1

Prediction of Thermo-physical Properties of Energetic

Materials by Similarity to Inert Substances

L.C. Yang∗

La Canada Flintridge, CA 91011

Thermo-physical properties under high temperature and pressure are difficult to measure for all materials. For energetic materials, the decomposition process further complicates the task. As a result there has been relatively little published experimental data for energetic materials, hindering analytical modeling and understanding of a number of phenomena of interest including combustion and the deflagration-to-detonation-transition (DDT). Previously1, qualitative trends were illustrated for key properties such as density, specific heat and thermal conductivity as a function of temperature and pressure using the data of refrigerant R113 and applied to the heat transfer near the heating layer in HMX combustion. Recently, further advancements2,

3 were made in the analytic predictions of critical properties, liquid density and vapor pressure of HMX, RDX and PETN. This paper refines and verifies this approach. Namely, it proves for the first time that there exist universal characteristics in the liquid and supercritical states which can be applied to the predictions of many key thermo-physical properties for all materials, energetic and inert alike. This principle is illustrated by comparing calculated results and published data for water, R113 refrigerant and HMX. Results of a detailed exploratory survey on universal features of thermo-physical properties are included in the appendix.

Nomenclature A = Frequency factor in the Arrhenius decomposition rate equation1-3, s-1

A’ = Pre-exponential factor in CC vapor pressure equation for liquid A’S = Pre-exponential factor in CC vapor pressure equation for solid A5 = A for dense gases or liquidsA6 = A5 including van der Waals’ potential effect A8 = A6 using CC equation for van der Waals’ potential effect (See Ref. 2) Am = A in liquid at Tm determined experimentally as described in Refs. 3 and 7 ai = Constant coefficients in linear regression fit, saturated vapor pressure vs. 1/T bi = Constant coefficients in quadratic regression fit, saturated vapor pressure vs. 1/T

CC = Clausius-Clapeyron saturated vapor pressure equation Cpm = Specific heat at constant pressure and at Tm CpL = Specific heat at constant pressure at T, Liquid CvL = Specific heat at constant volume at T, Liquid D = Diameter of a molecule D’ = Cubic root of average volume occupied by a molecule D’c = D’ at Tc and Pc

DDT = Deflagration-to-Detonation Transition EG = Average energy per molecule in gaseous phase EL = Average energy per molecule in liquid phase HE = High energy material, refers to HMX, RDX and PETN in this paper

FHH = Frenkel-Halsey-Hill equation2 HMX = Cyclotetramethylene tetranitramine, C4H8N8O8

kB = Boltzmann constant = 1.38 x 10-16 erg/oK

∗ Senior Member, AIAA

46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit25 - 28 July 2010, Nashville, TN

AIAA 2010-7006

Copyright © 2010 by L. C. Yang. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2

L = Latent heat of liquid vaporization at melting temperature Tm; Also Liquid state LT = Depth of van der Waals’ potential well from a molecular layer at temperature T

LS→G = Heat of sublimation of solid, the same as ΔES→G LL→G = Heat of vaporization of liquid, the same as ΔEL→G = EG – EL ≠ LT ; but = L @ Tm

LS→L = Heat of Fusion, the same as ΔES→L M = Molecular weight n = Exponent of distance z from a layer of molecules in FHH potential equation n1 = Exponent of (D/D’) as a factor for predicting the decrease of θ as a function of D’ P = Pressure

[P] = Parachor Pc = Critical Pressure Pm = Saturated Vapor pressure at melting temperature Tm p = Saturation Vapor pressure.

PETN = Pentaerythritol tetranitrate, C5H8N4O12R = Universal gas constant = 8.314 J/mol-oK ; Also used for correlation coefficient

R113 = Ethane,1,1,2-Trichloro-1,2,2-trifluoroethane; CCl2F-CClF2; CF 113; Freon 113 RDX = Cyclotrimethylene trinitramine, C3H6N6O6

S = Solid state t = Time T = Temperature Tb = Boiling Temperature Tc = Critical temperature Tm = Melting temperature V = Potential energy of a molecule at a liquid surface at TmV’ = Potential energy of a molecule at a liquid surface at T > Tm V0 = Depth of van der Waals’ (FHH version) potential well Vc = Molar volume at TcVm = Molar volume at Tmz = Axial coordinate perpendicular to a layer of molecules ν = Effective degrees of freedom including rotational and vibrational normal modes ρ = Density, g/cc ρ’ = Liquid density, g/cc ρ” = Gas density, g/cc ρc = Critical density, g/cc ρm = Liquid density at Tm , g/cc θ = Strength factor of FHH version of van der Waals’ potential γ = Surface tension, dynes/cm

I. Introduction

In recent studies1-3 of the deflagration-to-detonation (DDT) phenomenon in high energy materials, nearly all key thermo-physical properties at high pressures and temperatures were needed for the analytical formulations and assessments. Since the required data are largely unavailable, alternate strategies had to be adopted. In Ref. 1, trends of density, specific heat and thermal conductivity as a function of temperature and pressure observed in inert materials were used to qualitatively extrapolate the limited available data for HMX near the melting temperature to the high pressure and temperature regime. This method allowed successful estimation of parameters for a ~50-μm-thick supercritical layer adjacent to the melting layer and the density of the surrounding gas required for effective transient inertial confinement of the reacting supercritical layer. Both of these features are important in the understanding of DDT. However, in subsequent studies of DDT reaction kinetics2 and the characteristics of the Arrhenius frequency factor3 A, quantitative knowledge of the following thermo-physical properties is necessary:

1. The critical properties Tc, Pc and ρc. Limited reports4,5 provide some estimated values of these parameters for HMX and RDX based on empirical formulas. This paper provides additional assessments and justifications for using these formulas.

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2. The density in the liquid state for the entire temperature range Tm < T < Tc. Reference 3 has proposed that the density is the main parameter for determining the Arrhenius frequency factor A. This paper provides detailed formulation and discussions on use of the Frenkel-Halsey-Hill2 version of van der Waals’ potential to calculate this density.

3. The saturation vapor pressure as a function of T for the entire temperature range Tm < T < Tc. Because the high frequency factor required in DDT exists in a high density fluid state in which the molecular separation distance is small, it is difficult to accurately estimate A based on molecular collision theory. This information has to be indirectly calculated from the vapor pressure. References 2 and 3 emphasized the fact that the vaporization process is closely related to the molecular collision process and shares a similar pre-exponential factor with that used in the decomposition (which is also activated by molecular collisions). It was found that the quadratic polynomial provides an accurate fit of ln P as a function of 1/T. In fact, a good fit can be achieved by using only three data points: (Pm , Tm), (Pb , Tb) and (Pc , Tc). This greatly simplifies the calculation procedure.

This paper focuses on these three topics. In addition, a limited-scope exploratory study on the universal features of thermo-physical properties that are in common for all materials (including high energy materials), was performed ( Appendix 1), and helped guide the derivations/formulations developed in the sections that follow.

