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Heat-initiated oxidation of aluminum nanoparticles

Dissertation
Author: Richard Clark
Abstract:
This dissertation uses multi-million atom molecular dynamics simulations to determine the key mechanisms which control the oxidation of aluminum nanoparticles (ANPs) due to uniform heating. The primary simulation studied is the oxidation of a 46 nm ANP uniformly heated to 1100 K (the melting temperature of aluminum metal is 933 K). In this system, 3 stages are identified, corresponding to simulation times of 0 to 50 ps (Stage 1), 60 to 110 ps (Stage 2), and 120 to 1000 ps (Stage 3). Radial analysis, fragment analysis, and other simulation tools are used to determine the controlling mechanisms in each stage, as well as determining the cause of each stage transition. Stage 1, initiated by the ANP temperature exceeding the melting temperature of the aluminum core, is seen to be a confined burning stage. Oxygen penetrating from the core-shell interface reacts exothermically with the core metal, releasing local heat. When this heating drives the local temperature of the shell to exceed the melting temperature of alumina (the shell), a sudden increase in the number of oxidation reactions is seen. The development of asymmetry in the shell and increased uptake of oxygen from the exterior also support the conclusion that melting of the shell initiates Stage 2. Local heating continues until the shell temperature reaches a critical temperature (T = 3000 K), at which point aluminum atom ejections from the exterior of the shell into the surrounding oxygen begin. These ejections lead to direct oxidation outside the ANP and the formation of oxygen-rich clusters. Both internal and external oxidation continue throughout the remainder of Stage 3, leading to complete oxidation of the particle. After having determined the basic mechanisms in the 46nm simulation, the effect of initial temperature on the oxidation stages is studied. Three 26nm ANP simulation systems, heated to 1100 K, 1200 K, and 1400 K, are analyzed using the tools developed in the preceding study. By comparing these systems, it is determined that many of the features identified in the model are strongly dependant on the initial temperature of the nanoparticle (including the time delay to the onset of each stage and the rate of oxidation in stages 2 and 3), while others have a relatively small dependence on the initial temperature provided (including the initial rate of heating, the rate of aluminum ejections in stage 3, and the overall temperature of a system of a given size). Finally, the tools and models developed in this study are applied to the case of a flash-heated ANP, providing a comparison point to aid in the understanding of this more complicated reaction. From this, it is determined that the initial heating of flash-heated ANPs is dominated by a mechanical mixing mechanism resulting from the pressure-driven volume expansion of the flash-heated core. This volume expansion significantly increases the chemical reaction zone of the core-shell oxidation, resulting in the very fast release of energy seen in the flash-heating cases.

TABLE OF CONTENTS

ACKNOWLEDGEMENTS………………………………………………………

ii

LIST OF TABLES………..………………………………………………………

vi

LIST OF FIGURES……….………………………………………………………

vii

ABSTRACT……………………………………………………………………….

x

Chapter 1

Introduction…….………………………………………………….

1

1.1

Therm ites and the Aluminum Nanoparticle (ANP)... . ……………………

2

1.2

Experimental Studies into ANPs…..……………..……………………….

3

1.3

Computational Studies into ANPs………………..……………………….

5

1.4

The Melt Dispersion Mechanism..… ……………..………………………

6

1.5

Motivation of this Study ………....……….………………………………..

7

Chapter 2

Molecular Dynamics Methods……………………………………..

9

2.1

Overview of Molecular Dynamics.……………….……………………….

9

2.2

Velocity - Verlet Algorithm.……….……………….

………………………

10

2.3

Pairwise Forces……………………..……………..……………………….

14

2.4

Cut off Radius and Cell Decomposition.…………..

………………………

15

2.5

Boundary Conditions…….....……………………..……………………….

16

2.6

Analytical Techniques...………………………….. . ………………………

16

2.7

System Manipulation Techniques..…………………...…………………...

18

Chapter 3

Validation of Poten tial………………………….………………….

