generally involve the decomposition of the initial reactant molecules into more stable

(usually gaseous) products through various simple or complex reaction pathways. De-

pending on the initial conditions, the resulting reaction (or combustion) wave could

either propagate subsonically and is said to be in the flame propagation regime, or

supersonically and is termed to be a detonation. In either case however, the reaction

rate usually exhibits a sudden, rapid increase at a certain critical (ignition) tempera-

ture, after which the reaction becomes explosive [2, 4]. This dissertation will involve

the discussion of reactions that strictly fall into the flame regime category.

In 1967, researchers in the Soviet Union came upon a new phenomenon while

studying combustion processes in gasified systems, which they termed as self-propagating

high temperature synthesis (SHS)1 [5]. Following local ignition, SHS involves the

exothermic reaction between two or more reactants that are initially in the solid

1Note that the observed SHS reaction was actually gasless.

2


phase, yielding a new solid product or compound. The low cost of the SHS process,

its short duration, the high purity of the synthesized products, and the capability

of directly controlling the products’ size, shape, and properties, provided a gateway

to various industrial applications and this in turn helped further propagate scientific

research into solid state reactive materials. It is important to note, however, that the

reactants in the SHS reaction are not limited to those that are initially in the solid

state, but could also include gases and liquids [6]. Moreover, the presence of oxygen

is not a necessary component for the reaction’s initiation and progress, as is the case

in this thesis where the reactions that will be discussed can occur in vacuum.

Typically, SHS was carried out using loose micron-sized powders, or powders

pressed into pellets [6]. However, the complexity of the microstructural geometry of

powders posed a challenge for the development of theoretical models that could prop-

erly describe the reaction occurring between the particles and within the compacts.

So in 1976, foils consisting of alternating layers of reactants were proposed as an alter-

native means to studying gasless combustion [7]. In addition to offering a simplified

geometry for studying reactions, multilayered foils also established larger surface con-

tact areas between the reactants with less interface contamination, and allowed for

a better control over the spacing and layering of the reactants, and thus for a better

reproducibility of the experimental observations [8]. Since Nickel/Aluminum systems

have been the focus of much of the research that has been done on intermetallic

formation reactions for over 20 years [7, 9–17], mainly due to the ease of fabricating

3


the multilayers and the attractive physical and mechanical properties of the resulting

synthesized products [18, 19], we take advantage of this currently existing vast body

of work, and adopt the Ni/Al system in this dissertation for the purpose of modeling

multilayered reactive metals.

1.1 Ni/Al Nanolaminates

1.1.1 Multilayer Fabrication

Reactive Ni/Al multilayers are comprised of alternating layers of Ni and Al, fabri-

cated using either physical vapor deposition such as magnetron sputtering [8, 14, 20–

22], or mechanical techniques such as rolling [23–28]. The individual layers can range

in thickness from a few nanometers to tens of microns, with each sample typically

consisting of anywhere between a few to thousands of layers. This results in a total

multilayer thickness that is in the micron range for sputter deposited foils, and in the

millimeter range for the mechanically processed ones [19, 29].

Contrary to mechanical processing techniques which produce non-uniform, ran-

dom layering of the materials, vapor deposition usually allows for a uniform deposition

and offers a precise control over the desired spacing between the reactants [29]. In

this thesis, we consider only sputter deposited nanolaminates, where the individual

layer thickness is usually in the nanometer range (thus the designation), and the total

foil thickness is in the micrometer range.

4


Another feature of the multilayers is a premixed region present at the interface

between two adjacent reactant layers. This thin intermixed layer inevitably forms

during the deposition process, where the kinetic energy of the impacting atoms gets

dissipated as heat, thus facilitating the favorable mixing of Ni and Al atoms and

leading to the formation of a thin solid premixed 2 layer at the interface [30]. Note also

that surface atoms are usually less tightly bound than the ones in the interior, and this

further facilitates mixing at the interfaces. Even though the thickness of the premix

cannot be accurately determined experimentally, but only estimated [8,29,31–34], its

effect on the reaction can be investigated in a controlled manner by first minimizing its

presence through cooling the substrates during the deposition [19], and then annealing

the foils at low temperatures for different periods of times [8]. The latter allows the

premix to grow to varying degrees depending on the annealing time.

1.1.2 Reaction Basics

The reaction between Ni and Al is an exothermic reaction, meaning that it is ac-

companied by an overall release of a certain amount of energy in the form of heat and

is characterized by having a negative reaction enthalpy (∆Hrxn). Endothermic reac-

tions, on the other hand, require an overall input of energy as a driving force for the

reaction to occur, and are characterized by having a positive enthalpy of reaction [35].

The exothermic nature of the NiAl formation reaction is a direct consequence of the

2Throughout the thesis, unless otherwise noted, we will be dealing with stoichiometric composi-tions with a 1:1 atomic ratio of Ni and Al.

5


fact that the NiAl product has a lower potential energy compared to that of both Ni

and Al in their separate states. Thus, the formation of a chemical “bond” between

Ni and Al atoms leads to a more stable compound, and the excess energy is released

as heat. In this sense, the reaction is thermodynamically favorable and should occur

spontaneously. In nanolaminates, another driving force for Ni and Al to mix are the

steep concentration (or chemical potential) gradients present between the alternating

nano-scaled layers [36–38]. Therefore, even in the absence of the attractive inter-

atomic forces, their is a spontaneous tendency for the Ni and Al atoms to diffuse

into their neighboring layers in an attempt to achieve a system with a higher entropy

state. Note however, that had the reaction been endothermic in such a way so as the

overall increase in enthalpy overcame the increase in entropy, then mixing would not

have been favorable under standard conditions [35].

However, when Ni and Al are initially in their respective solid states, a certain

activation energy barrier has to be overcome before the reaction can occur. In this

scenario, the activation energy represents the minimum amount of energy that is

needed in order to perturb the solid crystal lattice structures in such a way so as

to permit some atoms to break free and diffuse, either in the form of intermittent

jumps or continuously [39]. Under standard pressure and temperature conditions, the

probability of having sufficiently large thermal fluctuations is negligible, making the

diffusion process extremely slow and causing the reaction to be kinetically hindered.

This explains why a given initial amount of energy input is required before the Ni/Al

6


reaction can actually occur.

1.1.3 Reaction Initiation and Self-Propagation

Reactions in the Ni/Al nanolaminates can be initiated using a localized, mo-

mentary external stimulus, such as an electric spark, a laser pulse, or a mechanical

impact [19]. This energy (or heat) input speeds up the interdiffusion process in the

region where it was applied, leading to the reaction of Ni and Al, the formation of the

NiAl product, and the release of heat. The heat released eventually diffuses to the

neighboring cold regions of the foil, and the cycle of events repeats. In this manner,

a self-propagating reaction front gets established, which moves along the length of

the foil until all the reactants are consumed. When the individual reactant layers

are thin (i.e. in the range of tens of nanometers), large front velocities are frequently

observed. For instance, for the Ni/Al system, self-propagating reaction fronts with

speeds exceeding 10 m/s have been reported [20, 21, 29]. If, on the other hand, the

initial momentary stimulus is not applied to a local region of the foil, but rather ho-

mogeneously over the entire length (or longest dimension), then atomic diffusion gets

intensified over all space and the entire foil ignites simultaneously. This is termed as

a homogeneous reaction.

For both self-propagating and homogeneous reactions, there is usually a minimum

(critical) threshold for the amount of heat that needs to be input before the reaction

can take off [31, 40, 41]. However, it is not necessary for at least one of the reactants

7


to melt before the reaction can become self-sustaining [19]. After the spark has

been removed, if the rate of heat losses, either to the environment or through heat

conduction within the multilayer (and away from the reaction zone), is faster than

the rate at which heat is being generated, then quenching occurs and no reaction is

observed. When no quenching occurs, the propagation of the flame front, in most of

the cases that we will be concerned with in this dissertation, is usually independent

of the initial ignition conditions. In other words, the domain lengths over which the

flame front propagates are usually sufficient for the dissipation of all memory effects

that could be caused by the initially imposed stimulus.

In the self-propagating reaction front scenario, a stable reaction front is mainly

characterized by its average velocity, thermal width, reaction width, and reaction

temperature. These measures will be defined and discussed in more detail in the

chapters to follow, but in general, they depend on a number of different factors [8,

23, 29, 42–49], including ambient conditions, layer thickness, material composition,

and on the microstructure or uniformity of the layering. For instance, a general

trend [29, 42] exhibited by the Ni/Al nanolaminates is a monotonic increase in the

average front velocity as the layer thickness decreases, due to the associated decrease

in the atomic mixing time-scale, up to a point where the trend flips and the velocity

starts decreasing. The latter is affected by the premixed layer thickness, which tends

to reduce the maximum reaction temperature and as a result the flame front velocity.

A variety of different in situ experimental techniques have been implemented in

8


order to resolve and monitor the spatial and temporal microstructural features of self-

propagating reaction fronts. These techniques include x-ray microdiffraction, x-ray

reflectivity, and dynamic transmission electron microscopy (DTEM) [38,49–51]. They

are also usually complemented by nanocalorimetry, differential scanning calorimeter

(DSC), and pyrometry measurements [19, 29, 52–54] for extracting thermodynamic

information such as heat capacity, heat of the reaction, reaction temperature, and

sequence of phase formations.

1.1.4 Scientific Motivations and Applications

Recent studies of multilayered materials have been motivated from both a funda-

mental science perspective and by potential applications. The former have particu-

larly aimed at taking advantage of the fact that these materials offer a unique setting

for analyzing phase transformations under rapid heating (up to 108 K/s) and large

compositional gradients [14–16,37,38,55–61]. Unresolved questions include the effects

of these extreme conditions on phenomena such as the sequence of phase formations

and the final product microstructure, nucleation, the morphology and stability of the

reaction front, and the mode of interatomic diffusion.

From the applications perspective, reactive multilayers have been used in various

areas including joining, brazing, sealing, and ignition of secondary reactions [5, 20,

30,40,62–73]. These applications have motivated studies aiming at improving funda-

mental understanding of the underlying reaction dynamics, and consequently tuning

9


the reaction properties.

As has been mentioned above, the reactions taking place in reactive multilayers

occur under extreme conditions, and involve processes that span a wide range of length

and time scales. This poses demanding challenges on experimental and theoretical

attempts that aim at unraveling the underlying fundamental physical mechanisms,

many of which are simply impractical to realize without insights from computational

modeling. Furthermore, computational models are also useful in yielding predictive

information that could aid the design and synthesis of new materials, thus cutting

on time and production costs. Consequently, this makes the task of understanding

reactive materials a multifaceted effort that couples experiments with computational

and theoretical models.

In this thesis, we utilize computational models in conjunction with experimental

measurements for the purpose of elucidating some of the underlying physics that is

associated with the reactions in Ni/Al nanolaminates, for providing information that

is not directly accessible experimentally, and for developing more reliable models

that are capable of encompassing and reproducing a variety of observed (some yet

unexplained) phenomena.

1.2 Outline

The dissertation is comprised of five main chapters organized as follows:

10


Chapter 2 introduces the continuum model formalism that is used for simulat-

ing the transient reaction dynamics in Ni/Al nanolaminates. It then moves on to

describing the mechanism implemented to reduce the continuum model in order to

overcome the stiffness in the equations, and achieve an enhancement in the compu-

tational efficiency. It finally provides a short overview of the numerical scheme used

in the computations for solving the governing equations of the reduced model.

Chapter 3 [74] generalizes the reduced continuum model described in chapter 2

to account for a variable, anisotropic thermal conductivity tensor. It includes the

derivation of generalized thermal transport models, and a systematic analysis of the

role and ramifications that such generalizations may have on the predicted flame front

structure and dynamics.

Chapter 4 [75] includes a multiscale inference analysis aimed at constructing a

generalized atomic diffusivity law for incorporating into the reduced model derived

in chapter 3, which would endow it with the capability to simultaneously capture a

disparate range of phenomena over a wide temperature range. It involves molecular

dynamics computations performed in order to gain insight into the dependence of the

atomic diffusivity on temperature under adiabatic conditions. The MD analysis is

then used to guide the construction and implementation of a new composite diffusivity

law based on information gained from macroscale experimental measurements.

Chapter 5 extends the modeling formalisms for single multilayers towards explor-

ing reactions occurring in layered particle networks. Due to the high dimensionality

11


and the high computational costs associated with simulating such systems, a further

reduction of the generalized reduced model developed in chapter 4 is attempted. At-

tention is focused on a quasi-1D chain of particles, and regimes are determined under

which spatial homogenization on the particle level would be valid.

12

Chapter 2

Methodology

2.1 Multilayer configuration

The system under consideration is a nanolaminate consisting of geometrically

flat1, alternating layers of Ni/Al with a 1:1 atomic ratio. Unless otherwise noted, the

layers are assumed to be initially separated by a thin solid premixed region, but are

otherwise compositionally pure. The thickness of each bilayer is λ = 2(1+ γ)δ where

2(δ − w) is the thickness of an Al layer, 2γ(δ − w) is the thickness of a Ni layer,

γ ≡ ρAl

ρNi

MNi

MAl

1In reality, the deposited layers are not perfectly flat, but rather exhibit a certain degree ofinterface roughness as seen in the Transmission Eelectron Micrograph images (see Fig. 15 in [45]).The effects of these geometrical surface oscillations on the reaction dynamics have been numericallyinvestigated in [45], and were found to be negligible for our particular system under consideration.Therefore, we can safely adopt the flatness assumption in our model formulation.

13

CHAPTER 2. METHODOLOGY

is the ratio of the atomic densities of Al and Ni respectively, ρAl and ρNi are the

densities of Al and Ni respectively, and MAl and MNi are the corresponding atomic

weights. The thickness of the premixed region is 4w. A representative schematic

of a single Ni/Al bilayer is shown in Figure 2.1. Note that the schematic is not

meant to represent a repeating structural unit, but is shown to simply illustrate the

arrangement and the thickness of each layer in a given bilayer.

Numerous computational studies have aimed at characterizing the velocity of self-

propagating reaction fronts in multilayered materials, and their dependence on mi-

crostructure and composition. In [8, 42], a simplified continuum analytical model

was implemented that assumes constant thermophysical properties and that the heat

released by the reaction is deposited near the flame temperature. Subsequent com-

putational studies have aimed at systematically relaxing the simplifications of the

analytical model, and have incorporated more elaborate models to account for prop-

erty variation, phase change effects, heat losses, as well as the dependence of heat

release on composition [31, 41, 43–47, 76–78]. Below, we mainly focus on discussing

the latter only, specifically the models developed in [31, 41, 47, 76], as they suffice to

help establish a foundational basis for the remainder of this dissertation.

14


!"

#"(⊥)

$%&'"

$%"

&'" 2(δ − w)

2γ(δ − w)

λ = 2(1 + γ)δ4w

Figure 2.1: 2D schematic of an unreacted (1:1) Ni/Al bilayer separated by a thinNiAl premix region. The total thickness of the bilayer is λ = 2(1 + γ)δ, where theAl, Ni, and premix layers have individual thicknesses of 2(δ−w), 2γ(δ−w), and 4w,respectively.

15


2.2 Continuum Model

Besnoin et al. [47] developed a continuum model for the multilayered system out-

lined in section 2.1, where the processes of atomic mixing and heat release in the

multilayer are described in terms of a coupled system of partial differential equations

expressing the evolution of a conserved scalar and enthalpy fields:

∂C

∂t= ∇ · (D(T )∇C) , (2.1)

ρ∂h

∂t= −∇ · q +

∂Q

∂t. (2.2)

The dimensionless conserved scalar, C, hereafter referred to as “concentration”, quan-

tifies the degree of atomic mixing; it varies between −1 ≤ C ≤ 1, and is defined such

that C = 1 is pure Al, C = −1 is pure Ni, and C = 0 is pure NiAl. The atomic

diffusivity, D, is assumed to be symmetric (i.e. does not depend on atomic iden-

tity), independent of concentration, and obeys a single Arrhenius law in the entire

temperature range characterizing the reaction, namely according to:

D(T ) = D0 exp

(

− Ea

RT

)

(2.3)

where T is temperature, R is the universal gas constant, D0 = 2.18 × 10−6 m2/s is

the pre-exponent, and Ea = 137 kJ/mol is the activation energy. The values of D0

and Ea are obtained as best fits based on experimental measurements of the velocity

of axially-propagating fronts [42].

It is important to note that Eq. (2.1) implicitly assumes that the reaction is

diffusion-limited, or in other words, that the time it takes for the reactants to react

16


upon encounter and form the product is almost instantaneous when compared to the

time it takes them to first diffuse towards each other.

In Eq. (2.2), h denotes the specific enthalpy, ρ is the mean density, and q is the

conductive heat flux given by Fourier’s law. Q is the heat released by the reaction

and is assumed to exhibit a quadratic dependence on concentration [43, 47]:

Q(C) = −∆HrxnC2 (2.4)

where ∆Hrxn is the (negatively signed) change of enthalpy (reaction enthalpy).

By making the additional assumptions that (i) the thermal conductivity, κ, is

isotropic and independent of temperature, and (ii) exploiting the separation of length

and time scales over which atomic and thermal diffusion occur2, Eq. (2.2) is further

simplified to (over a single bilayer for a 2-D system):

∂H

∂t= κ

∂2T

∂x2+

∂Q

∂t(2.5)

where κ is a constant mean thermal conductivity of the reactants, H is the layer-

averaged enthalpy,

∂Q

∂t= −ρcp∆Tf

∂C2

∂t, (2.6)

ρcp is the mean heat capacity per unit volume, ∆Tf is the temperature increase due

the reaction enthalpy, and

C2 =1

δ

∫ δ

0

C2(x, y, t)dy . (2.7)

2The thermal diffusivity is typically several orders of magnitude larger than the atomic diffusivity,which consequently makes the thermal front thickness (on the order of microns) also several orderof magnitude larger than the length scales over which atomic diffusion occurs (on the order ofnanometers). Thus, the temperature across a single bilayer can safely be assumed to be homogeneous.

17


Note that in defining the layer averages in Eqs. (2.6)–(2.7), the domain of interest

has been restricted to half the thickness of an Al layer, where y = δ coincides with

the centerline of the Al layer. This simplification exploits the symmetry of the flat,

periodic arrangement of the multilayer.

Finally, as discussed in [47], effects of melting of the reactants and products are

taken into account, whereby the temperature field, T , is recovered from the enthalpy

field, H , through a complex relationship that involves the heats of fusion of the

reactants and products according to:

T =

T0 +H/ρcp if H < H1

TAlm if H1 < H < H2

TAlm + (H −H2)/ρcp if H2 < H < H3

TNim if H3 < H < H4

TNim + (H −H4)/ρcp if H4 < H < H5

TNiAlm if H5 < H < H6

TNiAlm + (H −H6)/ρcp if H6 < H

(2.8)

where TAlm = 933K, TNi

m = 1728K and TNiAlm = 1912K denote the melting tempera-

tures of Al, Ni, and NiAl, respectively, HAlf , HNi

f , and HNiAlf , are the corresponding

heats of fusion (per unit mole), β ≡ C/(1 + γ) represents the fraction of pure (un-

18


mixed) Al, ∆HAlf ≡ ρAlHAl

f /MAl, ∆HNif ≡ ρNiHNi

f /MNi, ∆HNiAlf ≡ ρHNiAl

f /MNiAl, ,

H1 ≡ ρcp(TAlm −T0),H2 ≡ H1+β∆HAl

f ,H3 ≡ H2+ρcp(TNim −TAl

m ),H4 ≡ H3+βγ∆HNif ,

H5 ≡ H4 + ρcp(TNiAlm − TNi

m ), H6 ≡ H5 + (1− C)∆HNiAlf .

2.3 Model Reduction

Despite the simplicity of the continuum model described in the previous section

and its relatively good agreement with experimental observations in terms of predict-

ing average flame front propagation velocities [31], it is still computationally expen-

sive. This is due to the stiffness of the governing equations (2.1)&(2.5) caused by the

steep dependence of atomic diffusion on temperature, and the wide range of time and

length scales associated with atomic diffusion, thermal diffusion, and the multilayer’s

geometrical configuration [41].

In an attempt at overcoming the stiffness associated with the equations, Salloum

and Knio [31] developed a reduced reaction formalism, resulting in a computational

model that is substantially more efficient than that based on the continuum approach.