II. Critical Properties Reference 4 proposed three empirical equations (Eqs. 1, 3 and 6 below) for calculation of the critical properties of many substances:

[ ] 25.1)0.11377.0( += PVc (1) Where Vc = M/ρc is in units of cc/g-mol, and parachor [P] is defined as a function of surface tension γ (dynes/cm), liquid density ρ’ and gas density ρ” (both are in units of g/cc):

[ ] "';'"'

4/14/1

ρρργ

ρργ

>>≈−

=MMP (2)

[P] can also be calculated from the number of atom types and atomic bonds4. A short cut to estimating Vc can be made from Fig. 1-11 in Appendix 1 which indicates that D’c/D = (Vc/Vm)1/3 = (ρm/ρc)1/3 ≅ 1.42, with a standard deviation of 0.0477. This parameter appears to be a universal physical constant. This approach is beneficial because γ is not a parameter commonly available in databases for high energy materials. A recent reference on parachor and surface tension can be found in Reference 6. Reference 4 further proposed to use the following formulas to calculate Tc from Tb :

159027.1 +×= bc TT (3) The same reference indicated that Eq. 3 is applicable to halogen- and sulfur-free substances, and other than aromatics and naphathenes. Short cuts with reasonable accuracies (R2 = 0.9746) can be used from Appendix Fig. 1-7 as follows,

Tc = 1.4419×Tb + 25.013 (4)

A less accurate short cut (R2 = 0.6797) can be obtained from Appendix Fig. 1-8 as follows,

Tc = 1.8398×Tm + 91.173 (5) It appears that Eqs. 4 and 5 are applicable to a broad species of molecules. Equations 3 and 4 are more accurate than Eq. 5. However, Tb is not available for high energy materials. It has to be extrapolated from the vapor pressure equation at Tm which inevitably introduces inaccuracy. Tc’s estimated by Eqs. 4 and 5 for HMX, RDX and PETN are

4

approximately 100oK higher than those reported in Ref. 3 which has accuracy on the order of ±45oK. The critical pressure Pc (in units of atmosphere) is proposed in Ref. 4 as follows, which is an empirical van der Waals’ equation.

8

8.20−×

=c

cc V

TP (6)

Reference 5 calculated the critical properties of HMX and RDX following equations in Ref. 4. The same equations were also used in Ref. 3 to calculate the critical properties of PETN. These results are summarized in Table 1. They represent the state-of-the-art values for these three common high energy materials. For reference in later parts of this paper, the heat of vaporization at Tm , L , is also tabulated.

Table 1 Critical Properties of HMX, RDX and PETN

HE Refer-ence Tm , oK L= ΔEL→G @Tm ,

kcal/mol Tb , oK Tc , oK

HMX M 548.16 27.6* 744* 927.16* RDX G & P 477.26 23.21 659.87* ~840.16* PETN G & P 416.06 23.64 597.97** 773.11** * Maksimov’s Values5; ** Calculated by the Author3; M: Maksimov5; G&P: Gibbs & Popolato7.

III. Selection of Case Examples for Study

The study reported in Appendix 1 indicates that the fundamental thermo-physical parameters are the molecular mass M, latent heat of vaporization at Tm , L, and the molecular structure. Note that there are inter-relationships between them, for example, L and M are correlated for most molecules and follow a universal functional dependence unless the molecular structure is drastically different (Fig. 1-19). In Appendix 1, it is identified that the long chain structure in hydrocarbons exhibit out-of-the-family properties. Other properties such as the specific heat, Tc , Tb , Molecular diameter D, and vapor pressure at Tm are derived parameters which depend on these fundamental parameters. Therefore, ideally, for simulation of the high energy materials, inert molecules with nearly identical fundamental parameters are of interest. This approach is viable because nearly perfect matches do exist. As an example, key thermo-physical properties of Bezo[ghi]perylene (C22H12)8, a polycyclic aromatic hydrocarbon, are remarkably close to that of HMX as shown in Table 2. Unfortunately, the substance is known to be potentially toxic9, therefore much of its other thermo-physical properties are not available. Its limited data shown in Table 2 do correlate well with that of HMX, RDX and PTEN as shown in Appendix 1.

Table 2 Comparison of Key Thermo-Physical Properties between Bezo[ghi]perylene and HMX

Molecule M Tm , oK Tb , oK L , kJ/mol

C22H128 276.33 551 773 112.5

HMX3 296.17 548.16 744 115.53

Not all available data covers sufficiently the data ranges of interest. This is particularly true for the ranges in high pressure and temperature which are important in the study of the DDT phenomenon. Ethane,1,1,2-Trichloro-1,2,2-trifluoroethane, CCl2F-CClF2 , a fluorocarbon for refrigerant application was chosen by Ref. 1 for the case study because it is the only species entry in the NIST Thermo-Physical Properties of Fluid Systems10 website database which has a relatively large molecular weight, pressure range of 200 MPa and temperature range above Tc . The trends of the properties were considered typical because they agree for the available data for many other substances. This approach was qualitatively successful as mentioned previously. One shortcoming of this molecule is its relatively low value of L thereby results in low values in Tm , Tb and Tc.

5

Thermo-physical properties of water are probably the most extensively studied of any material and are the foundation of thermodynamics. However, water is by no means an ordinary material. The maximum density occurs near 4oC and its unusually large dielectric constant and specific heat indicate that its unique molecular structure plays important roles in the liquid state. Most interestingly, it is one of a few substances which exhibit strong super-cool behavior (down to -42oC11). It has been observed that many physical properties, e.g., dielectric constant12 and vapor pressure13, vary continuously when it is cooling down to the super-cooled state, passing through Tm = 0oC (273.16oK). Measurements of crystallization rate14, 15 as a function of temperature in super-cooled water indicate that the specific heat is also likely to be a continuous parameter. The specific heat in super-cooled water in fact provides a negative thermal energy with respect to Tm which explains the observed rate of crystallization as a decreasing function of T. The most important aspect of the super-cooled state is that it provides proof that the molecular kinetics in the liquid is indeed governed by Maxwell-Boltzmann statistics which are based on the Kelvin temperature scale. This is the foundation of the collision frequency derivation performed in Ref. 2 for the frequency factor. Table 3 summarizes key thermo-physical properties of HMX, water and R113. Significant differences exist between these three substances. The proposal of using “common” and generalized features applicable to all substances is subjected to the observations in Appendix 1. There, it can be seen that for the critical properties, the liquid density in the temperature regime of Tm < T < Tc , and the saturation vapor pressure (in the same temperature regime), are indeed universal properties which correlate with M and L. Surprisingly, they are applicable to water, an apparent “outlier” (or one of the “worst” cases).

Table 3 Comparison of Key Thermo-physical Properties between HMX, R113 and Water

Molecule M Tm , oK L , kJ/mol Tb , oK Tc , oK Pc , bars ρc , g/cc HMX3 296.17 548.16 115.53 744 927.16 29.7 0.485 R11310 187.37 236.93 29.79 320.735 487.21 33.92 0.56

Water10 18.0153 273.16 42.785 373.12 647.069 220.64 0.322

IV. Liquid Density − Formulation To calculate the liquid density, it is necessary to explore the spatial functional form of the van der Waals’ potential which is balanced by the thermal energy of an individual molecule. The Frenkel-Halsey-Hill (FHH) equation (References are cited in Appendix 2 of Ref. 2) was developed for the description of the 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, where z is axial coordinate perpendicular to a layer of molecules:

( ) nzzV 1∝ (7)

The value of n is empirically determined to be 3. To adapt the formulas to our current derivation, the following form2 is used (with reference to the ground state at Tm) ,

( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛−= 33 /

11DzD

V θ (8)

Where θ is the strength factor of van der Waals’ potential, and D is the separation distance between two adjacent molecules which approximately equals the diameter of the molecule at T = Tm. Therefore, when z →∞,

LD

V == 3θ (9)

At T > Tm , z increases to a different and larger value D’, the potential changes to the corresponding new value:

6

Figure 1 Transition of Van der Waals’ Potential .

