andrew francis- modeling the melt dispersion mechanism for nanoparticle combustion

8/3/2019 Andrew Francis- Modeling the Melt Dispersion Mechanism for Nanoparticle Combustion

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MODELING THE MELT DISPERSION MECHANISM FOR NANOPARTICLECOMBUSTION

BY

ANDREW FRANCIS

A THESIS

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty

of Texas Tech University inPartial Fulfillment of

the Requirements forthe Degree of

MASTER OF SCIENCE

IN

MECHANICAL ENGINEERING

Valery Levitas

Chairperson of the Committee

Michelle PantoyaCo-Chair

Walt Oler

Fred HartmeisterDean of the Graduate School

December 2007


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Texas Tech University, Andrew Francis, December 2007

ii

ACKNOWLEDGEMENTS

There are truly too many people for me to mention everyone who has helped me

to complete this work. I would like to thank my friends and family for always supporting

me in all aspects of my life. To my wife, Marka, thank you for all of your love and

support, and for always being a Godly example in everything that you do. I am

extremely grateful to Dr. Michelle Pantoya for giving me the opportunity to continue my

intellectual growth and contributing so much to my education. Dr. Valery Levitas, I

really appreciate your patience with me and the time you have spent helping me with this

project. Most of all, I would like to thank Christ for His gift that makes life worth living.


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iii

TABLE OF CONTENTS

Acknowledgements ii

Abstract...... iv

List of Tables..... v

List of Figures........ vi

I: Introduction.. 1

1.1 Thermites..... 1

1.2 Nanocomposite thermites..... 2

1.3 Melt dispersion mechanism..... 5

II: Theory... 7

2.1 Equation for flame rate vs. particle parameters... 7

2.2 Distribution function for particle parameters....... 11

III: Results and Discussion 14

3.1 Effect of distribution of particles geometricparameters on flame propagation rate.. 14

3.2 Effect of passivation temperature (T0) and relative

particle radius (M) on flame propagation rate.. 18

3.3 Effect of alumina shell strength ( u), its distribution, and relativeparticle radius (M) on flame propagation rate.. 22

3.4 Effect of mixing particles with different relative particleradius (M) on flame propagation rate... 26

IV: Conclusions. 31

References.. 33


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iv

ABSTRACT

Thermite particles have long been known to increase in reactivity as they decrease

in size. However, during fast heating (106- 10

8K/s) of Al nanothermites, the diffusion

mechanism that explains micron size thermite reactions cannot explain the extremely fast

ignition times and much higher flame propagation velocities. A new mechanism known

as the melt dispersion mechanism has recently been introduced to explain the fast

oxidation of these Al nanothermites. A model has been created dependant upon key

parameters to predict the reactivity of Al nanothermites. In this study, flame propagation

velocities are statistically evaluated in terms of an integral that employs a probability

density function (pdf) for key parameters and a flame velocity equation dependent on

relative particle size (Al core radius divided by oxide shell thickness), oxide shell

formation temperature, and oxide shell strength. It is shown that flame propagation

velocity depends sensitively on relative particle size, relative particle size distribution,

oxide shell formation temperature, and shell strength. It is also dependant upon particle

size, and oxide shell thickness but not as sensitively. Both single and bimodal particle

sizes were studied. Combining smaller nanoparticles with larger nanoparticles in a

bimodal mixture significantly increases the flame propagation velocity as compared to a

composite consisting of only the larger particles. The results presented here suggest that

better reproducibility of the flame velocity may be achieved experimentally by selecting a

material with a narrow relative particle size distribution. A combination of increased

oxide shell formation temperature and increased oxide shell strength could be used to

maximize the flame velocity in particles with increased relative particle size.


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v

LIST OF TABLES

1. Material parameters at melt temperature T=Tm... 8

2. Experimental data corresponding with Figure 4...... 10


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vi

LIST OF FIGURES

1. Heat of reaction of Thermitesand High Explosives.... 2

2. Flame propagation speeds (burn rate) for nano-Al mixed

with MoO3 loose powders 4

3. Oxidation of micron (a) and nano (b) Al particles....... 6

4. Correlation between experimental data and melt dispersion model.... 10

5. Relative flame velocity as a function of relative

standard deviation for variations inR.............................................................................. 14


standard deviation for variations in ... 16


particle size for various ..... 17

8. Relative flame velocity as a function of relativestandard deviation for T0 = 300 K... 19


standard deviation for T0 = 400 K... 20





12. Relative flame velocity as a function of relativestandard deviation for T0 = 700 K... 21

13. Relative flame velocity as a function of relative standard deviation foru = th 23

14. Relative flame velocity as a function of relative standard deviation foru = th/1.1.. 23

15. Relative flame velocity as a function of relative standard deviation foru = th/1.2.. 24

16. Relative flame velocity as a function of relative standard deviation foru = 1.1th... 24


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vii

17. Relative flame velocity as a function of relative standard deviation foru = 1.2th....... 25

18. Relative flame velocity as a function of mass fraction for M2 = 40........ 28

19.Relative flame velocity as a function of mass fraction for M2 = 80.... 28

20.Relative flame velocity as a function of mass fraction for M2 = 150.......... 29