The development is based on a boundary layer analysis that enables one to transform

the partial differential equation in Eq. (2.1) into canonical form. It starts by (i)

assuming that atomic diffusion is dominant across the layers (in the y-direction), and

as before (ii) exploiting the separation of length and time scales over which atomic

19


and thermal diffusion occur, thus simplifying Eq. (2.1) into:

∂C

∂t= D(T )

∂2C

∂y2. (2.9)

This is followed by introducing a normalized “layer age,”

τ ≡∫ t

0

D(T )

δ2dt′, (2.10)

and a normalized spatial variable ξ ≡ y/δ, which allows Eq. (2.9) to be recast into

approximate canonical form:

∂C

∂τ=

∂2C

∂ξ2. (2.11)

Numerically integrating Eq. (2.11) allows us to directly extract C, and hence C and

C2, for a given value of τ . Moreover, using the same set of assumptions as be-

fore, the differential energy equation (2.2) can also be replaced with its volume- or

region-averaged form. Consequently, the resulting system of equations governing the

evolution of the reaction is expressed as:

∂τ

∂t=

D(T )

δ2(2.12)

∂H

∂t= − 1

V

∫

V

(∇ · q)dV +∂Q

∂t(2.13)

where H is the volume-averaged enthalpy, V is the volume of a computational cell

or region3 which is taken to be fixed and semi-closed (admits heat but not mass

3The volume of the cell in this case, or the area in the case of the layer-averaged formulation

20


diffusion),

∂Q

∂t= −ρcp∆Tf

∂C2(t)

∂t(2.14)

is the volume-averaged heat release term, ρcp is the mean heat capacity per unit

volume, ∆Tf ≡ ∆Hrxn/ρcp is the temperature increase due the reaction enthalpy,

and

C2 =1

V

∫

V

C2dV . (2.15)

The dependence of C and C2 on τ can be expressed in terms of a canonical solu-

tion, which is tabulated in a pre-processing step and made available to the computa-

tions [31]. The temperature, T , can be recovered as before using the relationship in

Eq. (2.8) .

According to the formulation outlined above, we can see that the advantages of

the reduced model over the detailed model are that it (1) eliminates the need of

using a fine scale mesh in order to resolve the process of atomic mixing occurring

at the nano-scales through replacing the evolution equation for C with an evolution

equation for τ , and (2) requires the computation of only global (average) spatial and

temporal reaction and thermal diffusion terms, thus reducing the stiffness inherent

in the governing equations of the detailed model and allowing for substantially larger

integration time-steps (this is true even when semi-implicit integration schemes are

used in the detailed model as in [45–47]).

discussed in the previous section, is defined such that largest dimension is ≪ the thermal frontthickness in order to ensure the validity of the temperature homogeneity assumption. Later on, asystematic mesh refinement study will be conducted in order to determine a suitable mesh size thatwould satisfy this requirement.

21


To check the validity of the reduced model and quantify the enhancement in com-

putational efficiency that it offered, Salloum and Knio [31, 41, 76] conducted various

numerical experiments comparing the results of both, the reduced and the detailed

models. The analysis indicated that the two models gave predictions of average ve-

locities of self-propagating fronts and ignition thresholds that were in agreement, and

that the required CPU time using the reduced model was up to 64 times less than

that needed by the detailed model. Moreover, they demonstrated the capability of

extending the reduced formalism to include transient multidimensional computations

in both uniform and heterogeneous multilayers — a feature that would have been

prohibitively expensive using the detailed model.

Having been tested and validated, the reduced continuum methodology will be

adopted as our starting point for the remainder of this thesis. In what follows, a brief

description of the numerical scheme (adapted from [41]) utilized for implementing the

reduced model is provided.

2.4 Numerical Scheme

The reduced model outlined in section 2.3 involves a coupled system of equations,

(2.12) and (2.13). Numerical solution of this system of equations is conducted using

a finite-difference scheme that is adapted from [41, 76]. Brief details are provided

below.

22


For simplicity, we focus on simple domains, rectangles in 2D or boxes in 3D, which

are discretized using a uniform grid. The cell sizes in the x, y and z direction are

denoted by ∆x, ∆y and ∆z, respectively. Field variables are discretized on cell cen-

ters, whereas fluxes are defined at cell edges. Consequently, with each computational

cell one associates the discrete state vector, (H, τ ), composed of the enthalpy, H ,

and the local age of the individual subregions contained in the computational cell, τp,

p = 1, . . . , N (where N denotes the number of subregions4). All other local physical

quantities can be readily obtained based on the local state vector. In particular, the

temperature, T , can be retrieved from the volume-averaged enthalpy, H , by inverting

a complex relationship involving the heats of fusion of the constituents (Eq. (2.8)) [76].

A second-order, conservative, centered-difference approximation is used to esti-

mate the conduction heat fluxes at the faces of each computational cell. This ap-

proximation transforms the governing equations (2.12) and (2.13) into a discrete

system of coupled ODEs. The resulting discrete system is advanced in time using

the mixed scheme introduced in [41]. This scheme combines exact treatment of the

source term appearing on the right-hand side of (2.13), with Runge-Kutta-Chebychev

(RKC) [79–83] treatment of the thermal and atomic diffusion terms. Additional de-

tails regarding the construction of the mixed integration scheme can be found in [41].

In all the 1D and 2D computations discussed in the subsequent chapters, adiabatic

conditions are imposed on all domain boundaries (unless otherwise noted). In these

4A region is defined as a subset of the computational domain whose dimensions are ≪ than thethermal front width, and a subregion is a subset of this region which has a uniform δ layering. Aregion could be comprised of one or more subregions.

23


computations, ignition is simulated by imposing an initial temperature profile within

the domain such that T (x, t = 0) = Ts for 0 ≤ x ≤ ws, where Ts and ws are the

spark temperature and width respectively. Beyond the spark region, the temperature

decreases linearly to the ambient temperature, T0, according to:

T (x, t = 0) = T0 + (Ts − T0)(2ws − x)/ws for ws ≤ x ≤ 2ws,

T (x, t = 0) = T0 for x ≥ 2ws

In the 3D computations, ignition is simulated by imposing a surface heat flux on

a portion of the domain boundary, which is maintained constant over a fixed time

period. At all other times, and all other locations, adiabatic boundary conditions are

used, as in the 1D and 2D computations.

As discussed in [76], the thin premix region is accounted for by setting the initial

value of the average composition to C(t = 0) = 1 − w/δ. In all the cases considered

(unless otherwise noted), a constant premix width w = 0.8 nm was prescribed, and

the computations are performed using a uniform Cartesian grid. Accordingly, for

δ > 12 nm, a cell size ∆x = ∆y = ∆z = 1 µm was used, whereas ∆x = ∆y =

∆z = 0.5 µm for smaller values of δ. These mesh resolutions were selected following

a systematic mesh refinement analysis that aimed to ensure that the predicted front

properties became essentially independent of cell size.

24

Chapter 3

Effects of Thermal Diffusion

3.1 Motivation

It is now well known that the average velocity of the front depends on a number

of factors [8, 23, 29, 42–48], including ambient conditions, layer thickness, material

composition, and on the microstructure or uniformity of the layering. The dynamics

of self-propagating fronts in multilayers can however be complex under certain condi-

tions, even when the layering (or microstructure) is essentially uniform. For instance,

evidence of oscillatory front motion has been predicted computationally [45,47] as well

as seen experimentally [84]. Recently, it has also been observed that self-propagating

fronts in multilayers can exhibit cellular [49], or spinlike features [85]. These transient

front dynamics can thus be inherently multidimensional in nature.

In addition to enabling simulations in two and three space dimensions, the reduced

25

CHAPTER 3. EFFECTS OF THERMAL DIFFUSION

reaction methodology developed in [31,41,76] has shown that by using this formalism

one is able to predict the behavior of multilayer composites that feature non-uniform

layering, particularly when the front behavior cannot be readily described using an

average layering [29, 76].

To reproduce the front velocity in nonuniform multilayers, a variable thermal

conductivity model was developed in [76] which accounts for the dependence of the

thermal conductivity on the local concentration, but otherwise ignores potential tem-

perature and directional effects. The effects of property variation with temperature

and concentration were in fact considered in the analysis of Gunduz et al. [86]. How-

ever, this analysis assumed that physical properties were isotropic, and was restricted

to quasi-1D axial front propagation. Consequently, the impact of the dependence of

thermal conductivity on concentration, temperature, and material layering is as of

yet not well understood. The present study addresses this issue, through a systematic

analysis of various thermal conductivity models.

To this end, we consider four different thermal conductivity models. The first

is a constant conductivity model, where following [45] the thermal conductivity is

approximated by the average conductivity of the reactants and taken to be fixed in

space and time. The second model is the concentration dependent thermal conduc-

tivity model developed in [76]. In this model, the thermal conductivity is treated as

isotropic, temperature effects are ignored, but the dependence on local composition

is directly accounted for using a simplified mixture rule. The third model generalizes

26


the second by accounting for the impact of the reactants’ layering. It is motivated

by experimental observations of nanostructured multilayers [87] which indicate that

the in-plane thermal diffusivity may be several orders of magnitude larger than the

normal (i.e. perpendicular to the layers) thermal diffusivity. Finally, the fourth model

generalizes the third by accounting for the temperature dependence of the individual

constituents.

In order to illustrate the role and ramifications each of these dependencies has on

the flame dynamics, we contrast predictions obtained using the four models on the

average front velocity in nanostructured Ni/Al multilayers. In doing so, the bilayer

thickness is systematically varied. Furthermore, we also contrast the results obtained

for Ni/Al and NiV/Al multilayered systems (V stands for Vanadium) in order to

briefly explore whether the substantial difference in the thermal conductivity of pure

Ni and relevant Ni alloys1 would lead to a substantial difference in reaction front

velocities.

In addition to analyzing the behavior of flame velocities, a systematic study is

conducted of the quasi-1D structure of the reaction front. This effort is motivated by

the fact that despite the availability of various studies focusing on self-propagating re-

actions in nanolaminates, several relevant features of the reaction front have not been

thoroughly investigated, including the dependence of thermal and reaction widths on

the bilayer thickness, as well as the variation of the composition with temperature.

1The NiV/Al multilayers are fabricated by vapor deposition using pure Al and NiV (7% wt V)targets [29]. Unlike pure Ni, the Ni93V7 alloy is non-magnetic, which offers advantages during thedeposition process.

27


Investigation of these features is further motivated by the development of advanced

diagnostic tools [38, 49] which are enabling direct observations into the front struc-

ture, as well as molecular simulations [88–91] which offer the possibility of developing

more elaborate reaction models. Specifically, a detailed characterization of the quasi-

1D front structure would be helpful for the purpose of planning measurements or

multiscale computations, and ultimately for validating or refining reduced models.

Finally, results from a limited number of 3D computations are outlined. These

are used to conduct a preliminary exploration of the role of the thermal model on

transient front motion in three dimensions.

3.2 Derivation of the generalized thermal

transport models

Our goal in this chapter is to study the effect of thermal diffusion on the dy-

namics of the flame propagation, and specifically on the role played by the thermal

conductivity. In all cases, the heat flux q is assumed to follow Fourier’s law, namely

q = −κ∇T (3.1)

where κ is the thermal conductivity. As mentioned above, new models are developed

in the present study that account for the effects of material heterogeneity (layering)

and the variation of thermal conductivity with temperature. Results obtained with

28


these new models are contrasted with predictions obtained based on earlier, simpler

models, namely those corresponding to a constant conductivity [47], or to an isotropic,

concentration-dependent conductivity [76]. For the sake of completeness, all models

considered in the present work are outlined below.

3.2.1 Constant κ

Original models of self-propagating reactions in reactive nanolaminates [42, 44,

45, 92, 93] have relied on a constant conductivity approximation. Presently, this is

implemented using an appropriate average conductivity of the reactants. For a Ni/Al

system, with a 1:1 ratio of the reactants, the thermal conductivity is expressed as:

κ = κ ≡ κAl + γκNi

1 + γ(3.2)

where κAl = 237 Wm−1K−1and κNi = 91 Wm−1K−1refer to the room-temperature

thermal conductivities of Al and Ni, respectively. (These estimates are discussed in

section 3.2.4). In the computations below, we shall also consider a variant of (3.2)

adapted to an NiV/Al system. In the latter case, the thermal conductivity is given

by:

κ = κ ≡ κAl + γκNiV

1 + γ(3.3)

where κNiV = 18.9 Wm−1K−1is the thermal conductivity of Ni(93%)V(7%) at room

temperature, see section 3.2.4.

29


3.2.2 Concentration dependent κ

In [76], an isotropic, concentration-dependent thermal conductivity model was

developed. The model considered Ni/Al multilayers, and the thermal conductivity

was expressed as:

κ = (κ− κNiAl)C + κNiAl (3.4)

where κ is given by (3.2) and κNiAl = 92 Wm−1K−1is the room temperature thermal

conductivity of NiAl, see section 3.2.4.

For NiV/Al multilayers, the mixture-based approach in [76] yields:

κ = (κ− κNiV Al)C + κNiV Al (3.5)

where κ is given by (3.3) and κNiV Al = 48.7 Wm−1K−1is the room temperature

thermal conductivity of NiVAl, see section 3.2.4. In the computations below, we

contrast results obtained for the different multilayer compositions using the κ models

given by (3.2)–(3.5), as well as more elaborate models developed below.

3.2.3 Direction-dependent κ

This section extends the concentration-dependent conductivity formulation above

by accounting for the anisotropy of the unreacted medium. Specifically, due to the

initial layering of the nanolaminate, and the differences in the thermal conductivities

of individual layers, transport rates may generally differ according to whether the

temperature gradient is parallel to the layers, or normal to the layers. This may po-

30


tentially affect the motion of the self-propagating front, particularly when propagation

occurs along multiple space dimensions, or normal to the layering [41].

As shown in Fig. 2.1, the total thickness of the bilayer (including the premix) is λ,

the Al layer has a thickness tAl = αλC, and the Ni layer has a thickness tNi = βλC,

where α and β represent the fractional thickness volumes of Ni and Al, respectively.

For nanolaminates with a 1:1 Ni/Al composition, we have:

α ≡ 1

1 + γ,

and

β ≡ γ

1 + γ.

The anisotropic character of κ will be assumed to be solely the result of the layered

configuration of the system. Thus, the direction-dependent κ matrix will be diagonal,

of the form:

κ =

κ‖ 0 0

0 κ⊥ 0

0 0 κ‖

where κ‖ designates the in-plane (x and z directions) conductivity, and κ⊥ designates

the thermal conductivity along the normal (y) direction.

For brevity, we focus our attention on Ni/Al nanolaminates and seek expressions

for κ‖ and κ⊥ by considering two scenarios, namely one in which heat is flowing along

the layers, and another where heat is flowing normal to the layers. The former case

31


is analogous to the situation of an electric circuit with resistors connected in parallel,

whereas the second corresponds to resistors connected in series. Accordingly, κ‖ can

expressed as the weighted sum of the individual thermal conductivities for Al, Ni,

and NiAl, according to:

κ‖ =κAltAl + κNitNi + κNiAltNiAl

λ.

Substituting the appropriate expressions for the individual layer thicknesses conse-

quently yields:

κ‖ = ακAlC + βκNiC + κNiAl(1− C) . (3.6)

When heat flows normal to the layering, continuity of the normal flux immediately

yields:

κNi∆TNi

tNi

= κ⊥∆Tλ

κAl∆TAl

tAl

= κ⊥∆Tλ

κNiAl∆TNiAl

tAl

= κ⊥∆Tλ

where ∆TNi, ∆TAl, ∆TNiAl, are the temperature differences across the Ni, Al, and

NiAl layers, respectively, and ∆T = ∆TNi + ∆TAl + ∆TNiAl is the overall tempera-

ture drop across the corresponding bilayer. Summing the above three equations and

substituting for the individual layer thicknesses, we get:

1

κ⊥=

βC

κNi

+αC

κAl

+1− C

κNiAl

(3.7)

32


The analysis above can be repeated for nanolaminates comprised of NiV/Al mul-

tilayers and one obtains:

κ‖ = ακAlC + βκNiVC + κNiV Al(1− C) , (3.8)

and

1

κ⊥=

βC

κNiV

+αC

κAl

+1− C

κNiV Al

. (3.9)

3.2.4 Direction and temperature dependent κ

It is well known [94–98] that the thermal conductivity may exhibit a strong de-

pendence on temperature. Consequently, we construct an extension of the direction-

dependent model of the previous section, namely by accounting for the variation of

the conductivity with temperature. The extension essentially consists in estimating

the temperature dependence of the individual conductivities appearing in (3.6–3.9),

namely Al, Ni, NiAl, Ni(93%)V(7%), and Ni(46.5%)V(3.5%) Al(50%). Below, we

seek correlations that approximate the corresponding conductivity values in a wide

temperature range that includes temperatures typically encountered in reacting front

computations, 298K ≤ T ≤ 2500K.

Temperature dependence of κAl

A comprehensive database for pure Al at temperatures both above and below the

melting point has been gathered from previous studies and scrutinized both empiri-

33


cally and theoretically by Touloukian et al. [99], who provide a table of recommended

thermal conductivity values. Expressions of κAl(T ) were obtained from best fits to

these recommended values. Since the thermal conductivity of Al exhibits a disconti-

nuity at the melting temperature, two separate fits had to be determined, one for the

solid state and another for the liquid state. Figure 3.1 depicts the data of Touloukian

et al. [99], within the temperature range of interest, along with the best fits for the

solid and liquid states, respectively:

κAl(solid)(T ) = aT 6 + bT 5 + cT 4 + dT 3 + eT 2 + fT + g T < Tm

κAl(liquid)(T ) = a′

T 4 + b′

T 3 + c′

T 2 + d′

T + e′

T ≥ Tm

(3.10)

where κ is in Wm−1K−1, Tm ≈ 933K, a ≈ 1.0496 × 10−14, b ≈ −3.7969 × 10−11,

c ≈ 5.5101 × 10−8, d ≈ −4.0679 × 10−5, e ≈ 0.015847,f ≈ −3.0421, g ≈ 460.18,

a′ ≈ −1.0232 × 10−13, b

′ ≈ 2.4885 × 10−9, c′ ≈ −2.287 × 10−5, and d

′ ≈ 0.073068,

and e′ ≈ 40.758.

Remark: Practical manufacturing processes can utilize high purity Al targets, or Al

alloy targets such as Al(1100) [95,99]. However, the amount of dopants or impurities

in the Al alloy targets is generally very small, and the thermal conductivity of de-

posited Al layers is expected to remain fairly close to that of pure Al. Consequently,

in the analysis below, we ignore potential effects of impurities or diluents on thermal

conductivity of Al, and consider exclusively the pure Al estimate (3.10) above.

34


200 500 1000 1500 2000 260080

100

120

140

160

180

200

220

240

260

Temperature (K)

Th

erm

al C

on

du

ctivity (

W/m

/K)

Touloukien data

Fit

Melting Temp.

Al

Figure 3.1: Thermal conductivity of pure Al as a function of temperature. Shown aredata from Touloukian et al. [99], along with the two best fits for the solid and liquidstates of Al. The melting temperature indicated in the plot is 933K approximately;R2 ≈ 0.9992 for the fit in the solid state, whereas R2 ≈ 0.9998 for the one in theliquid state region.

35


Temperature dependence of κNi and κNiV

Thermal conductivity values for pure Ni as a function of temperature were ob-

tained from [94]. The value of κNi at the Curie temperature, which was not given

in [94], was adapted from [99]. A best fit to the resulting data set was obtained, and

used to construct a suitable expression for κNi(T ). Two separate branches were estab-

lished in order to correctly capture the behavior of κNi(T ) below and above the Curie

temperature. The best fit for κNi(T ) below the Curie temperature, TC = 631K, is a

3rd order polynomial, while that for temperatures above TC is linear:

κNi(T ) = aT 3 + bT 2 + cT + d T < TC

κNi(T ) = a′

T + b′

T ≥ TC

(3.11)

where κ is in Wm−1K−1, a ≈ −3.4997 × 10−7, b ≈ 5.7741 × 10−4, c ≈ −0.38236,

d ≈ 162.93, a′ ≈ 0.021563, and b

′ ≈ 50.2632. The original data along with the

fits are shown in Fig. 3.2. Note that available experimental data for the thermal

conductivity of Ni does not extend beyond the Ni melting temperature, TNim = 1728K.

Consequently, the correlation (3.11) for κNi(T ), T ≥ TC , has simply been extrapolated

to temperatures exceeding the Ni melting temperature.

In contrast to pure Ni, experimental values of the thermal conductivity of Ni(93%)V(7%)

are generally lacking. Consequently, an approximate κNiV (T ) relationship was con-

structed, as outlined below.

36


200 500 1000 1500 2000 260060

65

70

75

80

85

90

95

100

105

110

Temperature (K)

Th

erm

al C

on

du

ctivity (

W/m

/K)

CRC data

Fit

Curie Temp.

Ni!Pure

Figure 3.2: Thermal conductivity of pure Ni as a function of temperature. Shownare the original data reported in [94] along with the best two fits for the data belowand above the Ni Curie temperature, TC ≈ 631K. The value of κNi at TC has beenobtained from [99]. R2 ≈ 0.9997 for T < TC , whereas R

2 ≈ 0.9999 for T ≥ TC .

