RDX Mixtures

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Thermodynamics of HMX Polymorphs and HMX/RDX Mixtures Philip C. Myint, and Albert L. Nichols Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b03697 • Publication Date (Web): 09 Dec 2016 Downloaded from http://pubs.acs.org on December 14, 2016

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Industrial & Engineering Chemistry Research

Thermodynamics of HMX Polymorphs and HMX/RDX Mixtures Philip C. Myint∗,† and Albert L. Nichols III‡ †Design Physics Division, Lawrence Livermore National Laboratory, Livermore, CA, USA ‡Materials Science Division, Lawrence Livermore National Laboratory, Livermore, CA, USA E-mail: [email protected]

Abstract We present thermodynamic models for the five most commonly studied phases of the energetic material octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX): liquid HMX and the four solid polymorphs α-, β-, γ-, and δ-HMX. We show results for the density, heat capacity, bulk modulus, and sound speed, as well as a phase diagram that illustrates the temperature and pressure regions over which the various HMX phases are most thermodynamically stable. The models are based on the same equation of state as in our recently published paper on another energetic material, hexahydro-1,3,5trinitro-1,3,5-triazine (RDX). We combine our HMX and RDX models together so that the equation of state can also be applied to liquid and solid mixtures of HMX/RDX. This allows us to generate an HMX/RDX phase diagram and calculate the enthalpy change associated with a few different kinds of phase transitions that these mixtures may undergo. Our paper is the first to present a single equation of state that is capable

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of modeling both pure HMX and HMX/RDX mixtures. A distinct feature of HMX is the strongly metastable nature of its polymorphs. This has caused some ambiguity in the literature regarding the relative order of the six possible solid-solid transition (α-β, α-γ, α-δ, β-γ, β-δ, and γ-δ) temperatures. A related issue is whether the HMX phase diagram should have an α phase region. We show that there are only four possibilities for the relative order, and explain why the available data indicates that the only order in which there is an α phase region is by far the most plausible one.

Introduction Energetic materials are substances that may undergo rapid, exothermic reactions when subjected to an external stimulus, such as heating. An active area of research in energetic materials examines cookoff, which pertains to their behavior as they are heated. 1–3 Research on cookoff has been motivated by past incidents involving accidental fires on navy ships. They are particularly susceptible because munitions on these ships tend to be tightly packed in storage rooms and located near living quarters. One such incident involving the USS Forrestal killed 134 people and injured an additional 161 in 1967. 4 It has been noted that most of the casualties were not due to the initial fire caused by an accident, but by the chain reaction of explosions induced by cookoff of munitions near the fire. The thermal response of energetic materials may be studied by placing samples inside a sealed container and heating them on time scales ranging from microseconds to days until they explode. The heating induces reactions that form product gases. The pressure increase from the gas production further accelerates the reactions and may lead to an explosion that ruptures the container walls. In typical laboratory setups, the pressure can become as high as a few kilobars before the explosion occurs. Numerical simulations of these setups require accurate equations of state for the product gases and for energetic materials in their unreacted (liquid and solid) forms. Equations of state for the unreacted forms, especially those that 2 ACS Paragon Plus Environment

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are applicable to mixtures of energetic materials, are relatively undeveloped compared to their counterparts for the product gases. The main goal of this work is to develop an equation of state for the unreacted forms of the energetic material octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX). Specifically, we are interested in modeling the five phases of HMX that have been observed near ambient pressures: liquid HMX and the four solid phases α-, β-, γ-, and δ-HMX. These solid phases are often referred to as polymorphs because they represent different crystal structures of the organic compound. Two additional polymorphs (ǫ- and φ-HMX) have been observed at pressures above 12 GPa (120000 bar), 5,6 but these pressures far exceed the limit for the equation of state, which is targeted towards conditions relevant to cookoff. HMX is commonly used in rocket propellant formulations. 7 The pressures encountered in rocket engines are comparable to those in cookoff experiments. Solid-solid transitions from one polymorph to another may affect the performance of HMX through a number of ways, such as by causing fracture formation. 8 The fracturing is a result of the volume change that occurs in these first-order phase transitions. HMX also has numerous military applications, including explosive formulations of weapons. The thermodynamically stable form at ambient conditions is β-HMX. This is the polymorph of greatest interest to the military because it has the highest energy density and is the least sensitive to external stimuli. In contrast, the other polymorphs have been considered at one time or another to be dangerously sensitive materials. 9 Their unexpected appearance could therefore lead to serious accidents. Compared to substances like water, HMX has relatively large uncertainties regarding the true values for the enthalpy change and temperature associated with its phase transitions. The large variability is indicative of kinetic effects that arise due to strong nucleation and growth barriers between the polymorphs. These barriers inhibit phase transitions and allow metastable phases to remain present for days and even weeks. 10–12 Several authors have studied the nucleation and growth of HMX phases through experiments and continuum3 ACS Paragon Plus Environment