21.Relative flame velocity as a function of mass fraction for M2 = 1000........ 29

22. Relative flame velocity as a function of mass fraction for M1 = 20........ 30


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1

CHAPTER I

INTRODUCTION

1.1 Thermites

A traditional thermite reaction is an exothermic reduction-oxidation reaction

between a metal (e.g., Al, Mg, B, Ti, and Zr) and a metallic or non-metallic oxide (e.g.,

Fe2O3, CuO, MoO3, and C2F4) [21]. The aluminum (Al) particles in a thermite are

covered by a thin but growing oxide shell () on the order of nanometers (10-9 m) in

thickness. For the reaction to take place, oxygen from the oxidizer must diffuse through

the oxide shell to the metal, and particles from the metal must diffuse to the oxidizer. As

a result, traditional thermite reactions are rate limited by the diffusive rates of the fuel and

oxidizer through the oxide shell. This reaction is relatively slow when compared to the

reaction of a high explosive (HE). A high explosive reaction is also a reduction-

oxidation reaction, but does not depend on diffusion rates and instead is controlled by the

energy required to break atomic bonds. The reaction of a HE occurs so fast that it

detonates, creating a supersonic shock wave that compresses the unreacted material in

front of the shock wave increasing the temperature to the point of ignition [10]. The HE

cyclotetramethylene-tetranitramine (HMX) detonates at the velocity of 9.1 km/s which is

more than three times the speed of sound in HMX [9][14]. The unconfined thermite 1-3

m Al+MoO3 is shown to have a flame propagation velocity of 4.1 m/s, 10

3

times less

than the speed of sound in Al [22][11]. Consequently, the time for a traditional thermite

to completely react is on the order of seconds in contrast to that of a HE, which is on the


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2

order of picoseconds (10-12

s) [7][5]. However, the amount of energy released by a

traditional thermite reaction is larger than that in a HE reaction. Figure 1 shows the heat

of reaction for three traditional thermite reactions and three common HE reactions.

0 1000 2000 3000 4000 5000

Al/Bi2O3

Al/Fe2O3

Al/MoO3

TNT

RDX

HMX

Heat of Reaction

cal/g

cal/cc

Figure 1. Heat of reaction of Thermites [6]and High Explosives [19].

1.2 Nanocomposite thermites

Nanocomposite thermites, sometimes referred to as metastable intermolecular

composites (MICs), are of great interest because they give off large amounts of energy

compared to HEs, provide much faster energy release when compared to traditional

thermites, and allow for greater control over material homogeneity and material

properties [5]. The reactivity of thermites, evaluated by ignition delay time (tig) and


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3

flame propagation velocity (C), increases as the diameter of the Al fuel particle

decreases. When the diameter of the Al particle is reduced from the traditional 1-100 m

range to the 20-120 nm range, the flame propagation velocities change from the order of

cm/s to 0.9-1 km/s, and the ignition delay times are 1000 times faster [1][20][3][17].

While undergoing fast heating (106- 10

8K/s), the rise time (tf) for temperature

and pressure in a nanocomposite thermite, which characterizes the reaction time, is on the

order of 10 s [3][2]. For self-diffusion coefficients (D) of oxygen and Al in -alumina

at 800-950C, D = 10-19 and 10-18 cm2/s respectively. The diffusion length

nmxDtl fd 510)62(2 == , which is five orders of magnitude smaller than the oxide

shell thickness [12]. If diffusion was the mechanism through which this reaction takes

place, the reaction time would have to be 1010

times slower, or the time of diffusion

would have to be 1010 times faster.

There is also other evidence that shows the diffusion mechanism which explains

microcomposite thermite reactions does not hold true for nanocomposite thermites. First,

for diffusion-controlled oxidation, the flame propagation rate is proportional to one over

the square of the particle diameter (d-2

) [4]. However, for particle diameters smaller than

80 nm, flame speed is independent of particle size (See Figure 2) [3].


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4

Figure 2. Flame propagation speeds (burn rate) for nano-Al mixed with MoO3 loose

powders [17]

Second, for diffusion controlled oxidation, ignition delay time is a power function

ofd. For particle sizes d < 120 nm, ignition delay time is independent of particle size [8].

Third, the flame propagation rates for microthermites increase with sample density, but

for nanothermites it decreases [16].

