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1 American Institute of Aeronautics and Astronautics

Frequency Factor in Arrhenius Decomposition Kinetics for Insensitive Energetic Materials

Lien C. Yang*

La Canada Flintridge, CA 91011

A fundamental limitation of the Arrhenius decomposition kinetics equation has been the lack of detailed physical descriptions of the frequency factor. In this paper, the frequency factor, an important parameter in fluid state behavior, is analyzed using molecular collision statistics. Parameters including molecular size, off-line collisions and Van der Waals’ potential are examined. It is shown that, in order to correctly formulate the frequency factor in dense fluid systems such as supercritical states, the traditional “mean free path” used for low density gaseous systems must be replaced with a “mean free gap” concept. Using a “hard spherical molecules” model, a functional relationship between the frequency factor (≤ ~1014 s-

1) and the mean free gap is established and correlated with the pressure, temperature and density of the fluid. The Clausius-Clapeyron equation is then successfully derived using statistical mechanics for the extrapolation of frequency factors (~1019 s-1) in liquids at melting temperatures to supercritical states for several explosives: HMX, RDX and PETN. Abrupt Deflagration-to-Detonation-Transitions can occur in the supercritical states when these high frequency factors abruptly become in effect if the average spacing between molecules falls below ~18% of the molecular diameter and the collisions switch into a high frequency factor mode (possibly due to the formation of “dimer” structures). Finally, the Frenkel-Halsey-Hill 1equation is applied to estimate the various effects of Van der Waals’ potentials on the frequency factor.

Nomenclature a = Pressure correction constant in Van der Waals’ equation for real gases A = Frequency factor in the Arrhenius decomposition rate equation, s-1

A’ = Pre-exponential factor in CC saturated vapor pressure equation A1 = A for low density gases A2 = A1 including molecular diameter effect A3 = A1 including oblique collision effect A4 = A1 including Van der Waals’ potential effect A5 = A for dense gases or liquids A6 = A5 including Van der Waals’ potential effect A7 = A6 using FHH equation for Van der Waals’ potential effect A8 = A6 using CC equation for Van der Waals’ potential effect Am = A at Tm determined experimentally in Refs. 9-12 and Ref. 18

AAT = Accelerated aging test b = Volume correction constant in Van der Waals’ equation for real gases

CC = Clausius-Clapeyron saturated vapor pressure equation Cp = Specific heat at constant pressure D = Diameter of a molecule D’ = Cubic root of average volume occupied by a molecule

DDT = Deflagration-to-Detonation Transition E = Energy per molecule in gaseous or liquid phase, E = Average value of E. E1 = In-line electrical field of donor dipoles lying in a plane Ea = Activation energy in the Arrhenius rate equation, kcal/mol or ergs/molecule

FHH = Frenkel-Halsey-Hill equation h = Planck Constant = 6.625 x 10-27erg-s; Also, the height of a right triangular pyramid.

HMX = Cyclotetramethylene tetranitramine, C4H8N8O8 HNS = Hexanitrostilbene, C14H4N6O12

* Senior Member, AIAA

45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit2 - 5 August 2009, Denver, Colorado

AIAA 2009-5190

Copyright © 2009 by L. C. Yang. . Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


k = Reaction rate constant in the Arrhenius decomposition rate equation kB = Boltzmann constant = 1.38 x 10-16 erg/oK L = Latent heat of vaporization at melting temperature; Also Liquid state L’ = Potential energy equivalent value of L in the molecule-to-molecule dipole interaction l = Mean Free Path in low density gases m = Mass of a molecule M = Molecular weight n = Number of molecules per unit volume of un-reacted material at a given time t

NA = Avogadro constant nm = n at melting temperature no = Initial value of n p1 = Electrical dipole moment of dipole field donor molecule p2 = Electrical dipole moment of dipole field receptor molecule P = Pressure; Also as partial vapor pressure in FHH equation Pc = Critical Pressure Pm = Saturated Vapor pressure at melting temperature Po = Saturation Vapor pressure at a given temperature

PETN = Pentaerythritol tetranitrate, C5H8N4O12 r = Separation distance between two molecules; Also, cylindrical radial coordinate R = Universal gas constant = 8.314 J/mol-oK = 1.988 cal/mol-oK

R113 = 1,1,2-Trichloro-1,2,2-trifluoroethane refrigerant RDX = Cyclotrimethylene trinitramine, C3H6N6O6

S = Solid state; Also used for Entropy in Appendix 4 t = Time T = Temperature Tb = Boiling Temperature Tc = Critical temperature Tm = Melting temperature v = Velocity of a molecule v = Average value of v

vd 3 = Differential volume element in velocity phase space = v2sinθdθdφdv (spherical) V = Potential energy of a molecule V3 = Repulsive potential between two molecules in Lennard-Jones equation

Vave = Average local potential energy of a molecule in FHH equation formulation Vmax = Maximum local potential energy of a molecule in FHH equation formulation VD+λ = V at equilibrium position V∑ = Sum of total potential energy due to all dipoles on a liquid free surface v’ = Net velocity after deduction of L’, molecule-to-molecule v” = Velocity corresponding to the total energy including the Van der Waals’ potential v’” = Net axial velocity after deduction of L, in fluid va = Velocity corresponding to Ea , ½ mva

2 = Ea va’ = Effective value of va after deduction of L’ w = Net total velocity after deduction of L, = (v2-2L/m)1/2

z = Axial coordinate perpendicular to a layer of molecules α = Electrical or electronic polarizability of a molecule β = Factor for converting the effective void volume into ε

ε = Maximum depth of penetration of a molecule into the surrounding interstitial void formed by a plane of molecules

ε1 = In-Line electrical field of the donor dipole ε’ = Equivalent or average depth of interstitial void in a plane of molecules ν = Average total degrees of freedom including rotational and vibration normal modes ω = Translational degree of freedom of kinetic energy θ = Polar angle in velocity vector (phase) space (spherical coordinates) θ = Van der Waals’ potential strength in FHH equation θ’ = Limiting value of θ which allows the axial kinetic energy of a molecule to exceed L φ = Azimuth angle in velocity vector (phase) space (spherical coordinates)


λ = Average interstitial separation distance between two molecules (edge-to-edge) ρ = Density, g/cc ρc = Critical density ρm = Liquid density at Tm

ρo = Crystalline density of HMX

I. Introduction Since its introduction about one century ago, the Arrhenius rate constant equation has been widely adopted in many technical fields for the analysis of reaction kinetics. While it has provided adequate descriptions in agreement with many measured data, the success of this equation has been largely empirical. This should be considered remarkable as this simple equation, as usually presented does not have a rigorous analytic proof1:

kndtdnn −==

•

(1)

RTEaAek /−= (2)

⇒ [ ]tAe

o

RTaE

enn/−−= (3)

The assumption of a 1st order reaction rate (i.e., n& ∝ n in the first power) is reasonable; the concept of activation energy (Ea) is easy to envision as threshold energy; and the exponential factor is common in statistical mechanics. However, the physical meaning of the constant A, often called the “frequency factor” due to its unit measurement in Hertz, is not well defined or intuitive. In gas phase reaction kinetics, A may be interpreted as the “rate of molecular collision.” However, this interpretation is questionable in the solid state. For this reason, A is also widely referred just as the “pre-exponential factor.” Therefore the author has suspected for some time that the unusual abrupt deflagration-to-detonation transition (DDT) phenomenon may have been originated from the characteristics of A because all other related physical processes such as heat transfer and pressure confinement do not support an abrupt discontinuity in the reaction pressure and velocity of propagation. This is the primary motivation of current study. Two attempts to conceptualize the “frequency factor” A was made early in the last century without success1, 2: The first was a study of chemical reaction kinetics by Max Trautz and William Lewis in the years 1916-18. Statistical analysis based on a Maxwell distribution was applied to a simple two-component reversible reaction: A + B ↔ AB. However, this case is not directly applicable to the simple irreversible decomposition of an energetic material in which A→ B + C + D + …. . In addition, there is a serious discrepancy between the observed reaction rates and theoretically calculated values using their framework. Therefore, it has been generally accepted that not all collisions are effective in the reaction--there may be a relationship between a favorable molecular orientation and the reaction rate, i.e., the existence of a steric factor. The second concept was the "transition state theory" of chemical reactions, formulated by Wigner, Eyring, Polanyi and Evans in the 1930s. In this approach, the difference in Gibbs’ thermal energy replaces Ea thereby introducing linear temperature dependence in the value of A. As pointed out in Refs. 1 and 2, this procedure introduces potentially counterproductive complications because Gibbs’ function is explicitly dependent on temperature and pressure. Recently, the Arrhenius equation has been successfully used by the author in combustion analysis of HMX based Nonel tubes3,4, accelerated aging testing (AAT)5 and the DDT phenomenon in insensitive high explosives (mainly HMX)6. However, the results were not entirely convincing due to the fact that a constant value of A obtained near the melting temperature was used. Without a clear account of how A varies with the temperature, pressure and density, several important problems of interest (e.g., the detonation transition and the adiabatic compression effect on gas decomposition in Ref. 6) could not be analyzed with certainty. A comprehensive tabulation of the Arrhenius constants for HMX was compiled in Ref. 6 (Section VII, Table 3). Several interesting features were noted in these data, and they are briefly summarized below as they pertain to the current study:

1. Ea of gaseous7, 8 HMX appeared to be significantly smaller than that of pure liquid HMX9-12. 2. Both Ea and A for HMX in organic solvent7 appeared to be smaller than that in pure liquid HMX9-12.


