[American Institute of Aeronautics and Astronautics 44th AIAA/ASME/SAE/ASEE Joint Propulsion...

1 American Institute of Aeronautics and Astronautics

Deflagration-To-Detonation-Transition (DDT) in Detonating Energetic Components

Lien C. Yang*

La Canada Flintridge, CA 91011

Under temperatures >1500oK and pressures > 100 MPa, a thin supercritical gas layer of thickness ~50 μm can form on an insensitive high explosive (HMX) surface. Confinement induced fast explosion in ~10 ns produces pressure > 1.0 GPa and facilitates the DDT. Causes of the failure to observe DDT in conventional combustion experiments are analyzed and interpreted. Good confinement by a flyer-plate can by-pass the deflagrating phase for efficient detonation initiation. Prompt detonation in laser initiation is due to in-depth heating, and, at short wavelengths, direct bond breakage. Rapid DDT of lead azide is due to its explosion before the reaction’s rate-limiting liquid-state transition can occur. Applicability of Arrhenius kinetics in heat induced decomposition is reviewed. Self-heating and adiabatic compression effects are assessed. Trends of thermophysical properties of large organic molecules under high pressures and temperatures are illustrated.1

Nomenclature

A = Frequency factor in the Arrhenius rate equation, s-1

Ai = A for the ith process in a multi-process reaction A’ = Time dependent energy density per unit area in the reaction zone (Eq. 10) B = Coefficient of stress-induced growth of the reaction zone (Eq. 10) C = Mean square velocity of molecules in gaseous phase c = Mean velocity of molecules in gaseous phase Cp = Specific heat at constant pressure CP = Pentaammine(5-cyano-2H-tetrazolato-N2)cobalt(III)perchlorate BNCP = Tetraammine-cis-bis(5-nitro-2H-tetrazolato-N2)cobalt(III)perchlorate D = Diameter of a molecule DBX = Copper(I)5-nitrotetrazolate E = Energy per unit volume in gaseous phase •

E = Rate of energy release per unit volume of HMX due to decomposition Ea = Activation energy in the Arrhenius rate equation, kcal/mol (Ea)i = Ea for the ith process in a multi-process reaction Eo = Explosion energy per unit mass of HMX E1 = Total energy in a laser pulse ΔHm = Enthalpy of melting per mole ΔHs = Enthalpy of sublimation per mole HMX = Cyclotetramethylene tetranitramine, C4H8N8O8 HNS = Hexanitrostilbene, C14H4N6O12 k = Reaction rate constant in the Arrhenius rate equation K = Thermal conductivity, cal/cm-s-oK K1 = K of the condensed (solid/liquid) phase K2 = K of gaseous phase kB = Boltzmann’s constant l = Thermal penetration depth or liquid layer thickness M = Molecular weight of HMX = 296.17 n = Number of molecules per unit volume of un-reacted material at a given time t ni = n for the ith process in a multi-process reaction no = Initial value of n P = Pressure Pc = Critical Pressure PETN = Pentaerythritol tetranitrate, C5H8N4O12

* Senior Member, AIAA

44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit21 - 23 July 2008, Hartford, CT

AIAA 2008-4628

Copyright © 2008 by L. C. Yang. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


R = Universal gas constant = 8.314 J/mol-oK = 1.988 cal/mol-oK R113 = 1,1,2-Trichloro-1,2,2-trifluoroethane RDX = Cyclotrimethylene trinitramine, C3H6N6O6

= Rate of burning t = Time Δt = Reaction completion (either vaporization or decomposition) time t1 = Decomposition completion time (n/no = 0.01 or 0.001) tr = An arbitrary reference time T = Temperature Tb = Boiling temperature Tc = Critical temperature Tf = Flame temperature Tfm = Maximum value of Tf Tm = Melting temperature Ts = Surface temperature of liquid HMX under burning V = Propagation velocity in DDT tube experiment v1 = Speed of sound in condensed HMX v2 = Speed of sound in gaseous HMX W = Laser power absorption per unit volume in insensitive high explosive, w/cc x = Axial position in a DDT tube experiment xo = Thickness of the supercritical HMX layer adjacent to the condensed phase x1 = Boundary growth in the condensed HMX phase adjacent to the HMX reaction zone x2 = Boundary growth in downstream confinement adjacent to the HMX reaction zone φ = Laser energy absorbed per unit area, or energy released in reaction zone per unit area δ = Thickness of laser absorption layer γ = Specific heat ratio Γ = Energy to pressure correlation coefficient, P = Γ*E κ = Diffusivity, cm2/s κ1 = κ in the condensed HMX phase adjacent to the HMX reaction zone κ2 = κ in the downstream confinement adjacent to the HMX reaction zone ρ = Density, g/cc ρc = Critical density ρo = Crystalline density of HMX ρ1 = Average ρ in the condensed HMX phase adjacent to the HMX reaction zone ρ2 = Average ρ in the downstream confinement adjacent to the HMX reaction zone σ = Stress generated in the condensed phase (liquid or solid) HMX τ = Laser pulse duration

I. Introduction

Detonation in insensitive high explosives is a phenomenon characterized by high pressures, propagation speeds and reaction rates. In order to initiate it with a low level stimulus, a transition is usually required. Over decades of experimentation, techniques have been adopted for the reduction and control of the transition for greater application reliability. Examples are the lead azide-based explosive train, exploding bridge-wire, exploding foil and flyer plate. All of these are intended to provide a higher-level starting point for the initiation. Considerable progress has been achieved in the understanding of these initiation mechanisms in relation to the detonation analyses based on the Chapman-Jouguet theory1. However, DDT initiated by a low-level and slow-rate stimulus, such as heat, remains poorly understood, and its very existence reduces the good safety intent of adoption of insensitive high explosives. Because of cook-off is a common accident scenario, DDT has been under intensive study in recent decades. Figure 1 briefly illustrates the classical work of Bernecker and Price2-6. Fig. 1a) shows the DDT tube used in their experiments. It is essentially a sealed steel cylinder (except Ref. 6 where plastic tubes were used), containing an explosive column ~ 1.0 cm in diameter and ~30 cm long, using heavy walls for robust confinement. The tube was loaded with insensitive high explosives of variable composition, loading densities, porosity and particle size. The initiation was by a B/KNO3 pyrotechnic igniter sometimes with a booster (“loader”) added. In spite of the fact that B/KNO3 is not optimal for a fast pressure rate or high gaseous output, DDT was produced reliably in the set up.


200, 8.25E-03

450, 230

12.16, 0.00001323

430, 114

440, 132

-50

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0 200 400 600t , us

x , m

m

t , μs

Convective Burning

Deflagration

Detonation

DDT Onset

Post Convective

Burning200, 8.25E-03

450, 230

12.16, 0.00001323

430, 114

440, 132

-50

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x , m

m

t , μs

Convective Burning

Deflagration

Detonation

DDT Onset

Post Convective

Burning

a) b)

1.32E-05, 0.0068 114, 0.1118

8.25E-03, 0.1118

132, 28.4230, 28.4

0.001

0.01

0.1

1

10

100

1.E-05 1.E-03 1.E-01 1.E+01 1.E+03

x , mm

P , G

Pa

DeflagrationConvective Burning

Detonation

DDT Onset

1.32E-05, 0.0068 114, 0.1118

8.25E-03, 0.1118

132, 28.4230, 28.4

0.001

0.01

0.1

1

10

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1.E-05 1.E-03 1.E-01 1.E+01 1.E+03

x , mm

P , G

Pa

DeflagrationConvective Burning

Detonation

DDT Onset

c) 230.00, 7.730.02, 0.49

1.3E-05, 2.2E-06

8.3E-03, 8.5E-05

132.00, 7.73

114.00, 0.49

1.E-06

1.E-04

1.E-02

1.E+00

1.E-05 1.E-03 1.E-01 1.E+01 1.E+03

x , mm

v , m

m/u

sV

, mm

/μs

x , mm

Deflagration

Convective Burning

Detonation

DDT Onset

230.00, 7.730.02, 0.49

1.3E-05, 2.2E-06

8.3E-03, 8.5E-05

132.00, 7.73

114.00, 0.49

1.E-06

1.E-04

1.E-02

1.E+00

1.E-05 1.E-03 1.E-01 1.E+01 1.E+03

x , mm

v , m

m/u

sV

, mm

/μs

x , mm

230.00, 7.730.02, 0.49

1.3E-05, 2.2E-06

8.3E-03, 8.5E-05

132.00, 7.73

114.00, 0.49

1.E-06

1.E-04

1.E-02

1.E+00

1.E-05 1.E-03 1.E-01 1.E+01 1.E+03

x , mm

v , m

m/u

sV

, mm

/μs

x , mm

Deflagration

Convective Burning

Detonation

DDT Onset

d)