The inability of diffusion controlled oxidation to explain the reactivity of

nanocomposite thermites indicates that other models are needed to help explain

experimental observations. One model presented by Levitas et al is described in terms of

a dispersion mechanism and shows potential for explaining experimental observations

associated with nano-particle combustion. The objective of this study is to explore how

500

700

900

1100

0 50 100 150

Al Particle Diameter (nm)

BurnRate(m/s)

Independent of particlediameter

500

700

900

1100

0 50 100 150

Al Particle Diameter (nm)

BurnRate(m/s)

Independent of particlediameter


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5

varying particle parameters impact the operational regime for the melt dispersion

mechanism.

1.3 Melt dispersion mechanism

For relatively slow heating rates (~103

K/s) nanothermites behave like

microthermites, reacting slowly, signifying that the reaction is rate limited by diffusion.

For both micro and nanothermites the oxide shell grows as metallic and oxidizer

molecules diffuse through the oxide shell and react with their counterparts. In both cases,

either the oxide shell breaks before melting or slow damage to the oxide shell leads to

slow flow of the liquid Al through cracks in the oxide shell inhibiting oxide spallation

[18].

Fast heating of nanothermites (106 108 K/s) creates substantial internal stresses

due to the difference in thermal expansion of Al and Al2O3. Also, the melting

temperature of nano-Al decreases with particle diameter. This allows the Al core of the

fuel particles to begin to melt during fast heating [11]. The volume of Al increases by

6% when it melts [23]. The pressure inside the Al particle causes tensile hoop stress (h)

in the oxide shell. Due to the small thickness of the oxide shell (1 8 nm), it is almost

defect free and therefore its ultimate strength (u) approaches the theoretical maximum

strength of alumina (th) estimated at 11.33 GPa [11]. The combination of large volume

change, differences in thermal expansion coefficients, and high oxide shell strength

causes high pressures within the Al core (1-2 GPa) [11]. When -h = u the oxide shell

spallates causing an imbalance in the pressure within the Al particle and exposed surface.


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6

The pressure at the surface of the Al sphere immediately following spallation is estimated

to be 10 MPa due to gas pressure and surface tension, while internally the pressure is still

1-2 GPa [11]. This results in an unloading wave that propagates to the center of the

particle creating a tensile pressure of similar value. This tensile pressure disperses

molten atomic scale clusters of Al radially outward at a velocity of 100 250 m/s [11].

The clusters then react quickly, on the order of picoseconds (10-12

s), with the oxidizer

because they no longer have an oxide shell [11].

Aluminum meltsbefore shellfracture

Small size moltenaluminum clustersdisperse from anunloading wave at highvelocity

b) Nano:

a) Micron:

Initial alumina shell and aluminumcore

Growingalumina shell

fractures inalumina shell

Healeddefectivealumina shell

Characteristic time

Spallatingalumina shell

Figure 3. Oxidation of micron (a) and nano (b) Al particles. Adapted from [11].


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7

CHAPTER II

THEORY

2.1 Equation for flame rate vs. particle parameters

Levitas et al showed that the major parameter controlling the reactivity of nano-Al

thermites is the volume fraction of Al melt (f) at the time of oxide shell spallation, which

occurs when the maximum tensile stress at the internal surface of the shell reaches the

ultimate tensile strength of the oxide, -h = u. The maximum tensile stress (h) in the

alumina oxide shell at r=R was derived by Levitas et al [11] for the case of a two-layer

sphere with an internal radius R and external radius R~

using elasticity theory. The first

term in Equation 1 represents the stress due to the thermal expansion of solid Al, and Al

melting in the particle core. The second term represents the stress caused by surface

tension at the aluminum-alumina interface. The third term accounts for the surface

tension in the oxide shell at the alumina-gas interface and the pressure of the gas on the

oxide shell.

))2(32()2()2(4))(2(6 212122

2122

3

21212

3

hRH

KKGKGmRp

RH

KGm

H

KKGm gii +++

++

++

=

[1]

where

))()()1()(( 11011m

m

m

m

s

m

si fTTfTTfTT +++= , [2]

)( 022 TTi = , [3]

sm KffKK 111 )1( += , [4]


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8

MRRRm /11/1/~

+=+== , [5]

))1((43 23

1221

3 KmKGKKmH ++= . [6]

Here Mis the relative Al core radius,R/,Kis the bulk modulus, G is the shear modulus,

is the linear thermal expansion coefficient, f is the volume fraction of Al melt in the

particle core, Tmis the Al melting temperature, T0is the oxide shell passivation

temperature, 3 m is the volumetric expansion during the melting of Al,1

and 2 are the

surface tensions at the aluminum-alumina interface and alumina-gas interface that

appears during the reaction, gp is the pressure of the gas outside the oxide shell,

subscriptss and m are for the solid and melt phases of Al, and subscripts 1 and 2 are for

Al and Al2O3, respectively. The values of the parameters in Equation 1 are given in

Table 1 [11].