D’

D’ = D + λ

( )

( )233 '//'11' DDDDD

V ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

θ (10)

Note that the factor (D/D’)2 is introduced to account for the decrease in θ due to the decrease in surface dipole moment density2 as the spacing between molecules increases. The energy balance equation is proposed as follows2,

( )mB TTkDD

DDLV −=⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

2'''

52 ν (11)

Or, ( ) ( ) MB

TDDDDk

LT +⎟⎟⎠

⎞⎜⎜⎝

⎛−= 52 /'

1/'12

ν (12)

Where ν is an “average degrees of freedom”. The features and meaning of this constant will be discussed later in the manuscript. Critical Temperature Tc In Eq. 12, if we temporarily assume ν has a constant value, then there exists a maximum T corresponding to Tc at

357.125'

3 ==D

D c (13)

and Tc = 0.651*L/ν + Tm (14)

Eq. 14 can be used to calculate ν if Tc is known. The reason that it has a different appearance as compared to Eq. 5 is because L is an increasing function of Tm as shown in Appendix Fig. 1-6. The value of ν also increases as a function of L as shown in Appendix Fig. 1-13. The choice of the value for n is actually a quite difficult task. In Ref. 2 Appendix 2, it has been shown that for small z values, n ≅ 3. On the other hand, for large z values, n ≅ 4 was obtained (Ref. 2 Appendix 2, Eq. 34). In reality, n is a continuously varying parameter as a function of z. The likely final value of n at Tc is n = 6 as predicted by the van der Waals’ potential, because 1.42-6 = 0.122 gives a potential energy comparable to the linear kinetic energy of the molecule at Tc . This concept is schematically illustrated in Fig. 1. However, for liquid density, which has small z values, n = 3 provides accurate estimate in spite of the predicted D’c/D value of 1.36 at Tc is slightly smaller than the 1.42 shown in Appendix Fig. 1-11. The new assumption that θ decreases as a function of D’ needs be evaluated. If we replace the exponent 2 by n1 for the (D/D’) factor, Eq. 10 has the following form:

( )( ) 1'/

/'11' 33

nDDDDD

V ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

θ (15)

Values of n1 can be calculated as a function of D’ or T by using Eq. 11 (Change exponent of D/D’ term from 5 to [3 + n1]). Figure 2 and 3 plot calculated n1 as a function of D’/D and T for R113 using the best fit value of ν = 9 for

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NIST data (and the numerical values of ρ and T provided in the data used for calculations)10. It can be seen that n1 indeed has a value of 2 at large D’/D and T.

Similar plots for water are shown in Figs. 4 and 5. Notably, n1 significantly deviates from 2 until larger D’/D and T are reached indicating complex molecular interactions exist in small molecular spacing and low temperature.

Effective Degrees of Freedom The right hand side of Eq. 11 contains two necessary assumptions. First, the thermal energy is assumed to be proportional to T - Tm . This is equivalent to assuming a fictitious liquid state (or super-cooled liquid state) in which the energy is equivalent to that of the solid state below Tm (extrapolated to zero value at 0oK). This enables one to by-pass the need of complex calculation using solid-state properties. Second, the “effective degrees of freedom” ν is used. If one writes the full form for this energy term, it has a form similar to the “internal energy”: EL = ∫CpdT . Figs. 6 and 7 show the relationship between the van der Waals’ potential well depth LT and the liquid internal energy EL for R113 and water. For reference purpose, the gas internal energy EG and the latent heat of vaporization EG – EL , are also shown. The latter is different from LT which is based on the following equation suggested in Eq. 10:

232

' −−

⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

DDLLL m

T ρρ (16)

It is clear that if the liquid internal energy EL is used in the energy balance, its intersection with LT at point A would produce a too low Tc value. In order to obtain the correct Tc corresponding to point B, a lower effective internal energy (plotted as straight line corresponding to a constant ν) is required. In other words, not all degrees of freedom in the liquid systems contribute to the energy balance. Some degrees of freedom deeply imbedded inside the molecular structure do not directly participate in the equilibrium between the liquid and gaseous phases. This

Figure 2 n1 versus (D’/D) for R113. Figure 3 n1 versus T for R113.

T , oK

Figure 4 n1 versus (D’/D) for water. Figure 5 n1 versus T for Water.

T , oK

8

“absence” behavior of some degrees of freedom has been addressed in previous studies: In the derivation of statistical version of CC equation in Ref. 2, only 3-D linear kinetic dynamics was used, and in the discussion of DDT in Ref. 3, a transition from 3-D to 1-D dynamics (perpendicular to the supercritical gas layer) was suggested. Because the detailed behavior of this “effective degrees of freedom” is not known, there is no direct proof that it is a constant value. However, an observation of existing data of specific heats indicates that for the majority part of temperature range away from Tc , this is a reasonable assumption. Figs. 8 and 9 plot the specific heats of liquid R113 and water as a function of temperature in the unit of R/2 so that the degrees of freedom can be directly read from the charts. It can be seen that the changes of Cv and Cp in the majority part of the temperature range are reasonably small to warrant an approximately constant average specific heat (Especially for Cv).

Notably, the specific heat values are much higher than the best fit value of ν ≅ 9. The specific heat of liquid remains an interesting topic. For the examples shown above, their counterparts in the gaseous state have much lower values indicating a possibility that more degrees of freedom are active in the liquid state than that in the gaseous state.

V. Liquid Density − Results

Figure 10 plots the calculated results for R113. The agreement with NIST’s experimental data10 is excellent in the entire liquid regime. The results deviate significantly from NIST data10 in the saturated vapor regime. Therefore, the results in vapor regime are invalid for this case study and cases that follow. The root causes of this inaccuracy is the fact that both the choice of n = 3 and the concept of molecular layer are not valid in the gaseous phase. The liquid regime results for water shown in Fig. 11 are not as good fit as that for R113 when compared to the NIST data10. This is expected because water has a large variation in n1 as shown in Figs. 4 and 5. Fig. 12 shows the results for HMX. The agreement with Bedrov’s experimental data16 and the DIPPR equation17, both near Tm , are considered

E , k

J/m

ol

LT

EL , LiquidEG , GasEG - EL

E , k

J/m

ol

LT

EL , LiquidEG , GasEG - EL

Figure 6 Energy Diagram for R113 (EL and EG are from Ref. 10).

Figure 7 Energy Diagram for Water (EL and EG are from Ref. 10).

E , k

J/m

ol

LT

EG - EL

EG , GasEL , Liquid

E , k

J/m

ol

LT

EG - EL

EG , GasEL , Liquid

Figure 8 Specific Heats as a Function of T for Liquid R113 (Data from Ref. 10).

Figure 9 Specific Heats as a Function of T for Water (Data from Ref. 10).