Figure x a) DDT Tube Experiment; Illustrative Behaviors: b) x-t Curve; c) P-x Curve; d) V-x curve

Steel Confinement

BKNO3 Igniter

In-Situ Instrumentation Probes

Explosive Column

x0 L

Steel Confinement

BKNO3 Igniter


Explosive Column

Steel Confinement

BKNO3 Igniter


Explosive Column

x0 L

a)

Figure 1 a) DDT Tube Experiment; Illustrative Behaviors: b) x-t Curve; c) P-x Curve; d) V-x Curve


Figure 2 Test Arrangements for 20-ns Laser Initiation Experiments in Ref. 12

Stee

l Witn

ess

Plat

e

Cylindrical Steel Sleeve

Explosive Column

0.0508-mm-Wide Streak Camera View Slit

Borosilicate Window

Laser

Stee

l Witn

ess

Plat

e

Cylindrical Steel Sleeve

Explosive Column

0.0508-mm-Wide Streak Camera View Slit

Borosilicate Window

Laser

Two types of in-situ instrumentation were used. An ionization probe recorded the arrival of the high temperature “convective burning” flame and a wire strain gauge recorded the arrival of the pressure wave. It was noted that the former was advanced ahead of the latter, termed as “post convective propagation.” Figs. 1b) thru d) plot the x-t, P-t and V-t relationships as measured. The pressure plot was initially qualitative due to uncertainties in the gauge response calibration. The P-t plot shown was constructed by the author by applying the theoretical relation that P ≈ ¼*ρV2. These data provided characteristics of the latter portion of DDT in terms of convective burn, low velocity detonation and steady-state detonation. However, due to the long time duration and the limitations in the recording system (both with regards to sample rate and physical resolution), the earlier deflagration portion of the tests were not well-defined. The plots shown for the small x regime were constructed by the author from the conventional burn rate data of HMX by Boggs7, 8 in which the rate was approximately (mm/μs) = 7.6467*10-04*P(GPa) - 3.0243*10-

06. Using the improved data acquisition systems currently available, such as the Nicholet Odyssey9, longer time and higher sample rate recordings would likely provide a more defined time record similar to that acquired in the flyer plate impact experiments, (e.g. Reference 10). In Bernecker and Price’s experiments, the most important conclusions regarding DDT were: 1) The onset point was very close to the ignition point at the igniter to explosive interface; and 2) The onset pressure was below 200 MPa; 3) As the density of the explosive increased, the period of convective burn decreases, thus supporting the generally accepted concept that a higher explosive density facilitates DDT for a shorter transition time frame.

However, these results also created a serious discrepancy because in Boggs’ experiment, no detonation was ever observed up to the maximum test pressure of ~340 MPa. The data there followed the trends as indicated by the dashed arrows in the Fig. 1 instead. Interestingly, detailed DDT records covering the full range of the phenomenon were available from a series of un-related experiments. In the early 1970’s, Menichelli and Yang11-13 performed experiments on the initiation of insensitive high explosives by a 20-ns duration Q-switched ruby laser (Test configuration is showed in Fig. 2). The study focused on prompt or instantaneous initiation. Therefore, a portion of the DDT results observed were not further analyzed. Recent review and analysis of these data found that the transitions were typical of those seen by Bernecker and Price, but on a much smaller physical scale.

Some streak camera records extracted from Ref. 12 are shown in Figure 3 which covers PETN and RDX at moderate densities of 1.52 to 1.64 g/cc. Redlines are added to highlight the wave front propagation. Figures 3a), 3b) and 3d) demonstrate the reduction of convective burn region. Figures 3c), 3e) and 3f) show that the region was reduced to non-existence, but a deflagration region confined to the millimeter-wide explosives is recognizable. Records for further progressive reduction of deflagration toward a prompt or instantaneous initiation are reported in Ref. 13 and therefore they are not duplicated here. Fig. 3e) is the only record which was not overexposed by lighting. In order to preserve the detonation wave width data which essentially corresponded to the 0.0508-mm wide camera view slit, the x-t contour is not highlighted with redline. The region marked “A” in Fig. 3f) is not a true “hyper-detonation velocity” phenomenon. It resulted from the deep penetration of the laser light into the explosive. Both of these features are important in supporting the discussions in the sections that follow. The additional markings shown in the figure stand for aluminum film deposited window (Al) or plain window (Pl). These were two variations of window configurations used for the test which appeared not to produce significantly different results in initiation.

II. Criteria for Detonation The following rules of thumb on the conditions for detonation to occur and to be sustained are identified. They are based on current theoretical formulations on detonation phenomena and experimentally observed facts that may provide some insights on DDT in HMX. It will be shown in latter sections that by meeting these criteria, a high-pressure and fast-rate explosion can be created in the rather low-pressure and slow-rate combustion process. Velocity of Propagation The observed velocity is on the order of several km/s. This is orders of magnitude faster than the velocity that


can be sustained via heat transfer in normal models for the conductive burn. Figure 4 shows the heat transfer characteristics of HMX in combustion. In standard heat transfer textbooks, the thermal penetration depth into the material for a “liquid layer”, l, as it is often called in combustion models, equals (κ*Δt)0.5, where κ is the thermal diffusivity and Δt is the reaction completion time. Note that at this point of discussion, Δt could be either for a complete vaporization or complete decomposition. The regression or burning rate, is equal to l/Δt = (κ/Δt)0.5. The value of κ for crystalline HMX is calculated from thermal properties provided in Ref. 14. Due to uncertainty in this parameter as will be discussed later, two values covering a large range are also plotted. It is clear that the

a)Al

a)Al

b)

Al

b)b)

Al

d)

Al

d)d)

Al e)Pl

e)Pl PETN Al

f)

A

PETN Al

f)

PETN Al

f)

A

Figure 3 Selected Streak Camera Records of 20-ns Laser Initiation from Ref. 12

0

0.2

0.4

0.6

0.8

1.0E-09 1.0E-08 1.0E-07 1.0E-06

t , s

xo

, u

m

0

0.2

0.4

0.6

0.8

e ,

J/

cm^

2

0.23

0.47

0.70

0.94

l, μ

m

φ, J

/cm

2

κ = 5x10-3 cm2/sκ = 1x10-3 cm2/sκ = 5x10-4 cm2/s

Δ t , s

0

0.2

0.4

0.6

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1.0E-09 1.0E-08 1.0E-07 1.0E-06

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xo

, u

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e ,

J/

cm^

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/cm

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10

100

1.E-09 1.E-08 1.E-07 1.E-06

t , s

V ,

m/

s

0.1

1

10

100κ = 5x10-3 cm2/sκ = 1x10-3 cm2/sκ = 5x10-4 cm2/s

r•

Δ t , s

0.1

1

10

100

1.E-09 1.E-08 1.E-07 1.E-06

t , s

V ,

m/

s

0.1

1

10


0.1

1

10

100

1.E-09 1.E-08 1.E-07 1.E-06

t , s

V ,

m/

s

0.1

1

10



r• r•

Δ t , s

Figure 4 Characteristics of Conductive Burning of HMX: a) Liquid Layer Thickness, l, as a Function of Characteristic Heating Time, Δt ; b) Burning Rate, , as a Function of Δt.

b) a)


rates are too small as compared to the known detonation velocity of HMX. The maximum rate on the order of 20 cm/s reported by Boggs corresponds to a Δt of approximately several microseconds and a l of several micrometers. Current detonation theory is based on sonic propagation. The stress wave propagates at supersonic speed. Its associated high stress compresses and dissipates energy15, 16 into the explosive. The resultant temperature rise affects a fast in-situ energy release at the wave front, furthering the propagation rate to a level higher than that corresponding to the normal sonic phenomenon. The short run-distance (< 1.0 cm) detonation initiation threshold stress level for 1.89 g/cc HMX is available in Reference 14. The apparent level of ~4.41 GPa is applicable in the low-velocity detonation regime. The true threshold for long run-distance detonation is likely on the order of 1.0 to 2.0 GPa. These levels are approximately ten times higher than that required for the onset of DDT. In other words, in order for the pressure in deflagration to develop to the low-velocity detonation threshold stress level, an explosion event inside the deflagration process is required. Current study confirms that this indeed is the case. Reaction Zone Width, xo As stated previously, the width of the steady state detonation front is on the order of 0.0508 mm as seen in the streak camera records. This is in agreement with theoretical models for HMX, e.g., by Yang and Do15, 16. The width in a low-velocity detonation regime is not appreciatively different. In theoretical modeling, there is an apparent “pulse broadening” in this regime. However, the finding may be an artifact because the high peak of the stress wave is attenuated by energy dissipation, as mentioned in the last paragraph, resulting in an apparently more flat pulse. The original pulse width may be essentially the same. Reaction Time The reaction completion time, t1, is approximately xo/V, where V is the detonation velocity shown in Fig. 1d). As an example, by using V equals 9.11 km/s for the steady-state detonation velocity and 3.943 km/s for the low velocity detonation of 1.89 g/cc HMX14, one finds a t1 of 5.56 ns and 12.88 ns respectively. Assessment of whether the Arrhenius decomposition kinetics can support such high rates is discussed in a later section. Energy Content in the Reaction Zone Along with the zone width, this condition specifies the density of the explosive in the zone. The total energy per unit area in the zone is φ = ρxo*Eo, where Eo is the explosion energy per unit explosive mass. Eo equals 6197 J/g for HMX17. The values of φ calculated by assuming xo = l are plotted in Fig. 4a). The significance of this parameter is apparent when compared with our previous work on laser-generated stress in confined thin metal films18 which is briefly summarized in Figure 5. It can be seen that for Δt ≈ 20 ns, φ > 12 J/cm2 is required in order to generate a peak stress on the order of 2.0 GPa, necessary for successful shock initiation of explosives. This is over one order of magnitude higher than φ shown in Fig. 4a). In other words, the detonation propagation can not be sustained by the