Table 1. Material parameters at melt temperature T=Tm [11]

Substituting Equation 1 into oxide shell fracture criterion -h = u , one obtains the

equation for the volume fraction of melt necessary to cause fracture of the shell.

02 =++ CBfAf [7]


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9

where

22

3)2(6 KMGmKA m += , [8]

us

m

g

KmGKMKGTKKMm

KGpMKmB

)34()()2(6

)32)(2)1((

2

3

222

3

222

2

++++

+++=

[9]

)))1((43(

))2(32)(2)1((

)2(4)2(6

2

3

2

3

2

2222

2

221

3

22

3

KmKGmKKM

KGKGKpMm

KGmKTGKMmC

SSu

SSg

S

++

++++

++=

[10]

Here K is the difference in bulk moduli between liquid and solid Al, SK is the bulk

modulus of solid Al, T is the difference between the melting temperature of Al, Tm, and

the passivation temperature, T0, at which the initial oxide shell was formed, and is

the difference in linear thermal expansion coefficients between solid Al and alumina [13].

The flame velocity can be expressed in terms of the volume of the melt as follows

AACBBVfVV 2/4( 2maxmax +== for 10 < f , [11]

maxVV = for f =1,

where Vmax is the maximum velocity (950 m/s) that can be achieved in the experimental

setup under study [13]. Levitas et al used Equations 7 through 11 to plot the relative

flame velocity (V/Vmax) as a function ofMfor different values ofu, beginning with u =

th = 11.33 GPa. Figure 4 shows a comparison between these theoretical values and

experimental data given in Table 2 that was previously obtained.


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10

Table 2. Experimental data corresponding to Figure 4 [13]

Figure 4. Correlation between experimental data and melt dispersion model [13].

It is evident that the experimental data correlates well with the melt dispersion

theory.

Diameter Shell Thickness Average flame velocity

nm nm M m/s

44 2 10.00 950

50 1 24.00 789.3

80 2 19.00 948

110 1.5 35.67 721

121 2 29.25 765

120 4 14.00 947


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11

2.2 Distribution function for particle parameters

This study explores the effects of varying key parameters within the melt

dispersion mechanism. This analysis is based on probability density functions (pdfs) for

the particle size distribution (p), oxide shell thickness distribution (t), and oxide shell

strength distribution (s). The equations for these normal distributions used for this

analysis are given by

=

2

2

1exp

2

1),(

MM

M

Mxxp

, [12]

=2

2

1exp

2

1),(

xxt , [13]

=

2

2

1exp

2

1),(

u

xxs , [14]

wherex is the integration variable, and M , , and are the standard deviations

corresponding to parameters M, , and u, respectively. These pdfs give the probabilityp

dx, t dx, ands dx that a given particle with a parameterx will lie in the interval dx aboutx.

The integral of each pdf fromx = to is unity. However, the only valid values

for the parameters M, , and u used in our analysis are positive (>0), so each of the pdfs

can only be integrated from 0 to . Each pdf can be multiplied by a variable that

depends onx. The variables of interest in this analysis are the total volume of the

spherical Al core (v) and the volume of the Al core that is molten at oxide shell spallation

(vm).


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12

33

3

4Mv = [15]

vffMvm

== 33

3

4 [16]

Integration with respect tox from 0 to then returns the mean value of variables v and

vm.

dxxvxpvp )()(0

= [17]

dxxvxpv mmp

)()(0

= [18]

dxxvxtvt )()(0

= [19]

dxxvxtv mmt )()(

0

= [20]

dxxvxsvs )()(0

= [21]

dxxvxsv mms )()(

0

= [22]

Here superscriptsp, t, ands correspond to parameters M, , and u, respectively.

Dividing the average volume of molten Al at oxide shell spallation by the average total

volume of the spherical Al core ( vvm ) gives the average fraction of Al melt ( f ) needed


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13

to cause oxide shell spallation which is equivalent to relative flame velocity, V/Vmax,from

Equation 11. Dividingmv by v also cancels any deviation from unity caused by making

0 the lower limit of integration.

maxVVvvfp

m

pp == [23]

maxVVvvft

m

tt == [24]

maxVVvvfs

m

ss == [25]

Equation 23 can be manipulated to account for relative flame velocity varying due

to distributions of bothR and . In order to do this, both the numerator and the

denominator of Equation 23 is multiplied by the pdf of oxide shell thickness distribution

(t(y, ), shown in Equation 13. A new integration variable (y) is introduced to designate

the integration variable related to .. The equation, shown in Equation 26, must then be

integrated twice.