37


We start by estimating the thermal conductivity of NiV at room temperature. To

this end, we rely on measured values of the thermal conductivity at room temperature

of the NiV/Al multilayers in the normal direction; these range between 35 and 50

Wm−1K−1 [100]. Using (3.9), taking k⊥ to be the mean of the reported experimental

range (≈ 42.5Wm−1K−1), substituting for the corresponding value of κAl(298K) ≈

237.05Wm−1K−1, and neglecting the contribution of the thin NiVAl premix region i.e.

setting C = 1, one deduces an approximate value for κNiV (298K) ≈ 18.93 Wm−1K−1.

Using the room temperature value above, an approximate κNiV (T ) is then con-

structed by assuming that the temperature dependence of the thermal conductivity

of NiV is similar to that of pure Ni above the Curie temperature. This results in:

κNiV (T ) = aT + b [Wm−1K−1] (3.12)

where a ≈ 0.021563, and b ≈ 12.5072. This approximation appears to be reasonable

because the Curie temperature of Ni(93%)V(7%) is lower than 298K [101–103]. An

additional rationalization is based on the observation that the thermal conductivity

of Ni(90%)Cr(10%) exhibits a linear dependence on temperature with a slope close

to that of pure Ni [99] above TC .

Temperature dependence of κNiAl and κNiV Al

Relationships for κNiAl(T ) and κNiV Al(T ) were constructed based on an experi-

mental study of Terada et al. [97], which reports measurements on the variation of

thermal conductivity with temperature for different NiAl alloys. (See Fig. 5 in [97].)

38


Note that the experimental data in [97] does not include measurements for the exact

composition commonly encountered in vapor-deposited NiV/Al multilayers, namely

Ni(46.5%)V(3.5%)Al(50%) [29]. Consequently, we have relied on measurements re-

ported for a similar composition, namely Ni(48%)V(2%)Al(50%), which appears suit-

able given that the small difference in V content is not expected to have a substantial

impact on the thermal conductivity of the alloy.

The relationships for κNiAl(T ) and κNiV Al(T ) were obtained as best fits to the

data [97]; they are respectively expressed as:

κNiAl(T ) = aT 4 + bT 3 + cT 2 + dT + e T ≤ 1100K

κNiV Al(T ) = a′

T 4 + b′

T 3 + c′

T 2 + d′

T + e′

T ≤ 1100K

(3.13)

where κ is in Wm−1K−1, a ≈ 1.8283× 10−10, b ≈ −4.6435× 10−7, c ≈ 3.6225× 10−4,

d ≈ −0.066, e ≈ 90.4, a′ ≈ −3.5907 × 10−10, b

′ ≈ 1.2284 × 10−6, c′ ≈ −0.0016,

d′ ≈ 0.9019, and e

′ ≈ −110.4937.

The experimental data from [97] along with the best fits for NiAl and NiVAl are

shown in Figs. 3.3 and 3.4, respectively. As can be noticed from the figures, the

thermal conductivity fits do not extend over the whole temperature range that is

of interest in reacting front computations, since the measurements reported in [97]

were limited to about 1200K, and the polynomial expressions above do not provide

suitable estimates of κ beyond this range. However, as noted by Terada et al. [97],

the variation of the thermal conductivity of NiAl and NiVAl becomes insignificant

39


at high temperatures. Therefore, the correlations reported in (3.13) are limited to

T ≤ 1100 K, and the thermal conductivities of κNiAl and κNiV Al for T > 1100 K, are

approximated by the corresponding values at T = 1100 K.

Remark: As for Ni, available experimental measurements of the thermal conductivities

of NiAl and NiVAl do not extend above the corresponding melting temperatures.

Potential variations in κ at or above the melting temperature have consequently been

ignored.

Summary

Having obtained the necessary expressions of κ(T ) for the various compounds,

one can incorporate the effects of direction and temperature dependence simply by

substituting these expressions into(3.6)–(3.9). Figure 3.5 shows the estimated values

of the thermal conductivity in Ni/Al multilayers, whereas Fig. 3.6 shows estimated κ

values for NiV/Al multilayers. In both cases, the estimate corresponding to a constant

thermal conductivity approximation is also depicted.

40


200 300 400 500 600 700 800 900 1000 1100 120088

90

92

94

96

98

100

102

104

106

108

Temperature (K)

Th

erm

al C

on

du

ctivity (

W/m

/K)

Terada data

Fit

NiAl

Figure 3.3: Thermal conductivity of stoichiometric NiAl as a function of temperature.Shown are the data reported by Terada et al. [97] along with the best fit for the data.R2 ≈ 0.9955.

41


200 300 400 500 600 700 800 900 1000 110010

20

30

40

50

60

70

80

90

100

Temperature (K)

Therm

al C

onductivity (

W/m

/K)

Terada data

Fit

NiVAl

Figure 3.4: Thermal conductivity of Ni(48%)V(2%)Al(50%) as a function of temper-ature. Shown are the data reported by Terada et al. [97] along with the best fit forthe data. R2 ≈ 0.9993.

42


298 600 1000 1400 1800 2200 250080

90

100

110

120

130

140

150

160

170

180

Temperature (K)

Therm

al C

onductivity (

W/m

/K)

parallel direction

normal direction

C = 0

C = 0.4

C = 1

C = 0.2

C = 0.7

Figure 3.5: Dependence of κ on temperature, direction, and concentration. The solidand dashed line plots correspond to thermal conductivity of pure Ni/Al multilayersalong the axial (x and z) and normal (y) directions, evaluated using (3.6) and (3.7)respectively. Curves are generated for different values of C, as indicated. The (N)symbol refers to the value of κ for the constant conductivity model computed using(3.2).

43


298 600 1000 1400 1800 2200 250040

60

80

100

120

140

160

Temperature (K)

Therm

al C

onductivity (

W/m

/K)

parallel direction

normal direction

C = 0.4

C = 0.2

C = 0

C = 0.7

C = 1

Figure 3.6: Dependence of κ on temperature, direction, and concentration. Thesolid and dashed line plots correspond to thermal conductivity of NiV/Al multilayersalong the axial (x and z) and normal (y) directions, evaluated using (3.8) and (3.9)respectively. Curves are generated for different values of C, as indicated. The (N)symbol refers to the value of κ for the constant conductivity model, computed using(3.3).

44


3.3 Results

In this section, we provide a comparative analysis of results obtained using the

different thermal conductivity models outlined above, for both Ni/Al and NiV/Al

multilayers. We base our analysis on three main characteristics of the flame front,

namely (i) the average thermal width, (ii) the average reaction width, and (iii) the

average axial velocity. We start first by defining each of these three properties and

the means by which they are calculated. We then move on to presenting results of

the quasi-1D computations which involve comparing the constant, concentration de-

pendent, and concentration and temperature dependent conductivity models. This

analysis is then extended to normally-propagating fronts [41], based on performing

two sets of nominally 2D computations: one for a flame propagating along the layers

(x-direction), and another for a flame propagating normal to the layers (y-direction).

We restrict this to (i) the concentration and direction dependent, and (ii) concentra-

tion, direction and temperature dependent κ models, since the other two models do

not include direction dependence and are thus insensitive to propagation direction.

Finally, we present preliminary results from 3D computations using the direction and

temperature dependent κ models which show unsteady propagation dynamics that

are reminiscent of recent experimental observations [85, 104].

45


3.3.1 Front Properties

The structure of quasi-1D reaction fronts is characterized in terms of the thermal

and reaction widths. Letting x denote the propagation direction, the thermal width

of the front is defined in terms of the gradient of the temperature profile. Specifically,

the gradient profile is first computed, and the standard deviation, σT , of the resulting

profile is used as a measure of the thermal width. The procedure is based on first

defining the thermal mean, xT , according to:

xT =

∫

x∂T

∂xdx

∫

∂T

∂xdx

(3.14)

and then computing σT from:

σ2T =

∫

(x− xT )2∂T

∂xdx

∫

∂T

∂xdx

(3.15)

The reaction width is defined in a similar fashion. Specifically, we first compute the

reaction mean,

xR =

∫

x∂Q

∂tdx

∫

∂Q

∂tdx

(3.16)

where ∂Q/∂t is the local heat release term, and then deduce the reaction width, σR,

from:

σ2R =

∫

(x− xR)2∂Q

∂tdx

∫

∂Q

∂tdx

(3.17)

Since in some cases the front propagates in an unsteady fashion, in the results we re-

port averages (arithmetic means) taken at four separated time instants, corresponding

46


to flame positions that are appreciably away from the domain boundaries.

As discussed in [31], the average axial front velocity is simply deduced by fitting

a linear curve to (xR(t), t) data pairs.

3.3.2 Axial Front Structure

In this subsection, we focus on a quasi-1D geometry, and contrast predictions

obtained using three thermal conductivity models. The analysis covers a wide range

of bilayer thicknesses, specifically δ = 12, 24, 48, 72, 120, and 300 nm, and in addition,

contrasts results for Ni/Al and NiV/Al multilayers.

Figure 3.7 shows σT plotted against δ, for the constant, concentration dependent,

and temperature dependent κ-models. Shown are results for both Ni/Al multilay-

ers and NiV/Al multilayers. It can be noticed that for small bilayers, δ ≤ 48nm,

the predictions of the different models are very close to one another, but flare out

slightly as δ increases. In the range considered, the σT versus δ curves appear es-

sentially linear, and the predicted thermal widths for NiV/Al are consistently lower

than those for Ni/Al. This appears consistent with the observation that the overall

thermal conductivity values of the NiV/Al multilayers are smaller than those of Ni/Al

multilayers.

Figure 3.8 shows predicted values of σR plotted against δ. Shown are results

obtained using the constant, concentration dependent, and temperature dependent

κ-models, for both Ni/Al and NiV/Al multilayers. As for σT , the reaction width also

47


12 24 48 72 120 200 3000

20

40

60

80

100

120

140

! (nm)

"T (µ

m)

#!Constant (Ni/Al)

#!Concentration (Ni/Al)

#!Concentration+Temperature (Ni/Al)

#!Constant (NiV/Al)

#!Concentration (NiV/Al)

#!Concentration+Temperature (NiV/Al)

12 24 48 720

5

10

15

20

25

30

35

Figure 3.7: Thermal width of the front versus δ. Curves are generated for Ni/Al (solid)and NiV/Al (dashed) multilayers, using the constant, concentration-dependent, andconcentration and temperature dependent κ models, as indicated. Inset shows a zoomin on the region of δ = 12− 72 nm.

48


exhibits an essentially linear dependence on δ, and lower values of σR are predicted for

NiV/Al multilayers than for Ni/Al multilayers. In the present case, however, the re-

action widths for the temperature dependent κ-model appear to be smaller than those

for the constant and concentration dependent κ-models whose values are comparable.

In addition, differences between the outputs of the different models becomes apparent

at δ = 24 nm, i.e. at smaller bilayers than for σT . Absolute differences become more

pronounced as δ increases, though the ratio of σR for the constant κ-model to that

of the temperature dependent κ-model retains a constant value of approximately 1.4

throughout the range of δ’s considered. The same trend is observed for both the

Ni/Al and NiV/Al multilayers. It is also interesting to note that for NiV/Al multi-

layers having small bilayer thickness, the time resolved TEM measurements of Kim

et al. [49] reveal a very sharp concentration contrast. Specifically, for a 2:3 Al/NiV

multilayer with 25nm bilayers, the TEM micrographs suggest a transition between

unmixed and essentially mixed states on the order of microns. This appears to be

consistent with the σR predictions for the smallest values of δ considered.

Figure 3.9 shows the average axial flame velocity as a function of δ, on a log-

log scale, for the Ni/Al and NiV/Al multilayers using the same three κ-models. In

accordance with previous theoretical, numerical, and experimental observations [29,

31,41,42,76,93] in the present cases the average flame velocity decreases as δ increases.

This is expected since we have focused on a multilayer regime where δ ≫ w, i.e. the

bilayer thickness is substantially larger than the premix width. Consistent with prior

49


12 24 48 72 120 200 3000

5

10

15

20

25

30

35

! (nm)

"R

(µ

m)

#!Constant (Ni/Al)

#!Concentration (Ni/Al)

#!Concentration+Temperature (Ni/Al)

#!Constant (NiV/Al)

#!Concentration (NiV/Al)

#!Concentration+Temperature (NiV/Al)

12 24 48 720

2

4

6

8

Figure 3.8: Reaction width of the front versus δ. Curves are generated forNi/Al (solid) and NiV/Al (dashed) multilayers, using the constant, concentration-dependent, and concentration and temperature dependent κ models, as indicated.Inset shows a zoom in on the region of δ = 12− 72 nm.

50


observations, for the same thermal conductivity model, the predicted average axial

front velocities for NiV/Al multilayer are smaller than the corresponding values for

Ni/Al multilayers. It is also interesting to note that, by virtue of the logarithmic

scaling in Fig. 3.9, the results indicate that whereas the relative differences between

the predictions of the different thermal conductivity models are similar in the entire

range of bilayers considered, the absolute differences become smaller as δ increases.

Figure 3.9 also indicates that the temperature-dependent model leads to ve-

locity predictions that are lower than experimental measurements at the same bi-

layer [29, 76], though it employs the more elaborate representation of the thermal

conductivity. This suggests that the semi-empirical constants describing the depen-

dence of the atomic diffusivity on temperature, see Eq. (2.3), would generally need

to be re-calibrated. Results of such an analysis will be reported elsewhere.

Additional insight into the behavior of the average front velocity can be gained

by examination of Fig. 3.10, which depicts the average axial flame velocity versus

σR, for both Ni/Al and NiV/Al multilayers. The same data are used as in Figs. 3.8

and 3.9. The results appear to be in agreement with simplified theoretical analyses [42,

92, 93], which also predict an inverse proportionality between mean axial velocity

and reaction width. Consistent with earlier observations, at fixed σR, the velocity

predictions for NiAl multilayers are generally higher than for NiVAl multilayers. For

fixed composition, the predictions of the constant κ model are slightly larger than

those of the concentration dependent model, with a larger drop observed as the effects

51


12 24 48 72 120 3000.2

0.3

0.4

0.6

0.9

1.3

2.0

3

4.0

5.5

7.0

! (nm)

Va

vg (

m/s

)

"!Constant (Ni/Al)

"!Concentration (Ni/Al)

"!Concentration+Temperature (Ni/Al)

"!Constant (NiV/Al)

"!Concentration (NiV/Al)

"!Concentration+Temperature (NiV/Al)

Figure 3.9: Average, 1D, axial flame velocity versus δ. Curves are generated for Ni/Al(solid) and NiV/Al (dashed) multilayers using the constant, concentration-dependent,and concentration and temperature dependent κ models, as indicated. In all cases,w = 0.8 nm.

52


of temperature dependence are incorporated. The results also indicate that though

the front velocity is strongly affected by the reaction width, the effects of composition

and thermal transport play an important role as well. This can be appreciated by

the relative spread among the curves depicted in the figure.

An additional means of analyzing the quasi-1D flame structure consists in plot-

ting profiles of the concentration, C, against temperature, T . This is particularly

useful in identifying the expected degree of mixing at a given temperature, and con-

sequently the particular phase or regime at which the reaction is locally taking place.

Figure 3.11 shows the normalized average scalar concentration, C/C0, as a function

of the normalized temperature T/Tf , where C0 = 1 − w/δ is the initial average

concentration and Tf is the adiabatic flame temperature. Curves are generated for

Ni/Al multilayers for all values of δ considered in the analysis, using the constant,

concentration-dependent, and temperature-dependent κ-models. (Results obtained

for NiV/Al multilayers exhibited similar trends and are omitted.) Remarkably, the

curves obtained for all values of δ appear to essentially collapse onto each other. This

illustrates that, for the fairly wide range of bilayers considered in the analysis, from

the perspective of the present simplified formalism, the dynamics of adiabatic self-

propagating fronts sample a very small region in concentration-temperature space,

and that there are very small differences between results obtained for different bilay-

ers.

The present predictions thus have important implications concerning the design of

53


0.9 1.3 2 3 5 7 10 15 20 30 400.2

0.3

0.4

0.6

0.9

1.3

2

3

4

5.5

7

!R

(µm)

Va

vg (

m/s

)

"!Constant (Ni/Al)

"!Cocentration (Ni/Al)

"!Concentration+Temperature (Ni/Al)

"!Constant (NiV/Al)

"!Cocentration (NiV/Al)

"!Concentration+Temperature (NiV/Al)

Figure 3.10: Average, 1D, axial flame velocity versus σR. Curves are generated forNi/Al (solid) and NiV/Al (dashed) multilayers, using the constant, concentration-dependent, and concentration and temperature dependent κ models, as indicated.The same data points as in Figs. (3.8) and (3.9) are used.

54


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T/Tf

C/C

0

!= 300 nm

!= 120

!= 72

!= 48

!= 24

!= 12"!Constant

"!Concentration

"!Concentration+Temperature

Figure 3.11: Normalized average scalar concentration, C/C0 versus the normalizedtemperature T/Tf . The scatter plot depicts results for Ni/Al multilayers, obtained fordifferent values of δ using the constant, concentration-dependent, and concentrationand temperature-dependent κ models, as indicated. Note that the temperature bandobserved on the C = 0 axis includes superadiabatic overshoots that are associatedwith transient front propagation [45].

55


physical or computational experiments aiming at constructing simplified or reduced

models of mixing and heat release rates. Ideally, such models would provide predic-

tions for all possible values of concentration and temperature, or at least in a suitably

wide subset of the normalized concentration / normalized temperature unit square.

In this regards, the present predictions indicate that experiments on self-propagating

adiabatic fronts are not likely to sufficiently sample the resulting phase space, and

that repeated experiments with different bilayers may not necessarily shed additional

light on global mixing rates. Consequently, one should carefully consider scenarios

in which heat losses may play an important role, and/or involving different reaction

regimes. Formation reactions under spatially homogeneous conditions [31, 53, 105]

appear to be well suited for this purpose.

3.3.3 Normally-Propagating Fronts

In this section, we briefly examine the dynamics of normally propagating fronts.

Two-dimensional calculations are used for this purpose, based on small aspect ratio

domains in which the larger dimension is normal to the axis of the planes of the

bilayers. We restrict our attention to Ni/Al multilayers, and to bilayers in the range

24 nm≤ δ ≤120 nm. For this setup and front propagation regime, it is anticipated

that the effects of the directional dependence of the thermal conductivity would be

emphasized. Consequently, we will compare results from all four thermal conduc-

tivity models: the constant, concentration dependent, concentration and direction

56


dependent, and concentration, direction and temperature dependent κ-models. (In

the following, the latter is simply referred to as temperature-dependent model.) In

addition, we will compare the resulting predictions to earlier results obtained for ax-

ially propagating fronts. Figures 3.12 and 3.13 respectively show the reaction width

and the average front velocity as a function of δ. Plotted are curves obtained for each

of the four different κ models mentioned above.

We first note that for the constant κ model and the concentration-dependent κ

model, the thermal conductivity is isotropic. Consequently, in the context of the

present reduced reaction formalism, the simplified formulation cannot distinguish be-

tween normal and axial front propagation, and the predictions for the two propagation

modes must thus be identical. This is in fact verified in the computations. As a re-

sult, the curves for the constant and the concentration-dependent κ models are not

labeled, since they pertain to both axial and normal front propagation. This is not

necessarily the case for the direction and temperature-dependent thermal conductiv-

ity models, in which the effect of material layering is specifically accounted for. In

this situation, axial and normal fronts may have different structures. As shown in

Figs. 3.12 and 3.13, the differences between axial and normal propagation velocities

are most pronounced for the direction-dependent model.

Examination of the results further indicates that the reaction width predictions

for the constant and concentration-dependent κ models are very close. On the other

hand, the average front velocity for the constant κ model is systematically higher

57


24 48 60 72 90 100 1200

2

4

6

8

10

12

14

! (nm)

"R

(µ

m)

#!Constant

#!Concentration

#!Concentration+Direction(Axial)

#!Concentration+Direction+Temperature(Axial)

#!Concentration+Direction(Normal)

#!Concentration+Direction+Temperature(Normal)

Figure 3.12: Reaction width versus δ. Curves are generated for axial and normal frontpropagation in Ni/Al multilayers, using the (i) constant, (ii) concentration dependent,(iii) concentration and direction dependent, and (iv) concentration, direction andtemperature dependent κ models, as indicated.

58


24 48 72 1200.5

0.7

1

1.5

2

3

4.5

! (nm)

Vavg (

m/s

)

"!Constant

"!Concentration

"!Concentration+Direction(Axial)

"!Concentration+Direction+Temperature(Axial)

"!Concentration+Direction(Normal)

"!Concentration+Direction+Temperature(Normal)

Figure 3.13: Average front velocity versus δ. Curves are generated for axial andnormal front propagation in Ni/Al multilayers, using the (i) constant, (ii) concentra-tion dependent, (iii) concentration and direction dependent, and (iv) concentration,direction and temperature dependent κ models, as indicated.