heat transfer mechanism alone. The mass contained in the thin liquid layer formed by melting is just too small for creating a high stress level. The true value of φ required for detonation is higher than the indicated 12 J/cm2 because the acoustic impedance of glass is higher than that of the explosive and its products, resulting in a higher stress.

Ruby Laser

X-Cut Quartz Shock Gage

Thin Metal Film

Glass Window

Blue Glass Laser Absober

Ruby Laser

X-Cut Quartz Shock Gage

Thin Metal Film

Glass Window

Blue Glass Laser Absober

0

0.5

1

1.5

2

2.5

0 4 8 12Laser Fluence , J/cm^2

Peak

Pre

ssur

e , G

Pa

Films with High Light Absorption or

High Melting Temperature

Films with Low Light Absorption or Low Melting Temperature

φ , J/cm2

0

0.5

1

1.5

2

2.5

0 4 8 12Laser Fluence , J/cm^2

Peak

Pre

ssur

e , G

Pa

Films with High Light Absorption or

High Melting Temperature

Films with Low Light Absorption or Low Melting Temperature

φ , J/cm2

Figure 5 Twenty-Nanosecond-Duration Q-Switched Laser Shock Experiment. a) Test Schematic. b) Shock Stress as a Function of Laser Fluence, φ.

b) a)


Table 1 Critical Parameters According to Maksimov19

Parameter HMX RDX Tm , oC 280 205

ΔHm @ Tm, kcal/mol 16.7 ±1.0 9.0 ± 0.5 Tb , oC 471 ± 37 391 ± 33

ΔHs , kcal/mol 44.3± 0.7 31.5 ± 0.5 Tc , oC 654 ± 47 567 ± 42

Pc , MPa 2.97 ± 3 3.72 ± 4 ρc , cc 0.503 ± 0.059 0.485 ± 0.061

III. Thermal Properties of HMX

HMX has been widely used in applications and studied extensively for DDT. Therefore, ample crystalline HMX data exist for use in our current DDT case study. It is needed as inputs to the analyses of heat transfer, pressure ballistic and reaction rate in the latter sections. The applicability and limitations of these data are discussed below. Critical Properties The information was mainly contributed by Maksimov19 through theoretical calculations. It is briefly summarized in Table 1 along with that for RDX, a compound closely related to HMX. The properties resemble those commonly observed for organic compounds with large molecular weights. The rather low values of Tc and Pc indicate that a supercritical state does exist in conventional conductive burn experiments. This was a fact that Yang and Do20, 21

identified in late 1990’s in their analysis of non-electrical tube explosive transfer systems (commonly named “Thin Layer Explosive”, TLX). Unaware of Maksimov’s work at the time, they used the known physical parameters of the system to construct a model with an HMX surface temperature of 800oC and a reaction completion temperature of

609.7oC. The latter is close to Maksimov’s Tc. The model was successful in predicting the critical HMX particle size required for successful tube operation. Therefore, our previous work can serve as an independent verification of these critical parameters. Density A best fit equation for Bedrov’s22 data is as follows. The extrapolated value at ambient temperature is lower than the accepted value of 1.89 g/cm3. The measurements were obtained under saturated vapor pressure.

Therefore the equation is of limited usefulness at higher pressures. Appendix 1 indicates that under pressure of 100-200 MPa, ρ approaches the value of liquid at Tm. ρ (g/cm3)= -0.00065*T (oK) + 2.00456 550oK < T < 800oK (1) Thermal Conductivity A best fit equation for Bedrov’s22 data combined with that in Ref. 14 is as follows. The measurements were obtained under saturated vapor pressure. The equation cannot be directly used for higher pressures. K (cal/cm-s-oK) = [-0.0120*T (oK) + 15.2237]*10-4 550oK < T < 800oK (2) Specific Heat at Constant P An equation for low temperature data from ref. 14 is as follows. Data from Koshigoe-Shoemaker-Taylor23 gives a similar result. Extrapolation of the data for the liquid and supercritical regime using this equation is inaccurate. Cp (cal/g-oK) = 0.231 + 5.5x10-4*T (oC) 37oC < T < 167oC (3) Sonic Velocity Zang’s data24 provides for the velocity in β-HMX as a function of the crystalline structure from which an average value can be assessed to be ~2.74 km/s. Extrapolation of the data into high temperature and pressure regime is reasonable. Trends for Supercritical Regime In order to provide some rough guidelines for the estimation of the properties at high temperatures and pressures, a case example using existing data from R113 refrigerant is included in Appendix 1. The general trends which can be applied to HMX are discussed there.

IV. Model Scenario Description The concept of the model is simple: Under high temperatures >1500oK and pressures > 100 MPa, a thin supercritical gas layer of thickness ~50 μm can form on the surface of an insensitive high explosive such as HMX. Confinement-induced fast decomposition in ~10 ns produces an explosion pressure > 1.0 GPa facilitating the DDT. It is illustrated in Figure 6 for several different confinement configurations. The case for detonation is intent for both


Solid

/Liq

uid

b) SCZ CZ

Solid

/Liq

uid

b) SCZ CZ

Solid

/Liq

uid

b) SCZ

Solid

/Liq

uid

b) SCZ CZ

Solid

/Liq

uid

a) DZ CZ

Solid

/Liq

uid

a) DZ CZSo

lid/L

iqui

d

c) RRZ

Solid

/Liq

uid

c) RRZ

Solid

/Liq

uid

d) RRZSo

lid/L

iqui

dd) RRZ

Solid

/Liq

uid

d) RRZSo

lid/L

iqui

d

Stee

l Flye

r

e) RRZ

Solid

/Liq

uid

Stee

l Flye

r

e) RRZ

LegendsHMX Molecules

Small Molecules (H2O, CO2 , etc.)

DZ Dark Zone

RRZ Rapid Reaction Zone

CZ Combustion Zone

SCZ Super Critical Zone

LegendsHMX MoleculesHMX Molecules

Small Molecules (H2O, CO2 , etc.)

DZ Dark Zone

RRZ Rapid Reaction Zone

CZ Combustion Zone

SCZ Super Critical Zone

Figure 6 HMX Reaction Scenarios: a)Low Pressure Combustion; b) High Pressure Combustion in Nitrogen Gas; c) Detonation; d) DDT; e) Flyer Plate Initiation

steady-state and low-velocity detonations. The analyses for three key issues, namely, the existence of a supercritical gas layer, the effect of confinement and the reaction kinetics, will be presented in the sections that follow. The general methodology illustrated here for HMX can be applied to other explosives if their properties are known. Porosity is a more complex problem which requires additional work in the future. Therefore, it is not included in the model. For simplicity, the condensed phase including solid and a very thin liquid layer is shown as a single block.

In this figure, the lack of confinement in case a) and good confinement by the detonation products in case c) and by the flyer plate in case d) are apparent. However, it is implicit that under the same high pressure, case b) has poorer confinement and case d) has better confinement due to the difference between the low molecular weight of


the nitrogen gas used in the combustion experiments shown in Fig. 6b) and the heavy HMX molecules presented in the gaseous phase in the DDT configuration shown in Fig. 6d). However, the heavier HMX molecules also provide a lower speed of sound which partially offsets its dynamic confinement capability via mechanical impedance, ρ2v2.