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scale models. 13–18 Smilowitz et al. 15 have found that nitroplasticizer, a compound commonly found in plastic binders of some HMX-based formulations, acts as a solvent for HMX crystals at higher temperatures and thereby provides the HMX with lower energy nucleation sites. Their β-HMX samples that were wetted with nitroplasticizer completed the transformation to δ-HMX at 436 K. The latter is the stable form at high temperatures and atmospheric pressure. In contrast, their pure β-HMX samples did not fully complete the transition even at 445 K. These results show that some of the variability in the observed properties may be due to differences in formulation that subsequently get magnified by the kinetic barriers. The α-β transition has been a source of confusion in the literature. Studies carried out over 50 years ago have observed that β-HMX crystallizes to α-HMX if it is dissolved in a solution that is heated. 9,19,20 The α-β transition temperatures reported in these studies are far lower than the temperatures for any other HMX phase transition. This suggests that as β-HMX is heated, it first undergoes a transition to α-HMX before the α-HMX eventually transforms to δ-HMX. However, no study has experimentally observed the α-β transition to occur in the absence of a solvating medium, despite the fact that there are many studies on the α-δ transition, both with and without a solvent. Furthermore, several investigators who have examined HMX phase transitions report results for the β-δ transition only, with no mention of other phase transitions. As a result, there is ambiguity in the literature regarding the thermodynamic stability of α-HMX. Brill and Karpowicz 21 have performed activation energy measurements and Levitas et al. 17,18 have developed theoretical nucleation models to explain the observed predominance of the β-δ transition. Our work complements these studies by addressing the following questions from a thermodynamic perspective: if β-HMX is heated starting from ambient conditions, does it first undergo an α-β transition or does it directly transform to δ-HMX? What are the implications of each of these two possibilities? Our thermodynamic models for HMX are based on the Peng-Robinson equation of state (EOS), 22 which is the same EOS used in our previous study 23 to model two other en4 ACS Paragon Plus Environment

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ergetic materials — hexahydro-1,3,5-trinitro-1,3,5-triazine (RDX) and 2,4,6-trinitrotoluene (TNT) — including liquid and solid mixtures of the two compounds. Our RDX/TNT paper is the first to have applied this well-known EOS to energetic materials in their unreacted forms. An advantage of the Peng-Robinson EOS is its versatility; it provides a simple, unified framework to model both pure components and mixtures. We take advantage of this versatility by combining our HMX and RDX models together so that the equation of state can also be applied to HMX/RDX mixtures. The HMX/RDX system presents a greater challenge than the RDX/TNT system does because the existence of HMX polymorphs allows for the possibility of multiple different HMX/RDX solid phases. Nevertheless, it is important to understand these mixtures because RDX obtained from the Bachmann process, which is the most common method for producing military-grade RDX, generally contains significant HMX impurities of between 4 to 17% by mass (5 to 21% by moles). 24 We organize this paper into three major sections. The first section focuses on pure HMX. We begin this section by describing how the models for the five HMX phases are developed from experimental density and vapor pressure data. We then discuss our results for several HMX phase transitions and present our predictions for the phase diagram. We also show results for the heat capacity, bulk modulus, and sound speed. The second major section focuses on HMX/RDX mixtures. This section includes phase diagrams for the HMX/RDX system at atmospheric pressure and at higher pressures. In addition, we present calculations for the enthalpy change associated with a few different kinds of phase transitions that HMX/RDX mixtures may undergo. The third major section addresses the ambiguity mentioned above regarding α-HMX. We clarify this issue by analyzing experimental phase transition data. We conclude with a summary of our work and suggestions for future studies. Certain details have been omitted in this paper for the sake of space. We refer the reader to our earlier paper for details on the Peng-Robinson EOS, and to the Supporting Information for supplementary figures and tables referenced throughout this work. 5 ACS Paragon Plus Environment