19

3.1

Potential Energy Formalism…….……………………….…………………

19

3.2

Aluminum Metal Voter - Chen EAM Potential..……………………………

20

3.3

Validation of the Aluminum Potential ……..……..……………………….

25

3.4

Alumina Potential: Vashishta 3 - body Potentia l…..…………..……………

27

3.5

Validation of the Alumina Potential...……………..………………………

31

3.6

Oxygen Potential……....…………………………..……………………….

33

3.7

Crossover Function (Aluminum to Alumina Transition)…………………..

33

Chapter 4

Creating Initial ANP Systems..…………………….… ……… ……

37

4.1

Preparation of Nanoparticles……………………….………………………

37

4.2

Preparation of Alumina…………………..…………………………………

38

4.3

Preparation of Amorphous Nanoparticles..…..……..……………………...

40

4.4

Summary of Nanoparticle Systems…….…………..………………………

40

4.5

Efficien t Loading of Nanoparticles Systems onto Multiple Processors……

42

v

Chapter 5

Uniform Heating………………...…………………..…………….

48

5.1

Outline of Simulation..……...……………………….……………………

48

5.2

Preparation of Aluminum Nanoparticle…..……………………………….

49

5.3

Initiation St age…………………………...…..……..……………………..

51

5.4

Identifying Stages of Reaction....………..…………..…………………….

52

5.5

Confined Burning Stage.….……………...................……………………..

54

5.6

Ejection Stage.….………………………...................……………………..

65

5.7

Aluminum Nanopartic le in NVE – Overall Behavior.…………………….

69

5.8

Summary of 46nm Uniform Heating Simulation.......……………………..

70

Chapter 6

Effect of Initial Temperature on Heat - Initiated Oxidation of ANPs.………………………..…...…………………..……………

71

6.1

Differing Initial Temperature s..…………………….……………………..

71

6.2

Identification of Stages…………………..………………………………...

72

6.3

Stage 2 Onset: Shell Melting ..……

……………………………...…..…...

76

6.4

Summary of Differing Initial Temperatures..............……………………...

82

Chapter 7

Summary…………………...….. .…………………..………………

84

References…………………………………………………….……………………

86

Appendix

Application to Flash Heating...………………….. . ………………..

90

A.1 Introduction to Flash Heating……………………….……………………..

90

A.2 Flash - Heating: T=3000K Case…………………………………………….

91

A.3 Stage 1: Increasing Reaction Zone……....…..……..……………………...

97

A.4 Stage 2: Oxygen Penetration and Ejections...............……………………..

99

A.5 Comparison to 6000K and 9000K Simulations...…...……………………..

102

A.6 Summary of Flash - Heating………………………………………………..

105

vi

LIST OF TABLES

Table 3.1 Parameters used for Al - O interaction in the Vashishta

3 - body Potential (from [7])………………………………………………..

30

Table 3.2 Validation of Alumina Potential (from [7]) .…………………………..

31

Table 4.1 Simulated Aluminum Nanoparticles………………………… …………

41

Table 6.1 Onset Time of Stage 2 and Stage 3 for Three Different Initial

Temperatures……..………………………………………………………...

75

vii

LIST OF FIGURES

Figure 1.1 Picture of Melt Dispersion Mechanism by Levitas, et al [31]…………

7

Figure 3.1 Graph of the ρ( r) term in the Voter - Chen Aluminum potential………..

21

Figure 3.2 Graph of the φ (r) term in the Voter - Chen Aluminum potential……….

21

Figure 3.3 Graph of the F( ρ ) term in the Voter - Chen Aluminum potential……….

25

Figure 3.4 Cohesive Energy vs. Volume for F CC Al Metal using Voter

EAM potential………………………………………………………………

26

Figure 3.5 H(ev) vs. T for Aluminum using Voter EAM potential [7]……………

27

Figure 3.6 Validation of g(r) for Bulk Alumina [7]……………………………….

31

Figure 3.7 Validation of Melting Temperatur e of Alumina [7] …………………..