59


than for the concentration-dependent κ model. This indicates that the velocity of the

self-propagating front is not a simple function of reaction width, and that the thermal

transport model may generally affect both the structure and the average velocity of the

reaction front. Due to the complex dependence of the thermal conductivity on local

composition and layering, and due to the substantial non-linearities of the reaction

model, it is generally not possible to establish general rules for the dependence of the

front properties on the details of the transport model. On the other hand, specific

trends may be identified, as further discussed below.

Comparing the predictions of the concentration-dependent and the concentration

and direction-dependent κ models, one notes that the resulting predictions are the

same for the case of axial front propagation. This trend is expected because for the

axial propagation regime thermal gradients normal to layering do not develop in the

presently considered configurations. Specifically, they do not arise in a quasi-1D set-

ting for axial propagation, whereas they remain negligibly small in 2D under adiabatic

conditions. Consequently, the close agreement between the axial front predictions of

these two models may be explained by noting that the thermal conductivity in the

axial direction is the same in both models.

However, for normal front propagation the predictions of the concentration-dependent

κ model, and the concentration and direction-dependent κ model are no longer iden-

tical. In this case, one observes that the reaction width and the average front velocity

both drop when the effect of material layering is accounted for. This trend may also

60


be anticipated, since overall thermal transport normal to the layers would be more

effectively limited by the layers having highest thermal resistance.

Figures 3.12 and 3.13 also indicate that accounting for variations of the thermal

conductivities with temperature leads to appreciable drop in both the reaction width

and average front velocity. Furthermore, with the temperature-dependent κ model

there is close agreement between the predicted reaction velocities for normal and axial

front propagation. A similar trend is observed for the reaction widths. Thus, for the

presently considered system and setup, the effects of layering appear to be dominated

by the effect of thermal conductivity variation with temperature.

3.3.4 3D computations

This section briefly discusses preliminary 3D computations that implement the

presently developed generalized thermal conductivity models. We consider reactive

Ni/Al multilayers with different bilayer thicknesses, namely δ = 75 nm and δ = 6 nm.

Ignition was simulated using a localized surface heat flux that is imposed over a small

time interval [76]. Adiabatic conditions are assumed at all other times and locations.

Figure 3.14 shows instantaneous distributions of the temperature at the surface

of Ni/Al multilayer having δ = 6 nm. The computations were performed using the

temperature-dependent κ model. As can be seen in the different panels, the tempera-

ture distribution exhibits the presence of fingers that extend along the reaction front.

Observations of resolved animations (not shown) of the time-dependent temperature

61


field indicate that these fingers propagate in a transverse direction with respect to

the overall progress of the reaction front.

Figure 3.15 shows snapshots of the heat release rate at the surface of Ni/Al mul-

tilayer having δ = 75 nm. The computations were performed using the direction

dependent κ-model. For t ≤ 462 µs, the reaction front appears to propagate in a

uniform, steady fashion, essentially in a cylindrical fashion away from the origin.

However at later times, the two ends of the front at the domain boundaries speed

up and overtake the tip of the front, eventually creating a cusp (t = 687 µ s). The

cusp is eventually consumed, as the ends of the front touching the boundary appear

to weaken.

It is interesting to note that the transient motions observed for Ni/Al multilayers

with thin and thick bilayers bear resemblance to the recent experimental visualizations

of McDonald et al. [85,104]. In particular, the latter have revealed the occurrence of

spin-like reaction for thin bilayers, and of cellular structures for thick bilayers. The

transient structures observed in the present 3D computations appear consistent with

these observations. Unfortunately, direct comparison with measurements is not yet

possible, due to various simplifications invoked in the present computations, particu-

larly the neglect of heat losses.

On the other hand, it is interesting to note that when a constant thermal con-

ductivity model is used, the transient structures illustrated in Figs. 3.14 and 3.15 are

no longer observed. Specifically, with a constant κ model the flame front appeared

62


Figure 3.14: Instantaneous distributions of surface temperature. The 3D compu-tations are performed for Ni/Al multilayer with δ = 6 nm, using the temperaturedependent κ model over a domain size of (Lx ×Lz × Ly) = (1 mm ×1 mm ×10µm).Ignition was simulated by imposing a surface heat flux for a short duration over asmall square region centered at 0.1

√2 mm from the origin.

63


Figure 3.15: Instantaneous surface heat release rate profiles. The 3D computationsare performed for Ni/Al multilayer with δ = 75 nm, using the direction dependent κmodel over a domain size of (Lx × Lz × Ly) = (1 mm ×1 mm ×2µm). Ignition wassimulated by imposing a surface heat flux for a short duration over a small squareregion centered at 0.1

√2 mm from the origin.

64


to propagate in a uniform and steady fashion. Thus, the present results suggest

that manifestation of transient features in self-propagating front are modulated by

thermo-diffusive phenomena. Further investigations are evidently needed in order

to determine the origin of these transient features, and to characterize how their

properties depend on mixing, thermal transport, and heat losses.

65

Chapter 4

Inference of Atomic Diffusivity

4.1 Motivation

As mentioned in chapter 2, continuum approaches to modeling reaction propaga-

tion in multilayers generally rely on a simplified, phenomenological description of the

intermixing process, namely using a scalar composition field, and for the dependence

of the heat released on composition. The evolution of the composition (or conserved

scalar) is typically described in terms of a Fickian (or quasi-Fickian) process, governed

by a temperature-dependent diffusivity. The latter is assumed to follow an Arrhe-

nius law, and the pre-exponent and activation energy parameters that appear in the

corresponding expression, are usually calibrated as best fits based on experimental

measurements of the velocity of self-propagating fronts [8, 42]. With this approach,

computational models have proven to be effective at capturing the dependence of the

66

CHAPTER 4. INFERENCE OF ATOMIC DIFFUSIVITY

front velocity on composition and microstructural parameters.

However, recent experiments on low-temperature ignition and the subsequent evo-

lution of homogeneous reactions in multilayered materials have revealed that the

prevailing rates of intermixing are not consistent with those predicted by the global

Arrhenius fit of the atomic diffusivity. These investigations include the measurements

of Fritz [19], who examined the ignition of multilayer foils using electric currents and

hot plates, as well as nanocalorimetry experiments [106, 107] that have focused on

characterizing the evolution of an essentially homogeneous reaction within a single

bilayer. Consequently, it appears that the calibration of the atomic diffusivity based

on velocity observations would not lead to a diffusivity law that is valid throughout

the temperature range characterizing the reaction.

This chapter is motivated by a desire to address the above limitation. Specifically,

the questions we aim to address are: Would it be possible to combine the information

gained from low-temperature ignition experiments and from nanocalorimetry mea-

surements in order to refine the phenomenological description of intermixing rates

in continuum models? Provided this is possible, would models using the resulting

diffusivity law be able to simultaneously capture the evolution of homogeneous reac-

tions at low and intermediate temperatures, as well as the velocity of self-propagating

fronts?

Our approach towards these objectives is based on combining all sources of avail-

able information. We focus our attention on the extended reduced model formalism

67


discussed in chapter 3. To guide the model development, molecular dynamics com-

putations are performed in order to gain insight into the dependence of diffusivity on

temperature. Our approach for the microscale investigation is based on generalizing

the mixing measure formalism introduced by Rizzi et al. [91] for isothermal systems

to adiabatic conditions. Guided by the results of the MD analysis, we then carry

out the construction and implementation of a new composite diffusivity law based on

information gained from macroscale experimental measurements.

4.2 Atomistic Simulations

Following [91], MD simulations have been performed using LAMMPS [108]. Unless

otherwise noted, a single Ni/Al bilayer with a total thickness λ ≈ 8 nm, and δ ≈

2.34 nm is simulated by distributing Ni and Al atoms in a rectangular domain of size

≈ 53 A× 81 A × 53 A in the x, y, and z directions, respectively. Periodic boundary

conditions are imposed in all three directions in order to simulate intermixing in an

infinitely long (in the x and z directions) bilayer that periodically repeats along the

y axis. The atoms are initialized in an FCC lattice structure with the Al atoms

occupying the region 0 ≤ y ≤ 46.9 A and a lattice spacing of ≈ 4.07 A, and with the

Ni atoms occupying the region 48.8 ≤ y ≤ 78.8 A and a lattice spacing of ≈ 3.53 A.

The total number of atoms used in this arrangement is 16212, of which 8112 atoms

are Al, thus leading to a Ni/Al atomic ratio that is close to 1:1. A snapshot of the

68


initial system configuration is shown in Figure 4.1.

The atomic interaction potential used in the MD computations is based on the

embedded atom method (EAM) [109,110], where we specifically implement the Ni/Al

EAM/alloy potential developed by Mishin [111] and compiled by Becker et al. [112]

into a tabulated format that can readily input into LAMMPS.

The atoms are assigned initial velocities using a random Gaussian distribution

corresponding to a temperature T = 300 K, and the equations of motion are inte-

grated thereafter in three main stages: an equilibration stage, a heating stage, and

an adiabatic stage. A time step ∆t = 0.005 ps is used throughout the computations.

The first stage allows the whole system to relax to equilibrium at the desired ini-

tial conditions, and is run for a sufficiently long time using an isobaric-isothermal

NPT ensemble with pressure and temperature set to P = 0 bar and T = 300 K,

respectively. This is followed by a heating stage, also using an NPT ensemble, during

which the entire system is homogeneously heated from T = 300 K to an average

temperature T = 700 K at a rate of 0.8 K/psec. The heating stage brings the system

to a temperature that is sufficiently high for mixing to initiate within a reasonable

time-scale during the final adiabatic stage, but at the same time, still sufficiently low

so that almost no mixing occurs during the heating process. The final stage involves

integrating the system under constant pressure, constant enthalpy conditions (NPH

ensemble) with the pressure set again to P = 0 bar. During this stage, mixing of Ni

and Al atoms occurs, which is accompanied by an increase in the global average tem-

69


Figure 4.1: Snapshot of the initial configuration of Ni (white) and Al (green) atomsin the MD system at time t = 0. The arrangement corresponds to a Ni/Al bilayer oftotal thickness λ = 8 nm and δ = 2.34 nm.

70


perature of the closed system due to the exothermic nature of the mixing process. The

integration is carried out until a state of complete mixing is achieved which, for the

given system size, required around 4.4 million NPH integration steps. In both NPT

(NPH) ensembles, the temperature (enthalpy) and pressure of the system are main-

tained close to the desired mean values using the Nose-Hoover formalism [113–115]

with a relaxation timescale of 2 ps.

4.2.1 Coarse graining

The output of the adiabatic MD simulation runs allows us to monitor the evolution

of the global average kinetic temperature of the system, along with the evolution of

the instantaneous position of each atom. However, the parameter that we are seeking

to extract is D(T ), which is an effective diffusivity parameter that does not depend on

space nor on the particle identity. Consequently, this requires us to perform a coarse-

graining step in order to estimate D(T ). For this purpose, we adopt the mixing

measure formalism developed by Rizzi et al. [91], which relies on first calculating

the instantaneous cumulative distribution function (CDF) of each atom type, defined

according to:

Fi(y0, t) ≡ ni(y ≤ y0; t)

Ni

, i = Ni, Al, (4.1)

where ni is the number of atoms of type i whose position in the y-direction at time

t lies in the range 0 ≤ y ≤ y0, with 0 ≤ y0 ≤ λ. Ni is the total number of atoms of

type i, which is a constant for our NPT and NPH ensemble simulations. The Ni and

71


Al CDFs are then used to define an instantaneous mixing measure, M , such that:

M(t) ≡ 1

2

∫ λ

0

∣

∣

∣

∣

∂FNi

∂y− 1

λ

∣

∣

∣

∣

+

∣

∣

∣

∣

∂FAl

∂y− 1

λ

∣

∣

∣

∣

dy (4.2)

where ∂FNi/∂y and ∂FAl/∂y are the instantaneous probability distributions of the

Ni and Al atoms respectively, and 1/λ is the probability distribution corresponding

to a homogeneous, ideally-mixed system. In our MD simulations, the system starts

out in an unmixed state corresponding to compositionally pure layers, and eventually

relaxes towards equilibrium given by a fully-mixed state. Thus, according to the

definition in (4.2), M(t) evolves from an initial value of M(t = 0) = 1 to a value of

M = 0 when complete mixing is achieved.

The above coarse-grained formalism allows us to describe the evolution of the

microscopic dynamics using a measure, M , that is equivalent to the continuum average

scalar concentration, C. Consequently, through a direct mapping of M(t) to the

continuum concentration evolution equation given by equation (2.1), one can infer

the dependence of the diffusivity, D, on temperature. Details of the mapping and

inference procedures are outlined below.

4.2.2 Extracting D(T )

It has been shown in [31] that by (i) assuming that atomic diffusion is dominant

across the layers (in the y-direction), and (ii) exploiting the separation of length and

72


time scales over which atomic and thermal diffusion occur, Eq. (2.1) simplifies to:

∂C

∂t= D(T )

∂2C

∂y2(4.3)

and can be recast into approximate canonical form:

∂C

∂τ=

∂2C

∂ξ2(4.4)

where ξ ≡ y/δ is a normalized spatial variable, and τ is the normalized stretched

time variable defined in Eq. (2.10). For an initially unmixed system, the solution of

the canonical equation (4.4) can be expressed in terms of a Fourier-sine series of the

form [31]:

C(ξ, τ) =n=∞∑

n=1

2

anexp

(

−a2nτ)

sin (anξ) (4.5)

where

an =(2n− 1)π

2

Using equation (4.5), we can obtain the first moment of the concentration, C(τ),

according to:

C(τ) =

∫ 1

0

C(ξ, τ) dξ =1

δ

∫ δ

0

C(y, τ) dy

=

n=∞∑

n=1

2

a2nexp

(

−a2nτ)

(4.6)

As mentioned previously, our aim is to first map M(t) from MD to the concentration

evolution equation, which has now been simplified into equation (4.3). Consequently,

we need to transform the evolution equation for C into an evolution equation for M .

73


In order to achieve this, knowing that M is a measure that is equivalent to C, we

average equation (4.3) over δ in the y-direction to obtain:

1

δ

∫ δ

0

∂C

∂tdy =

1

δ

∫ δ

0

D(T )∂2C

∂y2dy

Since C and ∂C/∂t are both continuous and bounded, we can exchange the order of

integration and the time-derivative, which yields:

∂

∂t

(

1

δ

∫ δ

0

C dy

)

=1

δ

∫ δ

0

D(T )∂2C

∂y2dy

Finally, evaluating the right hand side and noting that M(t) ≡ C(t), we get:

∂C

∂t=

∂M

∂t= − D(T )

δ

∂C

∂y

∣

∣

∣

∣

y=0

(4.7)

Differentiating equation (4.5) with respect to ξ we have:

∂C

∂y

∣

∣

∣

∣

y=0

=1

δ

∂C

∂ξ

∣

∣

∣

∣

ξ=0

=1

δ

n=∞∑

n=1

2 exp(

−a2nτ)

(4.8)

In order to complete our analysis, we still need to compute τ . To this end, we

note that since M(τ) is to be identified with C(τ), we can use Eq. (4.6), along

with our knowledge of M(t) from MD, in order to appropriately interpolate for the

corresponding value of τ . Equation (4.7) can then be directly used for our main

74


objective of extracting D(T ), namely according to:

D(T ) = − δ

∂M

∂t∂C

∂y

∣

∣

∣

∣

y=0

= − δ2

∂M

∂tn=∞∑

n=1

2 exp(

−a2nτ)

(4.9)

Note that one cannot compute τ from the definition given by Eq. (2.10), since the

diffusivity D depends on temperature which, in our NPH MD simulations, is a time

dependent variable. Moreover, D(T ) is an unknown that we are trying to estimate.

4.3 Results

In this section, we start by applying the MD computations described in section 4.2

to analyze the behavior of homogeneous reactions under adiabatic conditions, and im-

plement the formalism outlined in sections 4.2.1 and 4.2.2 in order to infer the the

atomic diffusivity, D(T ). As further discussed below, when incorporated into the re-

duced model, the MD predictions of D(T ) lead to estimates of self-propagating front

velocities that exhibit large discrepancies with experimental observations and previ-

ous computational estimates. Consequently, the qualitative trends resulting from the

MD analysis are used in conjunction with macroscale measurements in order to con-

struct a suitable composite atomic diffusivity correlation. These include observations

obtained from homogeneous current ignition experiments [19], nanocalorimetry ex-

75


periments [106, 107], and self-propagating reaction front velocity measurements [29],

which respectively inform us on the behavior of intermixing rates at low, intermediate,

and high temperatures.

4.3.1 MD analysis

Integrating the equations of motion in an NPT and NPH ensemble allows us to

predict the time evolution of the instantaneous positions of all the atoms in the system,

along with that of the average thermodynamic properties such as temperature. Using

this information in conjunction with Eqs. (4.1) and (4.2), we can obtain the time

evolution of the mixing measure M(t). To investigate the potential dependence of

the inferred diffusivity on the bilayer thickness, and consequently the suitability of

the quasi-Fickian model to describe the evolution of the mixing measure, we consider

two MD systems with δ = 2.34 nm and δ = 4.78 nm.

Figure 4.2 shows instantaneous CDF profiles obtained for δ = 2.34 nm. Plotted

are curves corresponding to the initial time, t = 0, and to a time, t = 2.2× 104 psec,

during which the system is approaching the equilibrium, fully-mixed state. Ideal

mixing, corresponding to a uniform distribution of the Al and Ni atoms, is depicted

using a dashed line. Fig. 4.3 shows the evolution of the mixing measure, M , during the

equilibration, heating, and adiabatic stages. The evolution of the average temperature

of the system is shown in Fig. 4.4. Note that by the end of the NPT heating stage

(t = 0.3×104 psec), the mixing measure has dropped by less than 3% (Fig. 4.3). Thus,

76


most of the mixing occurs during the NPH stage, which enables us to characterize

interdiffusion and heat generation rates under adiabatic conditions, in essentially the

entire range of M .

77


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y/λ

Fi(y

,t)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

y/λ

Fi(y

,t)Al

Ni

Ni

Al

(a) t = 0 psec (b) t = 2.2× 104 psec

Figure 4.2: Cumulative distribution functions (CDF) of Nickel (red) and Aluminum(blue), computed at (a) t = 0, and (b) t = 2.2 × 104 psec. The dashed line y = xcorresponds to the asymptotic limit of a completely mixed system.

78


0 0.5 1 1.5 2 2.5

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (psec)

M

Equilibration

Heating

Adiabatic

2.15 2.2 2.25 2.3

x 104

0

0.1

0.2

0.3

0.4

0.5

Figure 4.3: Mixing measure versus time for an MD system with δ = 2.34 nm. Thecurves depicts the evolution of M(t) during the initial equilibration stage, the rapidheating stage to T = 700 K, and the adiabatic stage. Inset provides an enlarged viewof the late stages of the computations, during which the Ni structure collapses.

79


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

x 104

200

400

600

800

1000

1200

1400

1600

1800

Time (psec)

Te

mp

era

ture

(K

)

2.15 2.2 2.25 2.3

x 104

1400

1500

1600

1700

1800

Adiabatic starts

Heating starts

Figure 4.4: Average temperature versus time for an MD system with δ = 2.34 nm.The curves depicts the evolution of T (t) during the initial equilibration stage, therapid heating stage to T = 700 K, and the adiabatic stage. Inset provides an enlargedview of the late stages of the computations, during which the Ni structure collapses.

80


Based on the evolutions of temperature and mixing measure, Eq. (4.9) is used

to infer the atomic diffusivity D(T ). The results are plotted in Fig. 4.5 for both

δ = 2.34 nm, and δ = 4.78 nm. Note that the inference is restricted to the NPH stage

only. As can be seen from Fig. 4.5, the results for δ = 2.34 nm and δ = 4.78 nm

are in close agreement for most of the temperature range, with small but noticeable

differences at high temperatures. Specifically, the curve for δ = 2.34 nm exhibits

a discontinuous jump as the system approaches equilibrium, whereas the curve for

δ = 4.78 nm follows a smoother rise. Overall, however, the diffusivity predictions are

weakly dependent on the selected system size.

Note that the inferred diffusivity values exhibit a scatter in the temperature range

T ≈ 770−880K. This coincides with the region in Fig. 4.4 around which a sudden dip

in the average global temperature is observed. Visualization of the MD simulations

reveals that by this time, some of the Ni atoms have diffused into the Al region, leaving

a narrow block of structured (solid) Al that is just a few atoms thick. The sudden

decrease in temperature and the scatter in the inferredD values occur around the time

that the thin Al block loses its initial structure. We also relied on visualization of the

MD computations to examine the origin of the sudden jump in the inferred D values

for δ = 2.34 nm around T = 1570K. The simulations revealed that this coincides with

the collapse of the remaining Ni structure, after which complete mixing is reached

very rapidly. This also coincides with the switch in the curvature of the curve for

the mixing measure (inset in Fig. 4.3) and the sudden rise in temperature (inset in

81


600 800 1000 1200 1400 1600 1800

10−16

10−15

10−14

10−13

10−12

10−11

10−10

10−9

10−8

10−7

Temperature (K)

D (

m2/s

)

δ = 4.78 nm

δ = 2.34 nm

Figure 4.5: Inferred diffusivity versus temperature. Plotted are curves generated forMD systems with δ = 2.34 nm (blue) and δ = 4.78 nm (black). The temperaturerange corresponds to the adiabatic stage in Figure 4.4. The solid curves correspondto approximations obtained as best fits to the inferred D(T ) values obtained forδ = 2.34 nm along three separate branches.