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HMX Vapor pressure and density The first step in applying the Peng-Robinson equation of state to HMX/RDX mixtures is to develop a model for each of the five HMX phases (α-, β-, γ-, δ-, and liquid HMX). In its most fundamental form, this EOS expresses the difference between the Helmholtz energy F of a substance and the Helmholtz energy F ig that the substance would have if, hypothetically speaking, all intermolecular interactions in it were removed so that it exists as an ideal gas: √ " # F − F ig 1 + (1 + 2)bρ a √ ln = − ln(1 − bρ) − √ . N RT 2 2bRT 1 + (1 − 2)bρ

(1)

Here, N is the number of moles, ρ is the molar density, R is the gas constant, and T is the temperature. We derive all other properties, such as pressure, heat capacity, and chemical potential, by applying basic thermodynamic relations to (1). Intermolecular interactions and non-zero molecular volumes are represented by the parameters a and b, respectively. These parameters serve as adjustable variables that are fitted to experimental data. We represent each HMX phase with a different set of a and b. The γ phase is not a true polymorph, but rather is a hydrate with a stoichiometric ratio of four HMX molecules per water molecule. 25 Although it is the least studied solid phase of HMX, there is still enough experimental data available on γ-HMX for us to determine its a and b parameters in the same manner as with other HMX phases. Just like in our previous work, 23 we calibrate a and b of the HMX phases with experimental vapor pressure and density data. For some phases it is helpful to also consider phase transition data, particularly the temperature and enthalpy change ∆h associated with the transition. Figure 1 shows Clausius-Clapeyron fits to vapor pressure measurements from a number of authors, along with our results for δ-HMX. We have fit the a and b of δ-HMX


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to reproduce the vapor pressure measurements of Taylor and Crookes. 26 We use their data because their paper clearly states that δ-HMX is being studied (not all papers clearly state which polymorph is being studied), and because their measurements are close to the results from another study (Cundall et al. 27 ), which suggests a degree of reproducibility. 10

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360

390

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450

480

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Figure 1: Vapor pressure of HMX vs. temperature. The solid curve represents our model for δ-HMX. The other curves are Clausius-Clapeyron fits to experimental data. Unfortunately, the other vapor pressure curves in Figure 1 cannot be directly used to determine a and b for the other HMX polymorphs because doing so would lead to phase transition temperatures that are not consistent with results from the literature. Table 1 lists β-δ phase transition temperatures from numerous experimental studies. Tables S1, S3, and S5 in the Supporting Information present the temperatures for the α-δ, γ-δ, and α-β transitions, respectively. All of these first-order phase transitions are accompanied by an endothermic enthalpy change ∆htrans that can be detected through differential scanning calorimetry (DSC). Table 1 illustrates some of the issues we have discussed in the Introduction regarding the metastable nature of the HMX polymorphs. The endothermic event is not sharply-peaked at a single temperature, but rather is spread out over a range of temperatures. The metastability also leads to a dependence of the phase transition temperature on the DSC heating rate, as well as on the size and morphology of the sample particles. Similar 7 ACS Paragon Plus Environment