32

Figure 3.8 Specific Heat Comparison using Alumina Potential [7] ………………

32

Figure 3.9 Weighting Function on Al with Proximity to Oxygen ………………..

35

Figure 4.1 Pair Distribution Function for Created Amorphous Alumina …………

39

Figure 4.2 Visual Definition of the Nanoparticle Size, D, and Shell

Thickness, S …………………. ……………………………………………..

41

Figure 4.3 Maximum Percent Load on a node as a function of

Relative ANP Size ……... ………………………………………………….

44

Figure 5.1 Snapshots of the ANP Interface during Initial Thermalization ……….

50

Figure 5.2 Radial Analysis of Oxygen Content During Heating ………………….

51

Figure 5.3 Global Temperature of System During NVE Simulation ……………..

52

Figure 5.4 Rate of Heat Generation by System (slope of 5 - 3) ……………………

53

Figure 5.5 Radial Analysis of Oxygen Fraction during Stage 1 of Reaction ……..

55

Figure 5.6 Snapshots of a Slice through Center of ANP with Core Al atoms

Hidden at 0 ps and at 50 ps………………………………………………...

56

viii

Figure 5.7 Decreasing Radiu s of Core - Shell Interface Boundary ………………..

57

Figure 5.8 Increasing Number of Al 2 O 3 Fragments in Stage 1 and 2 ……………..

58

Figure 5.9 Ratio of Oxygen to Aluminum in Shell and in All Fragments ……….

59

Figure 5.10 Increase in 2 Types of Fragments over Time (Al 2 O 3 and AlO 4 ) …….

60

Figure 5.11 Radial Analysis of Temperature during Stage 1 ……………………..

61

Figure 5.12 Snapshots of a Slice through the Center of ANP with Core

Al Atoms Hidden at 60 ps and 110 ps... ……..…………………………….

62

Figure 5.13 Maximum an d Average Shell Thickness over Time …………………

63

Figure 5.14 Proposed Flowchart of Mechanisms leading to Shell Failure ………..

64

Figure 5.15 Snapshot of Central Slice at t=400 ps ………………………………..

65

Figure 5.16 Fragment Analysis of System during Stages 1 - 3 …………………….

66

Figure 5.17 Dramatic Increase in Al atoms outside the shell coinciding with

onset of Stage 3 …..………………………………………………………..

67

Figure 5.18 Radial Analysis of Temperature during Stage 1 - 3 …………………..

68

Figure 5.19 Snapshots of Central Slice of ANP with Core Al Hidden

at 0 ps and 800 ps ………………………………………………………….

69

Figure 6.1.a Overall System Temperature over Time for 3 Cases ………………...

73

Figure 6.1.b Rate of Temperature Change (i.e. Heating Rate) over Time

for 3 Cases ……..……..……..……..……. .……..……..……..……..……..

74

Figure 6.2 Radial Temperature Profiles comparing T=1100 K and

T=1400 K Cases over first 150 ps …..……………………………………..

76

Figure 6.3 Fragment Analysis of the T=1400 K Case over the first 200 ps ………

77

Figure 6.4 Al Ejections int o Environment ………………………………………..

79

Figure 6.4.b Shifted Graph of Al Ejections into Environment ……………………

80

Figure 6.5 Al - Rich Fragments over Time ………………………………………..

81

ix

Figure 6.6 Al - Poor Fragments over Time ………………………………………..

82

Figure A.1 Images of Central Slice of T=3000K 46nm Flash - Heating

Simulation at time t=0 ps. Colored by species (left) and

temperature (right) ………..……..……..……..……..……..………………

91

Figure A.2 Images of Central Slice at Various Times (0ps, 50ps, 100ps,

150ps, 200ps) ……………………… ……………………………………..

93

Figure A.3 Temperature over Time of T=3000K 46nm flash - heating

simulation ……..……..……..……..……..……..……..……..……..………

95

Figure A.4 Rate of Temperature Increase in T=3000K 46nm flash - heating

simulation ……..……..……..……..……..……..……..…….. ……..………

96

Figure A.5 Expansion of Outer Radius of T=3000K 46nm Particle over Stage 1. Dashed line indicates transition from Stage 1 to Stage 2 ………………….