82


Fig. 4.4).

To verify the MD-predictions of D(T ), an approximate fit of the individual data

plotted in Fig. 4.5 was first obtained for incorporating into the continuum model.

Attention was focused on the results obtained for δ = 2.34 nm, and the MD-based

results were interpolated using three separate branches, respectively approximating

the data during the initial rapid rise, the monotonic increase after Al loses its initial

structure, and the constant cluster of data observed after the collapse of Ni. The

resulting composite fit is also depicted in Fig. 4.5. Note that in constructing the

composite fit we ignore the scatter in D(T) in the temperature range T = 770−880K.

In order to carry out our verification step, and check whether we can recover the time

evolution of the mixing measured observed in the MD simulations (Fig. 4.3) using our

smoothed D(T) approximations, we (i) insert the fits forD(T ) into Eq. (2.12), and (ii)

integrate the resulting equation using the temperature evolution obtained from the

MD simulation (Fig. 4.4). Fig. 4.6 contrasts the resulting reduced model prediction

for M(t) with the results obtained from the MD simulations. As can be seen from

the figure, the reduced model solution using the D(T ) fits successfully reproduces

the MD-computed evolution of the mixing measure, thus lending confidence in our

inference scheme and in the resulting smoothed composite fit.

In addition to inferring D(T ), we have also relied on the MD computations to

explore the dependence of the reaction heat on the mean concentration (or mixing

measure). Recall that the original formulation of the continuum model relied on a

83


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (psec)

M

MD

Continuum

Figure 4.6: Mixing measure versus time for the MD system with δ = 2.34 nm, underhomogeneous, adiabatic reaction conditions. The blue curve corresponds to the MDdata shown in Figure 4.3 during the adiabatic stage, whereas the red dashed curvecorresponds to predictions using the reduced continuum model with the approximateD(T ) fits depicted in Figure 4.5. Note that the continuum model does not take intoaccount the initial premixing that had occurred in MD during the heating stage, andinstead starts from a purely unmixed state M(t = 0) = 1.

84


linear Q(C) relationship [43, 44], which was later replaced with a quadratic relation-

ship [45]. To shed light on the nature of the relation between the reaction heat and the

degree of mixing, we analyzed the evolution of the average potential energy in the MD

computations. Because the system is closed during the adiabatic NPH stage, and the

pressure is held at P ≈ 0 bar, the evolution of the potential energy of the system can

be directly associated with the heat of mixing. In Fig. 4.7, we plot the instantaneous

average potential energy of the system against the instantaneous mixing measure M

during the NPH stage of the MD simulation. As shown in the figure, the poten-

tial energy appears to follow a quadratic dependence on M at high and low values,

namely in the range M > 0.86 and M < 0.27, whereas it varies essentially linearly

with M in a large intermediate range. As discussed in [45], experimental observations

of reaction heats reveal a quadratic trend at low M , and a linear trend at high values.

Thus, the presently observed trend in the MD computations at higher values of M

is unexpected, and may be affected by the fact that, at those corresponding values,

the MD system is near the point when Al loses its initial structure. Overall however,

the MD computations indicate that over a substantially large interval, the potential

energy of the system depends linearly on M . This motivates us to re-examine the

impact of the correlation between Q and M , specifically whether a linear or quadratic

relationship would yield better agreement with macro-scale observations.

The validity of the MD-based correlation for D(T ) was examined by testing

whether the reduced model, that incorporates this correlation, is capable of correctly

85


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6.4

−6.35

−6.3

−6.25

−6.2

−6.15x 10

4

M

PE

(eV

)

Figure 4.7: Average potential energy of the MD system with δ = 2.34 nm versus themixing measure, M , during the adiabatic phase.The M values correspond to thoseshown in Fig. 4.3 during the adiabatic stage.

86


capturing average velocity of self-propagating reaction fronts. Unfortunately, all at-

tempts at directly using the atomic diffusivity correlations inferred from MD resulted

in continuum predictions that failed to correctly reproduce experimental observations,

specifically leading to computed velocities (not shown) that are several folds faster

than the measurements. Such discrepancies are a clear indication that the MD-based

correlations significantly overestimate intermixing rates. This represents a substantial

drawback for the present MD computations, especially since continuum formulations

using an empirically calibrated diffusivity law [31,41,76] can successfully capture the

velocity of self-propagating fronts, as well as their dependence on the bilayer thick-

ness. An additional drawback of the MD computations is that the predicted increase

in temperature due to the reaction, ∆Tf , exhibited in Fig. 4.4 also underestimates

the values computed based on reaction heats. Note that, even when the lower ∆Tf

predicted by MD is accounted for when using the MD D(T ) correlations, the resulting

continuum predictions still tend to markedly over-estimate the average front veloci-

ties. These shortcomings constitute a substantial hurdle facing the use of the present

MD computations for the purpose of formulating or calibrating continuum models

that can reproduce experimental observations with sufficient fidelity.

In an attempt to identify a possible source for the errors mentioned above, we have

briefly examined whether the MD-inferred diffusivity values are considerably affected

by the details of the inter-atomic potentials that govern the evolution of the system.

To this end, the inference of the diffusivity law was repeated based on MD simulations

87


that implement the more recent version of the Ni/Al EAM/alloy inter-atomic poten-

tial [116]. This, however, led to atomic diffusivity values that are even higher than

those depicted in Fig. 4.5, and consequently to even larger discrepancies between the

associated front velocity predictions and experimental observations. While the exer-

cise demonstrates that the details of the inter-atomic potential can have substantial

impact on the predicted average properties of the system, no attempt was made in

this effort at refining the MD potentials, or at developing a more sophisticated MD

model that would, for instance, account for additional factors such as the presence of

impurities, the impact of heat losses, and the presence of an initial premix region.

Considering the shortcomings of the MD inferred atomic diffusivity values, we

have opted alternatively to exploit merely the qualitative trends provided by the MD

computations regarding the variation of D with temperature. Specifically, the MD

computations reveal that the diffusivity could increase rapidly and even exhibit dis-

continuities at certain critical temperatures. Therefore in what follows, we explore

using these trends in order to formulate a generalized diffusivity law, and to infer the

corresponding parameters by relying on experimental observations of homogeneous

current ignition, nanocalorimetry, and reaction front velocities. Moreover, as men-

tioned above, based on the largely linear dependence of Q on C suggested by MD,

we look into whether a linear or quadratic relationship is a more suitable assumption

to use with the newly generalized diffusivity law.

88


4.4 Macroscale Information

4.4.1 Low Temperature Regime

In this subsection, we aim to infer atomic diffusivity values at the low temperature

end. We rely on the homogeneous current ignition experiments of Fritz [19] for this

purpose. The experiments were conducted in air using sputter-deposited (1:1) NiV/Al

multilayers with δ ≈ 56 nm and total thickness of approximately 19µm. Ignition was

triggered by driving, along the foil length, a constant current pulse with a current

density of 111 kA/m2. The current was maintained for a 5 msec interval and then

abruptly turned off. This resulted in essentially uniform heating of the foil to a

temperature of about 543 K. Following the Joule heating stage, the foil was allowed

to react freely. The temperature on the foil surface was monitored during the entire

process using an optical pyrometer with a response time of 40 µs and a temperature

sensitivity range of T = 473−1273 K. Assuming that the atomic diffusivity exhibits an

Arrhenius dependence on temperature, the corresponding values of the pre-exponent,

D0, and activation energy Ea, have been inferred numerically based on the measured

evolution of the surface temperature.

To this end, the reduced continuum model given by Eqs. (2.12)–(2.14) was imple-

mented in order to simulate the system response under homogeneous, non-adiabatic

conditions. The latter arises because the experiments were carried out in open air,

leading us to account for convective heat losses. This was modeled using a heat trans-

89


fer coefficient, which was considered to be equivalent to that of a hot horizontal flat

plate facing upwards [19]. Note that on the order of the length and time scales con-

cerned, radiative heat losses are negligible in comparison with convective heat losses,

and were therefore ignored.

The finite-difference numerical scheme developed in [74] was adapted for solving

the resulting system of differential equations. The impact of the Joule heating was

modeled using an equivalent volumetric source term, which resulted in an essentially

the same rise in the foil temperature as was observed in the experiments. In our

implementation of the reduced model, we accounted for the presence of Vanadium

in the multilayers by matching the experimentally measured reaction heats [29, 100],

namely ∆Hrxn = 1200 J/g. Accordingly, the nominal temperature change due to the

reaction is estimated using ∆Tf = ∆Hrxn/cp, where cp is the foil mean specific heat

capacity. The average concentration was initialized using C(t = 0) = 1−w/δ, with a

constant premix width of w = 0.8 nm [29,76]. Finally, based on the MD observations

reported in the previous section, we used a linear dependence of the reaction heat on

the mean concentration.

In order to determine the Arrhenius diffusivity parameters, D0 and Ea, a non-

linear gradient-free multivariate direct search optimization technique based on the

simplex algorithm was implemented. The technique aims at minimizing an objective

function given by the mean-squared error between the experimental temperature ob-

servations, Texp(t), and the temperature evolution predicted by the reduced model,

90


Tnum(t). The optimization is carried out iteratively, where a feedback loop is estab-

lished between the optimization code, which updates D0 and Ea, and the reduced

model code, which provides the temperature evolution, Tnum(t). The iterations are

carried out until the objective function converges to a desired minimum tolerance

value. The optimization procedure was constrained over temperatures T ≤ 800 K,

due to the low time resolution of the homogeneous current ignition experiments at

higher temperatures.

Figure 4.8 shows the observed experimental temperature evolution by Fritz [19]

(solid curve), along with predictions (dashed curve) using the reduced continuum

model with the optimized pre-exponential and activation energy values, D0 = 2.08×

10−7 m2/s and Ea = 92.586 kJ/mol. The impact of using a quadratic versus a

linear assumption for the heat released by the reaction on the resulting D(T ) values

is illustrated in Fig. 4.9. Note that the curves obtained using linear and quadratic

dependence are in close agreement for temperatures smaller than about 500 K, but

that they then diverge noticeably as the temperature increases. Also plotted for

comparison in Fig. 4.9 is the D(T ) correlation obtained by Fritz [19], namely using an

Arrhenius relationship with D0 = 5.58×10−9 m2/s and Ea = 78.9 kJ/mol. Noticeable

differences from the present predictions can be observed at low and high temperatures.

To gain insight into the differences between the predictions, one should first recall

that during the period of the electric discharge, the Joule heating dominates the total

energy released; this can be appreciated by the fact that the rate of the temperature

91


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035200

400

600

800

1000

1200

1400

1600

1800

2000

Time (sec)

Te

mp

era

ture

(K

)

Experimental

Reduced Model

0.005 0.01 0.015 0.02 0.025 0.03

550

600

650

700

750

800

Low Temperature

Regime

Figure 4.8: Temperature evolution with time for a homogeneous reaction regime ina NiV/Al multilayer with δ = 56 nm. The blue curve corresponds to experimentalobservations by Fritz [19], while the red dashed curve corresponds to predictions (trun-cated at T = 800 K) using the reduced continuum model with optimized pre-exponentand activation energy values, D0 = 2.08× 10−7m2/s and Ea = 92.586 kJ/mol. Insetprovides a zoom into the optimized region following the heating stage.

92


300 350 400 450 500 550 600 650 700 750 80010

−24

10−22

10−20

10−18

10−16

10−14

10−12

Temperature (K)

D (

m2/s

)

Q (C ) ∝ C

Q (C ) ∝ C 2

Q (C ) ∝ C 2

Original correlation by Fritz with

Figure 4.9: Comparison between the D(T ) correlation obtained by Fritz [19] withD0 = 5.58×10−9m2/s and Ea = 78.9 kJ/mol (red dashed curve), and those obtainedusing the reduced continuum model with optimized pre-exponent and activation en-ergy values, D0 = 2.08 × 10−7m2/s and Ea = 92.586 kJ/mol (solid black curve)and D0 = 5.176 × 10−8m2/s and Ea = 88.796 kJ/mol (green dashed curve) whenassuming either a linear or a quadratic Q(C), respectively.

93


rise drops dramatically as the current is switched off. As a result, the diffusivity esti-

mates obtained at temperatures lower than the temperature reached by the system at

the end of the electrical discharge (543 K) are effectively extrapolated values from the

range that is actually inferred by the analysis. Thus, it is only appropriate to com-

pare predictions at the higher temperatures. Focusing first on the present estimates,

one notes that with a linear dependence of Q on C, higher diffusivity estimates are

obtained than with a quadratic dependence. This is expected since, for an essentially

unmixed system, the rate of change of Q with C is smaller for a linear dependence

than it is for a quadratic dependence. Turning now our attention to the predictions

of Fritz, we first note that the analysis in [19] ignored the presence of Vanadium, and

consequently used a higher reaction heat, and used a different procedure that simply

aimed at optimizing D0 with a fixed value of the activation energy. Not surprisingly,

in the range T > 550 K, the inferred diffusivities obtained in [19] reveal lower val-

ues than the current reduced model predictions, and the nature of the discrepancy

appears to be largely due to the different reaction heats. Combined, the present expe-

riences underscore the fact that the inferred estimates may be substantially affected

by the details and fidelity of the underlying physical model.

94


4.4.2 Intermediate Temperature Regime

We now focus on inferring the atomic diffusivity in the intermediate temperature

range, relying on nanocalorimetry measurements [106] for this purpose. The experi-

ments were conducted using a nanocalorimeter comprising a stack of nanolayers de-

posited onto a silicon nitride membrane. The stack had a surface area of 0.5×6 mm2,

and was composed of a 3 nm thick Ti layer, a 50 nm thick Pt layer on the top side

of the membrane, and a single (1:1) Ni/Al bilayer with δ ≈ 15 nm embedded in be-

tween two 10 nm thick alumina layers on the bottom side. The stack was heated in

vacuum using a 20 msec capacitive discharge, and the power input into the stack was

measured every 0.01 msec by recording the instantaneous current and voltage drop.

Concurrently, the temperature on the Pt surface was monitored throughout the pro-

cess using a temperature versus resistance calibration curve. To ascertain that the

reaction was complete as a result of the 20 msec (non-constant) current pulse, the

stack was subsequently triggered multiple times using the same nominal electric dis-

charge, which resulted in no observable temperature increase beyond that caused by

the Joule heating.

In order to extract D(T ), the temperature and power input measurements pro-

vided by the nanocalorimetry experiments are used in conjunction with the thermal

model developed by Vohra et al. [107]. As explained in [107], the inferred values of

atomic diffusivity are only reliable within the sensitivity range of the nanocalorime-

try technique implemented in the experiments and where the rate of chemical heat

95


release is an increasing function of time, namely 720 ≤ T ≤ 860 K. For the sake of

completeness, a brief outline of the thermal model is given below. Further details on

the model can be found in [107].

The thermal model as developed in [107] revolves around the assumption that the

nanocalorimeter stack (including the silicon nitride membrane) can be analyzed using

a lumped model. The model exploits the construction of the device and its operating

conditions, and consequently ignores convective and conductive heat losses. Due to

the small stack thickness, however, radiative heat losses are retained. Furthermore,

since the inferred values adopted are limited to a temperature range falling below the

melting temperature of Al, melting effects need not be accounted for. Accordingly, on

the timescale of the experiment, the temperature within the nanocalorimeter stack is

governed by the following, simplified, volume-averaged energy equation:

(ρcp)s Vs

dT

dt= V ∂Q

∂t+∆φI − σ (ǫtAt + ǫbAb)

(

T 4 − T 40

)

(4.10)

where (ρcp)s and Vs respectively denote the mean specific heat and volume of the

stack. The three source terms on the right-hand-side of Eq. (4.10) correspond to

the rate of heat released by the reaction, the Joule heating, and the radiation loss.

The Joule heating term is expressed in terms of the current (I) and voltage drop

(∆φ) across the stack; both values are obtained from the experimental measurements.

The last term term represents gray body radiative heat losses, derived under the

assumption that losses from the sides of the stack are negligible compared to those

from the top surface and bottom surfaces. Here, σ is the Stefan-Boltzmann constant,

96


ǫ is the emissivity, A is the surface area of the stack, the subscripts t and b refer to the

top and bottom surfaces of the stack respectively, and T0 is the ambient temperature.

The first term on the right-hand-side of Eq. (4.10) represents the volumetric rate

of heat released by the reaction, where V refers to the volume of the Ni/Al bilayer.

In the present implementation, following the MD observations, we postulate a linear

dependence of the Q on C and thus express the specific heat release rate as:

∂Q

∂t= −ρcp∆Tf

∂C(t)

∂t(4.11)

However in this case, because there is no Vanadium present in the Ni layer, we esti-

mate ∆Tf based on the experimentally measured reaction heat ∆Hrxn ≈ 1378 J/g [42].

Note that in the original formulation of [107], Q was assumed to depend on the sec-

ond moment C2, as shown in Eq. (2.14). In particular, in inferring D(T ), Vohra et

al. [107] used the second moment of the concentration to obtain τ , which they sub-

sequently input into Eq. (2.12) to compute D(T ). On the other hand, in the present

formulation, we will instead implement Eq. (4.6) to appropriately interpolate for τ ,

which we then use in Eq. (2.12) to extract D(T ). This represents the main difference

between the approach in this study and that of Vohra et al. [107], while all other

aspects of the model and of the inference procedure remain the same.

Figure 4.10 shows the inferred atomic diffusivity as a function of temperature

obtained using the modified thermal model above. The results show that, within the

temperature range of interest, D can be closely approximated by a quadratic fit, as

depicted in the figure inset (the fit in the inset has been plotted by taking the natural

97


logarithm of the quadratic interpolant). The peculiar behavior of this local fit may

be due to potential variation of the activation energy and/or the pre-exponential

factor in the corresponding temperature range. It may also be affected by the fact

that the nanocalorimeter experiments reveal a complex heat release curve admitting

double peaks, associated with different exotherms. Of course, the representation of

such details is outside the scope of the reduced model paradigm, that attempts to

represent heat release based on the evolution of a scalar mixing measure. Despite this

limitation, the predicted decreasing trend in D for T ≥ 900 K is consistent with the

observation that in this temperature range the observed heat release rate becomes a

decreasing function of time (cf. Figure 4 in [107]).

The results also show that the present estimates, obtained using a linear depen-

dence of Q on C, are noticeably higher, by almost a factor of 2, than those obtained

in [107] using a quadratic dependence instead (not shown). This is expected as C

has a slower decay rate than C2, which would consequently require higher diffusion

rates for the same rate of temperature increase. This is also consistent with the trend

observed earlier in Fig. 4.9, which showed that using a quadratic Q(C) correlation

one obtains lower D(T ) estimates than with a linear correlation.

Figure 4.11 overlays the D(T ) predictions using the Arrhenius diffusivity param-

eters optimized in section 4.4.1 for 298 ≤ T ≤ 800 K, and the quadratic fit function

shown in Fig. 4.10 for 720 ≤ T ≤ 860 K. One can notice from the figure that the pre-

dictions of the low and intermediate temperature regimes result in fairly consistent es-

98


300 400 500 600 700 800 900 10000

0.5

1

1.5

2

2.5

3

3.5x 10

−13

Temperature (K)

D(m

2/s)

1.2 1.3 1.4

x 10−3

−31

−30.5

−30

−29.5

−29

1/T (K −1)

ln(D

)(m

2/s)

Intermediate Temperature

Regime

Figure 4.10: Inferred diffusivity, D, versus temperature, T . The estimates rely onthe experimental measurements of [106] for a nanocalorimeter incorporating a Ni/Albilayer with δ = 15 nm and a variant of the thermal model developed in [107]. Insetshows that the inferred D(T ) data in the temperature range of interest does notexhibit an Arrhenius relationship when plotted as ln(D) versus 1/T , and that ratherthe natural logarithm of a quadratic fit, shown by the red solid curve, would be moreappropriate.

99


timates of D(T ) over most of the overlap region. In particular, unlike the correlations

inferred by Fritz [19] and Vohra et al. [107], there does not appear to be appreciable

discrepancies between the present estimates inferred from homogeneous ignition and

nanocalorimetry experiments. This lends confidence in the current approaches and

underscores the importance of accounting for the energetics in a consistent fashion.