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satisfactory. Therefore this approach meets the objective for providing density prediction in the entire liquid regime as required for calculation3 of frequency factor A8. The failure of valid prediction of ρ-T relationship by Eq. 11 in the vapor phase has no impact on the calculation of frequency factor A5 because it is carried out by the hard sphere model which does not depend explicitly on this relationship2. The results for liquid RDX shown in Fig. 13 does not agree with the DIPPR equation17. However, the latter data appears to be physically impossible because the ρm should be close to the crystalline density of RDX (1.799 g/cc) instead of ~1.2 g/cc as predicted by the DIPPR equation. The cause of this discrepancy is unknown. A linear equation obtained from the calculation which covers the low temperature regime near Tm is shown on the figure for the interested readers. The results for PETN are shown in Fig. 14 without comments because there is no data available to the author for comparison. Overall, the formulation is considered successful in predicting the liquid density for HMX, RDX and PETN and it is applicable to other high energy materials if their Pm and L data are available and critical properties are calculated by Eqs. 1- 6.

ρ, g

/cc

T , oK

ρ, g

/cc

ρ, g

/cc

T , oK

ν ≅ 9.5

ρ, g

/cc

T , oK

ρ, g

/cc

T , oK

Figure 11 Calculated Density as a Function of T for Water and Compared with Reference Data10.

ν ≅ 9.5

Figure 10 Calculated Density as a Function of T for R113 and Compared with Reference Data10.

10

ρ, g

/cc

ρ, g

/cc

Figure 12 Calculated Density as a Function of T for HMX and Compared with Reference Data16, 17.

ν ≅ 24

ρ, g

/cc

ρ, g

/cc

Figure 13 Calculated Density as a Function of T for RDX and Compared with Reference Data 17.

ν ≅ 21

11

VI. Saturation Vapor Pressure Historical Approach The Clausius-Clapeyron vapor pressure equation2 (CC) was derived from macroscopic thermodynamic relationships using Gibbs free energy and under two assumptions3: 1) Vapor density is much smaller than that of the liquid; and 2) Vapor density is sufficiently low so that the ideal gas equation of state is applicable. These assumptions are valid near Tm for common substances and the CC equation provides the following saturation pressure-temperature relation:

RTLeAp −= ' (17)

Or, 01ln a

Tap += (18)

However, when T increases from Tm toward Tc, the assumptions are not applicable3. Inaccurate or invalid prediction of the vapor pressure result. The quantity L in Eq. 17 is the latent heat of vaporization. In Ref. 2, L used in the derivation of the statistical mechanical version of vapor pressure is defined as the “depth of the van der Waals’ potential well” instead of the “latent heat of vaporization” (illustrated in Fig. 6 of Ref. 3 as EG - EL). These two quantities are equal to each other at T = Tm , because there, EG – V0 ≅ 0, and EG – EL ≅ L. At T other than Tm , L(T) = LT ≠ EG(T) – EL(T). The distinct difference between these two energy concepts is clearly seen in Figs. 6 and 7 where EG(T) – EL(T)→ 0 as T → Tc , But ( ) 0' 2 ≠= −DDLL cT as T → Tc. One way to mend this deficiency is to fit the experiment data using Eq. 18 above over a temperature range from Tm to a higher T, the so called August empirical vapor pressure equation18. The limitation of this equation is obvious for both theoretical derivations and practical applications: it cannot be extrapolated to Tc . The common form of the August equation is as follows,

TBAp −=10log (19)

ρ, g

/cc

ρ, g

/cc

Figure 14 Calculated Density as a Function of T for PETN .

ν ≅ 21

12

The August equation has no applicability to our current objective as no p-T data is available for high energy materials. An improved equation formulated in late 19th century, the commonly called Antoine equation18, has the following form:

TC

BAp+

−=10log (20)

When T is in Kelvin temperature units, the factor C has negative values. Therefore, superficially, the equation appears to suggest a shifting of the reference zero temperature to a higher temperature other than the Kelvin absolute zero. This approach has no justification at this time and therefore its validity is highly questionable. The applicability of Antoine equation also has a limited temperature range18 and is not universally applicable for the full temperature range of Tm < T < Tc . A detailed discussion on this issue is seen in a recent paper19. Nevertheless, the Antoine equation currently remains the primary state-of-the-art methodology. Values for the constants in Eq. 20 for many substances are tabulated in the NIST Chemistry WebBook20 under “Phase Change Data.” The vapor pressure of water has been studied extensively and many empirical vapor pressure equations have been reported in the literature21. The most well known one is the Goff-Gratch equation22 which has a very complex form and can not be generalized for applications to other substances. In addition, its range of applicable temperature is also limited. Therefore, development of an alternate and generalized vapor pressure equation is necessary for application to energetic materials. It must be pointed out that because of the lack of vapor pressure at temperatures other than near Tm for high energy materials, Tb values listed in Table 1 were calculated by a crude extrapolation in which constant values of A’ and L acquired at Tm via Eq. 17 are used for the extrapolation to obtain Tb corresponding to p = 1 bar via Eqs. 17 and 18 using constant values of a0 and A’ as follows (This procedure was used in Ref. 5 for HMX and RDX and in Ref. 3 for PETN by the author),

m

m RTLpAa +== ln'ln0 (21)

In addition, pm is usually extrapolated from lower pressures (p < pm and T < Tm) by the solid sublimation vapor pressure equation (This procedure was used in Refs. 5 and 23 for HMX and RDX and in Ref. 3 for PETN by the author):

mGSGS RTLSm

RTLS eApeAp →→ −− =⇒= '' (22)

This double extrapolations procedure inevitably introduces errors in the Tb estimation. For this reason, the accuracy of this particular calculation methodology will be evaluated and reported in the next two sections. One encouraging fact is that the quantity LS→G , unlike its counterpart LL→G , is not very temperature sensitive because the solid density (thereby the potential well depth) is nearly constant compared to that in the liquid. Therefore estimate by Eq. 22 above is reasonably accurate. Development of an Alternate Vapor Pressure Equation It was accidentally found by the author that the natural logarithm of saturated vapor pressure p of materials can be accurately described by a quadratic function of 1/T via regression fit of data (a.k.a polynomial fit or least square fit):

01

22ln b

Tb

Tbp ++= (23)

13

In fact, the same equation can be accurately determined by using merely three data points: (Tm , Pm), (Tb , Pb) and (Tc , Pc), which is designated as the “3-point quadratic regression fit” and used in the next two sections. This good result may be attributed to the fact that Tb’s for substances are usually located in the middle region of the temperature range between Tm and Tc . Thus it yields a more accurate quadratic regression fit (This may need more rigorous mathematical proof than the empirical evaluation here). The approach is of interest as Tb can be extrapolated by Eqs. 21 and 22 and Tc and Pc can be calculated by methods illustrated in Section II. Equations for Water Table 4 summarizes calculated coefficients of Eqs. 18 and 23 for water using three data sources: Handbook of Physics24, NIST10, and Wikipedia25. Using the coefficient of determination R2 as a criterion (1-R2 < 2.5×10-5), the linear regression fit using Eq. 18 gives good results (August equation like) in the solid regime but poor results in the liquid regime. The quadratic regression fit using Eq. 23 gives good results in both solid and liquid regimes. The set of results using Wikipedia data is of interest because it includes the super-cooled regime. The results are relatively less accurate as expected because of the broader temperature range and the uncertainties of the data in the super-cooled regime. The results for the “3-point quadratic regression fit” are also excellent because it produces essentially the same coefficient values as in the quadratic regression fit using full data points. However, its accuracy can not be judged by the coefficient of determination as R2 = 1 by the default of only using three points. The accuracy is better evaluated with respect to the true experimental data as shown in Figs. 15 and 16 which also include the extrapolation approach using Eq. 21. The error introduced by using Eq. 21 to predict p is approximately 20% at Tb = 100oC. Conversely, the error introduced by using these equations to predict Tb is ΔTb ≅ 20%*RTb

2/L ≈ 5.41oK.