3. Nitrogen gas was ineffective as a background pressurization medium for affecting HMX decomposition to achieve DDT13, 14.

4. “A” of HMX in the gaseous phase7, 8 , ~1012 s-1, is orders of magnitude smaller than that in the liquid phase9-12 near the melting temperature Tm , 5x1019 s-1.

5. Both Ea and A may be affected by decomposition products15, i.e. they are smaller in the initial decomposition but increase to higher values in the later stages of decomposition.

6. Experimentally measured values of A = 5x1019 s-1 and Ea = 52.7 kcal/mol at Tm for HMX9-12 can support the ~5 ns reaction time required for detonation of this explosive at elevated temperatures in the supercritical regime, if the same value of A is in effect6.

7. The essential issue for DDT thus appears to be how the low rate in a low-density, superheated state reaches the high rate required for detonation or how such superheated material attains the required same density as that at Tm, i.e., ρm. It could be due to the heating during decomposition, adiabatic compression, or a combination of both mechanisms.

In the following work, some of these phenomena are interpreted and formulated in a general way that may allow calculations to match the experimental data more accurately.

II. Gaseous State Unlike the AAT which usually takes place in the solid state involving complex interactions (phonon mechanism, impurities, crystalline imperfections and granular surface conditions), decomposition in either the liquid or gaseous state is relatively simple: there must be molecular collisions. Mean Free Path Concept The description of this simple concept can be found in many textbooks on gas dynamics16 and is briefly illustrated in Fig. 1, which assumes that the molecules are spherical. For Fig. 1 a), the mean free path l is defined as follows,

nDtvDntv

22 21

2 ππ==l (4)

The factor 2 is introduced by transformation from absolute velocities to relative velocities in order to correct an error introduced by the model assumption that all molecules subjected to the collision have fixed positions in space.

Decomposition Activated by Simple Collision of Identical Molecules (Fig. 1a) In order to explore assumptions and mathematical limitations, it is necessary to first examine the simpler case. That is, upon a single collision between two molecules of energetic material, decomposition will occur if the relative kinetic energy involved in the collision is equal or above Ea. The formulation below follows well-established methodology in statistical mechanics. Only linear kinetic energy is considered, i.e., using the Maxwell velocity distribution rather than the more general Maxwell-Boltzmann distribution. Contributions from rotational energy and vibration modes are not included. This greatly simplifies the problem because in complex molecules of an energetic material, the degrees of freedom are large, in

D

D

2D

D

D

2D

d

D

D + d

d

D

D + d

b)

Figure 1 Molecular Impact: a) In Molecules of Identical diameter; b) In Molecules of Different Diameters.

a)


the range of 20 to 40. This is reflected in their larger specific heat, Cp on the order of 20-50 R as compared to 3/2 R to 7/2 R in simpler molecules. The mathematical conventions in Ref. 17 are used.

( ) vdvfdn r3= φθθ dddvvvd sin23 =r

( ) Tkmv

B

BeTk

mnvf 223

2

2−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

π; vt l=Δ ; aa Emv =2

21 (5)

Let us assume that in the initial state, no decomposition has occurred. At a time of avt l=Δ later, all molecules in “dn” with velocity greater than va would have undergone a single inelastic collision and decomposed. Otherwise, the collisions between molecules with relative velocities lower than va are instantaneous and elastic (i.e., hard spheres) and there is no cumulative effect for the consecutive collisions of molecules with energy below Ea, i.e., no transition state effect. Therefore the total number of decompositions in “dn” is;

( ) TkEaBv

Tkmv

B

Ba

a

B enETkdvveTk

mndn −∞ − −

=⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫ 2

12

122

/3

22

4 22 ππ

π

Here and in later derivations, we have used the unique feature of the integral of a negative exponent function:

( ) ( ) ] ( ) ] ( ) ( )

( ) ( ) ( )........."'

...."'

aaa

xa

x

xx

xx

xx

x

xgxgxg

exgdxexgexgexgdxexg a

a

aa

a

>>>>

≅+−−−= −∞

−∞−∞−−∞

∫∫

Using HMX molecule at Tm (Numerical data from Ref. 18) as an example,

( ) ( ) ( ) ( ) 3.4816.54810*65.2"''

4

≈°°

===KK

TkExgxgxgxgB

aaaaa

Therefore the decomposition rate is:

( ) TkEaB

a

BaeETmknv

dndtdn −−== 21

2321 π

ll

One thus obtains for the frequency factor (Note that R = NAkB ; NA = Avogadro constant):

( )

( )

23

23

21

21

21

21

21

23

2

2

1

4

4

21

TPEDkm

nEDTmk

ETmkA

a

aB

aB

−−

−

=

=

= −

π

π

πl

by using Eq. 4 (6)

The last step, substituting n with n =P/kBT is from a standard statistical calculation,

( ) ( )

TnkdvevTk

mdmn

vdvfmvvP

BTkmv

B

B =⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

−∞

∫∫

∫24

0

2

0

2

3

22

3

2sincos4

cos2cos

πθθθπ

θθπ

v

(7)

Using 31sincos

2

0

2 =∫ θθθπ

d ; and17 252

83

0

4 −=−

∞

∫ απα dxex x , where Tkm B2/=α


Eq. 6 is an expected result in the framework of the collision model: The collision frequency increases with the density of the gas (or pressure at a given temperature). The critical role of l in relation to the collision frequency is more clearly illustrated in an alternative derivation in which we directly integrate the individual decomposition rate for each velocity increment:

( ) ( )

( )

( ) ( )

23

23

21

21

21

21

21

23

21

23

22

/3

2

21

32

22

4

421

212

4

44

TPEDkm

nEDTmkETmkA

eETmkn

dvveTk

mn

dvvvfvdvvvvfdtdn

aB

aBaB

TkEaB

v

Tkmv

B

Ba

a

B

−−

−

=

==

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

==

−

−−

∞ −∫

∫∫

π

ππ

π

ππ

ππ

l

l

l

ll

(8)

An identical result is obtained. Mathematically, this is due to the fact that the exponential factor in the integral is dominant at energies near Ea. Thus the use of va*v2 or v3 in the integrant makes very little difference. However, this mathematical process implies that the high collision rates-- and therefore the high decomposition rates of molecules with higher velocities-- are maintained essentially constant in the same period of the time duration l/va, i.e., they are not consumed and cease to exist in the domain “dn” under consideration. The only way this can be physically true is if we assume that re-establishing the equilibrium velocity distribution is a very fast process, resulting in continual regeneration of the population of high velocity molecules. The relaxation process may therefore be important to consider in the decomposition process, especially given that collision is a very fast process in the sub-nanosecond regime (Typical values of l is 10-6 cm and v is 105 cm/s for vapor phase HMX). It is important to point out that in Eqs. 6 and 8, we have assigned the n associated with l to A instead of modifying Eq. 2 to become a second-order rate equation, because n has a different physical meaning here which is not really consumable. In other words, once the molecule is decomposed, it will be replaced with the by-product molecules which will provide “comparable” collision efficiency6. Reaction growth requires this assumption and more detailed analysis will be further explored in future work. Decomposition Activated by Collision of Molecules of Different Sizes (Fig 1b) From Fig. 1 b), it can be readily seen that,

ndDtvdDn

vt2_

22

12

1

_

)(22

2)( +=

+=

ππl

Therefore, 2

41

12 1* ⎟⎠⎞

⎜⎝⎛ +∗=

DdAA (9)

As an example, in the HMX combustion under nitrogen background pressure, with d = 3.6 x 10-8 cm and D = 6.4 x 10-8 cm, A2 has a value equal to 0.61 A1. Therefore, the decomposition is less effective than that in pure HMX gas at the same pressure. Oblique Collision


The concept of an oblique collision is illustrated in Fig. 2. There is an important difference between it and the “steric factor” effect already mentioned. In the latter, the collision is in-line but occurs at an “insensitive”, or “wrong” spot on the molecule thereby making the collision ineffective for initiating the decomposition. Our current concept of oblique collision actually pushes the problematic one step further towards quantification of the “effectiveness” of a particular collision.

For the oblique collision case, the frequency factor is derived as follows. While the reduction of the frequency factor due to geometry is small at ¼, the reduction due to factor kBT/Ea is large as mentioned previously. There is a good reason for tentatively not including this effect in the derivations that follow: the experimentally measured value or empirically determined value of A can only provide an average value of all possible collision alignments thus may have already included an “average” presentation of this effect.

( )

( )

TkE

a

B

kEaa

v

Tkmv

B

v

Ba

Ba

a

B

a

eE

TkA

deEEA

ddvevTk

mdn

vdvfvdtdn

−

−−

∞−

∞

∗∗=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

∫

∫ ∫ ∫

∫

41

1

0

cos2

11

0 cos

2

0

23

3

cos

22

22

23

cossincos

2

2sincos

cos

π

π

θθθθ

φπ

θθθ

θ

θ

θ

π

θ

r

l

(10)

a

B

ETk

AA ∗∗= 41

13 (11)

Effect of Van der Waals’ Potential The Van der Waals’ potential is a reversible energy. Therefore, in a low density gas, it appears as an elastic collision and does not impact the Maxwell distribution. However, in the close proximity of a collision contact point, it increases the molecular velocity, and therefore the decomposition excitation effectiveness, if both molecules

θθ

b)a)

Figure 2 a) In-Line Collision; b) Oblique Collision. Red Arrows Indicate the Direction of Collision. The blue Arrow Indicates the Effective Direction for Decomposition Excitation.


involved in the collision have strong potential to each other as illustrated in Fig. 3 a) below. On the other hand, if one of the molecules subjected to the collision has weak potential (low dipole moment), the velocity of the oncoming molecule will have no appreciable change as illustrated in Fig. 3 b). This latter situation accurately describes a nitrogen gas pressurized HMX combustion system. There, the nitrogen molecule has a low potential as indicated by its L ≅ 1.4 kcal/mol as compare to that of HMX, L = 24.89 kcal/mol18. The full formulation of the calculated frequency factor is shown in Eq. 12. The nature of Eq. 2 is degenerated in A and Ea. Therefore a net effect of enhancing the reaction rate can be manifested either in a reduction of Ea or an increase in A, as illustrated in Eq. 12.