97

Figure A.6 Increasing Surface Area of Core - Shell Boundary in T=3000 K

46nm ANP over Stage 1 …… ……………………………………………...

98

Figure A.7 Radius of Core - Shell Boundary during Stage 2 showing

Oxygen Penetration into Core ……………….. ……………………………

100

Figure A.8 Radial Oxygen Fraction in Ejection Part of Stage 2

for T=3000 K 46nm ANP …………………………………………………

101

F igure A.9 Temperature vs. Time for 46nm Flash - Heating Simulation ………….

103

Figure A.10 Rate of Temperature Increase vs. Time in 46nm Flash - Heating

Simulations ……..……..……..……..……..……..……..……..……..……..

104

x

ABSTRACT

This dissertation uses multi - million atom molecular dynamics simulations to determine the key mechanisms which control the oxidation of aluminum nanoparticles (ANPs) due to uniform heating. The primary simulation studied is the oxidation of a 46 nm ANP uniformly heated to 1100 K (the melting temperature of aluminum metal is 933 K). In this system, 3 stages are identified, corresponding to simulation times of 0 to 50 ps (Stage 1), 60 to 110 ps (Stage 2), and 120 to 1000 ps (Stage 3). Radial analysis, fragment analysis, and other simulation t ools are used to determine the controlling mechanisms in each stage, as well as determining the cause of each stage transition. Stage 1, initiated by the ANP temperature exceeding the melting temperature of the aluminum core, is seen to be a confined burn ing stage. Oxygen penetrating from the core - shell interface reacts exothermically with the core metal, releasing local heat. When this heating drives the local temperature of the shell to exceed the melting temperature of alumina (the shell), a sudden in crease in the number of oxidation reactions is seen. The development of asymmetry in the shell and increased uptake of oxygen from the exterior also support the conclusion that melting of the shell initiates Stage 2. Local heating continues until the she ll temperature reaches a critical temperature (T = 3000 K), at which point aluminum atom ejections from the exterior of the shell into the surrounding oxygen begin. These ejections lead to direct oxidation outside the ANP and the formation of oxygen - rich clusters. Both internal and external oxidation continue throughout the remainder of Stage 3, leading to complete oxidation of the particle.

xi

After having determined the basic mechanisms in the 46nm simulation, the effect of initial temperature on the oxid ation stages is studied. Three 26nm ANP simulation systems, heated to 1100 K, 1200 K, and 1400 K, are analyzed using the tools developed in the preceding study. By comparing these systems, it is determined that many of the features identified in the mode l are strongly dependant on the initial temperature of the nanoparticle (including the time delay to the onset of each stage and the rate of oxidation in stages 2 and 3), while others have a relatively small dependence on the initial temperature provided ( including the initial rate of heating, the rate of aluminum ejections in stage 3, and the overall temperature of a system of a given size).

Finally, the tools and models developed in this study are applied to the case of a flash - heated ANP, providing a co mparison point to aid in the understanding of this more complicated reaction. From this, it is determined that the initial heating of flash - heated ANPs is dominated by a mechanical mixing mechanism resulting from the pressure - driven volume expansion of th e flash - heated core. This volume expansion significantly increases the chemical reaction zone of the core - shell oxidation, resulting in the very fast release of energy seen in the flash - heating cases.