Table 4 Regression Fit Constants of ln[P (bars)] versus 1/T (oK) for Water

Data Source State Fit T b2 b1 b0 1-R2

HDBK of Phys.24

S L

-25→0oC -6129.44 17.328 5.78E-07

Q -46222.30 -5774.14 16.646 2.375E-08

L

L 0o↔100oC

-5203.28 13.977 0.00013 Q -231214.95 -3746.53 11.702 1.2E-07 L

0oC↔ Tc -4892.62 13.034 0.00066

Q -218294.12 -3803.63 11.763 1.4E-05 L

Tm, Tb, Tc -4954.96 13.129 0.00072

Q -219086.51 -3820.04 11.823 0

NIST10 L

L 0oC↔ Tc -4896.81 13.046 0.00068

Q -222915.87 -3778.41 11.731 1.1E-05 L Tm, Tb, Tc

-4954.76 13.129 0.00073 Q -222106.48 -3801.62 11.797 0

Wikipedia25

S L

-100↔0oC

-6147.52 17.453 8.8E-07 Q 10074.02 -6241.36 17.618 5.9E-08

L

L -5687.27 15.796 0.00011 Q -107076.24 -4689.84 13.514 1.1E-06 L 0↔100oC

-4885.98 13.024 0.00063 Q -221383.25 -3786.27 11.741 1.2E-05 L

-100oC↔Tc -5216.34 13.756 0.0017

Q -191206.18 -3952.41 11.949 2E-05

S: Solid; L: Liquid; L: Linear Regression; Q: Quadratic Regression.

Equations for R113 Table 5 and Figs. 17 and 18 summarize the results for R113 using p-T data from Ref. 10, in the same format as that used for water. The database does not provide any data in the super-cooled or solid regimes. It can be seen that the results are essentially the same as that for water, namely, quadratic regression gives good fit and the accuracy is

14

preserved in the “3-point quadratic regression” fits. The error introduced by using Eqs. 21 for extrapolated p is ~15% at Tb. Conversely, the error introduced by using this equation to predict Tb is ΔTb ≅ 15%* RTb

2/L ≈ 4.31oK. The inaccuracy introduced by using Eq. 21 to extrapolate Tb for high energy materials is similar. For example, L = 27.6 kcal/mol and Tb = 744oK for HMX, ΔTb ≅ 15%* RTb

2/L ≈ 6oK. However, the inaccuracy is more sensitive to L value. Using the accuracy of L to be ΔL = ± 1.0 kcal/mol estimated in Ref. 5 for HMX, ΔTb ≅ ΔL* L/T ≈ ±37oK as reported in the same reference. This tolerance is transferred to the estimated value of Tc with additional tolerance built-in Eqs. 3 and 4. Therefore, the tolerance for Tc of HMX has a rather large range of ±47oK as reported in Ref. 5.

T , oC

Erro

r

T , oCT , oC

Erro

r

Figure 16 Percentage Errors for Fig. 15 Reference to the Experiment Data24.

p

T , oC

p

T , oC

Figure 15 Saturated Vapor Pressure Plot Based on Different Equations and Experimental Data from Ref. 24 for Water.

15

Table 5 Regression Fit Constants of ln[P (bars)] versus 1/T (oK) for R113

Data Source State Fit T b2 b1 b0 1-R2

NIST10 L

L Tm↔Tb

-3604.14 11.273 0.00031 Q -192325.8 -2208.3 8.767 5.8E-09 L

Tm↔ Tc -3416.19 10.594 0.00068

Q -149821.27 -2499.94 9.257 2.3E-05 L

Tm, Tb, Tc -3464.12 10.695 0.0007

Q -147173.32 -2536.88 9.35 0 S: Solid; L: Liquid; L: Linear; Q: Quadratic; Tm= 236.95oK; Tb = 320.735oK; Tc = 487.21oK

Application of “3-Point Quadratic Regression Fit” to High Energy Materials The “3-point quadratic regression” method has been applied to HMX, RDX and PETN for calculation of saturated vapor pressure3 in the entire liquid regime Tm ≤ T ≤ Tc which was in turn successfully used to calculate the frequency

T , oK

p

T , oK

p

Figure 17 Saturated Vapor Pressure Plot Based on Different Equations and Experimental Data from Ref. 10for R113.

T , oKT , oK

Figure 18 Percentage Errors for Fig. 17 Reference to the Experiment Data10.

16

factor3 A8 . The inputs and calculated coefficients are reported in Ref. 3 (Table 4 there) and reproduced in Table 6 below for reference.

Table 6 Inputs and Results of Saturated Vapor Pressure 3-Point Quadratic Regression Fit

HE

Input Results

L , kcal/mol

Tm , oK

Pm , mmHg

Tb , oK

Pb , bars Tc , oK Pc ,

bars b2 , oK2 b1 , oK b0

HMX 27.6* 548.16 0.6 744* 1 927.16* 29.7* -2793340.4 -6018.991 13.132

RDX 23.21 477.26 0.871 659.9* 1 840.16* 37.2* -62944.016 -11156.91 16.978

PETN 23.64 416.06 0.126 598** 1 773.11** 23.79** -3181911.1 1071.234 7.107

* Maksimov Values5; ** Calculated by the Author3

VII. Summary and Conclusions

1) Substantial linear relationship exists between Tc and Tb; and between Tc and Tm . These universal relations can

be used to estimate Tc for the materials studied in this work. The results cover broader material types than that previously reported in Ref. 4. The relationship can be applied to energetic materials.

2) The ratio D’c/D = (ρc /ρm)-1/3 appears to be a constant for all species studied in this work. This potentially universal constant can be used to estimate Pc through a universal van der Waal’s equation-of-state reported in Ref. 4.

3) A universal functional relationship exists between the liquid density and temperature. The function contains the following parameters/elements:

a. L, the latent heat of vaporization at Tm . It is identified as the depth of the van der Waals’ potential well (FHH version) between a molecule and the surrounding molecular layer at Tm.

b. LT , the depth of the van der Waals’ potential well (FHH version) in the temperature range Tm ≤ T ≤ Tc , can be represented by LT = L*(D/D’)2.

c. FHH version of van der Waals’ potential with the distance exponent n = 3. d. An “effective degrees of freedom”ν , which is an average degrees of freedom in the liquid, is

relatively temperature insensitive and is smaller than that indicated by the molar specific heat. 4) L is the primary parameter for liquid thermo-physical properties of species studied in this paper. Because of L ∝

M, M also appears to be a strong parameter for the properties. However, the fine molecular structure has influences as well. This leads to case-by-case variability and the existence of “class of molecules”, e.g., noble gases, hydrocarbons, and etc. The values of L and Cpm of HMX, RDX, PETN and Bezo[ghi]perylene follow the trend of hydrocarbons in spite of very different molecular structures between them (HMX, RDX and PETN are not chain structured7 like in many hydrocarbons).