( )

( ) ( ) ( )

( ) ( ) ( )

( ) TkLa

TkLa

TkLEaB

TkLEaB

Tkmv

v B

v

BB

Ba

BaB

a

a

eELAeTPLEDkmA

eLEDTmkn

eLETmkndvevTk

mn

vdvfvdtdn

'1

'24

'2

'23

'

'

'1*)'(4

'4

'212

41

23

23

21

21

21

21

21

232

23

−=−=

−=

−=⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

−−

− −−

−−−−∞

∞

∫

∫

π

π

ππ

πll

r

l

(12)

Where, ( )'2'2' LEm

Em

v aaa −==

The value of L’ is comparable, but not equal to L, the heat of vaporization at Tm. An approximate estimate based on electrical dipole-to-dipole interaction in Appendix 1 gives L’ ≈ 0.637*L. This approach may also used in the assessments of contribution on the kinetics by a solvent if its dipole moment property is known.

III. Dense Gases When the gas density becomes very high to the extent that the volume occupied by molecules is a significant portion of the gas volume, the ideal gas equation-of-state PV= NART (where NA is the Avogadro constant) or P= nkBT will no longer be applicable. The real-gas equation-of-state in the context of Van der Waals’ potentials must then be considered16, 17:

a)

b)

Figure 3 Effect of Van der Waals’ Potential: a) Strong Potential Presented in Both Molecules; b) Weak Potential in One Molecule.


( ) NRTbVVaP =−⎟⎠⎞

⎜⎝⎛ + 2 (13)

Due to limitations in model accuracy, the correction constants “a” and “b” are usually determined empirically via experiments. Currently, this type of experimental determination is not all possible for high energy explosive materials. A modified model has to be adopted to effectively account for the collision phenomenon. Unit Molecular Cell Model A self explanatory picture of the model is illustrated in Fig. 4 in which a cubic cell of variable size corresponding to the average volume occupied by a single molecule is enveloping each spherical molecule.

Fig. 5 shows the concept of a gas equilibrium distribution with a mean gap or separation λ on the order of one third of D (as a convenient example for illustration only). It is clear that in this configuration, the mean free path concept is no longer realistic. Each molecule will collide with its surrounding molecules much more frequently than in a low density gas, and the relevance of the molecular diameter to the collision frequency has to be de-emphasized. The distance within which the molecule may travel can actually be larger than λ if close packing is allowed, i.e., by an additional distance ε as shown in Fig. 6. The magnitude of ε depends on the value of λ (and therefore on the density) and is illustrated in Fig. 6 a) and b) for the maximum allowed values (as examples only). Therefore the equivalent or effective total additional gap ε’ is estimated as follows, ( ) 3

1−= MND Amρ

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−+−=+

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

−=

2

2

33

4

'5236.01''

'2'

'

DDDDD

DDD

DD

βελ

πβε

λ

(14)

β = 1/8 is chosen because at D’ = D, it gives ε’ ≅ 0.0595*D, which is close to that calculated for the Kepler close packing of spheres19, 20, ε’ ≈ 0.0432 (Voids ≈ 1 – 0.74048 = 0.2595 which yields an approximate equivalent void depth of 0.2595/6 = 0.04325*D) as illustrated in Fig. 6 c).

D D’D D’

Figure 4 Unit Cubic Cell Concept.

Dλ

λλ

Dλ

λλ

Figure 5 Mean Free Gap concept.


Hard Sphere Results The frequency factor for this model is derived in the same way as in Eqs. 6 and 7, except that l is replaced by λ+ε’. Even though we have now introduced a “molecular plane,” the functional dependence on the polar angle θ and azimuth angle φ are not considered. This is a reasonable approach because in narrow gap geometry, the off-axial motions are greatly restricted. This assumption may introduce an up to a factor of two in overestimate of A5. The relative velocity used in Eq. 4 is not included here due to the fact that (λ + ε’) is referred to the equilibrium position rather than the positions of individual molecules being impacted. This assumption may introduce an up to a factor of

2 in underestimate of A5. ( ) ( )

( )

( ) aB

TkEaB

v

Tkmv

B

ETmkA

eETmkn

dvveTk

mn

vdvfvvdv

vfdtdn

Ba

a

B

21

23

21

23

22

3

2)'(

1

2)'(

12)'(

4

)'(

5

32

33

−

−−

∞ −

+=

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

=+

=

∫

∫∫

πελ

πελ

πελπ

λελrr

(15)

There is a major change in the derivation of the gas pressure. Not all the gas molecules in the ensemble are directly involved in the flux impact used in Eq. 7. We assume that only the layer of molecules adjacent to the gap is active in the collision and in producing the pressure, i.e., only n2/3 molecules per unit area are active in the impact for pressure. Also, the unit area for the pressure calculation is only a mathematical plane without the interstitial depth (ε) feature.

( )

( )TkDDD

TknnnDDTknTkn

veTk

mnvmvdddvP

B

BmBB

Tkmv

Bv

B

a

12

1

222

00

)'('

)'(/

2coscos2sin

31

31

32

32

32

22

/3

32

2

−−

−−−

−∞

−=

−=−==

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫∫∫

λ

πλθθφθθ

ππ

(16)

hh

εε

εε

Figure 6 a) Case of Small ε; b) Case of Large ε; c) Kepler Close Packing Configuration.

a) b) c)


This result is essentially similar to the volume correction in the Van der Waals equation (Eq. 13). By adopting this model, differences can be noted in predicted behavior, in both P and A, from the values expected based on Eqs. 6 and 7. Table 1 lists the differences in normalized variables. They are plotted in Figs. 7 a) and 7 b) respectively. These are expected results because at high densities, the finite molecular size causes an increase in the collision frequency as well as the pressure. The results of the two models agree with each other when D’/D > ~10.0.

Table 1 Comparison of Relative Magnitude between Ideal Gas and Hard Sphere Model Ideal Gas Hard Sphere Fig. No.

P/T ( ) 3/' −DD 12 )1/'()/'( −− −DDDD Fig. 7a)

A A1; ( ) 3'2 −DDπ A5; ( )( ) 12'5236.01'−

−+− DDDD ββ Fig. 7b)

From Eqs. 15 and 16, it can be seen that the functional relation between A and P/T is nonlinear. Therefore the correlation between them, i.e., assessing the functional dependence of A on P and T, has to be performed numerically. Fig. 8 presents a plot of ( )( ) 12'5236.01'

−−+− DDDD ββ versus

12 )1/'()/'( −− −DDDD . A continuous and smooth function is apparent and can facilitate the numerical solution procedure. To examine the specific values relevant to DDT, Fig. 9 plots A and P for HMX at T = 1000oK. In spite of the limited accuracy of the current model in the regime of 1.0 < D’/D < 1.05, the data demonstrates the following interesting features:

1. At D’/D = 1.01, A = 1.56 x1014 s-1 is far below the 5 x1019 s-1 reported in Refs. 9-12 and Ref. 18 which is required for achieving a reaction completion time of ~5*10-9 s in order to sustain a detonation6. Therefore, the growth toward the weak-detonation to steady-state-detonation-transition may require the assistance of a heating effect from the initial decomposition6.

2. There is a remote possibility that aided by the exponential factor RTEae− in Eq. 2, A = 1.56 x1014 may support nanosecond decomposition. However, the required T is calculated to be a minimum of 2500oK, and the corresponding pressure is ~ 1 to 12 GPa. These conditions may exist in steady-state detonation but are not supported by the heat transfer and inertial pressure confinement estimates established for DDT scenario in Ref. 6. With the newly acquired formulation of A, assessment 1 above will be revisited and reconfirmed.

3. At D’/D = 1.01, P = 5.16 GPa (51.6 kbar) which exceeds the shock initiation threshold of HMX, 4.4 GPa (44 kbar)18.

4. At D’/D ≅ 1.18, P ≅ 0.2 GPa (2.0 kbar), the predicted onset pressure for HMX DDT6, A ≅ 4.2 x1013 s-1, which is not very far from the calculated 1.56 x1014 s-1 for D’/D = 1.01. Therefore its growth to the weak detonation threshold may be achievable via the heating effect from initial decomposition.

5. The calculated values of A = 1012 to 1.56 x1014 s-1 (D’/D up to 10.0) are in agreement with results reported in Refs. 7 and 8. This indicates that the steric factor likely has no significant effects in the current problem.

While the heating effect (termed as “self heating effect” in Ref. 6) will be further evaluated, it may not be an absolute necessary condition for DDT. In Sections IV and V, it will be shown that when the average molecular spacing decreases to below approximately 0.18 times of the molecular diameter (i.e., D’/D < 1.18), the frequency

Figure 7 Comparison of Results between Ideal Gas and Hard Sphere Models: a) P/T vs. D’/D; b) A vs. D’/D.

a) b)


factor may abruptly switch to high values on the order of 1019 s-1 due to the possible formation of a “dimer” structure in which the effective collision distance between two molecules drastically decreases and the collision frequency drastically increases.