( ) ( )

( ) ( )dydxyx

ee

dxdyTyxfyxee

VVvvfy

M

Mx

u

y

M

Mx

II

m

IIII

M

M

===

0 0

33

5.0

2

)(

5.0

2

)(

0

0

33

0

5.0

2

)(

5.0

2

)(

max

3

4

22

),,,(3

4

22

2

2

2

2

2

2

2

2

[26]


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14

CHAPTER III

RESULTS AND DISCUSSION

3.1 Effect of distribution of particles geometric parameters on flame propagation

rate

Values of relative flame velocitywere plotted versus relative standard deviation

( M/M) for values ofMranging from 5 to 500 in Figure 5 using Equation 23.

( )

( )

===

0

33

5.0

2

)(

0

33

0

5.0

2

)(

max

3

4

2

),,,(

3

4

2

2

2

2

2

dxxe

dxTxfxe

VVvvf

M

Mx

u

M

Mx

p

m

pp

M

M

[23]

Figure 5. Relative flame velocity as a function of relative standard deviation for variations inR.

0.2 0.4 0.6 0.8 1

0.5

0.6

0.7

0.8

0.9

1

30

40

20

10

60

100

500

M=5

RelativeFlameVelocity(V/V

max)

Relative Standard Deviation (/)


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15

For this case, Tm =933.67 K, T0 = 300 K, =2 nm and u=11.33 GPa. The only

parameter that varies is M, which is assumed to have a standard normal distribution with

standard deviation m. Since Mis equal toR/, and is held constant,R alone varies.

The actual upper integration limit used to evaluate this equation and all subsequent

integrals with an upper limit of was the nominal value of the varying parameter plus

seven standard deviations (e.g. for this case M+7 M). Above this the effects were

negligible (below .00001%).

Figure 6 shows the relationship between V/Vmaxand relative standard deviation

derived from Equation 24.

( )

( )

===

0

33

5.0

2

)(

0

33

0

5.0

2

)(

max

3

4

2

),,,(3

4

2

2

2

2

2

dxxMe

dxTxMfxMe

VVvvfx

u

x

t

m

tt

[24]

Values ofTm, T0, u, and were kept the same as before, but this timeR was held

constant for each value ofM, and was assumed to have a standard normal distribution.


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16

Figure 6. Relative flame velocity as a function of relative standard deviation for variations in .

It is evident that for the same nominal value ofM, variations inR affect the

relative flame velocity more than variations in . This is due to the fact ifR varies, the

range ofMvalues is twice as large as when varies. For example, for relative standard

deviation of .5, whenR varies, the range ofMvalues is .5Mto 1.5M, but when varies,

Mranges from .75Mto 1.25M. Also, when is allowed to vary, it is well below the

critical value of equal to 7.7 nm below which the oxide shell strength approaches

maximum strength [11].

Equation 26 can be used to model relative flame velocity assumed to vary due to

distributions of bothR and . Simplifying Equation 26 results in

0.2 0.4 0.6 0.8 1

0.5

0.6

0.7

0.8

0.9

1

30

40

20

60

100

500RelativeFlameVeloc

ity(V/Vmax)

M=5 & 10

Relative Standard Deviation (/)

25


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17

dyyedxxe

dxdyTyxfyxee

VVvvfyMx

M

u

yMx

MII

m

IIII

M

M

===

0 0

32

)(

32

)(

0

0

33

0

2

)(

2

)(

max

2

2

2

2

2

2

2

2

3

2

),,,(3

2

[27]

Figure 7 shows how V/Vmax varies with Mfor various values of.

Figure 7. Relative flame velocity as a function of relative particle size for various .

Because of the small change in flame propagation velocity as increases beyond

the scope of the melt dispersion mechanism, relative flame propagation velocity is

assumed to vary uniformly at the average relative flame velocity between oxide shell

thicknesses = 2 nm and 200 nm (i.e. = 3.8 nm) above M=18.9. Below M=18.9,

50 100 150 200

0.5

0.6

0.7

0.8

0.9

1

Relative particle size (M)

Relativeflamevelocity(V/Vmax)

=2 nm

=3.8 nm

=200 nm

18.9

=3.8nm


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18

relative flame velocity is not affected by changes in . Based on these assumptions,

Equation 27 is simplified to

dyyedxxe

dyyeTxfxexe

VVvvfyMx

M

y

u

MxMx

MII

m

IIII

M

MM

+===

0 0

32

)(

32

)(

3

0

2

)(

0 9.18

0

32

)(

32

)(

max

2

2

2

2

2

2

2

2

2

2

3

2

),,8.3,(3

29.18

[28]

Reducing this equation results in

+

===

0

32

)(

0 9.18

0

32

)(

32

)(

max

2

2

2

2

2

2

9.18

),,8.3,(

dxxe

Txfxexe

VVvvfM

MM

Mx

u

MxMx

II

m

IIII

. [29]

which is independent of variations in .