5) The natural logarithm of p can be correlated accurately to 1/T in a quadratic regression fit for the full temperature range of Tm ≤ T ≤ Tc. It is illustrated by case studies of R113 fluorocarbon refrigerant and water. This may be a new alternate for the current standard format using the Antoine equation.

6) The quadratic regression vapor pressure equation in 5) above can be maintained with essentially the same accuracy by using only three data points: (Pm , Tm), (Pb , Tb) and (Pc , Tc). This procedure greatly facilitates the prediction of vapor pressure of high energy materials.

7) The method in 6) has been applied to calculate the vapor pressure of HMX, RDX and PETN which in turn has been used to estimate their frequency factors in Arrhenius decomposition kinetics3 in dense fluid state.

Appendix 1

An Exploratory Survey of Additional Universal Features in Thermo-physical Properties Background Thermal physics has many universal properties established centuries ago. Examples include: The Avogadro constant, Boltzmann constant, universal gas constant and mechanical energy to heat equivalent factor, all are universal constants; ideal gas law is a universal law for gases; and numerous others such as Dalton’s law, Henry’s law, Debye’s T3 law, Dulong and Petit’s law, Curie’s law and etc. are well known; Maxwell distribution is a

17

universal theory. These features are important in the understanding of thermo-physical properties of materials in solid and gaseous states. Universal features for the liquid state are relatively scarce. Ironically, the Clausius-Clapeyron equation and van der Waals’ equation are the only examples being used in Refs. 2 and 3 while the Trouton’s rule is not directly applicable (see later discussions). Due to the needs for several modeling studies, this topic has been recently surveyed by the author. Note the effort is far from exhaustive. But several new findings are of interest and proven to be useful in current modeling work. Therefore, it is documented here for interested readers. To assume that thermo-physical properties for common inert substances are applicable to predicting the properties of high energy materials may be an oversimplification. First, the properties are not all available. For the approximately twenty thousands of organic substances listed in the common handbooks26, perhaps only 10% of them are tabulated with both Tm and Tb . Recently, due to international safety regulations, the material safety data sheets (MSDS) have mandated the listing of information on Tm and Tb . This requirement has improved the knowledge on these two parameters. On the other hand, because the impact on environments and toxicity are difficult to establish, extra cautions are required in handling materials with unknown safety properties. This may hinder the physical experiments including the thermo-physical property measurements. For these reasons, comprehensive properties available in the public domain is limited to a couple of thousands substances and their availability is largely motivated by practical applications, notably, the hydrocarbons for energy applications, fluorocarbons for refrigerant applications and gases for cryogenic engineering. Database Available for the Survey NIST’s Thermo-Physical Properties of Fluid Systems website10 is a very comprehensive database in which 12 thermo-physical properties are available for both liquid and gaseous states in wide range of temperatures. It has 75 substance entries: All 5 noble gases, 30 fluorocarbon compounds for refrigerant application, 22 hydrocarbons and 16 simpler molecules, H2O, N2, H2, H2 (para), D2, O2, F2, CO, CO2, N2O, D2O, NH3, NF3, SO2, H2S, SF6, COS and Methanol (CH4O). Tabulated lower temperature values start at Tm except in a handful cases, being slightly higher than Tm . For the latter cases, latent heats of vaporization at the starting temperatures are used as L without significant error. Because of the values of L for entries in the NIST database are lower than 65 kJ/mole and much smaller than that of HMX, RDX and PETN (Tables 1 and 2), twelve molecular species with 71.3 kJ/mol ≤ L ≤ 102.2 kJ/mol were selected from the CRC Handbook of Physics and Chemestry26 to extend the range of study for this parameter (They are identified as “High L Molecules” in the plots). They are Triacetin, C9H14O6; 1-Nonanol, C9H20O; 1-Decanol, C10H22O; 1-Dodecanol, C11H24O; Dodecanenitrile, C12H23N; Tetradecanenitrile, C14H27N; Tetradecane, C14H30; 1-Tetradecanol, C14H30O; Pentadecane, C15H32; 1-Hexadecene, C16H32; Hexadecane, C16H34 and Heptadecane, C17H36 . This database lists ΔvapH @ T = 25oC only. Because Tm’s for these species are fairly close to 25oC, ΔvapH(25oC) can be used for L without significant error. Also, the CRC Handbook does not tabulate Tm and Tb for these materials. This information was obtained from MSDS available on manufacturer’s websites. Note that most of these molecules have chain structures, therefore, they may not be representative for polycyclic aromatic hydrocarbon molecules. A comprehensive database on enthalpy of sublimation of organic compounds is found in Ref. 27. This paper compiled data from 1910 to 2002. However, the tabulated temperatures for the measurements are variable and cannot be directly applied to our current analysis. For comparison, data listed in Tables 1 and 2 for HMX, RDX, PETN and Bezo[ghi]perylene are used “as is” in the plots that follow, i.e., not included for trend-line equation processing. Tm, Tb and Tc as a Function of M This correlation is of historical interest. It is well known that Tm and Tb of alkane hydrocarbons28 are a smooth increasing function of M29. A qualitative explanation for this phenomenon given in Ref. 29 is that a molecule with higher M contains more electrons which provide stronger van der Waals’ potential between molecules via the London force2. The total force is further enhanced by the size of the molecule: A larger molecular surface area produces stronger force (thereby higher Tb and Tc). Figs. A-1 thru A-3 plot the information extracted from the reference databases10. It can be seen that the increasing trend is true for hydrocarbons and noble gases, but not true for the fluorocarbons. HMX, RDX and PETN appear to follow the trend of the hydrocarbons. However, more data is needed in the range 200 < M < 350 in

T c, o K

T c, o K

Figure 1-1 Tc versus M for Reference Data10.

18

order to confirm this conclusion. Tm-M plot appears to show larger scattering than Tc-M and Tb-M plots.

Tm, Tb, and Tc as a Function of L Refs. 2 and 3 have identified L as a parameter of fundamental importance because it is the depth of van der Waals’ potential well between molecule and a molecular layer. Its value is subjected to decrease as T increases (via Eq. 16). However, at Tm , it has the maximum value and is the characteristic of the molecule-to-molecule interaction. Figs. 1-4 thru 1-6 plot the information extracted from the reference databases10, 26. Both Tc-L and Tb-L plots exhibit small variation, but are non-linear. In the case of Tb-L plot, there appears are two linear segments each with different slope. The Tm-L plot shows large variation between data points as expected because Tm is dominated by the complicated relationship LS→L = LS→G - L rather than L alone (LL→G = L @Tm). The fact that the same increasing functional relationship is maintained for all three plots indicates that the two parameters LS→L and L are interrelated. In order to accurately predict the trend behavior in Fig. 1-4, more data is needed in the range 60 kJ/mol < L < 120 kJ/mol. Also, Tm may be better correlated with LS→L , but this is not covered in the scope of this paper. Note that the relationship shown in Fig. 1-5 is different from the Trouton’s rule30. This rule deals with “enthalpy of vaporization at Tb” instead of L, the latent heat of vaporization at Tm .

Tc-Tm, Tc-Tb and Tb-Tm Plots

T b,o K

T b,o K

T m,o K

T m,o K

Figure 1-2 Tb versus M for Reference Data10, 26. Figure 1-3 Tm versus M for Reference Data10, 26.

T b, o K

T b, o K T m

, o KT m

, o K

Figure 1-5 Tb versus L for Reference Data10, 26. Figure 1-6 Tm versus L for Reference Data10, 26.