Effect of Van der Waals’ Potentials The presence of Van der Waal’s potentials between the molecules will change the kinematics from constant speed used so far to variable speed depending on the position of the molecules in relation to the potential distribution. The random movement of molecules makes this a very complex problem. In fact, in the framework of Boltzmann-Maxwell statistics, we rarely deal with problems with this type of potential, except in the case of a uniform gravity field for the “atmosphere law.” The general geometry of the concept as shown in Fig. 10 is the same as that for the case without the Van der Waals’ potential shown in Fig. 5. Following the methodology used so far, where the collision frequency is inversely related to the time of flight between collisions, the general path for formulation of the frequency factor is as follows. As in the preceding derivations, the v is the velocity perpendicular to the gap and at the equilibrium position D+λ as shown in Fig. 11; The velocity at the off-equilibrium position corresponding to V(z) is v”, which is different from v . Conservation of energy as expressed in the energy equation requires that:

( )2

1

2"

"

2

22

122

1

⎥⎦⎤

⎢⎣⎡ −+==

+=+

+

+

VVm

vdtdzv

VmvVmv

D

D

λ

λ

The time of flight ∆t for single collision is,

0

4

8

12

16

0 20 40 60 80 100

(D'/D-1)^-1 * (D'/D)^-2

(D'/

D-0

.87

5-0

.06

54

5(D

'/D

)^-

2)^

-1

D'/D = 1.5

[D’/

D-0

.87

5-0

.06

54

5*

(D’/

D)^

-2]^

-1

0

4

8

12

16

0 20 40 60 80 100

(D'/D-1)^-1 * (D'/D)^-2

(D'/

D-0

.87

5-0

.06

54

5(D

'/D

)^-

2)^

-1

D'/D = 1.5

[D’/

D-0

.87

5-0

.06

54

5*

(D’/

D)^

-2]^

-1

A ,

s-1A

, s-1

Figure 8 Correlation between Key Geometrical Parameters of A and P/T.

Figure 9 Calculated P and A as a Function of D’/D for HMX at T = 1000oK.

λ λ

Dλ

λ λ

Dλ

λ λ

D

Z

z = 0 z = D + λ z = 2D + 2 λ

λ λ

D

Z

z = 0 z = D + λ z = 2D + 2 λ

Figure 10 Mean Free Gap Concept with Van der Waals’ Potential

Figure 11 Equilibrium Configurations of Molecules in Mean Free Gap Concept


( ) dzVVm

vtD

DD∫

+

−

+ ⎥⎦⎤

⎢⎣⎡ −+=Δ

λλ

21

22 (17)

Therefore, ( )

( )

( )( ) dzVV

mv

eETkn

dvvdzVV

mv

eTk

mn

dvvtvfdtdn

D

DDa

TkEa

B

v D

DD

Tkmv

B

Ba

a

B

∫

∫∫

∫

+

−

+

−−

∞

+

−

+

−

⎥⎦⎤

⎢⎣⎡ −+

=

⎥⎦⎤

⎢⎣⎡ −+

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Δ=

λλ

λλ

π

ππ

π

21

21

21

21

22/3

22

224

4

2

2

2

2

2

(18)

( )( )

dvet

vvTk

mmnP

dzVVm

v

ETkA

TkmvD

B

D

DDa

aB

B2

0

2

26

22

3

32

21

21

21

"2

4

22

−∞

+

−

+

−

∫

∫

Δ⎟⎟⎠

⎞⎜⎜⎝

⎛≈

⎥⎦⎤

⎢⎣⎡ −+

=

ππ

π

λλ

(19)

Note that in Eq. 19, the expression for P is approximate because the polar angle θ is not incorporated in the formulation. In reality, there are two special cases in which we can neglect the Van der Waals’ potential contribution and by-pass these rather complex calculations: Case i , λ/D << 1.0, where VD+λ = VD ≈ 0 and Case ii, VD+λ – VD ≈ 0. In Appendix 3, it will be shown that due to the symmetry of the problem, under the Frenkel-Halsey-Hill (FHH) version of the Van der Waals’ potential, Case ii is indeed a good approximation.

IV. Liquid Insensitive High Energy Materials The values of A and Ea measured at Tm

18 actually have very limited applicability in predicting the detonation behavior of these materials. Using HMX as an example, at Tm = 548.16oK, the 10% reaction completion time calculated using Eqs. 1-3, is on the order of 45.6 s, being many orders of magnitude too slow for supporting the detonation. However, they are important for two reasons: First, these data are the only source available for the quantitative assessment of the decomposition kinetics, i.e., one has to achieve a quantitative understanding of the available data and extrapolate the findings to the supercritical state at higher temperatures and pressures.

The second reason is apparent from the derivations achieved so far. The frequency factor is a very sensitive function of the gap distance between the molecules, i.e., the density, and the current model is not totally accurate in the small gap regime (1.0 ≤ D’/D < 1.05). It is well known that at Tm, there is a volume expansion of ~ 3 to 4% in transition from solid to liquid. This translates to a gap dimension on the order of 0.5%*D or smaller, comparable to the lower boundary of the small gap regime of our interest here for a better accuracy.

Compared to solid-state and gaseous-state physics, current understanding of liquid-state behavior is quite primitive. The existence of a critical temperature Tc makes it complicated to study any thermo-physical properties in the liquid state6. In the temperature span from Tm to Tc, density decreases from ρm to ρc (which is about 1/3 to ¼ of ρm) and the heat of vaporization decreases from L to zero. Cp is an increasing function of temperature indicating that various vibration modes are incrementally excited. At low pressure, Cp exhibits a singularity behavior at Tc 6. The previous formulation and some additional derivations in the Appendices may help in dealing with the liquid state. The following liquid-state analyses are limited to refining the gap theory formulated so far. Two well-established


formulations will be used in the analyses that follow: The Frenkel-Halsey-Hill potential (FHH) equation for liquid surface attractive potential and the Clausius- Clapeyron equation (CC) for saturated vapor pressure. Calculation of the Frequency Factor (A) Using the Frenkel-Halsey-Hill Potential (FHH) Equation The FHH equation (References are cited in Appendix 2) was developed for the description of adsorbed molecular layer thickness as a function of temperature and vapor pressure. It states that the layer thickness is proportional to the potential energy V(z) in the following form:

( ) nzzV 1∝ (20)

, where value of n is usually set at 3. To adapt it to our current formulation, the following form is used,

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−= 33 /

11DzD

V θ (21)

Therefore, when z →∞,

LD

V == 3θ (22)

Considering the average kinetic energy level of liquid molecules at Tm corresponding to the potential at ∆z = λ above the minimum of the potential well in Eq. 21,

TkTkzDLz

zz

dzdV

BBDzDz

21

2433

≅=Δ=Δ⎥⎦⎤

⎢⎣⎡=Δ⎥⎦

⎤⎢⎣⎡−

==

ωθ (23)

Normally, ω = 3. However, due to the small gap geometry we can choose ω = 1, i.e., ignoring the contribution of motions in the other two lateral axes. Therefore,

LTDkz B

61

7 ==Δ λ (24)

Using Eq. 15, we obtain

( ) ( )

( ) ( ) aB

aBB

aB

ETkmDL

ETmkTDk

LETmkA

23

21

23

21

23

21

23

26

2621

77

−−

−−

×=

==

π

ππλ

(25)

The value of L is available for many insensitive high energy materials18

at Tm but not for Tm< T < Tc. Therefore, this formulation can only be applied for calculation at Tm. Note also that the original FHH equation was intended for layered molecules. Our current application has extended it to cover sub-layer thickness (average thickness). In this approximation, the fine details of ε and ε’ are not included. Calculation of the Frequency Factor (A) Using the Clausius-Clapeyron (CC) Equation The common form of the well-known CC equation for saturated vapor pressure17 is:

TkLo

BeAP −= ' (26)

If one uses the heat of vaporization L at Tm, the accuracy of this formula can be approximately maintained to temperatures higher than Tm , up to Tb (Refer to, for example, the well-established Po-T data for water). Further extrapolation beyond Tb with limited accuracy can be made up to Tc – T ≅ 60oK, but beyond which it becomes invalid. Here, we start off by assuming Po = Pm for an L value that corresponds to Tm . In Appendix 4, a derivation of the closed form expression of the same equation using standard statistical mechanics and the “free gap” concept is obtained as follows:


TkLBo

BTeknP −=8

32

λ (27)

Therefore, by combining Eqs. 26 and 27, one obtains,

TkAD

TknA

BB

'''1 2

83

2 ≅=λ

(28)

and ( ) ( )2

32

12

32

12

3

)(''221 2

88 Tk

ADEmETmkAB

aaB−− == ππ

λ (29)

Numerical Results for HMX, RDX and PETN The calculated values of λ7

-1, λ8-1, A7 and A8 for HMX, RDX and PETN are presented in Table 2 along with the

input data. It can be seen that the values of A are significantly larger than those shown in Fig. 9 by using Eqs. 14 and 15, and the values of λ are significantly smaller. These differences are discussed in the next section.