V. Existence of a Supercritical Gas Layer It has long been recognized in combustion experiments25 that there exists a “flame stand off” or a “dark zone” at the burning surface. It is easy to see that in a propellant combustion involving multiple ingredients, a diffusion layer is necessary to allow for the mixing of vapors from the ingredients prior to the combustion reaction. For a single ingredient burn, such as pure HMX, the existence of a diffusion layer is more difficult to realize because it is controlled by the heat balance and decomposition rate rather than the mixing of vapors. Carslaw and Jaeger26 described a heat-transfer model using a simplified two-phase system between liquid (subscript 1) and gases (subscript 2) with an initial temperature Tf in the gaseous phase:

t

xerfcKK

TKT f

12/1

222/1

11

2/122

1 2 κκκκ

−−

−

+= (4)

⎥⎥⎦

⎤

⎢⎢⎣

⎡+

+= −

−

−−

−

txerf

KK

KKTK

T f

22/1

22

2/111

2/122

2/111

2/122

2 21

κκκ

κκκ (5)

The numerical solutions of the equations for two special cases are showed in Figure 7. Case a) has a small K1κ1

-1/2/K2κ2-1/2 ratio of 1.19 (Based on the averaged properties by Eqs.1-3, K1 = 7.84*10-4 cal/cm-s-oK, κ1 =

1.29*10-3 cm2/s, K2 ≈ 4.1*10-4`cal/cm-s-oK and κ2 ≈ 4.9*10-4 cm2/s); and case b), with an artificial larger ratio of 4.0.

The temperature profile in HMX combustion experiments at pressures up to 7.0 MPa was measured by Zenin27 using miniature thermocouple probes. The profiles resemble those shown in fig. 7b). In order to assess the behaviors at higher pressures, the surface temperature, Ts , the dark zone thickness, xo , and the flame temperature, Tf , are extrapolated by power law to pressures up to 100 MPa (1000 atm.) in Figure 8. The results for Ts and xo are reasonable and in agreement with follow-up analyses reported in References 8 and 45. Over-prediction in Tf was observed. Therefore the inverse power law fit was adopted referring to the full developed flame temperature of Tfm = 3278oK reported in Ref. 25. Good prediction is obtained as shown in Fig. 8d). Fig. 8b) shows the dark zone width, xo , calculated from the approximately 1/e point of Zenin’s Tf – Ts curves by assuming they are exponential. By extrapolating to P = 100 MPa, xo = 0.042 mm is obtained. Comparable values can also be obtained from the heat conduction equation: ρo ΔHs = K1(Tfm – Ts)*(1-1/e)/xo, = 7.3 cm/s @ 100 MPa from Boggs’ data, and assuming that K1 ≈ 5.6*10-3 cal/cm-s-oK. However, this value selected for K1, while it appears to be reasonable, can not be verified. In order to form a supercritical film on HMX surface, Ts needs be equal or higher than the critical temperature Tc. Several factors might have led to lower values in xo and Ts: 1) as shown in Appendix 1, at higher pressures, the differences in K between the liquid and gas phases is small. Therefore the temperature profile likely resembles that shown in Fig. 7a) with a higher Ts. Even under a conservative boundary condition of Ts = Tc, this will cause a

0

0.2

0.4

0.6

0.8

1

-2 -1 0 1 2

x

T/To

t = t = 0.25 t = 4.0

T/T f

rt22 κx ,

tr

tr

tr

0

0.2

0.4

0.6

0.8

1

-2 -1 0 1 2

x

T/To

t = t = 0.25 t = 4.0

T/T f

rt22 κx ,

tr

tr

tr

0

0.2

0.4

0.6

0.8

1

-2 -1 0 1 2

x

Ts/T

f

t = t = 0.25 t = 4

T/T f

x , rt22 κ

tr

tr

tr

0

0.2

0.4

0.6

0.8

1

-2 -1 0 1 2

x

Ts/T

f

t = t = 0.25 t = 4

T/T f

x , rt22 κ

tr

tr

tr

a) b)

Figure 7 Temperature Profile at the HMX Liquid-Gas Interface. a) With K1κ1-1/2/K2κ2

-1/2 = 1.19 and κ1/κ2 = 2.63; b) With K1κ1

-1/2/K2κ2-1/2 = 4.0 and κ1/κ2 = 2.0.


T f, o C

c )o

oT f =

T f =

T f, o C

c )o

oT f =

T f =

Ts

, oC

o

o

a)

T s

T sT

s,

oC

o

o

a)

T s

T s

xo

xo

x o ,

mm

b )

xo

xo

x o ,

mm

b )

Tfm

–T

f,

oK

d)

T f

T f

R 2

R 2

o

oTfm

–T

f,

oK

d)

T f

T f

R 2

R 2

o

o

Figure 8 Extrapolation of Zenin HMX Conductive Burn Data: Power Fit, a) Surface Temperature, Ts, versus Pressure, P; b) Dark Zone Thickness, xo, versus P; c) Flame Temperature, Tf , versus P; Inverse Power Fit, d) Tfm –Tf, versus P.


significant increase in xo. 2) the value of xo plotted in Fig. 8b) was reduced by the author at the approximately 1/e point of the temperature differential which may have been overly conservative and led to a smaller value in xo. 3) Zenin’s experiment adopted the traditional method of test cell pressurization using a large amount of nitrogen gas which might have contributed to lower Tf and xo values. The inert nitrogen does not produce heat energy and thereby has a “dilution effect” on the overall flame temperature distribution. 4) Detailed analysis of fusion is not included in Eqs.4 and 5, or in previous work20, 21. The heat of fusion, if it had been included in the analysis, could have lowered the overall temperature in the system including Ts and Tf. This is an area needs further work.

VI. Effect of Confinement on Pressure Development in a Supercritical HMX Gas Layer The problem of stress generation in a confined thin material layer with energy input/deposition was addressed

effectively in Ref. 18. Here, we generalize the formulation for the special case involving DDT phenomena. This approach is on the border line between quasi-equilibrium and non-equilibrium situations. It is consistent with our current interest, where the heating is a quasi-equilibrium process, while the detonation is a non-equilibrium process. As stated previously, HMX, which has no porosity (i.e., a crystalline state), is used in the example. The stress development is governed by the following formula (The decomposition is assumed to be instantaneous),

21 xxxEx

XEx

o

oooo

++Γ

=Γ

=ρσ (6)

tvv

dtx1111

1 ρσ

ρσ

≈= ∫ = Recession of the HMX boundary interfacing with the layer (7)

tvv

dtx2222

2 ρσ

ρσ

≈= ∫ = (8)

The value of Γ is chosen to be 0.3 because in an equilibrium gaseous system, P = n*kB*T and E = 7/2 *n*kB*T. Therefore, P = 2/7 * E ≈ 0.3*E. This was empirically verified in the JWL28, 29 theory. The formulation of x2 for the case of detonation is different. It is governed by an adiabatic expansion of the detonation products;

( )⎟⎟⎠

⎞⎜⎜⎝

⎛+≈⎟⎟

⎠

⎞⎜⎜⎝

⎛+≈

⎟⎠⎞⎜

⎝⎛

+

≈= ∫∫∫ 21

22

120

2

12

212

222

txct

vdtt

xc

vdt

tcxxvv

dtxo

t

o

o

o ρσ

ρσ

ρ

σρσ (9)

In all cases, Eq. 6 has the following form,

oxB

A+

=σ

σ ' ⇒ 0'2 =−+ AxB oσσ ⇒ B

BAxx oo

2'42 ++−

=σ (10)

The input conditions for the four cases in Fig. 6b) thru e) are summarized in Table 2 and results are plotted in Fig. 9.

t , s

N2

12.8

8 nsσ

, GPa

t , s

N2

12.8

8 nsσ

, GPa

xo , cm

0.00

508

cmσ, G

Pa

N2

xo , cm

0.00

508

cmσ, G

Pa

N2

Figure 9 Pressure in HMX Reaction Zone for Different Scenarios. a) As a Function of Time, t, with Fixed Initial Zone Layer Thickness xo = 0.0508 mm and Instantaneous Reaction; b) As a Function of xo at t = 12.88 ns, the t1 for 3.94 mm/μs Low Velocity Detonation.

a) b)

Recession of confinement boundary interfacing with the layer, which can be inert gas, gaseous HMX or flyer plate