3.2 Effect of passivation temperature (T0) and relative particle radius (M) on flame

propagation rate

Another major parameter affecting flame velocity is oxide shell formation

temperature, T0. The effects of increasing T0 for different values ofMwere studied using

Equation 23.

( )

( )

===

0

33

5.0

2

)(

0

33

0

5.0

2

)(

max

3

4

2

),,,(3

4

2

2

2

2

2

dxxe

dxTxfxe

VVvvf

M

Mx

u

M

Mx

p

m

pp

M

M

[23]


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19

Relative flame velocity was plotted against relative standard deviation ofR for the

same values ofTm, , and u as before for values of T0 ranging from 300 K to 700 K.

Figure 8. Relative flame velocity as a function of relative standard deviation forT0 = 300 K.

0.2 0.4 0.6 0.8 1

0.5

0.6

0.7

0.8

0.9

1

30

40

20

10

60

100

500

M=5T0 = 300 K

RelativeFlameVelocity(V/Vmax)

Relative Standard Deviation (M/M)


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20



0.2 0.4 0.6 0.8 1

0.6

0.7

0.8

0.9

1

T0 = 400 K

30

40

20

10

60

100

500

M=5



0.2 0.4 0.6 0.8 1

0.7

0.75

0.8

0.85

0.9

0.95

1

T0 = 500 K

30

40

20

10

60

100

500

M=5




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21



0.2 0.4 0.6 0.8 1

0.92

0.94

0.96

0.98

1

T0 = 700 K

40

60

100

M=5 10 2030

500


RelativeFlameVelocity(V/Vm

ax)

150

300

200

0.2 0.4 0.6 0.8 1

0.8

0.85

0.9

0.95

1

30

40

20

10

60

100

500RelativeFlameVelocity(V/Vmax)

M=5


T0 = 600 K


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22

The relative flame velocity increases dramatically with an increase in T0. As a

comparison, for T0 =300 K, complete melting of Al begins for values of nmM 6.19 ,

but for T0 = 700 K, complete melting begins for nmM 4.112 . This is due to the

decrease in difference between T0 and the melting temperature of Al (Tm). This smaller

difference causes less pressure build up in the Al particle due to differences in inelastic

strains of Al and Al2O3 (see Equations 1, 2, and 3) and allows a higher percentage of Al

to melt before oxide shell spallation.

3.3 Effect of alumina shell strength ( u), its distribution, and relativeparticle radius(M) on flame propagation rate

The effects of variations in oxide shell ultimate strength, u, for different values

ofMwere studied using Equation 25.

( )

( )

===

0

33

5.0

2

)(

0

33

0

5.0

2

)(

max

3

4

2

),,,(3

4

2

2

2

2

2

dxMe

dxTxMfMe

VVvvf u

u

x

x

s

m

ss

25

Relative flame velocity was plotted against relative standard deviation ofu for

Tm=933.67 K, =2 nm, T0=300 K, and various values ofMin Figures 13 to 17.


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23

Figure 13. Relative flame velocity as a function of relative standard deviation foru = th.

Figure 14. Relative flame velocity as a function of relative standard deviation foru = th/1.1

0.1 0.2 0.3 0.4

0.5

0.6

0.7

0.8

0.9

1


Relative Standard Deviation (sg/sg)

25

40

10

60

100

500

20

M=5

u = th =11.33 GPa

16

0.1 0.2 0.3 0.4

0.4

0.5

0.6

0.7

0.8

0.9

1

RelativeFlameVelocity

(V/Vmax)


25

40

10

60

100

500

20

M=5

u = th/1.1

16


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24

Figure 15. Relative flame velocity as a function of relative standard deviation foru = th/1.2

Figure 16. Relative flame velocity as a function of relative standard deviation foru = 1.1th.

0.1 0.2 0.3 0.4

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



25

40

10

60

100

500

16

20

M=5

u = th /1.2

0.1 0.2 0.3 0.4

0.6

0.7

0.8

0.9

1

RelativeFlameVelocity(V/Vma

x)


40

60

10

100

500

1000

16

25

M=5

u

= 1.1th


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25

Figure 17. Relative flame velocity as a function of relative standard deviation foru = 1.2th.

The flame velocity also varies directly with u. When u increases, more pressure

is required to cause oxide shell spallation so more Al melts before -h = u. This increase

in pressure causes a greater imbalance of pressure in the Al sphere immediately following

spallation which produces larger tensile stress in the unloading wave and promotes

dispersion of any solid Al left in the core [12]. Particles with a ratio of M=1000 reach a

higher relative velocity than M=40 particles by increasing u by only 20%, from th to

1.2th. For higher values ofu, the relative flame velocity increases much more rapidly

with increasing M than it does at lower values ofu.