T c, o K

T c, o K

Figure 1-4 Tc versus L for Reference Data10.

19

Tb , oK

T c, o K

Tb , oK

T c, o K

Figure 1-7 Tc versus Tb for Reference Data10.

This series of correlations is of importance because Figs 1-1 thru 1-6 do not provide quantitative prediction of Tc and Ref. 4 utilizes linear extrapolation of Tc-Tb relation to estimate Tc . Figs. 1-7 thru 1-9 plot the information extracted from the reference databases10, 26. Good correlation is exhibited in the Tc-Tb plot despite inclusion of disparate molecular compounds in the single plot. The good result is expected because both temperatures are mainly controlled by a single parameter L. The Tc-Tm and Tb-Tm plots exhibited more scattering because a different factor LS→L controls Tm . For inert substances where Tb is accurately known, the Tc-Tb relation provides a better estimate of Tc . However, for high energy materials where Tb is not known and it has to depend on the extrapolation via Eqs. 21 and 22, the Tc-Tb and Tc-Tm give about same results for estimated Tc (R2 ≅ 0.65). It is noted that Tc data is not available for the “High L molecules”. Therefore, they are not available for the plots in Figs 1-7 and 1-8.

Clausius-Clapeyron Equation at Tm Figure 1-10 plots ln Pm as a function of –L/Tm. Other than hydrogen, deuterium and helium, a linear relationship is

observed. This finding argues that the CC equation shown in Eq. 17 is indeed a universal relationship.

Tm , oK

T c, o K

Tm , oK

T c, o K

Tm , oK

T b, o K

Tm , oK

T b, o K

Figure 1-8 Tc versus Tm for Reference Data10. Figure 1-9 Tb versus Tm for Reference Data10, 26.

lnP m

-L/Tm , No units

He

lnP m

-L/Tm , No units

lnP m

-L/Tm , No units

He

Figure 1-10 Plot of CC Equation at Tm for Reference Data10.

D’ c/

DD

’ c/D

Figure 1-11 D’c/D for All Species Cited in Ref. 10

20

The out-of-the-family behavior of helium is easy to understand: Under low pressure, helium does not solidify. The reason for deviation in H2 and D2 is unknown at this time. Significant deviations from the universal line are also seen for HMX, RDX and PETN indicating possible accuracy problems in Pm , L and Tm , or combination of them. Molecular Separation Distance Ratio (At Tc to That at Tm) The fact that Eq. 11 has been proven to be a universal relationship implies that the resultant calculated D’c/D by this equation (shown in Eq. 13) is likely to be a universal constant. Figure 1-11 plots this parameter for all species from the reference database10. Excluding the out-of-the-family data for helium (The smallest molecule), this parameter is indeed substantially constant with only slight increasing trend as a function of M. Values for HMX, RDX and PETN are considerably above the average line. However they are within the plus three standard of deviations. Interestingly, in an earlier report on critical properties of liquid metal31, it was reported that the critical density ρc is about four times less than that at Tb. However, the Tc are predicted to be three to four times higher than Tb , being quite different from the organic substances under our study. These qualitative features were successfully applied to the high temperature/pressure system in the study of laser generated stress waves32 by the author. Liquid metal systems and the associated latest advancements, the “law of rectilinear diameters” are out-of-the-scope of current work. An introduction to the latter topic can be accessed in Ref. 33. Effective Degrees of freedom, ν This interesting parameter is calculated by Eq. 11 for T = Tc for all species in the reference database10 (D’c/D = 1.36 is used. Results for D’c/D = 1.422 are similar). The ν-M plot in Fig. 1-12 exhibited only a weak dependence (slope effect is comparable to data scattering). A stronger dependence is seen in the ν-L plot in Fig. 1-13. The data falls in a range of 5 < ν < 12, indicating that some rotational and vibrational degrees of freedom are in effect. Values for HMX, RDX and PETN are much higher than the average trend. It may indicate that due to the instability that exists in these energetic molecules, more effective degrees of freedom are active. However, this is subject to further work, namely, more data is needed in the range of 200 < M < 350 and 60 kJ/mol < L < 120 kJ/mol in order to confirm this assessment. The ν value for helium falls below 3 corresponding to the 3-D linear velocities. This discrepancy is likely due to Tm and is not well defined for helium (helium does not solidify under low pressures).

Critical Pressure Several explorations of Pc were attempted and the results are presented in Figs. 1-14 thru 1-17. No conclusive findings can be made. However, excluding a few out-of-the-family cases, the data in general stays in a band of 20 to 60 bars indicating that this critical property is a common feature as predicted by the van der Waal’s interaction between molecules. The lack of quantitative equation for the universal description of this feature is not a surprising result, because the physical meaning of the quadratic term in the Eq. 23 is not explained, and molecular structure may have played a key role in this parameter in the critic temperature regime.

HeNe

CO2

A

Kr XeD2H2

H2ONH3

ν

HeNe

CO2

A

Kr XeD2H2

H2ONH3

ν

He

CO2 C3H2F6

D2H2

SF6

C4F8ν

He

CO2 C3H2F6

D2H2

SF6

C4F8ν

Figure 1-13 Plot of ν as a Function of L for the Species in Reference Database10.

Figure 1-12 Plot of ν as a Function of M for the Species in Reference Database10.

21

Categorized Universal features Limited numbers of categories are identified for the molecular species in the current study. Associated with the particular category, there exist some universal relationships. Similar behaviors have been known in the past for many thermo-physical parameters. As an example, it is known that CV of mono-atom molecules is 3/2*R and for diatomic molecules is 5/2*R and so on. Our results are as follows, 1. The values of Cp at Tm of hydrocarbons are higher

than other molecules (both are from the reference database10). Behaviors of HMX16, 17 and RDX17 appear to follow that of the hydrocarbon (Fig. 1-18).

2. Similarly, the values of L at Tm of hydrocarbons

are higher than other molecules (both are from the reference database10). Behaviors of Bezo[ghi]perylene, HMX, RDX and PETN (from Table 1) appear to follow that of the hydrocarbon (Fig. 1-19).

P c, b

ars

P c, b

ars

lnP c

(bar

s)ln

P c(b

ars)

Figure 1-14 Pc as a Function of L for the Species in Reference Database10.

Figure 1-15 lnPc as a Function of –L/Tc for the Species in Reference Database10.

P c, b

ars

P c, b

ars

lnP c

(bar

s)ln

P c(b

ars)

Figure 1-16 Pc as a Function of M for the Species in Reference Database10.

Figure 1-17 lnPc as a Function of M for the Species in Reference Database10.

Cpm

, R/2

Cpm

, R/2

Figure 1-18 Cpm versus M for Species in Reference Database10.

22

3. Again, the calculated values of D of hydrocarbons are larger than other compounds (both are from the reference database10). However, the values2 of HMX, RDX and PETN appear to follow that of the non-hydrocarbon species. This is not a surprising result because we know they do not have the chain structure7 commonly found in the hydrocarbons (Fig. 1-20).

The category of higher values of Cpm and L seem to indicate that there are more active degrees of freedom in atomic dynamics existing in the molecules of high energy materials and the hydrocarbons. Note that the hydrocarbons also contain high energy that can be released in combustion.