In what follows, we will incorporate the atomic diffusivity information extracted

from the homogeneous ignition and nanocalorimetry experiments in order to infer the

atomic diffusivity at high temperatures. Specifically, we will fix the behavior of D(T )

using the Arrhenius relationship with the optimized parametersD0 = 2.08×10−7 m2/s

and Ea = 92.586 kJ/mol at temperatures 298 ≤ T ≤ 724 K, the inferred quadratic

fit function for 724 ≤ T ≤ 860 K, and seek to infer an Arrhenius law at higher

temperatures based on experimental observations of the velocity of self-propagating

reaction fronts. Note that we truncate the low temperature regime curve at the point

where it intersects the intermediate temperature regime curve, so as not to introduce

a slight discontinuous decrease in D(T ) at the transition point T = 720 K.

100


300 400 500 600 700 800 90010

−24

10−22

10−20

10−18

10−16

10−14

10−12

Temperature (K)

D(m

2/s)

Low Temperature Regime

Intermediate Temperature Regime

720 725 7303.5

4

4.5

5

5.5

x 10−14

D0 = 2.08 x 10

−7 m

2/s

Ea = 92.586 kJ/mol

quadratic nanocal data fit

Figure 4.11: Combined extracted D(T ) values (on a semi-log scale) in the low andintermediate temperature regimes. The blue data points were plotted using the Ar-rhenius diffusivity parameters optimized in Figure 4.8, while the red circles wereplotted using the quadratic fit shown in the inset of Figure 4.10. Inset provides azoom near the overlap region between the D(T ) predictions (on a linear scale) usingthe outcomes of the low temperature regime optimization and that of the intermediatetemperature regime inference.

101


4.4.3 High Temperature Regime

Reactions in the self-propagating mode present a suitable avenue for simultane-

ously investigating intermixing rates in a wide temperature range. For Ni/Al multi-

layers under adiabatic conditions, the latter spans T = 298− 1912K. One means of

acquiring indirect information on the underlying intermixing rates is from measure-

ments of the average velocities of the propagating front. Traditionally, this has been

done using derived analytical expressions [8, 42, 92, 93] that relate the front velocity

to the Arrhenius parameters. However, such approaches are limited by the various

simplifying assumptions used to make the analysis tractable, which include ignoring

melting effects [47] and the variation of thermo-physical properties with temperature,

composition, and material heterogeneity [74]. In addition, analytical approaches are

typically based on the assumption that the atomic diffusivity follows an Arrhenius

law that holds over the entire range of reaction temperatures. As demonstrated by

the results above, the latter assumption is evidently not appropriate.

In this section, we explore an alternative route towards extracting atomic diffusion

information from velocity data, namely based on numerical modeling and optimiza-

tion techniques, rather than analytical expressions. To this end, we exploit veloc-

ity measurements from experiments performed by Knepper et al. [29], integrate the

knowledge gained from our low and intermediate temperature regime analyses into

the reduced continuum model, and consequently optimize for the diffusivity parame-

ters at the high temperature end. A brief description of the experiments and of the

102


numerical optimization approach is provided below.

The experiments were conducted on sputter-deposited (1:1) NiV/Al multilayered

foils, with a uniform bilayer distribution ranging from λ = 10 − 200 nm, a total

thickness of 5−36 µm, and a surface area of 1×2 cm2. The foils were placed in between

two glass slides and ignited at one end using a 30 V electric spark. Measurements

of the reaction front velocity were carried out by collecting the light emitted by the

passing front using a linear array of optical fibers that have been pressed against the

glass slides at known positions.

The reduced continuum model given by Eqs. (2.12)–(2.13) is implemented in order

to simulate a self-propagating reaction under adiabatic conditions. As has already

been shown in chapter 3, heat diffusion effects along the multilayer play a critical

role in this regime, and are thus taken into account using the generalized thermal

transport model derived in [74] (chapter 3). In particular, the impact of the presence

of Vanadium on both the reaction heat and the thermal conductivity is taken into

account. Because relatively thick multilayers are used, and in light of the fast propa-

gation time scales, heat loss effects are ignored [46]. Unless otherwise noted, we rely on

a linear correlation between Q and C, and thus implement the model in conjunction

with Eq.(4.11). However, for comparison purposes (discussed in section 4.5 below),

a limited number of experiments are also conducted using the quadratic correlation

in Eq. (2.14). The initial average concentration is set to C(t = 0) = 1− w/δ, with a

constant premix width of w = 0.8 nm [29,76]. The electric spark ignition is simulated

103


by imposing, within the computational domain, an initial temperature profile with a

given spark temperature and width, beyond which the temperature decreases linearly

to the ambient temperature, T0; see [74] (chapter 2) for additional details.

As discussed in section 4.4.2, we incorporate the information acquired on atomic

diffusion at low and intermediate temperatures by incorporating into Eq. (2.12), the

low and intermediate temperature correlations shown in Fig. 4.11. In order to infer

D(T ) at higher temperatures, we again assume an Arrhenius law for T > 860 K, and

implement the same optimization scheme employed in section 4.4.1 to optimizeD0 and

Ea. Note, however, that in this case the objective function that we aim to minimize

is given by the mean-squared error between the experimental velocity observations,

Vexp(δ), and the average flame front velocity predicted by the reduced model at each δ,

Vnum(δ). Thus, the procedure consists of (i) passing on a sample (D0, Ea) input vector

to the reduced model, (ii) solving the reduced model for each of the δ’s over which the

velocity was reported experimentally, (iii) passing on the Vnum(δ) output vector to

the optimization scheme, (iv) calculating the objective function, and (v) accordingly

updating the parameter pair (D0, Ea). Steps (i-v) are repeated iteratively until the

objective function falls below a desired minimum tolerance value, or until no further

optimization (minimization) is possible.

Figure 4.12 shows the experimental average velocity measurements as a func-

tion of δ, along with the reduced continuum model predictions using the optimized

pre-exponential and activation energy values, D0 = 2.56 × 10−6 m2/s and Ea =

104


102.2 kJ/mol for T > 860 K, concurrently with the optimized D(T ) correlations re-

ported in Fig. 4.11 at the lower temperatures. The blue dots correspond to the mean

of the experimental velocity measurements, while the error bars reflect the standard

deviation of repeated measurements from the mean value. Plotted are computational

results that were obtained using a mesh size of ∆x = 1 µm and a time step of

∆t = 10 ns, and a finer mesh with ∆x = 0.5 µm and a time step of ∆t = 5 ns.

Whereas the predictions are in close agreement with each other and with the exper-

imental results at the higher end of the of δ range, it can be noticed that results

obtained with the coarser mesh do not capture the drop in velocity at the smallest

bilayer, and tend to over-estimate the experimental observations for δ ≤ 17.72 nm

with a relative error ranging from 1% to 23%. This is consistent with mesh refine-

ment analysis in [74], where a 0.5 µm mesh was found to be necessary for accurately

resolving the reaction front properties at δ < 24 nm.

The familiar trend depicted in Fig. 4.12 for the dependence of the average velocity

on bilayer thickness has been discussed in prior studies [29, 42]. These have shown

that the average velocity initially increases as delta decreases from sufficiently large

values, due to the associated decrease in the atomic mixing time-scale. However,

as δ approaches the region where it becomes comparable to the premix thickness,

the velocity eventually reaches a maximum value, before switching to a decreasing

trend with a further reduction in δ. This behavior is associated with the appreciable

drop in the available reaction heat, and consequently in the reaction temperature, as

105


0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 602

4

6

8

10

12

14

δ (nm)

Velo

cit

y(m

/s)

Experimental

∆ x = 0.5 µm; ∆ t = 5 ns

∆ x = 1 µm; ∆ t = 10 ns

15 16 17 186.4

6.6

6.8

7

7.2

0 20 40 60 80 100 120 140 160 180

λ (nm)

Low + Intermediate +

High Temperature Regime

Figure 4.12: Average axial self-propagating flame velocities as a function of δ on thebottom axis and λ on the top axis. The blue dots and error bars correspond to experi-mental observations of Knepper et al. [29], whereas the open circles and red dots corre-spond to predictions using the reduced continuum model with optimized pre-exponentand activation energy values, D0 = 2.56 × 10−6m2/s and Ea = 102.1910 kJ/mol inthe high temperature range, concurrently with the optimized and inferred D valuesreported in Figures 4.8 and 4.10 at the lower temperatures. The open circles wereobtained using a mesh size of ∆x = 0.5 µm, whereas the red dots were obtained usinga coarser mesh of size ∆x = 1 µm. Inset shows the variation of the finer-mesh veloc-ity predictions when taking smaller δ increments around the region where a velocityplateau is observed.

106


the premix region starts to occupy a substantial fraction of the bilayer. To illustrate

these phenomena, we plot in Fig. 4.13 instantaneous temperature profiles for different

δ values. It can be seen that as δ increases, the maximum reaction temperature behind

the front increases to an asymptotic value, and that for δ = 3 nm the peak reaction

temperature is significantly smaller than that of the thicker δ’s.

Upon closer inspection of Fig. 4.12, we notice that it also reveals a plateau or

shoulder in the velocity curve, occurring between δ = 15.1 nm and 16.5 nm. As

shown more clearly by the inset in Fig. 4.12, the trend of the velocity predictions

along smaller δ increments around this region seems to exhibit a plateau in between

δ = 15.1 nm and δ = 16.37 nm, before switching back to a decreasing trend with a

further increase in δ. Even though the frequency of the current velocity measurements

is not high enough to allow detailed comparison with the numerical predictions, a

similar feature near δ = 18 nm might be present in the experimental curve. This

phenomenon again appears to be related to the variation of the maximum reaction

temperature with bilayer thickness. Specifically, it can be seen in Fig. 4.13 that,

whereas the peak reaction temperature generally increases with bilayer thickness, it

becomes essentially constant for δ ≥ 16.5 nm, i.e. when the ratio w/δ drops below

approximately 5%. Comparing the results for δ = 16.5 nm and δ = 15.1 nm, we

note that even though the diffusion time scale is larger for the former, the reaction

temperature is also higher, which acts to compensate for the drop in the intermixing

rate and leads to nearly the same front velocity.

107


0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1200

400

600

800

1000

1200

1400

1600

1800

x (mm)

Tem

pera

ture

(K

)

δ = 3.0 nm

δ = 5.3 nm

δ = 9.1 nm

δ = 12.1 nm

δ = 13.4 nm

δ = 15.1 nm

δ = 16.5 nm

δ = 17.7 nm

δ = 19.7 nm

Figure 4.13: Temperature profiles along the foil length (direction of front propagation)at the time instant t = 97.445 µs for δ values around the region where there is ashoulder (plateau) in the reduced model velocity predictions reported in Fig. 4.12.Also shown for comparison are the temperature profiles at the time instants t =55.645 µs for δ = 3 nm and δ = 5.3 nm, and t = 79.945 µs for δ = 9.1 nm.

108


To shed additional light on these phenomena, we have repeated the inference

analysis for the high-temperature (T > 860 K) diffusivity values using a premix

parameter w = 0.91 nm, while keeping the same D(T ) correlations as those for

w = 0.8 nm at the lower temperatures (see discussion below). The resulting optimal

values of the pre-exponent and activation energy, D0 = 1.91 × 10−6 m2/s and Ea =

97.103 kJ/mol, differed slightly from those obtained with w = 0.8 nm. Figure 4.14

contrasts the average velocity predictions obtained using the D(T ) correlations for

w = 0.91 nm, with those depicted in Fig. 4.12. It can be seen that with w = 0.91 nm,

the location of the velocity shoulder shifts towards a higher bilayer thickness, δ ≈

19 nm. Consistent with our previous characterization of the results obtained with

w = 0.8 nm, this shoulder also occurs when w/δ ≈ 0.05. Provided that the trends

in Fig. 4.14 can be verified experimentally, the present observations point to the

possibility of exploiting the location of the shoulder in the velocity data as a means

for indirectly characterizing the thickness of the premixed layer. This may provide

a useful addition to current approaches at estimating the thickness of the premixed

layer [8,29,34], which are mainly based on combining calorimetry measurements [32,

33] with assumed models for the composition profile and for the dependence of the

heat release on composition.

It is also interesting to note that the velocity plateau observed in the present

computations is reminiscent of a similar, yet unexplained, phenomenon seen experi-

mentally in Zr/Al systems [117, 118]. Specifically, these measurements also reveal a

109


0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 602

4

6

8

10

12

14

δ (nm)

Velo

cit

y(m

/s)

Experimental

Reduced w = 0.8 nm

Reduced w = 0.91 nm

0 20 40 60 80 100 120 140 160 180

λ (nm)

Figure 4.14: Average axial self-propagating flame velocities as a function of δ onthe bottom axis and λ on the top axis. The blue dots correspond to experimentalobservations by Knepper [29], while the open circles and red dots correspond topredictions using the reduced continuum model. The open circles correspond to theopen circle data points shown in Fig. 4.12 obtained using a premix width w = 0.8 nm.The red data points were obtained for a premix width w = 0.91 nm with a re-optimized pre-exponent and activation energy values, D0 = 1.91 × 10−6m2/s andEa = 97.103 kJ/mol in the high temperature range, concurrently with the optimizedand inferred D values reported in Figures 4.8 and 4.10 at the lower temperatures.The arrows highlight the points where a velocity plateau is exhibited in both cases,and the red line provides a guide for the eye.

110


velocity plateau that has a similar structure as that depicted in the inset of Fig. 4.12.

A more elaborate analysis of the underlying causes and influencing factors behind

the predicted Ni/Al velocity plateau, and the possible extensions of such analyses

towards the explanation of the observations in the Zr/Al multilayered systems (see

discussion in [118]) is currently underway, and will be reported as a follow-on work.

4.5 Discussion

Analysis of MD computations and experimental observations of homogeneous and

self-propagating reactions lead to the formulation of a composite D(T ) fit. The lat-

ter is plotted in Fig. 4.15, which also shows estimates of D(T ) that are directly

inferred from MD computations, and the global Arrhenius correlation originally in-

ferred using the analytical model in [42]. As discussed in section 4.4, the composite

fit combines atomic diffusivity estimates inferred from low-temperature ignition ex-

periments, nanocalorimetry experiments, as well as measurements of the velocity of

self-propagating fronts. These respectively yield information regarding the variation

of D at low, intermediate, and high temperatures.

Note that for the present system, the estimates obtained from ignition and nanocalorime-

try are consistent with each other, at least in regions where the corresponding temper-

ature ranges overlap. However, in the composite fit, the diffusivity exhibits a jump as

one moves from the intermediate temperature branch to the high temperature branch.

111


200 400 600 800 1000 1200 1400 1600 1800 200010

−30

10−25

10−20

10−15

10−10

10−5

Temperature (K)

D (

m2/s

)

MD

Low Temperature Regime

Intermediate Temperature Regime

High Temperature Regime

Original Correlation (Mann et al. 1997)

Figure 4.15: Final composite atomic diffusivity, D, values as a function of temperaturecombining results reported in Figures 4.8 – 4.12. Also shown for comparison are theD(T ) values inferred from the MD simulations reported in Figure 4.5 for δ = 2.34 nm,and the original global Arrhenius correlation obtained in [42] with D0 = 2.18 ×10−6m2/s and Ea = 137 kJ/mol .

112


This behavior is in qualitative agreement with observations made in the analysis of

the MD computations. It is interesting to note that in both the MD computations

and the composite fit inferred from macroscale observations, the rapid rise in D oc-

curs near but noticeably below the melting temperature of Al. While the occurrence

of this phenomenon is quite intuitive, additional work is clearly needed in order to

further investigate the underlying microscale mechanisms, and accordingly refine as

appropriate the inferred diffusivity values. Also note that the D(T ) values inferred

from MD are substantially larger than those corresponding to the composite fit. As

mentioned to in section 4.3, the use of diffusivity values directly inferred from MD in

the reduced continuum model would result in large over-estimates of the velocity of

self-propagating fronts. Consequently, additional refinement of the present MD sim-

ulations is clearly needed in order to enable a suitable quantitative characterization

of intermixing rates.

Motivated in large-part by the analysis of the MD predictions, we have relied on

reduced model computations based on a linear dependence of Q on C in the construc-

tion of the composite D(T ) fit. In order to lend further confidence in this approach,

the analysis of section 4.4 was repeated using a quadratic correlation, namely of the

form Q ∝ C2, and the quality of the resulting average self-propagating front velocity

predictions was assessed in terms of their capability to capture the experimental mea-

surements. This assessment (not shown) revealed that using a quadratic correlation

leads to a downward shift of the entire D(T ) composite curve, but unfortunately gave

113


rise to a larger global error in the average front velocity predictions in comparison

with the linear correlation. This suggests that a linear dependence is the more ap-

propriate form to implement in our reduced model. Nevertheless, despite the present

experience, it would still be worthwhile to explore a more elaborate fit that better ap-

proximates the qualitative trends in MD as well as experimental measurements. This

will be addressed in a follow-on work. In this context, it would also be instructive to

consider a more elaborate thermal model that also accounts for the variation of heat

capacities with temperature.

One should note that the present inference methodology is inherently impacted

by the details of the macroscale descriptions of the prevailing phenomena. Whereas

homogeneous ignition and nanocalorimetry experiments are essentially governed by

local mixing and heat release, self-propagating reaction fronts are in addition gov-

erned by thermal transport along the multilayers. In the present approach, informa-

tion gained from self-propagating reaction fronts is used to infer intermixing rates

at higher temperatures only. This restriction is motivated by the recent observation

in [74] which indicated that for self-propagating reactions evolving adiabatically, most

of the mixing occurs at higher temperatures (T ≥ 1000 K), irrespective of the value

of the bilayer thickness considered (12 nm ≤ δ ≤ 300nm). (The present experiences

are consistent with those in [74], but indicate that for very small δ’s most of the

mixing occurs at T ≥ 900 K). Thus, the velocity of the self-propagating fronts ap-

pears to be weakly sensitive to the details of the mixing rates occurring at the lower

114


temperatures. A case-in-point is the ability of previous models [31, 41, 42, 47, 76] to

capture the velocity of self-propagating fronts using a global Arrhenius correlation

for D(T ). In this context, the present experiences suggest that these approaches can

be successful in capturing the dynamics of self-propagating fronts while involving, at

the same time, substantial discrepancies in representing intermixing rates prevailing

at low temperatures. This is illustrated in Fig. 4.15, which contrasts the presently

inferred composite D(T ) curve with the original correlation from [42]. Whereas the

original correlation yields lower atomic diffusivity values across the entire temperature

range, largest relative discrepancies are observed at low and intermediate tempera-

tures. Consequently, it appears worthwhile to re-examine, in light of the present

findings, earlier computational attempts at characterizing phenomena that exhibit

high sensitivity to interdiffusion rates at low temperatures, including ignition and

shock initiation. Investigation of these mechanisms is the subject of ongoing work.

Finally, despite the various sources of uncertainty inherent in the various steps of

the analysis, one of the advantages of the present methodology is that the resulting

macroscale representation of interdiffusion rates enables us, for the first time, to

simultaneously capture experimental observations of low-temperature ignition, the

evolution of homogeneous reactions, as well as the velocity of self-propagating fronts.

This motivates further investigation of the validity of the present representations,

namely in order to reduce the impact of these uncertainties and accordingly refine

the model.

115

Chapter 5

Reactive Multilayered Particles

5.1 Motivation

As has been mentioned in the introduction, multilayers were introduced as an

alternative to reactive powders, since they offered a simplified geometry for theoretical

and experimental investigations, allowed for a better control of the reaction properties

and reaction products, and exhibited much faster self-propagating front velocities.

However, certain applications such as chemical time delays [119] or neutralization of

chemical and biological weapons [120], require that the reaction be sustained over long

durations of time or for it to have a small velocity of self-propagation. In powders,

the total reaction time can easily be reduced (though not precisely controlled) by

increasing the porosity and the size of the particles. With multilayers, on the other

hand, this could be achieved either by increasing the bilayer thickness or increasing

116

CHAPTER 5. REACTIVE MULTILAYERED PARTICLES

the foil length. The latter option is usually not practical to fabricate for large-scale

applications, and not suitable for small-scale ones. The former option also has certain

restrictions in that there exists a critical bilayer thickness beyond which rates of heat

losses start overcoming rates of heat generation, eventually causing the reaction to

quench. Consequently, front velocities that are slower than 1 m/s cannot usually be

realized in multilayered foils that are less than 100 µm in thickness.

Realizing that the lower reaction velocities in powders are partly due to the large

thermal contact resistance between the particles, Fritz et al. [121] were able to ex-

ploit this property in order to fabricate compacts of multilayered particles that have

a small propagation velocity. These consisted of NiV/Al layers deposited using DC

magnetron sputtering onto square nylon mesh substrates with 50 µm fiber diameters.