1

CHAPTER 1

INTRODUCTION

Using multi - million atom m olecular dynamics (MD) simulations, I will explore the interplay of the mechanisms which give rise to the burning of an aluminum nanoparticle in oxygen. I will first provide an overview of the current studies into aluminum nanoparticles, both the existing experimental results as well as summarizing the relevant computational and theoretical work to date. In Chapter 2, I will review the various methods and tools used in the simulation and analysis of these simulation results. Chapter 3 will focus more spe cifically on the mathematical basis describing the aluminum nanoparticle system. Specifically, a discussion of the inter - atomic interaction potential forms that have been tested, their validation, and the potential energy functions used in the main part o f this study. Chapter 4 describes the process of creating the core - shell aluminum nanoparticles used in the simulations. In Chapter 5, one simulation of an aluminum nanoparticle being uniformly heated in oxygen is presented along with detailed analysis o f the mechanisms controlling the particle’s evolution. From this simulation, a basic model is constructed which describes the primary mechanisms controlling the particle’s evolution during burning. Chapter 6 will apply this basic model to several simulat ions of aluminum nanoparticles uniformly heated to differing initial temperatures – focusing on the ways in which the initial temperature effects the mechanisms as well as identifying those features which are largely temperature independent. Finally, in A ppendix the

2

method developed for use in the uniformly heated ANPs is applied to the case of flash - heating ANPs, allowing greater insight into some of the factors at work in this complex and important system.

1.1 Thermites and the Aluminum Nanoparticle ( ANP)

Conventional thermite reactions (Al + Fe 2 O 3 ) have been used in the welding of railroad ties since 1898. In more recent times, the thermite reaction has been used as a cutting tool, a heat source, a high - temperature igniter, and an additive to propell ants and explosives [1]. Since the advent of nanotechnology, it has become possible to construct new materials which have the potential to more than double the energy release of the best molecular explosive. In a 2006 review paper, D. Dlott explained that while the development of nanotech - based explosives and propellants has the potential to allow greater control and energy output than current energetic materials, to take advantage of this emerging field would require “a fundamental understanding of the re lationships between structure and performance that does not yet exist” [2]. Responding to this need, many experimental and computational research groups have explored various aspects of the aluminum nanoparticle. In the following section, I will briefly summarize the findings of many of these groups before outlining how large - scale MD simulations may be used to aid in the understanding of this complex system.

3

1.2 Experimental Studies into ANPs

To prepare a molecular dynamics simulation of aluminum nano particles, it first is necessary to become familiar with experiments investigating the component bulk materials. Experimentalists have carefully measured the density [3], bulk modulus, specific heat [4], and elastic moduli [5] of solid bulk aluminum. The equation of state in the liquid phase has been studied at temperatures ranging from the melting point (T=933 K) up to 4000 K [3,6]. As the potential which accurately describes alumina in bulk has been well characterized by the Vashishta potential [7], our inclusion of experimental work into alumina will focus primarily on recent studies investigating the formation of oxide layers and the behavior of small alumina clusters.

Experimental work on conventional (as opposed to nanosynthesized) aluminum thermite mixtures has followed a variety of approaches. These studies tend to focus on the classical thermite reaction (aluminum + iron oxide), and are divided by several characteristics: the composition of the material, the process by which the reaction is initia ted, the way in which the detonation proceeds, and the experimental tools used to analyze the results. The most extensive experimental studies into aluminum thermite mixtures are performed on bulk materials. Although the traditional thermite reaction mix es aluminum with iron oxide [8 - 10], other oxidizers, in particular molybdenum oxide, are also used [11 - 14]. Detonation is initiated by heating the sample [15,16], using an electric discharge at one point [9], hitting the material with a shock wave [18], o r by focusing a laser on the material [19 - 22]. In response to these initiation methods, the thermite mixture either flares up and immediately dies out, has a partial detonation, or proceeds to complete detonation. For those studies focusing

4

on the latter of these results, the sustained burning/detonation cases, experimental work has been performed which examines the effect of the container holding the mixture (i.e. confined or open detonation) [11]. Regardless of the material studied or the method of in itiation used, there are a wide variety of analysis tools that are used to study the reaction. The methods used to observe the material during detonation include high - speed photography, pyrometry, and thermogravimetric analysis [14]. Additionally, method s such as X - ray diffraction, optical micrography, and SEM are used to characterize the products of the reaction once it is complete [24].