0.1 0.2 0.3 0.4

0.6

0.7

0.8

0.9

1

RelativeFlameVelocity(V/V

max)


u = 1.2th

40

60

10

100

500

1000

16

25

M=5


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26

3.4 Effect of mixing particles with different relative particle

radius (M) on flame propagation rate

The effect of mixing particles with multiple Mvalues (M1 and M2) was also

examined using the MDM model. The volume of the Al contained in the core of each

particle is given by

3

1

3

113

4MvA = [30]

3

2

3

223

4MvA = , [31]

where subscripts 1 and 2 represent particles M1 and M2, respectively. The total amount of

Al within the mixture is given by

AA vnvnv 2211 +=

[32]

where n represents the number of particles of each relative size in the mixture. Using

Equation 11 to find the volume fraction of melt required for oxide shell spallation, the

total amount of Al that melts within the mixture can be given by

),(),( 22221111 MfvnMfvnvAA

m +=

. [33]

Because of how extremely small these particles are, it is impractical to count the

number of each size of particle. Before mixing the particles with different Mvalues, the

mass of each can be determined and from that, one can find the total mass of Al involved

in the reaction along with the mass percent of each particle size. This can be

accomplished by using the following equations

)( 111111SSAA vvncmm +== , [34]


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27

)()1( 222222SSAA vvnmcm +== , [35]

SSAA vv

cmn

1111

1 +

=, [36]

SSAAvv

mcn

2222

2

)1(

+

=

, [37]

where m is the total mass of all the particles, m1and m2 are the masses of particles with

relative size M1and M2,c is the mass fraction ofM1 particles, (1-c) is the mass fraction of

M2 particles, A

and S

are the densities of Al and Al2O3, respectively, for each relative

particle size. After determining v and vm,the volume fraction of Al melt at oxide shell

spallation is by definition

maxVVvvf m ==

. [38]

Relative flame velocity was plotted in Figures 18 through 21 as a function of the

mass fraction of the particle with the smallerMvalue (M1). For this case, Tm =933.67 K,

T0 = 300 K, u=11.33 GPa, and =2 nm forM1and =6 nm forM2. The density of Al

and Al2O3 are 2.7g/cm3

and 3.05g/cm3, respectively.


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28

Figure 18. Relative flame velocity as a function of mass fraction for M2 = 40.


0.2 0.4 0.6 0.8 1

0.7

0.75

0.8

0.85

0.9

0.95

RelativeFlameVelocity

(V/Vmax)

M2 = 40

30

40

25

35

M1=20

Mass Fraction M1 (c)

0.2 0.4 0.6 0.8 1

0.6

0.7

0.8

0.9


Mass Fraction M1 (c)

60

80

M2 = 80

30

45

M1=20


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29



0.2 0.4 0.6 0.8 1

0.6

0.7

0.8

0.9

RelativeFlameVelocity(

V/Vmax)

M2 = 150

50

100

35

150

M1=20

Mass Fraction M1

(c)

0.2 0.4 0.6 0.8 1

0.5

0.6

0.7

0.8

0.9

RelativeFlameVelocity(V/Vm

ax)

Mass Fraction M1

(c)

250

1000

M2 = 1000

40

100

M1=20


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30

Mixing particles that have lowerMvalues with particles that have higherM

values will greatly increase the mixtures flame velocity compared to the flame velocity

of the particles with the higherMvalue. Moore et al (2006) showed that a mixture of Al

particles that consisted of 70% particles withR=38 nm (M=8.3) and 30% particles with

R=2 and 10 m provided essentially the same flame velocity as 100% nanoparticles [15].

Figure 22 shows the relationship between the mixtures with M1=20 taken from Figures 18

through 21.


It is evident that as the percentage of smaller particles increases, the flame

velocity approaches the flame velocity of the smaller particles.

0.2 0.4 0.6 0.8 1

0.5

0.6

0.7

0.8

0.9

M1 = 20

M2=40

80

150

1000

Relative

FlameVelocity(V/Vmax)

Mass fraction M1 (c)


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31

CHAPTER IV

CONCLUSIONS

The effects of key parameters on relative flame velocity within the scope of the

melt dispersion mechanism were examined. Flame propagation velocities were

statistically evaluated using probability density functions for particle size, oxide shell

thickness, and oxide shell strength. It was shown that flame propagation velocity

depends sensitively on relative particle size, the relative particle size distribution, and

oxide shell formation temperature. It was also shown to depend upon particle size, oxide

shell thickness, and oxide shell strength, but not as sensitively. Variations in particle size

were shown to have a greater effect on flame velocity than variations in oxide shell

thickness. Mixing smaller nanoparticles with larger nanoparticles showed to increase the

mixtures flame propagation velocity compared to the larger nanoparticles and allowed it

to approach the maximum flame velocity.