The sputtered multilayered coating was then broken into particles that matched the

size of the mesh elements (each had a width similar to the mesh diameter and were

three times as long) by bending the mesh under water. Loose compacts of these par-

ticles, with a consistent average packing density of 20% of the theoretical maximum

density, were then used in order to measure self-propagating front velocities. Compar-

ing the resulting average velocities in the particle compacts with those in continuous

multilayered foils that have similar average properties (such as bilayer thickness),

Fritz et al. were able to demonstrate that almost a two order of magnitude reduc-

tion in velocity (and as low as approximately 1 cm/s) can be attained using particle

compacts with the given particle dimensions and compact porosity.

117


Motivated by the above experiments, Sraj et al. [122] later extended the transient

reduced numerical model described in section 2.3 of chapter 2, in order to simulate

self-propagating fronts in idealized multilayered particles consisting of quasi-2D rect-

angular foils that are in thermal contact. The Ni/Al multilayered foils were assumed

to be all identical, with a bilayer thickness of λ ≈ 830 nm (δ = 250 nm) and a length

of 1 mm. The thermal conductivity of the foils was assumed to be homogeneous

and equal to the constant average given by Eq. (3.2), while thermal contact between

different particles was established using a reduced thermal conductivity value given

by the sum of the internal foil resistance and a constant thermal contact resistance

value. Moreover, the thermal contact resistance and contact areas were assumed to

be the same for all particles. The model was then used to mainly analyze the depen-

dence of the propagation velocity on the particles’ contact area and thermal contact

resistance. In accordance with the analysis of Fritz, an ignition delay was observed

as the flame crossed from one particle to the next, which consequently resulted in an

overall reduction in the average flame velocity. Furthermore, it was noticed that the

average velocity decreased with an increase in either the contact area (for zero thermal

contact resistance) or the thermal contact resistance (for a given contact area)1, and

that the impact of the thermal contact resistance was mostly prominent for contact

areas < 100 µm. Specifically, average velocity values close to those observed experi-

1Note that, for the case of perfect thermal contact, the velocity curve exhibited a monotonicallydecreasing trend as the contact area was increased. However, when a thermal contact resistancewas introduced, the trend became non-monotonic, starting with an increasing trend at low contactareas, before switching to a decreasing trend for contact areas > 100 µm.

118


mentally were attained at the smallest contact areas (10 µm) when contact resistance

(10−5 m2K/W) was accounted for.

In order to gain a more detailed understanding of the reactions occurring in mul-

tilayered particle compacts and to be able to fully explore their behavior, we need to

move from the idealized scenario presented above to simulating more realistic particle

networks. One drawback, though, is that the quasi-2D computations conducted by

Sraj et al. required a 1 µm discretizing mesh size, resulting in about 104 computa-

tional node points for each particle, which for a 3D model would lead to 107 degrees

of freedom. Moreover, for moderately thick bilayers, a reaction front would need on

the order of a msec in physical time to cross between particles with a computational

time-step of ∼ 10−7 secs. For relatively thin bilayers and large thermal contact re-

sistances, a much smaller time-step on the order of nsecs would be required, namely

to avoid numerical stability issues caused by the sudden discontinuity experienced by

the fast moving fronts at the particle interfaces. The high spatial dimensionality of

the system is further compounded by additional factors such as porosity, connectivity,

particle surface oxidation, thermal contact resistance, contact area, bilayer thickness,

particle size, particle shape, and particle orientation. As a result, using existing mod-

els for simulations of more realistic particle networks that contain at least hundreds

of particles in a 3D configuration could prove to be computationally very costly, if

not prohibitive.

One means of moderating the hurdles above would be to attempt to reduce the

119


dimensionality through spatially homogenizing the system on the level of a single par-

ticle. This would help avert the need to mesh over each particle, and thus bring about

a reduction of almost 4–7 orders of magnitude, in addition to easing the restrictions

on the size of the time-step. However, as with every averaging or filtering approach,

one also runs the risk of grossly misrepresenting the actual reaction dynamics, which

could greatly depend on a proper resolution of the small scales. A common remedy

to this problem is to upscale or coarse-grain the governing system of equations such

that the effects of the unresolved microscales on the macroscale dynamics are prop-

erly accounted for. Another simpler, though more restrictive, approach would be to

numerically identify certain regimes under which such a homogenization would be

valid or applicable. In this chapter, we will adopt this latter approach.

Numerous experimental [123–128] and numerical [128–139] studies have already

been carried out on reaction-diffusion phenomena in heterogeneous media. The ex-

perimental investigations have mainly focused on reactions occurring in either gasless

powder mixtures, or between a powder and a gas. On the other hand, the theoretical

and numerical studies have considered cases where one of the reactants is comprised of

particles or point sources embedded in a homogeneous medium of the other reactant,

or where the medium is composed of discrete reactive cells or point sources contain-

ing the homogeneously fully mixed reactants. Most of these studies have focused on

demonstrating the discrete nature of the reaction, and have tended to emphasize the

invalidity of the homogeneous [140] and the quasi-homogeneous [141] approximations

120


in such situations due to the fact that they fail to correctly capture crucial proper-

ties of the reaction propagation. In the homogeneous approximation, the reaction

kinetics depend only on a uniform distribution of the temperature and concentration

of the reactants, whereas in the quasi-homogeneous approximation, the reaction ki-

netics account for the presence of certain heterogeneities in the system using some

functional form, but are otherwise homogeneous over space. Numerical and analyt-

ical comparisons between the predictions of the discrete and quasi-homogeneous (or

homogeneous) models were examined, and certain criteria for the domains of applica-

bility of the homogenized versions were determined. These criteria are mostly based

on ratios of reaction to diffusion time-scales, ratios of reaction to heat transfer time-

scales, or ratios of length scales given by the front width (reaction and diffusion) to

the particle size or to the characteristic scale of the system’s heterogeneity.

Varma et al. [129], for instance, considered a solid-gas system of Ti-N2, which they

modeled as set of identical 2D square reaction cells with blunt edges, arranged in a

random fashion on a square lattice with a prescribed degree of porosity. Each reaction

cell contained both reactants in a fully pre-mixed state, and was in contact with its

nearest neighbors on the lattice along the flat edges (that is with a contact area

equal to half the cell’s diameter). The cells were also assumed to be immersed in a

homogeneous gaseous medium, and that both (the cells and the medium) had constant

thermo-physical properties, including a constant thermal contact resistance value at

the interface between two contacting cells. The domain was then discretized using a

121


square mesh with a size equal to half the reaction cell’s diameter, and used to solve

the coupled system of differential equations for the evolutions of temperature (melting

was not taken into account) and concentration. Only heat transfer by conduction was

considered and an effective constant thermal conductivity value, composed of a sum

of resistances in series/parallel, was used for calculating the heat fluxes. The reaction

rate was assumed to be zero below a certain fixed ignition temperature, beyond which

it proceeds at a constant fixed rate until all the reactants are consumed. A systematic

study was then conducted, analyzing the dependence of the reaction front structure

and velocity on the particle size and density. Based on these results, a parametric

map of density vs. particle size was constructed, outlining the regions of transition

between quasi-homogeneous, relay-race2, and percolation regimes. Their observations

led them to conclude that the main criterion which determines the transition between

the continuum and the discrete limit is the ratio of the reaction to the heat transfer

(or heat conduction) time-scales. Specifically, they argue [125, 128] that if trxn ≫

tconduction, then the reaction (and thermal) front would encompass several particles

such that the continuum approximation becomes justified, whereas in the opposite

limit, the width of the reaction front becomes comparable to the size of a single

particle which consequently necessitates a discrete representation of the dynamics.

Another study, based on a more basic model than that of Varma et al., was

conducted by Rogachev [130]. A quasi-1D linear chain of reaction cells was considered,

2Relay-race regime is described as a series of localized successive, but intermittent, rapid ignitions,separated by long induction (or heating) periods in between each ignition event.

122


with each cell containing both reactants in a fully mixed state. Melting events were

not taken into account, and heat transfer by conduction occurred between the reaction

cells only, while all other forms of heat transfer were ignored. The temperature inside

each cell was assumed to be uniform (i.e. temperature gradients were ignored) on

the basis that the internal thermal resistance of the solid (or reaction cell) is much

lower than the thermal contact resistance between two contacting cells. Thus, each

reaction cell was characterized by a temperature value and a degree of conversion

(or concentration) value, with the rate of heat conduction between touching cells

being proportional to the product of the temperature difference between the cells, a

fixed contact area, and fixed heat transfer coefficient. Contrary to Varma though, two

forms of the reaction rate were considered; both exhibited an Arrhenius dependence on

temperature, but one in a continuous fashion over all temperatures, and the other in a

step-like fashion such that the rate was zero below a certain ignition temperature, and

Arrhenius beyond that. The time evolutions of temperature, concentration, reaction

rate, and front velocity were analyzed as a function of a parameter representing

the ratio of the characteristic time-scales of heat transfer to reaction rate. Velocity

predictions of the numerical model for different values of the parameter were compared

to those given by Zeldovich’s analytical expression for the velocity, derived using the

homogeneous (continuum) medium approximation. It was noticed that deviations

from the homogeneous predictions became more prominent with increasing values

of the ratio of heat transfer time to reaction time, and that the propagation regime

123


depended on this ratio, along with the sensitivity of the reaction rate to the initial and

combustion (or flame) temperatures. Moreover, it was seen that for strongly activated

reactions, the front velocity is independent of the imposed ignition temperature (in

the step-wise reaction kinetics case) below a certain critical ignition value, whereas

a universal dependence was exhibited for ignition temperatures above this critical

value.

The conclusions arrived at by the above two investigations, which help sum up the

main viewpoints of most of the other studies referenced earlier, seem to emphasize

the fact that the prime factor in determining the validity of a spatially homogenized

medium concerns the ratio of the reaction time scale to the heat transfer time scale.

At first glance, it might seem as though this could also serve as the governing criterion

in our case as to whether or not we can safely spatially homogenize our system on

the level of a single particle. However, the previous approaches impose a number of

simplifying assumptions that are inconsistent with our model, and this renders their

established criterion not directly applicable to our multilayered particles, but rather

entails its re-examination for our specific case.

One source of incompatibility lies in the fact that both Rogachev and Varma

considered particles in which the reactants were initially present in a homogeneously

premixed state, thus atomic diffusion was not present as a limiting factor in their

reaction rate expressions. This is a major inconsistency with the case that we are

interested in, since the reactants in our particles exist initially as separate alternating

124


layers and we assume that the reaction rate is diffusion rather than reaction-limited.

In the case of Varma, the reaction was assumed to initiate at a specific ignition

temperature, after which it proceeds at a constant rate. Defining a fixed ignition

temperature value for the reaction cannot be accomplished in a unique manner and

is usually mostly appropriate for high activation energies, which is not a necessary

condition for us in view of the results presented in chapter 4. Whereas assuming

a constant reaction rate is synonymous to assuming that the reaction occurs under

isothermal conditions, and this seems to be at odds even with the conditions in

Varma’s study. In the case of Rogachev, an Arrhenius dependence on temperature was

assumed, but with the use of a single value for the activation energy and the diffusion

coefficient. This does not comply with our observations in chapter 4, specifically

when melting and intermediate phase formation events take place. Furthermore,

Rogachev did not include a dependence of the reaction rate on the degree of reactant

consumption, but rather imposed an abrupt discontinuous termination of the reaction.

This is particularly inconsistent with compacts comprised of layered particles.

While Varma and Rogachev, along with most of the other studies, undertook the

problem of characterizing the reaction’s heterogeneity on scales on the order of a

particle size, they ignored taking into account heterogeneities on scales smaller than

this, mainly based on the fact that the thermal resistance in the particle’s interior

is much lower than that across the contact interface. For self-propagating reactions,

especially those that take place in multilayered structures, such an assumption is not

125


always obvious, and in fact has clearly been shown to be violated for the particle sizes

examined by Sraj et al. [122]. Moreover, thermal diffusion effects within a particle

(which is different from thermal diffusion between particles) could have important

ramifications on the flame front macroscale dynamics (such as stability) as has already

been demonstrated in chapter 3. We should note though, that an exception was a

study conducted by Grinchuk et al. [138], who did consider thermal diffusion within

the particles, but apart from this, their study still suffers from some of the other

drawbacks and inconsistencies listed above.

A final point worth noting is that the criterion, put forward by the above men-

tioned works, is namely based on a comparison between the chemical reaction and

heat transfer time-scales. Whereas the latter time scale can easily be estimated a

priori, the reaction time-scale is, in most circumstances, difficult to represent using

a single constant value. This is due to the obvious fact that the reaction rate has a

non-trivial dependence on temperature, and in the case of multilayers, on the bilayer

thickness as well. As a result, the comparison cannot readily be employed as a bench-

mark for checking the validity of spatial homogenization in the presently considered

reactive media.

Given all of the above shortcomings, it seems reasonable to re-address the issue

of identifying the regimes under which a homogeneous particle approximation holds,

within the specific context, conditions, and assumptions of our fully generalized re-

duced reaction model presented in chapter 4.

126


5.2 Problem Formulation and Approach

We will start from the generalized reduced model presented in the previous chap-

ter, which consists of the following two coupled system of differential equations:

∂τ

∂t=

D(T )

δ2(5.1)

∂H

∂t= − 1

V

∫

V

(∇ · q)dV +∂Q

∂t(5.2)

where D(T ) is given by the composite fit inferred in chapter 4 (see Fig. 4.15). In

accordance with the observations and analysis of the pervious chapter, we will use a

linear dependence of the reaction heat on the mean concentration such that:

∂Q

∂t= −ρcp∆Tf

∂C(t)

∂t(5.3)

As has been remarked previously, the temperature, T , can be retrieved from the

volume-averaged enthalpy, H , by inverting the complex relationship (2.8) involving

the heats of fusion of the reactants and products [76].

Since the experimentally manufactured multilayered particles [121] consist of NiV/Al

layers, we will account for the presence of Vanadium through estimating ∆Tf based

on the experimentally measured reaction heat [29, 100], namely ∆Hrxn = 1200 J/g.

The heat flux, q, is again assumed to follow Fourier’s law. We will rely on the fully

generalized thermal transport model derived in chapter 3, which corresponds to an

anisotropic, concentration-dependent, and temperature-dependent thermal conduc-

tivity, κ, that also appropriately accounts for the presence of Vanadium.

127


As has been mentioned above, our aim is to extend this modeling formalism for

single multilayers towards exploring reactions occurring in layered particle networks.

However, because of the high dimensionality and the high computational costs associ-

ated with simulating such systems, we will attempt to further reduce the generalized

reduced model through imposing spatial homogenization on the level of a single par-

ticle, and to numerically identify the regimes under which such a homogenization

would be valid or applicable.

In our homogeneous scenario, particles have spatially uniform temperature and

concentration fields. Accordingly, we can set ∇ · q to zero and the governing system

of equations for a single multilayered particle becomes:

∂τ

∂t=

D(T )

δ2(5.4)

∂H

∂t=

∂Q

∂t= −ρcp∆Tf

∂C(t)

∂t(5.5)

Note that while Eqs. (5.1)–(5.3) correspond to evolution equations within each com-

putational cell of a single meshed multilayered particle, Eqs. (5.4)–(5.5) represent

evolution equations over the entire homogenized particle. Numerical solution of the

system of Eqs. (5.1)–(5.3) is conducted using the same scheme described in section 2.4

of chapter 2. The same technique is implemented for solving Eqs. (5.4)–(5.5), but

with each particle being the equivalent of a single computational cell.

Generalizing the above reduced meshed and homogeneous models to multiple par-

ticles, involves simply applying the same set of equations to each particle separately.

128


Following Sraj et al. [122], coupling between the particles is achieved through ther-

mal conduction between touching particles across a certain surface contact area. In

the meshed case, accounting for heat transfer across particle interfaces occurs via the

heat flux, q, but with a reduced κ value given by the sum of the internal particle

resistance (varies with time) near the interface and a specified (constant) thermal

contact resistance value, Rc. In the homogeneous case, the particles’ internal thermal

resistance is ignored and the heat flux includes only the thermal contact resistance,

Rc.

As we pointed out earlier, the high spatial dimensionality of systems involving mul-

tilayered particle networks is compounded by other complexities arising from factors

such as porosity, connectivity, particle surface oxidation, thermal contact resistance,

contact area, bilayer thickness, particle size, particle shape, and particle orientation,

in addition to the potential presence of variability in all of these properties. Thus,

to render the problem manageable, we introduce a number of simplifications by elim-

inating some of these sources of complexity. Specifically, we focus on an idealized

setup where we have a quasi-1D linear chain of rectangular multilayered particles

that are in full contact. Moreover, apart from heat transfer across the particles’ sur-

face contact areas, we assume that the reactions occur under adiabatic conditions.

This is in accordance with the vacuum conditions under which the experimental ve-

locity measurements are typically conducted, and the negligible effects of radiative

heat losses (for thick samples [46]). We also impose the condition that the first par-

129


ticle in the chain has been fully consumed (inert), and that it has an infinite supply

of heat through maintaining, across its entire length, a spatially uniform tempera-

ture given by the maximum adiabatic flame temperature of Tf = 1912 K. This inert,

continuously hot particle is employed for the purpose of initiating the reaction in the

neighboring reactive particle through heat conduction along the contact interface, and

ensuring that a self-propagating front gets established along the chain. Furthermore,

we analyze chains that are comprised of a sufficiently large number of particles in

order to obtain predictions that are essentially independent of the initiation process.

Investigating reactions occurring within rectangular particles, as opposed to spher-

ical for example, allows us to utilize a simple uniform cartesian mesh for spatially

discretizing the interior of the particles, and thus avoid the need to seek more sophis-

ticated meshing techniques or to transform our equations to a more suitable coor-

dinate system. Moreover, for homogeneous particles, we would not need to account

for additional sources of thermal resistance such as macro-constriction and spreading

resistances, which usually require elaborate, though approximate, analytical mod-

els [142–145]. On the other hand, considering the limiting case of a quasi-1D linear

chain of identical particles with full contact (no porosity), permits us to reduce the

space of parameters that we need to probe to merely three factors. In particular,

within this idealized scenario, the remaining degrees of freedom consist of the half-

layer thickness, δ, the thermal contact resistance, Rc, and the length of the particles,

Lx. After extracting the regimes of validity of the particle homogeneity assumption

130


as a function of these three parameters, or possibly in terms of some reduced di-

mensionless combination of them, the other parameters can later be systematically

introduced and their effects on the robustness of the predictions can be examined.

131


5.3 Results

In this section, we perform a comparative analysis between the meshed and

homogeneous computations in order to identify the regimes under which the ho-

mogeneous particle approximation is appropriate. Specifically, we seek to deter-

mine when we can safely replace the meshed system of equations (5.1)–(5.2) with

the particle homogenized equations (5.4)–(5.5), without sacrificing the fidelity of

the computational predictions regarding the macroscale reaction dynamics occur-

ring on scales ≥ to the size of a particle. As described in the previous section,

computations are conducted for a quasi-1D linear chain of rectangular multilay-

ered particles, in full contact, with different values of thermal contact resistance

Rc = 10−7; 3 × 10−7; 7 × 10−7; 10−6; 3 × 10−6; 7 × 10−6 and 10−5 m2K/W , particle

length L = 10; 50; 100 and 500 µm, and bilayer thickness corresponding to δ = 50

and 250 nm. Note that the values of Rc and L that we consider in this study, even

though not exhaustive, help span a wide range of possible (analytically estimated)

values that could typically be encountered in real contacting particles [143–145].

In the meshed scenario, each particle is described using the state vector(

C(x, t, n), T (x, t, n))

,

where x, t, and n represent position in the x-direction, time, and particle (ID) number

respectively, whereas in the homogeneous scenario, each particle is instead described

using the state vector(

C(t, n), T (t, n))

. Because we aim at being able to replace the

former description with the latter, we expect this approximation to become asymp-

totically valid as we approach the limit of vanishing temperature gradients within a

132


single particle. Therefore, we start first by characterizing the degree of temperature

heterogeneity in the particles using the meshed computations to see whether we can

identify some critical range or combination of Rc, δ, and L, where the temperature

variations in the interior of the particles become negligible. For this purpose, we de-

fine two measures that reflect the extent of temperature gradients within a particle,

given by:

Smax =max { Tmax(t, n)− Tmin(t, n) }

Tf − T0

(5.6)

< S(t) >=< Tmax(t, n)− Tmin(t, n) >t

Tf − T0(5.7)

where Tf and T0 refer to the adiabatic flame temperature and room (initial) tem-

perature, respectively. Expression (5.6) corresponds to the normalized peak value

of the instantaneous maximum temperature difference in a given particle, whereas

expression (5.7) corresponds to the normalized time average of Smax(t). Smax and

< S(t) > are calculated for each particle in the chain, but in our analysis we rely only

on values from particles that are appreciably away from the domain boundaries. In

all of the cases that we have analyzed, these measures become almost constant near

the center of the chain, so in our results we only report the values that correspond to

particles situated at the chain’s midpoint. Note also that, in order to avoid biasing

< S(t) > to low values, S(t) for a given particle is averaged between the times when

the reaction is about to transfer to and from the particle of interest.