Since the advent of nanotechnology synthesis techniques in the 1980’s, it has became possible to construct thermite mi xtures with feature sizes on the order of nanometers (each small chunk of aluminum in the mixture can be 20 - 200 nm in diameter). With this capability, thermite compounds can be made which yield faster burn rates, higher burn temperatures, more released en ergy, and other desirable features [2]. Some experiments which have treated these novel mixtures in bulk include [18]. Additionally, in an attempt to understand the role played by the component aluminum nanoparticles, several experiments have been perfor med which study the burning of single (or small clusters) of aluminum nanoparticles in various gas mixtures [26,27].

The above studies emphasize the hope of the field: that one day we may understand the controlling mechanisms of the thermite compounds to the point that we are able to tailor - make energetic materials where all parameters of the burn are controlled (from sensitivity to burn rate to burn temperature).

5

1.3 Computational Studies into ANPs

In addition to the many experimental studies perform ed in an attempt to gain an underlying understanding of aluminum nanoparticle detonations, many computational techniques have been applied to this problem as well. Density function theory has been used to calculate the energetics of small systems, includi ng small clusters of aluminum and oxygen [28] or local features, such as grain boundaries [29]. This work has provided an excellent reference point for understanding the underlying reactions, but the DFT method is incapable of handling the feature size as a whole (even the smallest consistently created aluminum nanoparticles at present are >20nm in diameter and have millions of atoms – well beyond the system size accessible to DFT). Coarser simulation techniques have been applied to the thermite bulk mate rial as a whole, including a study into the bulk iron oxide/aluminum by Brito, et al. [34] and a recent simulation study on copper oxide/aluminum by Gan, et al. [35]. However, these techniques are, as a trade off, only able to provide approximate bulk pro perties without giving any understanding of the mechanisms which give rise to the bulk behavior. Only recently have the available computational resources progressed to the point where it is possible to look at the features of realistically - sized particles

while maintaining the interaction resolution on the scale of atom - atom interactions. In one study, conducted in 1999 by Campbell, et. al., a Streitz - Mintmire potential was used to simulate the growth of an oxide layer onto an aluminum sphere’s surface. In this study, the authors show that not only does the computational method produce a stable oxide barrier (of 33 Å) in accordance with experimental evidence, but also

6

provides a much more complete description of many features beyond the ability of modern experimental methods to detect [30].

From this wealth of experimental and computational data, however, we still have yet to reach the goal of being able to explain how to tailor - make an energetic compound to give the proposed effect. The theoretical mode l that is currently presented as the closest to this goal is the “melt dispersion mechanism” (MDM).

1.4 The Melt Dispersion Mechanism

The melt dispersion mechanism was introduced by Levitas, et al, in 2006 [31]. This model seeks to explain the dominant me chanism in the detonation of an aluminum nanoparticle. It attempts to explain why the relationship between particle size and flame speed is not diffusion - controlled (inversely proportional to r 2 ) – but instead appears constant for small particles (40 - 100 nm). In 2007, the model was expanded to include supporting experimental data. In this study, aluminum nanoparticles were subjected to fast heating (10 8 K/s). The particles were found to react as soon as the aluminum exceeded the aluminum melting tempera ture 933 K [31].

The mechanism of detonation is described as:

1) aluminum core melts

2) pressure creates strain in oxide shell

3) shell fractures

4) unloading wave causes liquid aluminum to spallate

5) liquid reacts with oxidizer

Or by the picture [fr om 31]:

7

Fig 1.1 Picture of Melt Dispersion Mechanism proposed

by Levitas, et. al. [31]

1.5 Motivation of this Study

As is apparent from the above research, there is clear need in the scientific community for an “inside - look” at the aluminum nanopartic le during burning and detonation. Although extensive experimental work has given valuable insight into the subject, the size of the relevant features (nanometers), the speed at which the burn occurs (ps - ns), and the temperature scales reached during deton ation (>2500K) combine to make analysis of the underlying mechanisms extremely difficult. Computational techniques have been limited to DFT simulations for small clusters and bulk - material detonation profiling. Because one cannot observe the aluminum

8

na noparticle as it reacts, a number of theories have been presented to identify the controlling mechanisms. Unfortunately, because of the difficulty in obtaining anything but macroscale or after - the - fact experimental information about the system, it remains difficult to determine if any theory accurately reflects the inner mechanisms of the aluminum nanoparticle.