The results presented here suggest that better reproducibility of the flame velocity

may be achieved experimentally by selecting a material with a narrow relative particle

size distribution. It also suggests that a combination of increased oxide shell formation

temperature and increased oxide shell strength could be used to maximize the flame

velocity in particles with increased relative particle size.

Research in the field of nanothermites continues to be very promising and

suggests ways to approach the flame propagation velocities of high explosives with the

much safer, less sensitive, and much higher energy producing nanothermites. Continued

research and experimentation is needed to explore the full extent of the melt dispersion


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32

mechanism, optimize these Al nanothermite reactions, and extend this mechanism to

other nanothermites.


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33

REFERENCES

[1] Asay B. W., Son S. F., Busse, J. R., and Oschwald, D. M. Propellants, Explosives, and Pyrotechnics,29, 216 (2004).

[2] Bazyn, T., Krier, H., and Glumac, N. Combustion and Flame,145, 703 (2006).

[3] Bockmon, B. S., Pantoya, M. L., Son, S. F., Asay, B. W., and Mang, J. T.

Journal of Applied Physics, 98, 064903 (2005).

[4] Bruzostowski, T. A. & Glassman, I.,Heterogeneous Combustion,edited by H. G. Wolfhard, I. Glassman, and L. Green, Jr., Academic,

New York, (1964).

[5] Dlott, D. D.,Energetic materials: Initiation, Decomposition and Combustion,Elsevier, New York, 125-192,(2003).

[6] Fischer, S.H., Grubelich, M.C. Theoretical Energy Release of Thermites,

Intermetallics, and Combustible Metals.Proceedings of the 24thInternational

Pyrotechnics Seminar(1998).

[7] Granier, J. J., Combustion characteristics of A1 nanoparticles and

nanocomposite A1+MoO3 thermites.Doctoral dissertation, Texas TechUniversity (2005).

[8] Granier, J. J. and Pantoya, M. L., Combustion and Flame, 138, 373 (2004).

[9] Khler, J., and Meyer, R.,Explosives,Fourth Edition, VCH Publishers, New

York, (1993).

[10] Kuo, K. K.,Principles of Combustion, Second Edition, Hoboken, New Jersey:John Wiley & Sons, 734-779,(2005).

[11] Levitas, V. I.; Asay, B. W.; Son, S. F.; Pantoya, M. L.Journal of Applied Physics,

101, 083524, (2007).

[12] Levitas, V. I.; Asay, B. W.; Son, S. F.; Pantoya, M. L.Applied Physics Letters,89, 071909,(2006).

[13] Levitas, V. I., Pantoya, M. L., and Dikici, B., Melt Dispersion versus Diffusive

Oxidation Mechanism for Aluminum Nanoparticles: Critical Experiments andControling Parameters (in press).


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[14] Menikoff, R. and Kober, E., Compaction waves in granular HMX, Tech. Rep.LA-13546-MS, Los Alamos National Lab., (1999).

[15] Moore, K., Combustion behaviors of bimodal aluminum size distributions in

thermites. Master's thesis, Texas Tech University (2005).

[16] Pantoya, M. L. and Granier, J. J.,Propellants, Explosives, and Pyrotechnics30,53 (2005).

[17] Plantier, K. B., Pantoya, M. L., and Gash, A. E., Combustion and Flame, 140, 299(2005).

[18] Rai, A., Lee, D., Park, K., and Zachariah, M.,Journal of Physical Chemistry B108, 14793 (2004).

[19] Sanders, V.E., Busse, J.R., and Son, S.F. Environmentally Responsible

Percussion Primers for Small and Medium Caliber Ammunition.Proceedings ofthe 39thJANNAF Conference, Colorado Springs, CO (Dec. 2003).

[20] Son, S. F., W. C. Danen, W. C., B. S. Jorgensen B. S.,, B. W. Asay, B. W., J. R.

Busse, J. R., and Pantoya, M. L., Defense Applications of Nanomaterials,ACS Symposium Series 891, 227240 (2005).

[21] Wang, L.L., Munir, Z.A., & Maximov, Y.M.. Review thermite reactions:

Their utililzation in the synthesis and processing of materials,Journal ofMaterials Science, 28, 3693-3708, (1993)

[22] Watson, K., Fast Reaction of Nano-Aluminum: A Study on Fluorination versus

Oxidation, Masters Thesis, Texas Tech University (2007).

[23] Zinovev, V. E.,Handbook of Thermophysical Properties of Metals at HighTemperatures,Nova Science, New York, 140 (1996).


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