The effectiveness of a steel plate in confining the reacted HMX for a high stress generation is evident. In the critical regime for DDT, t = 12.88 ns and xo = 0.0508 mm, the stress calculated for the case of burn in nitrogen gas is indeed about 15% lower than that for the case of DDT. In addition, there is an important trend indicated in Fig. 9a). When t1 increases (e.g., due to a lower temperature, Ts), the peak stress decreases following the stress curves in the longer time regime. The Ts in the HMX combustion experiment is low on the order of 600oC as shown in Fig. 8a). Thus, the peak stress may occur in the 10-6 s time frame20, 21 and with a magnitude of ~0.1 GPa —a level that can not sustain detonation propagation. The value of xo is on the order of 0.042 mm as shown in Fig. 8b). Thus, according to Fig. 9b), a lower peak stress (~6.2%) is also indicated. All these factors might have contributed to the failure to produce detonation in this type of experiment. In this formulation, the highest stress value is ~3.5 GPa, about one order of magnitude lower than the established detonation pressure15, 16. This is because of the limitations in the quasi-equilibrium condition. For detonation, the relationships of P = n*kB*T and E = 7/2 *n*kB*T are no longer applicable. The energy and momentum become uni-directional instead of isotropic. In this case, the pressure would be 7-times higher, or 9-times if Hugoniot1 compression (~30% in volume) is included in the calculation, resulting in ~32 GPa. One might suspect that this transition from quasi-equilibrium to non-equilibrium behavior depends upon the aspect ratio of the thin supercritical

Table 2 Input Numerical Values for Results Showed in Fig. 9 Case ID b) c) d) e)

Description Combustion in Inert Gas (N2)

Detonation DDT Flyer Plate Initiation◊

Key Comments N2 does not add energy and is a poor confinement

Dense products provide good confinement

Dense gaseous HMX is good confinement

Best confine-ment

Legend

Formulas A’ = ooo Ex ρΓ A’

Inputs Γ = 0.3; xo = 0.0508 mm; Eo17

= 6197 J/g; ρo = 1.9 g/cc

Formulas

B =

tv11

1ρ

+ tv22

1ρ

MRTC 3

=

32γCv =

086.1/Cc =

B = tv11

1ρ

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

21 2

12

txct

v oρ

MRTC 3

=

32γCv =

086.1/Cc =

B =

tv11

1ρ

+ tv22

1ρ

MRTC 3

=

32γCv =

086.1/Cc =

B =

tv11

1ρ

+ tv22

1ρ

B

Inputs

M = 28 T = 1923.16oK

γ = 1.35 ρ1 = 1.9 g/cc ρ2 = 0.355 g/cc @ 200 MPa* v1 = 2.74 km/s**

M = 24.3***

T = 3278oK γ = 1.35 ρ1 = 1.9 g/cc ρ2 = ρ1xo/(xo+ct) v1 = 2.74 km/s**

M = 296.17 T = 1923.16oK γ = 1.35 ρ1 = 1.9 g/cc ρ2 = 1.88 g/cc @ 100 MPa* v1 = 2.74 km/s**

ρ1 = 1.9 g/cc v1 = 2.74 km/s** ρ2 = 7.9 g/cc v2 = 7.79 km/s

*Ideal Gas approximation, Limit Case.**Mean Value Assessed from Ref. 24.***Average Value from Ref. 25. Mean Value of Ts = 800oC and Tf = 2500oC. ◊ Confinement Duration = 2*Plate-Thickness/v2.

xo

Confinementρ2 , v2

Liquid HMX ρ1 , v1

xo

Confinementρ2 , v2

Liquid HMX ρ1 , v1


layer, i.e., the diameter to thickness ratio needs be at least 5.0, indicating a “critical diameter” on the order of 5*0.00508 cm = 0.25 mm (0.01 in.). The other possible cause of the transition behavior is related to the completion of decomposition. One also might suspect that the onset of low velocity detonation can proceed when decomposition reaches 90%. Thus the transition represents a process that allows the 90% decomposition to move onto and reach a higher level, e.g., 99.9% (See discussions in next section). It also should be noted that the accuracy of plots in Fig. 9, can be further improved by iterative calculation.

VII. HMX Decomposition Kinetics The first-order Arrhenius reaction rate formulation1 has been widely in use for the prediction of behavior of

explosives. It has the following simple form,

kndtdnn −==

•

(11)

RTEaAek /−= (12)

⇒ [ ]tAe

o

RTaE

enn/−−= (13)

where n is the number of molecules per unit volume at a given time, t; no is the initial value of n; k is the rate constant; Ea is the activation energy and A is the frequency factor or the pre-exponent factor. Several questions arise when applying this formulation to the extreme conditions found in an explosion (i.e. high pressure and temperature; and covering all states, solid, liquid and gaseous). High pressure can lead to changes in Ea 30, 31. But the extent is much less than that in A. Sometimes, A is interpreted as the frequency of collision of a gas molecule. By elementary gas kinetics, A = n*c*πD2 = (P/kBT)*c*πD2, Where D is the diameter of the molecule. Therefore, A is a function of both temperature and pressure. The situation in the liquid state is much more complex, to the extent that no clear theoretical guideline is available. In a solid, the frequency of inter-molecular interactions has to be evaluated through the vibration of the molecules, or the phonons. Furthermore, it is not clear whether a purely kinematic collision is adequate for describing the activation of the reaction because the detailed relative orientation of the specific molecular structure may be relevant in the process. In spite of the obvious difficulties in experimental evaluation, limited information on kinetic parameters for HMX decomposition has been established and reported in the literature. Table 3 is a brief summary of the results highlighting the range of temperatures, physical states, frequency factors and activation energies. Several comments in relation to the present DDT study are as follows,

1. Robertson’s and Roger’s data were gathered near the HMX melt temperature under which high density close to that of the solid state exists. The experiment used relatively high heating rates. Therefore some bias due to a self-heating effect may be present.

2. Qing and Beckstead’s data was reduced by assessing HMX combustion experiment burn rate data incorporated with reaction rate. The 500oC temperature reported was close to the HMX critical temperature. One can see that while their smaller value of Ea was still comparable to that reported by Robertson and Roger, the value of A had decreased by several orders of magnitude.

3. A further drastic decrease in the frequency factor, A, is shown for the HMX gas phase which supports the concept of treating A as frequency of molecule collision.

4. Burnham and Weese’s data are interesting in two aspects: a) slow heating rates < 1oC were used; and b) Arrhenius rate constants were found to be dependent on the un-reacted fraction which remained in the reaction. For the early part of the reaction, n/no = 0.9, and the constants were close to Qing and Beckstead’s values. For the part of the reaction near its completion, n/no = 0.1, and the constants were close to Robertson and Rogers results.

5. McGuire and Tarver simultaneously evaluated three HMX decomposition processes. The Arrhenius constants for the leading process were close to that found by Robertson and Roger. This is not surprising because the experiments were conducted at 250oC, which is close to the melting temperature of HMX.

6. Maksimov’s results were similar to those reported by Hensen. It is not clear why in both of these cases, low values of A on the order of 1012 were obtained.

7. Maksimov’s data for HMX in solution were comparable to Qing and Beckstead’s results for liquid HMX. This seems to indicate that HMX decomposition is insensitive to its surrounding species, molecules of HMX or those from the solvent (This may be applicable to the HMX reaction products as well).


Table 3 Arrhenius and Modified-Arrhenius Rate Constant parameters for HMX: Frequency Factor and Activation Energy

Rate Equation Case ID State* T , oC A , s-1 Ea , kcal/mol Remarks Robertson32 S-L 271 5x1019 52.7 By Manometrical Method Roger33-35 L 270-284 6.4x1018 51.3 DSC, 25-20oC/min.

G 245-275 3.16x1012 38.0 ± 2.8 By Manometrical Method S 183-230 1.58x1011 37.9 Maksimov36

Sol. 171-215 1.0x1015 44.9 In m-nitrophenol Solvent Hensen37 S, L, G 181.4-1726 5.77x1012 35.6 ±0.263 Universal for all T

S, L <500 8.6x1015 44.1 →HCN, NO2, H2O, N2 Qing /Beckstead38 G >>500 6.6x1011 4.0 → H2CN+N2O↔H2CNO+N2

2.54x1015 44.74 n/no = 0.9

RTEaAek /−=

Burnham /Weese39 S 230-250

1.0x1019 51.77 n/no = 0.1

DSC, Rate <1oC/min.,Sample Mass≈ 0.5 mg

1.41x1021 52.7 →H2C = N-NO2

1.93x1016 44.1 →CH2O + N2O

dn/dt =

( ) RTEii

iaenA /3

1∑

McGuire/ Tarver40 S, L 198.5-282.4

1.60x1012 34.1 →Final Products RTEaeAPk /92.0 −=

Hobbs41 S 170.8-256.8 2.905x1013 22.0 Incorporated with Pressure

50-100 8.95x108 26.22 M =1.83; q = 0.99 k = ( )[ ] RTEM aenqA /11 −−−

Burnham et al.42 S

175-200 7.71x1018 50.22 M = 0.676; q = 0.999

* S = Solid; L = Liquid; G = Gas; Sol. = in Solvent.


t 1, s

T , oK

t 1, s

T , oKFigure 10 Decomposition Completion Time, t1, as a Function of Temperature for Two HMX Densities with and without Self Heating

8. In recent reports by Hobbs and Burnham et al., modified Arrhenius rate constants were added with functions containing pressure or the residual un-reacted fraction along with the frequency factor. Therefore, their values of A summarized in the table can not be directly compared with others presented in the table.