Because of its “snapshot” nature, molecular dynamics (MD) is ideally suited to this problem. Unlike experiments or coarse simulation techniques, M D provides the opportunity to observe the interplay of mechanisms at an arbitrarily small time scale. The number of atoms in an aluminum nanoparticle (on the order of several million) is well within the range that may be modeled using a parallel computing cluster. Finally, the high temperatures generated by the aluminum nanoparticle burning can be addressed in MD by implementing a reduced time step and accurate interaction potentials (discussed in Ch 3).

This work will use large - scale MD simulations to investigate the mechanisms which govern the burning of aluminum nanoparticles, providing new insight into the behavior and control of these important systems.

9

CHAPTER 2

MOLECULAR DYNAMICS METHODS

2.1 Overview of Molecular Dynamics

Molecular dynamics is a deterministic simulation technique which follows the evolution of a system of atoms over time. The “system” is described by a time snapshot of the atomic positions, x( ), and velocities, v( ), at a particular time. Once an initial system is defined, th e MD code uses the calculated interatomic forces to update the system to a snapshot a short time later. With successive iterations of this procedure, the initial system may be propagated forward to an arbitrary time in the future. MD thus becomes a virtu al laboratory in which one may examine mechanical and chemical events that would be impossible to measure experimentally.

To create an MD code one must implement the following stages:

1. Create an initial snapshot of the system (the initial positions and v elocities).

2. Calculate the forces acting on each atom.

3. Update the particle positions and velocities to a time a small step into the future.

4. Output the positions, velocities, or other derived information about the system for later analysis.

4. Retur n to #2 until the MD simulation is complete.

How well the simulated evolution of the system reflects the behavior of the same system in reality is controlled by the implementation of steps 2 and 3. As they are

10

more specific to the particular system studi ed in this case, steps 1 and 2 will be described in future chapters (chapters 4 and 3, respectively). In the following, the method used to perform step 3 will be described.

2.2 Velocity - Verlet algorithm

Updating the particle positions and velocities to a time step slightly in the future is accomplished by first deriving a set of equations of motion for the system, and then discretizing with respect to time. The most popular method of doing this in atomic simulations is the velocity - Verlet algorithm. It is assumed in this method that the energies (and forces) associated with the system may be calculated from the system snapshot (x and v) at each time frame. The exact details of how this is done depend on the potential energy function used to describe the atomic interactions, which is presented in chapter 3. For the moment, we will take it as given that the force on a given atom F i , can be determined using only the positions of the atoms.

Forecasting the position and velocity of an atom, i, a short time, dt, into the future can be calculated using a Taylor expansion on both terms:

!

x ( t + dt ) = x ( t ) + dx dt dt + 1 2 d 2 x dt 2 ( dt ) 2 + ...

or using the definitions: !

v = dx dt ,

a = dv dt

!

x ( t + dt ) = x ( t ) + v ( t ) * dt + 1 2 a ( t ) * dt 2 + ...

!

v ( t + dt ) = v ( t ) + a ( t ) * dt + ...

11

Combined with Newton’s law:

!

x ( t + dt ) = x ( t ) + v ( t ) * dt + F ( t ) 2 m * dt 2 + ... v ( t + dt ) = v ( t ) + F ( t ) m * dt + ...

As such, by neglecting the higher order terms represented by the “…” (which will be very small as dt << 1), we c an construct the algorithm:

1) Calculate Force on atom

2) x < - x + v(t)*dt + F(t)/2m * dt 2

3) v < - v + F(t)/m * dt

4) Output New snapshot

If performed for each atom at every time step, this would yield a trajectory of the system over time. In this c ase, the accuracy of each update would be on the order O(dt 3 ) in the position and O(dt 2 ) in the velocity.