133


Figures (5.1) and (5.2) show Smax and < S(t) > as a function of Rc, plotted

for different values of L and δ. It can be noticed that, for a given δ value, both

measures exhibit an overall decrease with increasing values of Rc and decreasing

values of L. This is consistent with the fact that as the thermal contact resistance

increases, the rate of thermal conduction between different particles decreases, thus

allowing more time for heat within the particles to diffuse and smoothen out the

temperature gradients. Meanwhile, for a fixed Rc, the time required for the particles

to achieve a homogeneous temperature state decreases as L decreases, causing shorter

particles to admit lower temperature gradients than longer ones. Note that Smax helps

us to quantify the maximum possible instantaneous temperature difference that can

occur within a particle for the given conditions. This mostly takes place during front

formation and propagation in the interior of the particle, provided that the particle

is long enough for a (strong) front to get established. Increasing Rc helps decrease

the maximum temperatures attained in the flame, but up to a limited extent. This

explains why Smax eventually seems to asymptote as a function of Rc. Moreover

as has been demonstrated in chapter 3, since the reaction rate in multilayers with

δ = 250 nm is slower, and thus has a larger thermal (and reaction) width than

δ = 50 nm, Smax is smaller for the former than the latter for a given L value. The

small kinks that appear in the curves in Fig. (5.1) are due to the different time instants

at which this maximum temperature difference occurs. For example, for δ = 250 nm,

L = 100 µm, and Rc = 3 × 10−7 m2K/W, the maximum occurs during the early

134


heating stages of the particle (just before a front initiates) when heat has not yet had

enough time to diffuse to the neighboring cold regions of the particle, whereas for

Rc = 7 × 10−7 m2K/W, it occurs when the front initiates. In the former case, there

is usually a slight back-flow of heat into the previous reacted particle as soon as the

front initiates due to the small Rc value, and this causes the temperature gradients

to be maximized just before the front has fully initiated and not after. Such a slight

increase in Smax with Rc happens only for certain combinations of Rc, L, and δ, and

so is not observed in all of the curves in Fig. (5.1).

On the other hand, < S(t) > is more representative of the overall average range of

instantaneous temperature gradients that occur within a certain particle around the

time of its reaction, and emphasizes more the gradients that are sustained for longer

periods of times. Consequently, it exhibits lower values than Smax, is monotonic

as a function of Rc (no kinks), and asymptotes to much lower values at high Rc.

Furthermore, the curves for the two different δ’s almost overlap, with deviations

between the two becoming more noticeable as L increases. Since δ = 250 nm has a

lower propagation velocity than δ = 50 nm, larger gradients are sustained for longer

times in thicker multilayers and sufficiently long particles. Note however, that for

small L and Rc values, δ = 250 nm has lower < S(t) > values, again due to the

slower reaction rates and larger thermal and reaction widths.

135


0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rc × 10−7 (m2K/W )

Sm

ax

δ = 50 nm; L = 10µm L = 50 L = 100 L = 500 δ = 250 nm; L = 10µm

Figure 5.1: Normalized peak values of the instantaneous maximum and minimumtemperature differences in a given particle, Smax, as a function of thermal contactresistance Rc. Plotted are curves corresponding to different values of particle size, L,and half-layer thickness, δ. Solid lines with solid dots correspond to δ = 50 nm, whiledashed lines with open circles correspond to δ = 250 nm.

136


0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Rc × 10−7 (m2K/W )

<S

(t)

>

δ = 50 nm; L = 10µm

L = 50

L = 100

L = 500

δ = 250 nm; L = 10µm

Figure 5.2: Normalized time average of the instantaneous maximum and minimumtemperature differences in a given particle, < S(t) >, as a function of thermal contactresistance Rc. Plotted are curves corresponding to different values of particle size, L,and half-layer thickness, δ. Solid lines with solid dots correspond to δ = 50 nm, whiledashed lines with open circles correspond to δ = 250 nm.

137


Considering that short-lived, large temperature variations in the particles might

not have a substantial impact on the macroscale (scales ≥ L) reaction propagation,

we expect the < S(t) > measure to be a more suitable indicator of the actual degree

of temperature heterogeneity within the particles for a given set of parameters. Thus

by examining Fig. (5.2), we postulate that particles with δ ≥ 50 nm, L ≤ 100 µm,

and Rc ≥ 30×10−7 m2K/W can be considered to be fairly homogeneous and that for

such particles, we might be able to safely replace the meshed set of equations with

the homogenized set of equations for a more suitable but still reliable description

of the reaction dynamics. Note however, that the restriction to high Rc values for

ensuring temperature homogeneity can be successively relaxed for shorter particles

(L < 100 µm), while still maintaining small errors.

In order to check our postulation, we need to quantify and characterize the error

associated with using the approximate homogeneous particle equations instead of the

meshed system of equations for describing the reaction progress between the particles.

For this purpose, we define a relative time error measure given by:

terror =tmeshed − thomogeneous

tmeshed

(5.8)

where t corresponds to the time that has passed between when 70% of the previous

particle in the chain has been consumed and when the current particle of interest has

reached a 70% consumption level, while the subscripts correspond to the meshed and

homogenized particles respectively. Similar to the heterogeneity measures above, in

our analysis we rely only on t values from particles that are appreciably away from

138


the domain boundaries, and in our results we only report the values that correspond

to particles situated at the chain’s midpoint.

Figure 5.3 shows the relative time error as a function of Rc, plotted for different

values of L and δ. First we note that the error is always positive, meaning that the

reaction transfer from one particle to the next is always slower in the meshed than in

the homogeneous case. This is consistent with the fact that thermal diffusion within

the particles is absent in the latter computations, which leads to faster (or at best

equal) transfer rates. It can also be seen that the error decreases with an increase

in Rc and a decrease in L, with the exception of L = 500 µm for δ = 50 nm, and

approaches values less than 10% for L ≤ 100 µm and Rc ≥ 70× 10−7 m2K/W. This

is also consistent with the observations in Figs. (5.1) and (5.2). As expected, the

error trends exhibited seem to be more in agreement with those of < S(t) > than of

Smax. To further investigate this, we plot in Figs. (5.4) and (5.5) the correlations of

terror with Smax and < S(t) > for the different values of Rc, L, and δ. Note that the

data points in both figures correspond to those shown in Figs. (5.1)–(5.3). As can be

noticed from Fig. (5.4), there does not seem to be a very clear correlation between

Smax and terror as the same values of Smax appear to correspond to multiple and widely

different error values. Surprisingly though, Fig. (5.5) also does not seem to portray

a clear trend of the error as a function of < S(t) >. At first glance, it might appear

as though larger errors are associated with larger temperature heterogeneities within

a particle. However upon closer inspection, we notice that there is no one-to-one

139


correspondence between terror and < S(t) >. Two pronounced indications of this are

the curves for δ = 250 nm; L = 10 µm, and δ = 50 nm; L = 500 µm, in addition to

the observation that the remaining curves overlap initially, but then start to spread

out (most noticeably beyond < S(t) >= 0.1).

In view of the above observations, there does not seem to exist a unique, general

measure based on a critical degree of temperature heterogeneity (or homogeneity)

within a particle that would provide a criterion for predicting the reliability of the

homogenized approximation. Nevertheless, the peculiar trend of the δ = 250 nm;

L = 10 µm curve in Fig. (5.5) provides a hint to another potential determining factor.

Despite the fact that the S-measure for this curve is almost negligible, thus indicating

that the particle should in principle be equivalent to its homogeneous counterpart,

the relative time error increases from 10% to about 35% in an almost vertical fashion,

irrespective of the mean value of S. Realizing that the only remaining difference

between the two scenarios (meshed and homogeneous) concerns the internal thermal

resistance of the particles, which is taken into account in the meshed equations but

ignored in the homogenized ones, we examine next the effect of the particle’s thermal

conductivity, κ, on the measured error using a non-dimensional ratio of Rc and κ.

Figure (5.6) shows the dependence of the relative time error on the non-dimensional

ratio of the particle’s internal thermal resistance to its thermal contact resistance,

L/κRc, where κ is considered to have a constant value given by the average thermal

conductivity of NiV/Al [see Eq. (3.3)]. The trends in the figure resemble those in

140


0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Rc × 10−7 (m2K/W )

t error

δ = 50 nm; L = 10µm L = 50 L = 100 L = 500 δ = 250 nm; L = 10µm

100

101

102

10−2

10−1

100

Figure 5.3: Relative time error in the reaction progress between the meshed andhomogeneous computations as a function of thermal contact resistance Rc. Plottedare curves corresponding to different values of particle size, L, and half-layer thickness,δ. Solid lines with solid dots correspond to δ = 50 nm, while dashed lines with opencircles correspond to δ = 250 nm. Inset shows the same data points plotted on alog-log scale.

141


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Smax

terror

δ = 50 nm; L = 10µm

L = 50

L = 100

L = 500

δ = 250 nm; L = 10µm

Figure 5.4: Relative time error in the reaction progress between the meshed andhomogeneous computations as a function of Smax. Shown are curves corresponding todifferent values of L, Rc, and δ. Solid lines with solid dots correspond to δ = 50 nm,while dashed lines with open circles correspond to δ = 250 nm. Both sets of datapoints correspond to those shown in Figs. (5.1) and (5.3).

142


0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

< S (t ) >

terror

δ = 50 nm; L = 10µm

L = 50

L = 100

L = 500

δ = 250 nm; L = 10µm

Figure 5.5: Relative time error in the reaction progress between the meshed and ho-mogeneous computations as a function of < S(t) >. Shown are curves correspondingto different values of L, Rc, and δ. Solid lines with solid dots correspond to δ = 50 nm,while dashed lines with open circles correspond to δ = 250 nm. Both sets of datapoints correspond to those shown in Figs. (5.2) and (5.3).

143


Fig. (5.3), but with a re-scaled Rc axis. Note however that by virtue of the re-scaling,

the combined effect of both L and Rc on the error is being emphasized. Contrary to

Fig. (5.3), there now appears to be an overlap of the curves for δ = 50 nm; L = 10 µm

with δ = 250 nm; L = 10 and 50 µm, and δ = 50 nm; L = 100 µm with δ = 250 nm;

L = 500 µm. However, as has been seen with the mean S-measure above, deviations

between the remaining curves become noticeable past L/κRc ≈ 0.2. The trend by

which the curves coincide and deviate though, suggests that the ratio of resistances

is an important factor, but is not the only determining factor. For instance, if we

simultaneously examine the dependence of terror on L/κRc and L/σT (where σT rep-

resents the thermal front width for a given δ) as shown in the 3D plot in Fig. (5.7), we

notice that the curves that overlap in Fig. (5.6) also happen to have equal (or close)

values of L/σT . Note that we have assigned to σT for each δ, the values computed in

chapter 3 using the temperature dependent thermal transport model for NiV/Al [see

Fig. (3.7)]. However, attempts to re-scale L/κRc to account for the effect of σT did

not succeed in making the curves collapse, nor did attempts of rescaling the relative

time error.

This again leads us to suspect that it is either not possible to obtain a unified

scaling criterion that is capable of capturing the spectrum of different behaviors ex-

hibited by multilayered particles with different δ, Rc and L, or that there might still

be other missing factors that we need to incorporate into our analysis before we can

arrive at a universal dependence. For example, re-inspecting Figs. (5.6) and (5.7), we

144


0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

L/kRc

t error

δ = 50 nm; L = 10µm

L = 50

L = 100

L = 500

δ = 250 nm; L = 10µm

0 0.2 0.5 1 1.5 20

0.1

0.2

0.3

0.4

Figure 5.6: Relative time error, terror, as a function of the non-dimensional ratio ofthe particle’s internal thermal resistance, L/κ, to its thermal contact resistance, Rc.Shown are curves corresponding to different values of L, Rc, and δ. Solid lines withsolid dots correspond to δ = 50 nm, while dashed lines with open circles correspondto δ = 250 nm. Inset provides a close-up into the region of small error and high Rc

values.

145


05

1015

2025

0

10

20

30

40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

L/σTL/kRc

t error

Figure 5.7: 3D plot of terror as a function of L/κRc and the non-dimensional ratioof particle size to thermal front width, L/σT . Shown are curves corresponding todifferent values of L, Rc, and δ. Solid lines with solid dots correspond to δ = 50 nm,while dashed lines with open circles correspond to δ = 250 nm. σT = 100 µm forδ = 250 nm and 20 µm for δ = 50 nm. The terror 2D slice helps highlight the pointsat which the curves cross the 10% error threshold.

146


see that while particles with similar ratios of L/σT tend to have similar sensitivities of

the error on L/κRc, particles with larger ratios give rise to lower sensitivities (smaller

errors). This observation runs counter to what we might have expected, since the

smaller the size of the particles is, compared to the thermal front width associated

with the layering, the more homogeneous their interior temperature fields would be

and thus should relate more closely with the homogeneous reaction description. It

is possible that for long particles (such as L = 500 µm), the dominating factor is no

longer the ratio of internal to contact thermal resistance, but rather the velocity of

front propagation (which is a function of reaction rate and the internal thermal resis-

tance) in the meshed particles, compared to the rates of reaction and heat conduction

in the homogeneous particles. Thus in this scenario, particles that are sufficiently long

and can support a self-propagating front obey a set of criteria that are slightly dif-

ferent than those for shorter particles. This might explain why the relative error for

the case of δ = 50 nm; L = 500 µm exhibits an anomalous trend in Fig. (5.3), as

it has a ratio of L/σT that is much larger than all the other curves (see Fig. (5.7))

and by this, it could have already crossed into a regime that is governed by a slightly

different set of criteria than the others. However, it is not directly obvious as to how

one could rescale the relative error using all the various independent parameters that

it depends on, and in a manner such that the effects of each on the error become

more/less prominent at distinct times.

Despite all of the above shortcomings and drawbacks, we were nevertheless success-

147


ful in determining one regime under which the homogeneous particle approximation

would be valid. In accordance with the observations in Fig. (5.6), it appears that

multilayered particles that satisfy the criterion of L/κRc < 0.2, irrespective of δ, are

associated with errors as small as 10% or less, and thus can be safely homogenized

without running the risk of misrepresenting the reaction dynamics on scales that are

on the order of the size of the particles. Moreover, we have shown that simple scaling

arguments such as those employed by the studies referenced earlier, which are mainly

based on relations between particle size and rates of heat transfer, are not sufficient

for establishing a universal criterion of validity.

148

Chapter 6

Conclusions

This dissertation utilized multiscale modeling, in conjunction with experimental

measurements, in order to investigate reactions occurring in reactive Ni/Al nanolam-

inates, and to develop more reliable models that are capable of encompassing and

reproducing a variety of observed phenomena.

Chapter 2 introduced the continuum model formalism that is used for simulating

the transient reaction dynamics in Ni/Al nanolaminates. Using this formalism, the

model reduction mechanism developed by Salloum and Knio [31] was described. The

reduced reaction formalism was implemented to overcome the stiffness associated

with the governing system of equations, and thus result in a model that is more

computationally efficient. This was followed by brief details on the numerical scheme

used in the computations for solving the governing equations of the reduced model.

In chapter 3, generalized thermal transport expressions were developed that are

149

CHAPTER 6. CONCLUSIONS

suitable for incorporation into the multi-dimensional reduced model introduced in

chapter 2. The generalized expressions, which are extensions of the isotropic concen-

tration dependent thermal conductivity model developed previously in [76], account

for the effects of layering of the initial microstructure, and of temperature variation

of the thermal conductivity of primary constituents. The effects of thermal transport

properties were analyzed by contrasting the predictions of four transport models,

namely constant, concentration dependent, concentration and direction dependent,

and concentration, direction and temperature dependent thermal conductivity mod-

els. Expressions were developed for both Ni/Al and NiV/Al multilayers, and pre-

dictions were obtained for both axially and normally propagating fronts in a wide

range of bilayer thicknesses. In all cases, heat exchange between the multilayer and

its surroundings was ignored, except for the initiation stimulus.

The analysis of the computed results in chapter 3 showed that (i) the dependence

of the thermal and reaction widths on the thermal transport model was generally

more pronounced than that of the average front velocity, especially for thicker bilay-

ers, δ ≥ 48 nm, (ii) a systematic reduction of the average front velocity is observed

when the effects of concentration, layering, and temperature dependence of thermal

conductivities are incorporated. Though the front velocity predictions of the four dif-

ferent models do not exhibit large differences, a re-examination of current calibrated

values of atomic mixing parameters seemed to be warranted, (iii) the trends observed

for axially-propagating fronts in Ni/Al and NiV/Al multilayers were generally similar.

150


However, the thermal and reaction widths for NiV/Al multilayers were appreciably

smaller than those predicted for Ni/Al multilayers. Predicted NiV/Al front velocities

were also smaller than those of corresponding Ni/Al multilayers, (iv) when temper-

ature variation and layering effects were accounted for, the velocity and reaction

width predictions for axially and normally propagating fronts were very close. On

the other hand, when the variation of thermal conductivities with temperature was

ignored, axial fronts were predicted to have larger reaction width and front velocity.

Thus, the effects of temperature variations in thermal conductivity appeared to dom-

inate the impact of layering in the unreacted microstructure, (v) surprisingly, scatter

plots of normalized mean composition versus normalized temperature appeared to

collapse, even when data for axially propagating fronts from a wide range of bilayer

thicknesses was included. This suggested that, at least under adiabatic conditions,

observations of self-propagating reaction fronts may only sample a narrow region in

the concentration-temperature phase space, and (vi) preliminary 3D computations

performed using the generalized transport models exhibited transient front features

that are reminiscent of recent experimental observations, whereas for the same con-

ditions the constant κ model predicted steady, uniform front propagation. These

experiences suggest that thermo-diffusive phenomena may play an important role in

the manifestation of unsteady front features.

Chapter 4 included a multiscale analysis conducted in order to infer intermixing

rates prevailing during different reaction regimes in Ni/Al nanolaminates. The analy-

151


sis combined the results of molecular dynamics (MD) simulations, used in conjunction

with a mixing measure theory to characterize intermixing rates under adiabatic con-

ditions. When incorporated into the reduced reaction model, however, information

extracted from MD computations led to front propagation velocities that conflicted

with experimental observations, and the discrepancies indicated that our MD simu-

lations over-estimate the atomic intermixing rates. Thus, using only insights gained

from MD computations, a generalized diffusivity law was developed that exhibited a

sharp rise near the melting temperature of Al. By calibrating the intermixing rates

at high temperatures from experimental observations of self-propagating fronts, and

inferring the intermixing rates at low and intermediate temperatures from ignition

and nanocalorimetry experiments, the dependence of the diffusivity on temperature

was inferred in a suitably wide temperature range. Using this generalized diffusivity

law, we obtained a generalized reduced model that, for the first time, enabled us to

reproduce measurements of low-temperature ignition, homogeneous reactions at in-

termediate temperatures, as well as the dependence of the velocity of self-propagating

reaction fronts on microstructural parameters.

In chapter 5, the generalized reduced model developed in chapter 4 was employed

towards exploring reactions occurring in layered particles networks. A further reduc-

tion of the model was sought through identifying regimes under which a homogeneous

particle approximation holds. The limiting case of a quasi-1D linear chain of rect-

angular multilayered particles, in full contact, was considered. Moreover, apart from

152


heat transfer across the particles’ surface contact areas, it was assumed that the reac-

tions occurred under adiabatic conditions. Comparisons between the computational

results of the meshed and the homogeneous reduced model descriptions were per-

formed for different values of particle size, L, thermal contact resistance, Rc, and

half-layer thickness, δ. These revealed that measures based on characterizing the de-

gree of temperature variations within the particles are not sufficient for providing a

criterion that is suitable for predicting the reliability of the homogenized approxima-

tion. A more suitable criterion based on the non-dimensional ratio of the particle’s

internal thermal resistance to its thermal contact resistance, L/κRc, was established.

It was found that multilayered particles that satisfied the criterion of L/κRc < 0.2,

irrespective of δ, were associated with small relative computational errors, and thus

could be safely homogenized. However outside this region, we were not able to deter-

mine a unique measure that could be applied towards gauging the performance, and

thus the degree of reliability, of the homogeneous approximation.

153

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Vita

Leen Alawieh was born on September 1985 in Beirut, Lebanon, where she grew

up and attended the International School of Choueifat - Choueifat from 1989 until

2003. In 2006, she received a B.Sc. in Chemistry from the American University of

Beirut. Later that same year, she enrolled in the Physical Chemistry program at the

University of Texas at Austin as a Fulbright Scholar, where she joined the Center for

Nonlinear Dynamics in the Physics department and received a M.Sc. in 2009 for her

experimental work on fluidized beds. She then moved to Baltimore, MD, where she

joined the Mechanical Engineering Ph.D. program at the Johns Hopkins University.

177

multiscale modeling of reactive ni/al nanolaminates

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