The studies summarized above are not without areas of disagreement. However, they clearly suggest that the frequency factor is a very sensitive function of HMX density. The detailed relationship between these two variables is currently under development and will be reported in a future paper. Within the scope of the present DDT study, it is reasonable to use A = 1.0x1015 s-1 and Ea = 52.7 kcal/mole for HMX in the supercritical regime in which a low ρc = 0.503 g/cc is in effect. More importantly, the summarized data suggest that A = 5x1019 s-1 and Ea = 52.7 kcal/mole for HMX in: 1) liquid near Tm; and 2) supercritical regime under a pressure of ~100 MPa. In both case, a density near that of the solid, ρo ~ 1.88 g/cc is in effect. This density in a supercritical film controls the criticality of DDT.

The 99.9% reaction completion time, t1, as a function of temperature is plotted for these two cases in Fig. 10. It can be seen that for the critical density of 0.5 g/cc, t1 > 10-5 s for T< 1500oK and for a compressed supercritical state under P ~ 100 MPa, t1 = 5.6 ns can be achieved at T ≈ 1085oK and t1 = 12.9 ns can be achieved at T ≈ 1050oK, and both are fairly close to the upper bound estimated by Maksimov for HMX, Tc = 974.16oK. In other words, by heating supercritical HMX by merely ~100oK above Tc, sustained detonation may proceed if the gas layer is beyond 0.0508 mm thick. Both of creation of a thin supercritical layer and its subsequent “explosion” are the consequence of the fact that decomposition is a very sensitive function of the HMX density. It can be qualitatively shown that in the time frame of ~10 ns, the gases outside the layer do not decompose significantly so that a nominal temperature distribution as shown in Fig. 7 can be maintained. The “explosion” in the layer occurs when the density

reaches ~1.88 g/cc and the pressure has advanced to ~100 MPa. Detailed formulation is pending on the future successful derivation of the frequency factor as a function of HMX density. Note that in this full level decomposition, the validity of Eqs. 11 thru 13 rely upon an assumption that the rate equation and its Arrhenius constants remain the same in the presence of a large amount of decomposition products. Also, the choice of 99.9% reaction completion as the definition of t1 is quite arbitrary. The time would decrease by a factor of 3.0 if 90% were chosen as the definition.

VIII. Self Heating Effects This type of analysis has been widely performed to determine the ignition condition for explosives35, 43, 44. The standard starting point of the heat transfer analysis can be found in Carslaw and Jaeger26. The governing equation is as follows,

KE

tTT

•

−=

∂∂

−∇κ12 (14)

Where

•

E is the energy released per unit time and per unit volume. The temperature gradient term is important because the ignition processes usually involve a slow heating rate stimulus in a small local hot spot. Thus the temperature build-up towards the ignition point is largely controlled by heat transfer to the surroundings of the hot spot. Since in DDT, we are dealing with a much faster phenomenon, an adiabatic condition can be assumed as a first approximation, in other words, that no appreciable amount of heat is lost to the surrounding via conduction. Eq. 14 thus becomes,


KE

tT

•

−=

∂∂

κ1 ⇒

•

−=∂∂

− nMEtTC opρ , Mn=ρ (15)

⇒ oo Mn== ρρ , •

−=∂∂

− nEtTCn opo (16)

nAen RTEa /−•

−= (17) Two additional assumptions have to been incorporated in Eqs.16 and 17 in order to solve them numerically. First, ρ remains constant, equal to the original density of un-reacted HMX. This is a reasonable assumption, since the HMX is under good confinement. Second, Cp has the value of un-reacted HMX at elevated temperatures as estimated by extrapolation of Eq. 3. This approach is adequate for qualitative illustration purposes only. The volume of detonation gases for HMX17 is 927 cc/g , or 0.0414 mol/g which is close to the 0.0338 mol/g for HMX at Tm. The values of Cp for HMX gaseous products25 such as H2O, CO, CO2 and N2 are about 1.0-2.0 J/g-oK, close to that of HMX. Thus the approximate equivalence can be used for a rough estimation. Fig. 11 a) shows the remaining HMX fraction as a function of time at an initial temperature of 770oK and ρ =1.88 g/cc, both with and without self-heating. It can be seen that the inclusion of self-heating drastically shortens the decomposition completion time with an abrupt rate increase at ~1.8 x 10-7 s which is in phase with the abrupt temperature rise profile shown in Fig. 11b). The decomposition completion time, t1, evaluated at other temperatures with self heating for both ρ =1.88 g/cc and ρ =0.5 g/cc are included in Fig 10. The figure shows that even with the self heating, the 0.5 g/cc density HMX cannot react fast enough to facilitate DDT for initial T up to 1500oK.

It should be noted that the self heating of liquid HMX cannot actually occur in a real physical situation. The decomposition in this case will nucleate gas bubbles instead of uniformly heating up the entire liquid. This merely transforms the gas/liquid interface problem from the macroscopic scale to the microscopic case on the bubble surface. The analytical treatments of HMX bubbling phenomenon are discussed in Refs.45 and 46. Bubbles which are lower density, higher pressure localized regions may also form in the supercritical gas which is in a quasi-liquid state.

IX. Effect of Adiabatic Compression Adiabatic compression is important in the study of DDT in porous insensitive high explosives where the rise in the gas temperature in the pores under compression is generally considered to be a major contributor to the reaction kinetics. In the current study, since neither porosity nor particle size is considered, the only impact of compression is in the supercritical gas layer. Compression increases the density of the gas in the supercritical layer from low values < 0.5 g/cc toward ∼1.88 g/cc to facilitate DDT. The pressure rise in a DDT system appears to be partly an independent variable as it is determined by the internal geometry of the test device and input heat rate. Empirically, the rate in the deflagration regime is on the order of 6.8 x 105 MPa/s (10,000 psi/0.1 ms). According to the

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.E-08 1.E-07 1.E-06 1.E-05 1.E-04

t , s

Frac

tion

ArrheniusWith Self Heating

, n/n

o

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.E-08 1.E-07 1.E-06 1.E-05 1.E-04

t , s

Frac

tion

ArrheniusWith Self Heating

, n/n

o

Frac

tion,

n/n

o

T , o K

Frac

tion,

n/n

o

T , o K

Figure 11 Decomposition as a Function of Time at Initial T = 770oK: a) With and Without Self Heating; b) Same Data with Self-Heating and Correlation with Temperature Rise.

a) b)


elementary thermodynamic equation for adiabatic compression, T2/T1 = (P1/P2)1/γ - 1, γ = 1.35, P2 ~ 100 MPa, P1 = Pc = 2.97 MPa, T1 = Tc = 927.16oK, T2 = 2307oK would result. If this magnitude of temperature rise actually occurred, it would be the dominant driver of the decomposition process in the HMX supercritical layer. In reality, the rate of compression is much slower, in a nanoseconds time frame of the in-situ DDT decomposition. Additionally, it is very doubtful that adiabatic conditions can be satisfied in the microscopic thin supercritical layer. A detailed evaluation of the system is further hindered by the fact that the decomposition rate in the flame outside the supercritical layer is not known, due to the fact that the rate constants are pressure-and temperature-dependent. These relationships remain incompletely characterized. In any event, as has been indicated already, the uncertainty in the current model is only about 100oC on the low side. It is envisioned that adiabatic compression can bridge this possible small difference thereby facilitating the establishment of DDT.

X. Laser Initiation-- Theory

In-depth volumetric heating of insensitive high explosives occurs during laser initiation. For lasers with visible and near infrared wavelengths, the absorption of light is not due to electronic transitions inside the explosive molecules. Imperfections such as crystalline dislocation and impurities which are deposited in the explosive can effectively absorb light. The heating and ionization that follow can cause an avalanche effect and produce a plasma. It has been demonstrated that the presence of such low temperature plasma in combustion flames, can absorb electromagnetic waves efficiently in spite of their low degree of ionization 47. The mathematical description of laser initiation of HMX is quite similar to that described for the self-heating effect except that in this case the energy/power is externally provided with variable levels and time durations:

WtTCp −=∂∂

− ρ , δτφτ //1 == EW ⇒ δτφρ /−=∂∂

−tTCpo (18)

Figure 12 shows the results of numerically solving Eqs.17 and 18 simultaneously for two important practical cases of laser initiation at an initial temperature of 300oK, using an 100-μs-duration free running mode laser and a 20-ns-duration Q-switched laser at energy levels above the estimated initiation thresholds of ~10.2 mJ (in ~0.2 mm x 0.2 mm x 0.0508 mm laser spot volume in HMX). In spite of the temperature rises that are nearly linear with respect to time, the decomposition rates in terms of the abrupt n/no transition resemble the case of self-heating shown in Fig. 11a). This “promptness” in initiation, where the process follows the laser pulse is quite universal. Two other examples are: 1) Initiation of a lead-styphnate/lead-azide/HMX explosive train based micro-detonator by a 20-ns-duratuion Q-switched laser and a 400-μs-duration free running laser48; 2) Initiation of a zirconium/ammonium-perchlorate pyrotechnic-based squib by a 500-μs-duration free running laser49. Fast function times well with in the laser pulse duration were recorded.