Table 2 Comparison of Frequency Factors at Melting Temperature between Reference 18 and Current work

ID Parameter Explosives Notes Acronym HMX RDX PETN 1 M 296.17 222.13 316.15 2 Tm , oK 548.16 477.26 418.06

Ref. 18

3 ρm , g/cc 1.88 1.799 1.778 Solid Value 4 D , 10-8 cm 6.4 5.9 6.7 ( ) 3

1−= MND Amρ

5 L , kcal/mol 24.89 23.21 17.28 Ref. 18 6 L , 10-12 ergs 1.73 1.61 1.20 Per Molecule 7 Ea , kcal/mol 52.7 47.1 47.0 Ref. 18 8 Ea ,10-12 ergs 3.66 3.27 3.27 Per Molecule 9 Am, s-1 5 x1019 2.02x1018 6.3x1019

10 Pm , mmHg 0.302 0.871 0.1267 Ref. 18

11 A’ 3.34 x1012 4.88x1013 1.993x1011 Defined by CC Eq. 12 1/λ7 , cm-8 2.15 x109 2.49x109 9.08x107

13 A7 , s-1 2.05 x1015 2.63x1015 1.783x1016 Calculated by FHH

Potential, n=3

14 1/λ8 , cm-8 1.81x1011 2.58x1012 1.54x1010 15 A8 , s-1 1.73 x1017 2.72x1018 1.46x1017

Calculated by using CC Equation

Results using the CC equation for T > Tm for HMX are shown in Fig. 12 (T is calculated for a given D’ via Eq. 42 in Appendix 2). Based on the common features of ρ, L and A’ as function of temperature in the regime of Tm < T < Tc for common materials , the values of A8 in the region of 1.0 ≤ D’/D ≤ 1.08 are considered accurate because D’/D = 1.08 corresponds to Tb (Appendix 3). The values of A8 in the region 1.08 ≤ D’/D ≤ 1.18 could be considered marginally accurate because D’/D = 1.18 is still far away from the region of rapid decrease in density and in the heat of vaporization characteristic of temperatures near Tc. They all show only very minor variations. Results for D’/D > 1.18 are considered invalid because the values of the heat of vaporization differ drastically from L at Tm (as does A’) and Eq. 42 becomes inaccurate, resulting in a falsely increased trend of A8 as a function of D’/D. Therefore, for practical applications, the measured value of Am reported in Refs. 9 through 12 and Ref. 18 can be used in the regime of 1.0 ≤ D’/D ≤ 1.18 with reasonable accuracy. It is suspected that the same kinetics may hold under high detonation pressures on the order of tens of GPa (102 kbars) as reported in Ref. 21. In this case, the melted molecules would be forced to fill in the void left by the Kepler close packing19, 20. Note that the calculation for HNS, a very important explosive used in many current applications, is not performed due to lack of available data. The results using the CC equation are in greater agreement with the measured data than estimates using the FHH equation because the CC calculation uses experimentally measured vapor pressure data and thereby is less dependent on theoretical formulation. The existing differences between the measured frequency factor values


and those calculated by the CC equation could be due to inaccuracies in the measurement of the vapor pressure, the heat of phase transition data and the DSC9-11. It should be noted that, via Eq. 2, differences of a factor of about 102 in A would only translate to ~5 kcal/mol differences in Ea for HMX (i.e., error is <10%).

V. Discussion Relatively low values of A (up to ~1014 s-1) can be calculated for low density gases and dense gas systems. These values are expected given that both the average molecular velocities and the molecular spacing are known with reasonable accuracy. Thus the order of magnitude of collision frequency is well defined. At first glance, the unexpected high values of Am and A8 (~1019 s-1), and low values of λ8 when the spacing between molecules lies within the small D’/D range of 1.0 to 1.18, is a surprise. In Appendix 5, it is shown that the improvement of the hard sphere model by incorporation of the repulsive potential is insufficient for explaining this phenomenon. By using Eq. 2, an increase in A by a factor of 105 is equivalent to a 12.6 kcal/mol decrease in Ea. However, this approach is not supported by CC equation in which heat of vaporization of the liquid in the density range of 1.0 ≤ D’/D ≤ 1.18 has

relatively small variations. In addition, Ea is a quite stable parameter because it is originated from the characteristic of electronic bonds6. In Appendix 3, it is shown that the fine structure of the Van der Waals’ potential can create a narrow and localized potential well near D’/D = 1.0 (Figs. 19 b) thru d)). Thus, the formation of a “dimer” structure is feasible for reducing the effective free gap width and increasing the frequency factor. However, the width of this localized potential well, based on the z-3 law according to the FHH equation, is still relatively wide as shown in the figures, such that it also can not explain the small λ8 calculated by the CC equation. An additional clue is provided in Appendix 1. It shows that when the separation distance between molecules approaches D’/D = 1.0, and for some dipole distributions in the molecule, the value of n in Eq. 20 can be much larger than 3. For D’/D in this localized region of 1.0 ≤ D’/D ≤ 1.18 only, if one chooses n = 15, the width of this localized potential well could be comparable to the 10-11 to 10-12 cm as calculated for λ7 and λ8 in Table 2. The dipole distribution obviously involves the electronic structures in the molecules as indicated by the original work by F. London on the dispersion force22, 23. Evaluation of atomic effects using perturbation theory as adopted by F. London is outside the scope of the present work. What can be shown, however, using of the rule of thumb of quantum mechanics, in Heisenberg’s “uncertainty principle”, is that ∆E*∆t = h and ∆p*∆x = h. Using HMX as an example, at Tm, v ≈ 2.14*104 cm/s and E ≈ 1.14*10-13 erg, thus (∆t)-1 ≈ 1.72*1013 s-1 and ∆x ≈ 6.26*10-10 cm (~0.01*D). If Ea = 3.66*10-12 erg and va = 1.22*105 cm/s are used, (∆t)-1 ≈ 5.52*1014 s-1 and ∆x ≈ 1.1*10-10 cm (~0.0017*D) are obtained to indicate the domains within which our calculated results reside. In terms of the collision scenario, a frequency factor on the order of 1019 s-1 is noteworthy. It is orders of magnitude higher than the frequency of visible light (~6*1014 s-1) and is comparable to the transit time of an electrical field (speed = 3x1010 cm/s) across a distance of an HMX molecule (D ≅ 6.4*10-8 cm). The apparent ability of a system to abruptly switch from lower frequency factors on the order of 1013 to 1014 s-1 to a much higher value of 1018 to 1019 s-1 when the average molecule spacing D’/D falls into the range 1.0 to 1.18, can abruptly accelerate the deflagration decomposition towards the detonation. This supports the DDT scenario proposed in Ref. 6. The temperature and pressure correspond to this density criticality (D’/D ≈ 1.18) are: T is slightly above Tc and P ~ 0.2 GPa (2 kbars) and it can occur in a supercritical gas layer of ~50 μm in thickness6. The DDT is thus equivalent to a transition of frequency factor from low to high values. This remarkable transition is possibly due to the change from large-spacing (~λ5) free-flight molecular collision mode to more restricted or bonded small-spacing (λ8 << λ5) molecular collision mode which is a unique characteristic in a “dimer” structure.

VI. Summary and Conclusions

A8

, s-1

Region Invalid

Tb ≅ 744oK; D’/D ≅ 1.08

T ≅ 880oK; D’/D ≅ 1.18

A8

, s-1

Region Invalid

Tb ≅ 744oK; D’/D ≅ 1.08

T ≅ 880oK; D’/D ≅ 1.18

Figure 12 Frequency Factor Calculated by CC Equation for Liquid HMX as a Function of D’/D


1. The frequency factor in Arrhenius rate equation for low density gaseous system was derived using traditional statistical mechanics. Its dependence on molecular diameter, oblique collision and Van der Waals’ forces was demonstrated. Its dependence on pressure, temperature and gas molecule density was also formulated. The Arrhenius equation assumes that thermal equilibrium is maintained.

2. In this study, the frequency factor in the Arrhenius rate equation for dense gaseous systems was then derived using a new “mean free gap” concept in lieu of the “mean free path”. Its dependence on pressure, temperature and gas molecule density was also formulated. However, a maximum numerical value on the order of 1014 s-1 so obtained is too low to support detonation in high explosives. Again, the Arrhenius equation can be considered valid only if thermal equilibrium is maintained.

3. Experimentally determined frequency factors for the Arrhenius rate equation at melting temperatures for liquid phase insensitive high energy materials including HMX, RDX and PETN have been evaluated and confirmed using a newly derived statistical mechanical version of the Clausius-Clapeyron equation. This high value frequency factor Am on the order of 1019 s-1 is possibly due to the formation of a “dimer” state via a quantum mechanical effect when the average molecular spacing falls into the range below 0.18 times the molecular diameter. This high level of collision frequency can support detonation in high explosives.

4. At supercritical temperatures slightly above the critical temperature and under high pressures such that 1.0 ≤ D’/D ≤ 1.18, the frequency factor may switch from low values to high values and result in detonation. The abruptness of the switch-over coincides with the abruptness of transition commonly observed in DDT.

5. For practical applications, using of Am for A in the density range of 1.0 ≤ D’/D ≤ 1.18 provides a reasonable approximation for the high explosives under current study.

6. The Frenkel-Halsey-Hill z-3 equation was successfully applied in handling the Van der Waals’ potential in formulation of the frequency factors and assessment of a variety of thermo-physical properties.

Appendix 1

Van der Waals’ Potential Formulation It has been long recognized that the strength of this phenomenon is weak when compared to the electron-based

forces in a chemical bond. Based on the critical density observed for common materials, ρc ≈ ¼ ρm , a rough estimate is that the potential energy involved is proportional to r-6. Therefore it is natural to consider this force to be due to the electrical dipole-to-dipole interaction that exists between molecules. However, there are several shortcomings in this concept. Let us first consider the interaction between the permanent dipoles. The textbook formula23 for the far field interaction indicates that the potential is dependent on r-3. This approximation is far from applicable for the near field interactions of interest in our study. Fig. 13 illustrates a configuration of the simplest in-line dipole-to-dipole interaction. Fig. 14 a) shows the potential as a function of Z under the conditions of a constant dipole moment but with different dipole separation distances, d. Fig. 14 b) shows the slope of the potential which corresponds to the n value in z-n.