A more accurate algorithm is the Velocity - Verlet method. Starting from the Taylor expansion (and adding another term !

b ( t ) = d 3 x dt 3 into the explicitly stated te rms):

!

x ( t + dt ) = x ( t ) + v ( t ) * dt + F ( t ) 2 m * dt 2 + 1 6 b ( t ) * dt 3 + O ( dt 4 ) v ( t + dt ) = v ( t ) + F ( t ) m * dt + 1 2 b ( t ) * dt 2 + O dt 3 ( )

(the O() notation represents terms of order x or higher)

and performing a similar expansion in the reverse time direction for the position:

12

!

x ( t " dt ) = x ( t ) " v ( t ) * dt + F ( t ) 2 m * dt 2 " 1 6 b ( t ) * dt 3 + O ( dt 4 )

(set, in the above, the initial time t - > t+dt):

!

x ( t ) = x ( t + dt ) " v ( t + dt ) * dt + F ( t + dt ) 2 m * dt 2 " 1 6 b ( t + dt ) * dt 3 + O ( dt 4 )

Combining the x(t) and x(t+dt) expressions and solving for v(t+dt):

!

v ( t + dt ) = v ( t ) + F ( t ) 2 m + F ( t + dt ) 2 m "

#

$

%

&

'

* dt + 1 6 b ( t ) ( b ( t + dt ) [ ] * dt 2 + O ( dt 3 )

But as the expansion of b(t+dt) follows:

Full document contains 117 pages
Abstract: This dissertation uses multi-million atom molecular dynamics simulations to determine the key mechanisms which control the oxidation of aluminum nanoparticles (ANPs) due to uniform heating. The primary simulation studied is the oxidation of a 46 nm ANP uniformly heated to 1100 K (the melting temperature of aluminum metal is 933 K). In this system, 3 stages are identified, corresponding to simulation times of 0 to 50 ps (Stage 1), 60 to 110 ps (Stage 2), and 120 to 1000 ps (Stage 3). Radial analysis, fragment analysis, and other simulation tools are used to determine the controlling mechanisms in each stage, as well as determining the cause of each stage transition. Stage 1, initiated by the ANP temperature exceeding the melting temperature of the aluminum core, is seen to be a confined burning stage. Oxygen penetrating from the core-shell interface reacts exothermically with the core metal, releasing local heat. When this heating drives the local temperature of the shell to exceed the melting temperature of alumina (the shell), a sudden increase in the number of oxidation reactions is seen. The development of asymmetry in the shell and increased uptake of oxygen from the exterior also support the conclusion that melting of the shell initiates Stage 2. Local heating continues until the shell temperature reaches a critical temperature (T = 3000 K), at which point aluminum atom ejections from the exterior of the shell into the surrounding oxygen begin. These ejections lead to direct oxidation outside the ANP and the formation of oxygen-rich clusters. Both internal and external oxidation continue throughout the remainder of Stage 3, leading to complete oxidation of the particle. After having determined the basic mechanisms in the 46nm simulation, the effect of initial temperature on the oxidation stages is studied. Three 26nm ANP simulation systems, heated to 1100 K, 1200 K, and 1400 K, are analyzed using the tools developed in the preceding study. By comparing these systems, it is determined that many of the features identified in the model are strongly dependant on the initial temperature of the nanoparticle (including the time delay to the onset of each stage and the rate of oxidation in stages 2 and 3), while others have a relatively small dependence on the initial temperature provided (including the initial rate of heating, the rate of aluminum ejections in stage 3, and the overall temperature of a system of a given size). Finally, the tools and models developed in this study are applied to the case of a flash-heated ANP, providing a comparison point to aid in the understanding of this more complicated reaction. From this, it is determined that the initial heating of flash-heated ANPs is dominated by a mechanical mixing mechanism resulting from the pressure-driven volume expansion of the flash-heated core. This volume expansion significantly increases the chemical reaction zone of the core-shell oxidation, resulting in the very fast release of energy seen in the flash-heating cases.