XI. Laser Initiation—Experiments Limited additional experimental work50, 51 was performed in late 1980’s by using 10-30 ns duration laser pulses. This work verified Menichelli and Yang’s finding that detonation of PETN and RDX can be initiated directly.

Frac

tion,

n/n

o

T , o K

a)

Frac

tion,

n/n

o

T , o K

Frac

tion,

n/n

o

T , o K

a)

Frac

tion,

n/n

o

T , o K

b)

Frac

tion,

n/n

o

T , o K

b)

Figure 12 Laser Initiation at 300oK Initial Temperature: Molar Fraction and Temperature as Functions of Time. a) 100-μs Pulse with W = 5x 107 w/cc; b) 20-ns Pulse with W = 2.5x1011 w/cc.


HMX, which is a close relative of RDX, was also successfully detonated under these conditions. Threshold conditions, particle size and density effects were evaluated. Perhaps due to the limited laser energy on the order of < 1.0 J, HNS could not be initiated as had been reported by Menichelli and Yang. The most important new finding was that by using laser pulses with wavelengths in the ultraviolet range, 266 to 355 nm, detonation of unconfined PETN could be initiated. This indicates a bond breaking process in the molecule which by-passes the liquid phase, resembling heat initiation of lead azide (see next section). It is clear that HNS initiation and initiation studies using a picoseconds duration laser (mode locking) pulse are of interests for further work. Due to its higher melting temperature17 of 318oC and good temperature stability, HNS is widely used in today’s applications52. Significant differences in HNS properties from that described for HMX so far are anticipated. It does not exhibit DDT handily. In fact, Ref. 12 reported that under laser initiation, a ~3.0 cm-long HNS column was completely burned without a transition to detonation. One limiting factor could have been the fact that HNS is more opaque as compared with HMX and PETN, thereby preventing in-depth heating by the laser. Inefficiency in laser initiation resulted. Based on the analyses of the present paper, a shorter laser pulse may produce higher temperatures and shock pressures. Therefore a picoseconds laser could improve the laser initiation efficiency. Information on DDT-based detonators for initiation by a millisecond duration free-running mode laser pulse is still limited to what was reported by Yang and Menichelli53. The first significant development in this direction took place in a laser ordnance firing system54 for the U.S. Air Force Small Inter-Continental Ballistic Missile in which the optical detonator adopted a configuration similar to that showed in Fig. 2 except that the window described in Ref. 53 was implemented. In the early development, highly compacted PETN was used as the explosive charge. It worked well in preliminary bench testing. However, it was soon found that after nominal temperature cycling tests, low output was measured indicating a degrading DDT performance. The cause was attributed to the de-consolidation of the charge under repetitive temperature cycles had created a gap at the window/PETN interface. The problem might have been correctable by using more thermally stable explosives, e.g., HMX. However, due to inadequate time for further testing, a decision was made to adopt the CP explosive55 developed and demonstrated by the U.S. Department of Energy. The modified design performed very well as reported in Ref. 54. A number of new laser initiation applications have now been implemented. However, due to practical considerations, diode lasers with duration up to 10 ms and power on the order of several watts are adopted. These systems are for pyrotechnic initiation. Therefore the only means of achieving a detonation output in these systems is by way of a lead azide-based explosive train. An HMX based and long pulse duration laser initiated DDT detonator design is of interest for future improvements and applications.

XII. Primary High Explosives The discussions in the last section pointed out the need to address of the use and initiation of primary

explosives, because the lead azide-based explosive train is still overwhelmingly in use today. The explosive industry has sought to develop an alternative for lead azide, initially for safety reasons and currently due to concerns about environmental pollution56. In these efforts, one of the principal difficulties has been the fact that lead azide function is poorly understood compared to other insensitive high explosives. Therefore, there is a poor technical basis for developing a replacement.

The first unique feature of lead azide is that it is the only member of the metal azide family which will readily

detonate. For example, sodium azide which has been widely used in the automobile air bags is not detonable43, 44. This may be related to the fact that the liquid phase of lead azide does not exist as pointed out by Refs. 43 and 44. When exposed to heat, it explodes before the formation of rate-limiting liquid phase is possible. This indicates that the decomposition is originated in the atomic bond breaking in the molecule. Secondly, DDT in lead azide is very swift to such an extent that except for the study of single crystal reactions57, steady-state detonation of lead azide is considered to be instantaneous in initiation in spite of without confinement. Considering lead azide’s physical properties, there is nothing out-of-the-ordinary about it, e.g., its thermal conductivity58 has a value of 1.55 x 10-4 cal/cm-s-oC which is comparable to typical insensitive high explosives. Its low detonation velocity17 of ~5.3 km/s is attributable to the low explosion energy17 1.54 kJ/g and low specific gas production17 of ~308 cc/g. Lead azide’s density of ~4.8 g/cc is much higher than most insensitive high explosives which may contribute to its facile DDT, since a high dynamic confinement is realizable. In addition, the lead atom results in a high molecular weight component in post reaction products which is also favorable for confinement as discussed in Sec. VI. The key factor contributes to the detonation initiation under unconfined condition of lead azide, and PETN in the last section, is the


T , oK

t 1, s

T , oK

t 1, s

Figure 13 Decomposition Completion Time, t1, as a Function of Temperature for Lead azide and CP without Self Heating

known high shock initiation sensitivities of these two explosives which offset the much lower shock stresses would be predicted by Sec. VI-type of calculation for a completely unconfined condition.

Candidates for lead azide replacement share some of the features above, e.g., all contains a metal atom, with

explosion temperature around 320oC, and have no reported melting temperature. But all of them have low densities as summarized in Table 4.

Table 4 Key Properties of Lead Azide and Its Possible Replacements

ID\Property Metal Atom

Density , g/cc

Exothermal Onset , oC

Exothermal peak , oC

Detonation Velocity , km/s (Density, g/cc)

Lead Azide58 Pb 4.80 332 341 5.3 (4.6) CP59, 60 Co 1.98 289 333 6.8 (1.66) BNCP60 Co 2.03 269 301 3.2 (1.62) DBX61 Cu 2.59 329 337 Not Available

The Arrhenius rate constants of lead azide can be

approximately determined by using the limited decomposition time versus temperature data available in Ref. 58. This analysis shows: A = 1.309 x 1029 s-1 and Ea = 81.9 kcal/mol. The Arrhenius constants for CP are reported in Ref. 59 as A ~ 3.16 x 1016 s-1 and Ea ~ 47.8 kcal/mol for the initial decomposition in the 0< n/no < 40% regime. Using these values, 99% decomposition time, t1, as a function of temperature is plotted in Fig. 13. A very long decomposition time on the order of 0.1 s is calculated for T~ 600oK. This slow decomposition is due to the fact that a supercritical layer with higher temperatures can not be formed without going through the liquid phase. In order to arrive at a fast decomposition time of ~13 ns, high temperatures on the order of 870oK for lead azide and 1310oK for CP are needed. However, the rigid solid structure in these explosives can impede the diffusion of reaction products to facilitate a self-heating effect. With this feature, as has been discussed in Sec. VIII and shown in Fig. 11b), a rapid temperature rise to beyond 2000oK can

be established in very short time on the order of 10-8 s. Detonation initiation thus can be sustained. Guidelines for the composition of a new ideal primary high explosive material would appear to be that the

material: 1) has no melting prior to explosion; 2) has a high explosion temperature; 3) has a high density; 4) contains a heavy metal atom (s); and 5) has a good combination of frequency factor, A and activation energy, Ea for a fast reaction rate at moderately high temperatures. The last issue is illustrated in Fig. 13 where the rate for CP is much lower than that of lead azide. As a result, significant DDT was indeed observed in CP initiation as reported in Ref. 55 (The report inadvertently named convection burn as deflagration).

XIII. Summary and Conclusions

This paper has established fundamental principles and a model of the deflagration to detonation transition (DDT) phenomenon using HMX as an example:

1. DDT can occur in insensitive high explosives within a millimeter of the heating source. 2. A supercritical gas layer of ~0.005-cm-thickness can form on the surface of the explosive. Under pressures

of ~ 100 to 200 MPa and temperatures on the order of ~ 1500oK, fast decomposition in ~10 ns in this supercritical gas layer can generate an explosion pressure on the order of >1.0 GPa to initiate the DDT process.