It can be seen that in the domain of 1.0 < z < 1.5, both V and n are very sensitive non-linear functions of d. It should be noted that the conditions illustrated in this example may be more complex than they appear. A situation could exist in which n equals 3 according to the FHH equation at a large Z but changes into much larger values as Z becomes small. An induced electrical dipole-to-

-q

-q

+q

+q

Z

Z = 0

Z = z + d/2

Z = -d/2

Z = zZ = z - d/2

Z = +d/2

V = 2q2z/(z2-d2) - 2q2/z

•

-q

-q

+q

+q

Z

Z = 0

Z = z + d/2

Z = -d/2

Z = zZ = z - d/2

Z = +d/2

V = 2q2z/(z2-d2) - 2q2/z

•

Figure 13 In-Line Dipole-to-Dipole Interaction Geometry

D = 1.0

Figure 14 Plots for the Case Studies in Fig. 13: a) V vs Z; b) n vs Z

a) b)


dipole interaction is a more readily accepted concept because it gives the correct functional form as an r-6 potential using the electrical polarizability factor α23:

31

12zp

=ε 31

122

zpp α

αε == ⇒ 6

21

124

zppV α

ε == ; (p1 = p2) (30)

To proceed with the summation of all dipole effects under mutual electrical dipole induction, the most plausible physical geometry as shown in Fig. 15 is followed. Note that this method, using the concept of the average dipole moment per unit surface area, leads to a z-4 dependence instead of z-3 as predicted by the FHH theory in Appendix 2.

( ) 232211

2

zrpE

+= (31)

( ) 2322

112

2zrpEp

+==

αα (32)

( )322

21

214

zrppEV+

=×=α

; (p1 = p2) (33)

( ) ( ) 4

2212

0322

221

2

242z

Dprdzr

DpDrdrVV παπαπ

=+

== ∫∫∞

∑ (34)

παπα

22

6

21

6

21 L

DpL

DpV

Dz=⇒==

=∑ (35)

Therefore we obtain,

LLD

pVDz

*637.0246

21 ===

= πα

(36)

There is a third possible approach for the description of the Van der Waals’ interaction, also termed the “London dispersion dipole interaction.” Via quantum mechanical dispersion effects of the electron wave function, a transient dipole moment can form in the molecules22, 23 to support a molecule-to-molecule interaction. It has been reported that this process occurs in close proximity to the molecules and also follows the r-6 potential dependence.

Appendix 2 Frenkel-Halsey-Hill Potential (FHH)

The main purpose of this theory was to describe the amount of adsorbed gas film on an adsorbent surface. In the first few molecular layers, the much stronger potential of the adsorbent surface has impact on the functional relationship. However, as the film thickness increases, the effect of the absorbent surface diminishes and the description approaches that of a liquid film, i.e., the properties are essentially those of a bulk liquid. The value of n=3 in Eq. 20 is reported to empirically fit the most experimental data. Theoretical derivations using macroscopic free energy and saturation pressure Po have also proven the validity24-27 of n = 3 for a partial vapor pressure P:

( ) ( ) 3lnz

zVPPTk oBθ

==− (37)

Traditionally, gas adsorption experiments are carried out by using fine powdery absorbents for providing a larger adsorption surface area in order to facilitate the measurement resolution. In the author’s research on the superfluidity of adsorbed helium 4 films, a very sensitive cryogenic temperature operable quartz crystal microbalance technique was developed28, 29. In order to calibrate this instrument, the adsorption of common gases was measured. In this technique, the decrease of the crystal shear mode resonant frequency, -∆fmb is directly proportional to the mass of the molecular layer loading per unit area adsorbed on the crystal electrode discs.

Z

YR

rφ

X

z

p2

p1

Z

YR

rφ

X

z

Z

YR

rφ

X

z

p2

p1

Figure 15 Plane Distribution Geometry for Dipole Integration


( )3

1

ln ⎥⎦

⎤⎢⎣

⎡=∝Δ−

PPTDzf

omb

θ (38)

Therefore, the slope of a plot of ln[1/Tln(Po/P)] versus ln(-∆fmb) verifies/confirms that n = 3. Some key results29 in the thick film regime are briefly summarized in Table 3.

Table 3 FHH Potential Exponent in Thick Adsorbed Films29

ID Gas D , 10-8 cm T , oK State* fmb ,

MHz n Notes

1 A 3.8 89.930 L 2.69 2 77.167 L 2.49 3 O2 3.6 89.953 L 2.61 4 N2 3.6 77.160 L 2.83

Linear

5 77.160 S 7.77 6 Kr 4.1 89.990 S 8.42 Non-Linear

7 3He 2.2** 1.356 L 6.43 8 4He 2.2 4.190 L

24

4.59 9 N2 3.6 77.160 L 10 3.12

Linear

*L: Solid film; L: Liquid film. fmb: Microbalance Frequency. **Assumes to be nearly the same as that of 4He.

It can be seen that for the larger molecules such as O2 and N2, n ≅ 3.0 is valid. Helium behaves differently possibly due to its smaller molecular diameter. However, in the regime between the first two mono-layers and the bulk layering, n ≅ 3.0 was observed for 4He. Solid films behave very differently. Results were not as reliable due to a possible adsorption distribution problem resulted from temperature differentials that might exist in the apparatus. The FHH equation has been successfully used in deriving of Eqs. 20-24, and assessments of the Van der Waals’ potential characteristics in dense gases (Appendix 3). It can also be used for an estimation of the liquid density at different temperatures as shown in the following formulation, if Tc is known:

Since, ( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛−= 33 /

11DzD

V θ and 3D

L θ= (39)

( )( )233 '/

/'11' DDDDD

V ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

θ (40)

Therefore, ( )MB TTkDD

DDLV −=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

2'''

52 ν (41)

One thus obtains, ( ) ( ) M

B

TDDDDk

LT +⎟⎟⎠

⎞⎜⎜⎝

⎛−= 52 /'

1/'12

ν (42)

The density n = D’-3 can be numerically solved by adjusting the value of ν, the average degrees of freedom, until Tc, the vertical tangent point, is correctly obtained. Fig. 16 a) shows the result for HMX using the known Tm ≅ 275oK and Tc ≅ 926oK (from Ref. 30). Similarly, Fig. 16 b) shows the result for refrigerant R113 using Tm = 236.92oK and Tc = 487.21oK (from Ref. 31). The latter result is highly compatible with values post on the NIST website31 (Fig. 17) except in the temperature region very close to Tc. Note again that the factor (D/D’)2 is used to account for the decrease in dipole moment field strength as the space between molecules increases. This approach


also provides correct predictions of the downward trend in heat of vaporization as a function of temperature except in the temperature region very near Tc.

Tm = 236.92oK

Tc = 487.21oK

Tm = 236.92oK

Tc = 487.21oK

Figure 17 R113 Density as a Function of Temperature Data as Published on the NIST Website31.

T , oK

ρ, g

/cc Tm = 548oK

Tc = ~936oK

ν = 14

T , oK

ρ, g

/cc Tm = 548oK

Tc = ~936oK

ν = 14

T , oK

ρ, g

/cc

T , oK

ρ, g

/cc Tm = 548oK

Tc = ~936oK

ν = 14

ρ, g

/cc

T , oK

Tm = 236.92oK

Tc = 496oK

ν = 9

ρ, g

/cc

T , oK

ρ, g

/cc

T , oK

Tm = 236.92oK

Tc = 496oK

ν = 9

a) b)

Figure 16 Calculated Liquid Density as a Function of Temperature: a) HMX; b) R113


Appendix 3 Behavior of the Van der Waals’ Potential in Dense Gases

The coordinate scheme for the discussion is shown in Fig. 18. Two different reference levels for FHH potential with n = 3 are shown in Fig. 19 a). For clarity of illustration, the one corresponding to the following equations (V = 0, for z/D = ±1.0) is chosen for the subsequent graphs.

( ) 1/1'

−+= DDD λ (43)

( )( ) 2

331 /1/11 −+⎟⎟

⎠

⎞⎜⎜⎝

⎛−= D

DzDV λθ and

( )( ) 2

332 /1/22/

11 −+⎟⎟⎠

⎞⎜⎜⎝

⎛

++−−= D

DDzDV λ

λθ (44)

( ) ( )( ) 2

33321 /1/22/

1/12 −+⎟⎟

⎠

⎞⎜⎜⎝

⎛++−

−−=+= DDDzDzD

VVV λλ

θ (45)

Ranges for of interests are:

λ2+<< DzD , or equivalently, DDz /21/1 λ+<< (46)

Ranges of λ/D and z/D λ/D 0.1 0.2 0.3 0.4 0.5 0.6 z/D 1.2 1.4 1.6 1.8 2.0 2.2

In this formulation, only the attraction potential between molecules is considered. Discounting of the long-range repulsive potential (as differs from the short-range Lennard-Jones repulsive potential in Appendix 5) is necessary for the molecules to establish an equilibrium position. If the potentials act on one molecule by way of its adjacent molecules (one on each side as shown in Fig. 18) the scheme consists both of an attraction potential from one side and a repulsion potential from the opposite side (both due to dipole-to-dipole interactions), the equilibrium position corresponding to λ can not be maintained. In the case of permanent dipoles, this approach implies alternating dipole polarities in a dipole array. In the case of dispersion-induced dipoles, there is no specific polarity restriction because the process is independent of the multiple molecular interactions which are not synchronized in time, and the effects are instead superimposed in space.