Acknowledgements

The author acknowledges D. Roth of the California Institute of Technology, Millikan Memorial Library, for assistances in locating several of the references relevant to this work. Acknowledgement is also given to U.S. Dept. of Commerce, National Institute of Standards and Technology (NIST), for posting many relevant references on public websites including “Thermo-Physical Properties of Fluid Systems10” referenced extensively in the paper. This paper has been reviewed and edited by E. Yang of Univ. of Pennsylvania.

References

1 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. 2 Yang, L. C., AIAA-2009-5190, “Frequency Factor in Arrhenius Decomposition Kinetics for Insensitive Energetic Materials,” 45th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Denver, CO, August 2-5, 2009. 3 Yang, L. C., AIAA-2010-xxxx, “Relationship between Arrhenius Frequency Factor (A) Characteristics and the Deflagration-to-Detonation Transition (DDT),” 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Nashville, TN, July 25-28, 2010. 4 Meissner, H. P. and Redding, E. M., “Prediction of Critic Constants,” Industrial and Engineering Chemistry, Vol. 34, No. 5, May, 1942, pp.521-526. 5 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. 6 Escobedo, J and Mansoori, G. A.,”Surface Tension Prediction for Pure Fluids,” AIChE Journal, Vol. 42, No. 5, May 1996, pp.1425-1433. 7 Gibbs, T.R. and Popolato, A., “LASL Explosive Property Data,” Univ. of California Press, Berkeley, CA, 1980. 8 Heat of Vaporization at Tm can be found in National Institute of Standards and Technology website URL: http://webbook.nist.gov/cgi/cbook.cgi?ID=C191242&Mask=4#Thermo-Phase or http://www.dfmg.com.tw/member/chemical/cas/191-24-2.htm ; Tb can be found in http://en.wikipedia.org/wiki/Benzo(ghi)perylene . 9 Benzo[g,h,i]perylene (CASRN 191-24-2), http://www.epa.gov/iris/subst/0461.htm; http://www.ewg.org/sites/humantoxome/chemicals/chemical.php?chemid=100288 ; http://www.speclab.com/compound/c191242.htm ; http://msds.chem.ox.ac.uk/BE/benzo[ghi]perylene.html. 10 National Institute of Standards and Technology website URL: http://webbook.nist.gov/chemistry/fluid/ 11 Debenedetti, P. G., and Stanley, H. E.; "Supercooled and Glassy Water", Physics Today 56 (6), 2003, pp. 40–46.

L , k

J/m

olL

, kJ/

mol

M1/3D

, cm

He

Other Molecules

M1/3D

, cm

He

M1/3D

, cm

M1/3D

, cm

He

Other Molecules

Figure 1-19 L versus M for Species in Reference Database10.

Figure 1-20 D versus M1/3 for Species in Reference Database10.

23

12 Rusche, E. W., “Dielectric Constant of Water and Supercooled Water from -5oC to 25oC,” Dissertation, New Mexico State Univ., May, 1966. 13 Cantrell, W., Ochshorn, E., Kostinski, A. and Bozin . K, “The Vapor Pressure of Supercooled Water Measured Using Infrared Spectroscopy,” J. Atmos. Oceanic Technol., Vol. 25, 2008, pp. 1724-1729. http://www.phy.mtu.edu/~kostinsk/cantrell_ir.pdf 14 Yang, L. C., “Crystallization Rate of Supercooled Water in Cylindrical Tubes,” Thesis, New Mexico State Univ., May, 1966. 15 Yang, L. C. and Good, W. B., “Crystallization Rate of Supercooled Water in Cylindrical Tubes,” J. Geophysical Research, Vol. 71, 1966, p. 2465 and Vol. 71, 1966, p. 6147. 16 Bedrov, D, Smith, G. D., and Sewell T. D., “Thermal Conductivity of Liquid Octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) from Molecular Dynamics Simulation,” Chem. Phys. Letters, Vol. 324, 2000, pp. 64-68. 17 Design Institute for Physical Property Data (DIPPR®). http://dippr.byu.edu . Values for HMX and RDX are posted in Washburn, E. B. and Beckstead, M. W., AIAA-2004-3870, “Modeling Multi-Phase Effects in the Combustion of HMX and RDX, Tables 1 and 2,” 40th AIAA/ASME/SAE/ASEE Joint Propulsion Conference, Ft Lauderdale, FL, July 11-14, 2004. 18 http://en.wikipedia.org/wiki/Antoine_equation 19 Růžička, K., Fulem, M., Růžička, V., “Vapor Pressure of Organic Compounds. Measurement and Correlation,” http://www.capec.kt.dtu.dk/documents/overview/Vapor-pressure-Ruzicka.pdf cited in http://en.wikipedia.org/wiki/Vapor_pressure 20 NIST Chemical WebBook http://webbook.nist.gov/chemistry/ 21 http://cires.colorado.edu/~voemel/vp.html cited in http://en.wikipedia.org/wiki/Vapour_pressure_of_water as Web Page listing various vapour pressure equations 22 http://en.wikipedia.org/wiki/Water_vapor; http://en.wikipedia.org/wiki/Goff-Gratch_equation . 23 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. 24 Gray, D. E., “American Institute of Physics handbook,” 3rd Edition, McGraw-hill Book Co., N. Y., N. Y., 1972. 25 http://en.wikipedia.org/wiki/Water_(data_page) ; cited in http://en.wikipedia.org/wiki/Water under List of water Topics. The database assembled data from: Sonntag, d., and Heinze,D.: Sättigungsdampfdruck- und Sättigungsdampfdichtetafeln für Wasser und Eis. (Saturated Vapor Pressure and Saturated Vapor Density Tables for Water and Ice)(1. Aufl.), VEB Deutscher Verlag für Grundstoffindustrie, 1982, and Guildner, L. A.; Johnson, D. P.; Jones, F. E. , "Vapor Pressure of Water at Its Triple Point: Highly Accurate Value". Science 191 (4233): 1261, 1976. doi:10.1126/science.191.4233.1261. PMID 17737716 . 26 Lide, D. R., “CRC Handbook of Chemistry and Physics,” 73rd Edition, CRC Press Inc., Boca Raton, FL 33431, 1992-1993. “Enthalpy of Vaporization of Organic Compounds,” pp 6-110 to 6-111. 27 Chickos, J. S. and Acree Jr., W. E.,“Enthalpies of Sublimation of Organic and Organometallic Compounds. 1910–2001,” J. Phys. Chem. Ref. Data, Vol. 31, No. 2, 2002 pp. 537-698. 28 http://en.wikipedia.org/wiki/Hydrocarbons 29 http://en.wikipedia.org/wiki/Alkane 30 http://en.wikipedia.org/wiki/Trouton's_rule 31 Grosse, A. V.,“The Temperature Range of Liquid Metals and an Estimate of Their Critical Constants,” J. of Inorg. and Nuclear Chem., Vol. 22, Issue 1-2, Dec., 1961, pp. 23-31. 32 Yang, L. C., “Stress Waves Generated in Thin Metallic Films by a Q-Switched Ruby Laser,” J. Appl. Phys., Vol. 45, No. 6, June, 1974, pp. 2601-2608. 33 Law of rectilinear diameters, http://search.yahoo.com/search?p=law+of+rectilinear+diameters&ei=UTF8&fr=fp-yie8-s&xargs=0&pstart=1&b=1&xa=D5NL5_dSe9U_7XS7Xsmcrw--,1276027753