3. Study of heat transfer and physical confinement of the layer support this model scenario. 4. Failure to observe DDT in traditional combustion experiments on HMX at up to ~340 MPa of pressure, is

attributable to dilution effects from the inert gas (N2) used for pressurization. Such inert gases tend to reduce the temperature and confinement impedance in the layer.


5. The frequency factor in Arrhenius decomposition kinetics is a very sensitive function of the density of the explosive. For the HMX example used, the only reliable values available are those for the liquid state that approximate the melting temperature or for the compressed supercritical state where a density close to that of the solid is reached and that for the critical density, ρc. For regimes outside these two reliable data points, further studies are needed.

6. The self-heating effect during decomposition is in effect for solid and gaseous states but not the liquid state due to the formation of gas bubbles in the liquid.

7. The heating effect of adiabatic compression of the gaseous phase in relation to DDT appears to be a minor factor, but may have some contributions. Further studies are needed in this area.

8. The high detonation initiation efficiency observed in 20-ns-duration laser initiation experiments can be attributed to the in-depth heating by optical energy, and, at ultraviolet wavelengths, direct atomic bond breaking. Further work on HNS initiation and by picoseconds-duration laser pulses are of interest.

9. The detonation of lead azide-type primary high explosives is attributed to explosion prior to the point where the decomposition rate limiting liquid phase can be formed.

10. Guidelines for seeking replacement of lead azide are suggested. 11. Trends of thermophysical properties as a function of temperature and pressure in the supercritical regime

are illustrated by R113. Further work along these lines is needed.

Appendix 1

Trends of Thermophysical Properties of Large Molecules at High Temperature and Pressure The sources for a systematic listing of properties of materials at temperatures and pressures beyond their critical

points are scarce. For the obvious reasons of test difficulties and safety issues, they are essentially non-existent for energetic materials. The author has been interested in locating information on organic molecules with large molecular weight for an illustration of the trends of their properties as a function of temperature and pressure which could be beneficial to the projection of properties for insensitive high explosives. Limited information for some materials from commercial and industrial interests has been compiled and posted on a public website by NIST62. In this database, the heaviest molecule which has tabulated properties for pressures of up to 200 MPa is the R113 refrigerant. The piecewise data on density, ρ; specific heat, Cp; and thermal conductivity, K, are integrated into single charts by the author and shown in Fig. 14 for trend illustration. Included also are the derived properties: specific heat ratio, γ and thermal diffusivity, κ, which are not tabulated in the database.

Several observations on the data features can be briefly summarized as follows, 1. The general trends appear to be quite universal. Case studies on other materials, e.g., propane showed

essentially the same characteristics. The data for R113 stops at 523.56oK, only 36.35oK above its critical temperature. With the exception of the low pressure (≤4 MPa) portion, it can be extrapolated to an estimated 853oK because the propane data in Ref. 63 maintained the same trends to a temperature of 700oK, approximately 330oK above propane’s Tc = 369.85oK.

2. Cautions must be exercised in comparing data between different materials even though they have comparable molecular weights. The values of Cp for HMX and R113, ~68.4 cal/mol and ~40.6 cal/mol respectively, are much larger than the typical value of ~7.0 cal/mol for a small molecule which mainly consists of the translational and rotational kinetic energies. This indicates that a significant portion of the energy reflected in Cp is related to the atomic vibration in the molecule, i.e., it is very structure-dependent and is different from molecule to molecule. Similarly, density may depend upon the forces and compressibility between molecules which usually are very different for different molecule species.

3. The value of K under high pressures and temperatures shown as a decreasing function of temperature is different from that of a rarified gas which increases with the temperature as K ≈ 0.075McCp(γD2)-1. Therefore, it is anticipated that as T increases further to a certain value, K will start to reverse the trend, i.e., an increasing function of T resembling the gaseous (vapor) phase shown for 0.1 and 1.0 MPa of pressure in Fig. 14c).

4. By assuming the data similarity between HMX and R113, it can be seen that by extrapolating the data in Eqs. 1-3, from low temperature and/or low pressure to the critical temperature and ~100 MPa, approximate errors of -40%, +15% and -100% could be introduced for ρ, Cp and K respectively. The value for κ is relatively constant in this temperature and pressure regime. Similarly, γ = 1.35 appears to be a relatively constant average value.


R113 (TTE)

1,1,2-Trichloro-1,2,2-Trifluoroethane

CCl2F – CClF2

M = 187.36, ρ = 1.583 g/cc (20oC)

Tm = 236.95oK, Tb = 320.735oK

Tc = 487.21oK, Pc = 3.3922 MPa

ρc = 0.57 g/cc

0.00001

0.0001

0.001

0.01

200 300 400 500 600T , K

k , 1

0̂-3

m̂2/

s

200 MPa100 MPa50 MPa20 MPa4.0 MPa1.0 MPa0.1 MPa

T , oK

κ, 1

0-3 m

2 /s

Tc

0.00001

0.0001

0.001

0.01

200 300 400 500 600T , K

k , 1

0̂-3

m̂2/

s


T , oK

κ, 1

0-3 m

2 /s

Tc

-2

0

2

4

6

8

10

200 300 400 500 600

T , K

Rho

, m

ol/l


T , oK

ρ, m

ol/ l

Tc-2

0

2

4

6

8

10

200 300 400 500 600

T , K

Rho

, m

ol/l


T , oK

ρ, m

ol/ l

Tc

a) b)

0

0.02

0.04

0.06

0.08

0.1

200 300 400 500 600

T , K

K ,

W/m

-K


T , oK

K ,

W/m

-oK

Tc0

0.02

0.04

0.06

0.08

0.1

200 300 400 500 600

T , K

K ,

W/m

-K


T , oK

K ,

W/m

-oK

Tc 100

200

300

200 300 400 500 600

T , K

Cp

, J/m

ol-K


To 826.68 J/mol-K

T , oK

Cp

, J/m

ol-o

K

Tc100

200

300

200 300 400 500 600

T , K

Cp

, J/m

ol-K


To 826.68 J/mol-K

T , oK

Cp

, J/m

ol-o

K

Tc

1

1.2

1.4

1.6

1.8

2

200 300 400 500 600T , K

gam

ma


To 4.95

γ

T , oK

Tc

1

1.2

1.4

1.6

1.8

2

200 300 400 500 600T , K

gam

ma


To 4.95

γ

T , oK

Tc

T , oK

Tc

c)

f)

Figure 14 R113 Thermal Physical Properties as Functions of Temperature and Pressure: a) Phase Change; b) Density; c) Thermal Conductivity; d) Specific Heat at Constant Pressure; e) Thermal Diffusivity; f) Specific Heat Ratio.

d)

e)

5. It is of interest to extend the type of data cited in this appendix to higher temperature regimes up to T≈ 3000oK for R113, and other candidate material with M ≈ 300.

Acknowledgements

The author acknowledges the assistance of several people with locating many of the references relevant to this work: D. Roth of California Institute of Technology, Millikan Memorial Library; L. J. Granda of Northrop Grumman Inc.; M. W. Beckstead, and D. Smyth of Brigham Young Univ.; E. B. Washburn of U.S. Navy NAWC; S. Peiris of International Detonation Symposium; E. Yang of Univ. Pennsylvania. Credits also should be attributed to U.S. Governmental agencies for posting many relevant references on public websites: the Depts. of Navy and Energy, for Proceedings of International Detonation Symposia #12 and #13; the Dept. of Energy, for selected reports by LANL, LLNL and SNL; and Dept. of Commerce, NIST, for Thermophysical Properties of Fluid Systems. This paper has been reviewed and edited by E. Yang of Univ. of Pennsylvania and S. Yang of Case Western Reserve Univ.


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58 Engineering Design Handbook, Explosive Series, Properties of Explosives of Military Interest, AMC Pamphlet AMCP 706-177, AD-764 340, U.S. Army Materiel Command, January, 1971.

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60 Fronabarger, J., Schuman, A., Chapman, R. D., Fleming, W., Sanborn, W. B. and Massis, T., AIAA-95-2858, “Chemistry and Development of BNCP, a Novel DDT Explosive,’ 31th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, San Diego, CA, July 10-12, 1995.

61 Fronabarger, J. W. and Bichay, M., AIAA-2007-5132, “Environmentally Acceptable Alternatives to Lead Azide and Lead Styphnate,” 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Cincinnati, OH, July 8-11, 2007.

62 National Institute of Standards and Technology website URL: http://webbook.nist.gov/ 63 Vargaftik, N. B., Vinogradov, Y. K. and Yargin, V. S., Handbook of Physical Properties of Liquids and Gases – Pure

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