Because the hard sphere model is used, in Fig. 19 a), the domain 0 ≤ z/D ≤ 1.0 is assumed to be a “forbidden region”. Three cases for λ/D = 0.1, 0.3 and 0.5 are plotted in Figs. 19 b), c) and d) respectively. In these plots, the values of V(z/D) are in normalized units of θ/D3. The inclusion of the factor (1 + λ/D)-2 is needed to account for the decrease in dipole field strength as D’ increases, according to the same approach used in Appendices 1 and 2. The average and maximum values of V (Vave and Vmax) at the bottom of the potential well as a function of λ are plotted in Figs. 19 e) and 19 f) respectively. It can be seen that the variation of V across the range of interest in z/D is reasonably small, especially in Fig. 19 b) where 1.0 < z/D < 1.2. In the larger range of 0.2 < λ < 0.5, Vave and Vmax do not differ appreciably from one another. When the average thermal energy (~kBT) is higher than the entire potential energy distribution down the bottom of the potential well, the behavior of a molecule closely resembles a hard sphere model that does not include the effects of potential energy. Both A5 and λ+ε’ calculated by Eqs. 15 and 16 are good approximations. However, when kBT is smaller than Vmax, a molecule can be trapped in the localized potential well near D ≈ 1.0 to form a “dimer” system32, 33. This “short order” molecular structure was pointed out in Ref. 25 (without the specific details in terms of the potential function as they are reported here.). In a way, this is similar to the molecular interaction in solids. However, the binding energy here is only on the order of one electron-volt, much smaller than that in the bonds in solids. In other words, here the aggregation state is localized and transitory, and therefore a global ordering can not be maintained to support the “phonon” mechanism that commonly adopted for description of the solid state.

λ λ

D

Z

z = 0 z = D + λ z = 2D + 2 λ

Potential Energy

00.20.40.60.8

1

0 1 2 3 4z/D

Potential Energy

0

0.2

0.40.6

0.8

1

-4 -3 -2 -1 0z/D

V1 = 1 - (z/D)-3 V2 = 1 + (z/D)-3

Potential Energy Potential Energy

λ λ

D

Z

z = 0 z = D + λ z = 2D + 2 λ

λ λ

D

Z

z = 0 z = D + λ z = 2D + 2 λ

Potential Energy

00.20.40.60.8

1

0 1 2 3 4z/D

Potential Energy

0

0.2

0.40.6

0.8

1

-4 -3 -2 -1 0z/D

V1 = 1 - (z/D)-3 V2 = 1 + (z/D)-3

Potential Energy Potential Energy

Figure 18 Schematic Diagram of Coordinates and Individual Potentials in a Molecule at the Center due to Van der Waals Forces from Two Adjacent Molecules.


An interesting observation which may be relevant to our inference regarding the existence of high frequency factors in the supercritical state of high explosives (over that calculated by Eqs. 15 and 16, using the free flight collision model), is shown as follows. As pointed out in Appendix 1, due to dipole-to-dipole interaction the value of n in the z-n potential function may increase drastically as the separation distance between the molecules decreases.

V(z/

D)

z/D

V(z/

D)

z/D

Potential Energy as a Function of Molecular Position, A/D = 0.1

00.20.40.60.8

1

1 1.2 1.4 1.6 1.8 2

z/D

V (z

/D

V1V2V1 + V2V(

z/D

)

λ/D=0.1Potential Energy as a Function of

Molecular Position, A/D = 0.1

00.20.40.60.8

1

1 1.2 1.4 1.6 1.8 2

z/D

V (z

/D

V1V2V1 + V2V(

z/D

)

λ/D=0.1

Potential Energy as a Function of Molecular Position, A /D = 0.3

0

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8

z/D

V(z/

D V1V2V1 + V2V(

z/D

)

λ/D=0.3Potential Energy as a Function of

Molecular Position, A /D = 0.3

0

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8

z/D

V(z/

D V1V2V1 + V2V(

z/D

)

λ/D=0.3

Potential Energy as a function of Molecular Position, A /D = 0.5

0

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8 2

z/D

V(z/

D V1V2V1 + V2V(

z/D

)λ/D=0.5

Potential Energy as a function of Molecular Position, A /D = 0.5

0

0.2

0.4

0.6

0.8

1

1 1.2 1.4 1.6 1.8 2

z/D

V(z/

D V1V2V1 + V2V(

z/D

)λ/D=0.5

V ave vs Lamda

y = -5.1805x2 + 3.6249xR2 = 0.9562

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6

Lamda

V a

ve

Vave versus λ

λ

V ave

V ave vs Lamda

y = -5.1805x2 + 3.6249xR2 = 0.9562

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6

Lamda

V a

ve

Vave versus λ

λ

V ave

V max vs Lamda

y = -5.5001x2 + 3.9295xR2 = 0.9678

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6

Lamda

V m

ax

V max

Vmax versus λ

λ

V max vs Lamda

y = -5.5001x2 + 3.9295xR2 = 0.9678

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6

Lamda

V m

ax

V max

Vmax versus λ

λ

a) b)

c) d)

e) f)

Figure 19 a) Van der Waal’s Potential Definition; Potential Energy versus z/D: b) λ/D = 0.1; c) λ/D = 0.3; d) λ/D = 0.5; e) Vave versus λ/D; f) Vmax versus λ/D.

λ/D λ/D


The conditions for Fig. 20 are identical to those used to acquire the result in Fig. 19 b) except that n = 15, is now used. It should be noted that this is only applicable to the range where λ/D = 0.1 and 1.0 ≤ z/D ≤ 1.2, whereas for z/D ≥ 1.2, n = 3 is retained in order to maintain the regular FHH equation behavior. It can be seen that the depth of the local potential well near z/D = 1.0 is increased significantly over that shown in Fig. 19 b) for facilitation of the possible formation of a “dimer” system32, 33. In addition, Vmax is increased to 1.26, much higher than the 1.0 corresponding to L. In other words, it would allow the “dimer’ to form in the supercritical state. The resulting high frequency factor and decomposition rate would be capable of sustaining/supporting the DDT as predicted in Ref. 6. Although Figs. 19 and 20 depicts the physical configurations inside the fluid, the existence of a localized potential well and its possible width narrowing are applicable to the free surface of the fluid because they are intrinsic to the potential function characteristics. This is supported by the CC equation which describes the vaporization at the fluid surface. The same concept may also improve the FHH calculations in Eqs. 24 and 25. It should be noted that the dimer scenario discussed here appears to be a universal feature of all materials because of the universal applicability of the CC equation. Its occurrence and existence are common as manifested in the common vapor pressure phenomena.

Appendix 4 Derivation of the Clausius-Clapeyron Equation (CC)

In the 1830’s, the CC equation for saturated vapor pressure was derived from macroscopic thermodynamic parameters17:

VTL

VS

dTdP

Δ=

ΔΔ

=/ (47)

Since

21

RTL

dTdP

P

RTVP

=

≈Δ (48)

Therefore, RTLeAP −= ' (49)

This formula is identical in form to the Arrhenius rate constant in Eq. 2. Here, the heat of vaporization (L) can be viewed as “activation energy for vaporization”. For quite some time, therefore, it has appeared reasonable to the author to use this formula to derive the Arrhenius equation. In addition, vapor pressure data at Tm is readily available


00.5

1

1.5

2

1 1.2 1.4 1.6 1.8 2

z/D

V (z

/D) V1

V2V1 + V2V(

z/D

)

λ/D=0.1n = 15


00.5

1

1.5

2

1 1.2 1.4 1.6 1.8 2

z/D

V (z

/D) V1

V2V1 + V2V(

z/D

)

λ/D=0.1n = 15

Figure 20 Plot of Potentials in Eqs. 44 and 45 with λ = 0.1 and n = 15


for many insensitive high energy materials18 to support this exploration. Surprisingly, however, despite the similarities in form, the correlation between them was found to be very difficult, mainly due to the fact that the equation governs pressure rather than the vaporization “rate” which is the true counterpart of the Arrhenius equation. In principle, adapting Eq. 49 for derivation of the frequency factor would appear to require a complicated formulation of the equilibrium between vaporization from the liquid state and condensation of the vapor into the liquid with conservation of the number of molecules as well as of energy (including L). Using the “mean free gap” concept instead, a statistical mechanics version of the equation was successfully derived for the first time to the author’s knowledge. Therefore, it is documented in detail in this Appendix. The formulation follows exactly the same methodology as in the main text. However, two key assumptions are required: 1) the axial collision rate of the molecules at the liquid surface is assumed to be the same in both directions, either towards the inner surface of the molecules or in the out-bound direction towards the vapor phase, and 2) the fictitious boundary for molecules to “escape’ is purely theoretical, and therefore the fine structure included in ε is not considered. Given that the molecules strike at an invisible boundary at which V→0 with a terminal axial velocity of v”’,

Lvmvm−= θ222 cos

2'"

2 and

21

2cos'" 22 ⎟⎠⎞

⎜⎝⎛ −= L

mvv θ (50)

θ’ is then defined such that,

Lm

v 2'cos 22 =θ (51)

( )

( ) ( )

dvveLm

vTk

mmn

dvveTk

mLm

vmn

evTk

mvdLm

vdvmn

evTk

mvLm

vddvmn

veTk

mnvmvdddvP

Tkmv

BL

m

Tkmv

BL

m

Tkmv

BL

m

Tkmv

BL

m

Tkmv

BL

m

m

B

B

B

B

B

22

2

1

8

2

2

'

0

221

8

2'

0

22122

28

2222'

028

22

8

2

002

22

32/3

21

32

22

/3

21

23

32

22

/32

1

21

32

22

/32

1

21

32

22

/3

32

2

21

22

34

22cos34

2cos22cos4

2cos2cossin4

2cos'"2sin

−∞

⎟⎠⎞

⎜⎝⎛

−

−∞

⎟⎠⎞

⎜⎝⎛

−

−−∞

⎟⎠⎞

⎜⎝⎛

−∞

⎟⎠⎞

⎜⎝⎛

−∞

⎟⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛ −⎟⎟

⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤⎟⎠⎞

⎜⎝⎛ −−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ −=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ −=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∫

∫

∫∫

∫∫

∫∫∫

πλπ

πθ

λπ

πθθ

λπ

πθθθθ

λπ

πλθφθθ

θ

θ

θ

ππ

(52)

A transformation of variables can then be performed such that,

Lm

vw 222 −= and vdvwdw = (53)

The following simple closed form expression is obtained,


( )

( )TkL

B

TkL

BB

TkLTkmw

B

TkLm

wm

Bm

B

B

BB

B

Tekn

eTk

m

Tk

mn

dweewTk

mn

dwewTk

mmnP

−

−

−

−−∞

⎟⎠⎞

⎜⎝⎛ +−∞

−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∫

∫

8

8

24

08

224

0

1

8

32

25

21

232

1

25

21

32

2

232

1

25

21

32

22/3

32

283

3

2

3

2

234

λ

π

πλ

πλ

πλπ

(54)

Appendix 5 Repulsive Potential between Molecules

This appendix is included for completeness of the discussion. In addition, the constraints and limitations of the approach used are outlined for possible future improvements. As would be expected, inclusion of the collision repulsive force in stead of a “hard sphere molecule” reduces the effective mean free gap width between molecular collisions. However, the reduction is not large enough to account for the very small values of λ8 calculated by the CC equation. Therefore, it cannot contribute to the explanation of the extra high Am values measured by experiments. The usual treatment of the problem is expressed in the Lennard-Jones potential equation17, 32 (L-J):

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

612

rr

rrVrV oo

o (55)

The second term is the Van der Waals’ potential that we have previously discussed. The first term is an empirical description of the repulsive force from quantum mechanics following the “Pauli exclusion principle32.” This formulation was intended for simple molecule-to-molecule interactions and was successful in the description of the dimer phenomenon in Argon gas32, 33. To apply it to our current problem, the first step is to incorporate the FHH potential. Given that:

( ) ( ) ( )zVzVzV 13 += Where,

( )( )

( )( )

( )232

331 '/11'

/11 DD

DzLDD

DzDzV ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

θ (56)

The factor (D/D’)2 is necessary to account for the decrease in the effective dipole field strength as the density decreases. In order to correlate with the FHH equation, the following additional constraints must be imposed on the repulsive potential: 1) The result must predict the ρm which is only 3-4% smaller than that of the solid; and 2) The depth of potential well should be comparable with L. The best fit choice of V3 for conform these constraints is (in spite of the fact that the factor of 96 used in the exponent appears to be exceedingly large in value):


( )( )963

1Dz

LzV = (57)

Figs. 21 a) and b) show the modified L-J potential (as defined by Eqs. 56 and 57 and in units of L) in two distance ranges of interest. They show clearly that the larger value of D’/D results in a smaller heat of vaporization, as evidenced at the asymptotic potential values. Note that at the bottom of the potential well for D’/D = 1.0, the numerical fit is still not completely satisfactory: the potential is approximately 87%*L and minimum value of z/D ≈ 1.04 is larger than that predicted by ρm (z/D ~1.01). Figs. 21 c) and d) show a comparison of λ in a hard-sphere model and the modified L-J model respectively. It can be seen that at the density corresponding to D’/D = 1.0, λ5 >> λ8 , but the reduction is not sufficient to account for the orders of magnitude smaller values calculated by using the CC equation and measured in experiments. At a density corresponding to D’/D = 1.5, the difference between λ8 and λ5 is small, as expected.

Acknowledgements The author acknowledges D. Roth of the California Institute of Technology, Millikan Memorial Library for assistance in locating several of the references relevant to this work. This paper has been reviewed and edited by S. Yang of Univ. of Pittsburgh and E. Yang of Univ. of Pennsylvania.

V 3+

V 1

λ8

λ5V 3+

V 1

λ8

λ5

V 3+

V 1V 3

+ V 1

V 3+

V 1V 3

+ V 1

λ8

λ5

V 3+

V 1

λ8

λ5

V 3+

V 1

a) b)

c) d)

Figure 21 Modified L-J Potential Using Eqs. 56 and 57: a) z/D to 3.0; b) z/D to 1.5. Comparison of λ between L-J and hard-sphere Models: c) D’/D = 1.0; d) D’/D = 1.5.


References1 1 Arrhenius equation, http://en.wikipedia.org/wiki/Arrhenius_equation 2 Collision theory, http://en.wikipedia.org/wiki/Collision_theory

3 Yang, L. C. and Do, I. P., AIAA 99-2420, “Key Parameters for Controlling of Function Reliability in ‘Nonel Tube’ Explosive Transfer System,” 35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Los Angeles, CA, June 20-24, 1999.

4 Yang, L. C. and Do, I. P., “Nonelectrical Tube Explosive Transfer System,” AIAA Journal, Vol. 38, No. 12, December, 2000, pp. 2260-2267. 5 Yang, L. C., AIAA-2007-5138, “Correlation between the Accelerated Aging Test (AAT) and Real World Storage Temperature,” 43rd AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Cincinnati, OH, July 8-11, 2007. 6 Yang, L. C., AIAA-2008-4628, “Deflagration-To-Detonation-Transition (DDT) in Detonating Energetic Components,” 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Hartford, CT, July 21-23, 2008. 7 Maksimov, Y. Y., Apal’kova, V. N. and Solov’ev, A. I., “Kinetics of the Thermal Decomposition of Cyclotrimethylenetrinitramine and Cyclotetramethylenetetranitramine in the Gas Phase,” Russian J. of Phys. Chem, Vol. 59, No. 2, , 1985, pp. 201-202. 8 Jing, Q and Beckstead, M. W., AIAA-98-3222, “Influence of Condensed Phase Mechanism on HMX Temperature Sensitivity,” 36th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Cleveland, OH, July 13-16, 1998. 9 Roger, R. N., “Differential Scanning Calorimetric Determination of Kinetics of Systems that Melt with Decomposition,” Thermochim. Acta, Vol. 3, 1972, pp. 437-447. 10 Roger, R. N., “Simplified Determination of Rate Constants by Scanning Calorimetry,” Analytical Chemistry, Vol. 44, No. 7, June, 1972, pp. 1336-1337. 11 Roger, R. N., “Thermochemistry of Explosives,” Thermochim. Acta, Vol. 11, 1975, pp. 131-139.

12 Robertson, A. J. B., “The Thermal Decomposition of Explosives. Part II. Cyclotrimethylenetrinitramine and cyclotetramethylenetetranitramine,” Trans. Faraday Soc., Vol. 48, 1949, pp. 85-92.

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16 Jeans, J., An Introduction to the Kinetic Theory of Gases, Cambridge at the University Press, Cambridge, UK, 1940. 17 Reif, F., Fundaments of Statistical and Thermal Physics, McGraw-Hill Book Co., New York, NY, 1965. 18 Gibbs, T.R. and Popolato, A., “LASL Explosive Property Data,” Univ. of California Press, Berkeley, CA, 1980. 19 http://mathworld.wolfram.com/SpherePacking.html 20 http://mathworld.wolfram.com/KeplerConjecture.html 21 Hensen, B. F., Smilowitz, L., Asay, B. W., Dickson, P. M. and Howe, P. M., “Evidence for Thermal Equilibrium in the Detonation of HMX,” Proceedings of 12th Symposium (International) on Detonation, San Diego, CA, August 11-16, 2002, pp. 987-992. 22 London, F., Z. Physik 60, 245 (1930) and Z. Physik. Chemie, B11, 222 (1930). English translations in Hettema, H., Quantum Chemistry, Classic Scientific Papers, World Scientific, Singapore (2000). 23 Intermolecular Forces, http://en.wikipedia.org/wiki/Intermolecular_forces

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Condensation),” J. Phys. Chem., 1950, 54 (8), pp1186-1191. 27 Hill, T. L., J. Chem. Phys. 17, 668 (1949) and J. Chem. Education 26, 347 (1948). 28 Chester, M., Yang, L. C. and Stephens, J. B., “Quartz Microbalance Studies of an Adsorbed Helium Film,” Phys. Rev. Lett., vol. 29, No. 4, 24 July 1972, pp 211-214. 29 Yang, L.C., “Quartz Microbalance Studies on Unsaturated Helium Films,” Dissertation, University of California, Los Angeles, 1973.

30 Maksimov, Y. Y., “Boiling Point and Enthalpy of Evaporation of Liquid Hexogen and Octogen,” Russ. J. Phys. Chem., Vol. 66, No. 2, 1992, pp. 280-281.

31 National Institute of Standards and Technology website URL: http://webbook.nist.gov/ 32 “Lennard-Jones Potential”, http://en.wikipedia.org/wiki/Lennard-Jones_potential 33 “Dimer”, http://en.wikipedia.org/wiki/Dimer; Also, http://www.ams.org/notices/200503/what-is.pdf

[american institute of aeronautics and astronautics 45th aiaa/asme/sae/asee joint propulsion...

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