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remarks apply to the other transitions. Despite these complications, most of the studies have found that the β-δ transition temperature falls between 440 – 467 K. Table 1: Experimental results for the temperature of the β-δ phase transition. All of the studies except for two 7,8 have examined the behavior only at atmospheric pressure. We discuss these two studies in more detail in the next section. As explained in the text, we use the mean temperature obtained by McKenney and Krawietz 24 at a 1 K/minute heating rate to model the vapor pressure profile for β-HMX with the Clausius-Clapeyron equation (2). Reference Teetsov and McCrone 20 Hall 28 Krien et al. 29 Rylance and Stubley 30 Kishore 31 Landers and Brill 7 Landers and Brill 7 Karpowicz and Brill 8 Brill and Karpowicz 21 Koshigoe et al. 10 Koshigoe et al. 10 Quintana et al. 33 Herrmann et al. 11 Herrmann et al. 34 Hussain and Rees 35 Zeman 36 Saw 12 Saw 12 Lee et al. 37 Weese et al. 38 McKenney and Krawietz 24 McKenney and Krawietz 24 Xue et al. 39

Temperature (K) Notes 430 ± 1 Crystallized in solution 460 ± 3 440 – 456 444 457 – 467 438 (1 bar) Slow heating 32 459 (100 bar) – 489 (1180 bar) 3 µm samples 471 (460 bar) – 513 (2080 bar) 3 µm samples > 438 453 Crystalline 460 Powdered 467 10 K/min 442 – 472 0.5 K/min – 50 K/min 454 – 466 0.5 K/min – 10 K/min 463 20 K/min 457.4 (onset) and 462.0 (peak) 20 K/min 438 (coarse) Complete after 4 hrs 443 (fine) Not complete after 8 hrs 468 10 K/min 463 – 471 (peaks) 1 K/min – 10 K/min 468.3 – 472.0 5 K/min 462.1 – 462.8 1 K/min 459 (onset) and 483 (end)

None of the vapor pressure curves in Figure 1 intersect in the range 440 – 467 K, and for this reason, none of them can be used to fit a and b of β-HMX. The temperature at which two curves representing different polymorphs intersect is the triple point where the two polymorphs are in equilibrium with each other and with the vapor phase. The temperature of the triple point is also the polymorphic transition temperature Ttrans at the triple point 8 ACS Paragon Plus Environment

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pressure. Because of the relatively weak effect of pressure on the phase behavior, the normal transition temperature Ttrans,n (i.e., the value of Ttrans at atmospheric pressure) is virtually identical to the triple point temperature. Rosen and Dickinson 40 have measured the vapor pressure of β-HMX. Their extrapolated Clausius-Clapeyron curve intersects that of Taylor and Crookes at 552.8 K, and that of Cundall et al. at 510.1 K. Despite the seemingly close overlap of Cundall et al. with Taylor and Crookes, their profiles intersect at 415.5 K. The δ-HMX curve of Brady et al. 41 does not intersect any of the others. Thus, none of the vapor pressure profiles in the figure would lead to an appropriate β-δ transition temperature. In order to obtain a reasonable β-δ transition temperature, we infer the vapor pressure profile of β-HMX from experimental data on the enthalpy change ∆h0trans at atmospheric pressure. The vapor pressure P of β-HMX at a temperature T can be approximated by the Clausius-Clapeyron equation P ∆hvap ln = Pδ R

1 Ttrans,n

1 − T

!

,

(2)

where R is the gas constant, (Ttrans,n , Pδ ) represents the triple point for β-HMX, δ-HMX, and gaseous HMX, and ∆hvap is the β-HMX enthalpy of vaporization at the triple point. In this equation, we have used the fact that the normal transition temperature Ttrans,n is essentially the same as the triple point temperature. We denote the triple point pressure as Pδ because we find this pressure by calculating the vapor pressure of δ-HMX at Ttrans,n . The β-HMX enthalpy of vaporization ∆hvap is approximately equal to

∆hvap = ∆hvap,δ + ∆h0trans ,

(3)

where ∆hvap,δ is the δ-HMX enthalpy of vaporization. For reasons that we explain in the section on HMX/RDX mixtures, we choose Ttrans,n to be 462.5 K, roughly the average of McKenney and Krawietz’s range 24 for DSC heating at a rate of 1 K/minute. At this tem9 ACS Paragon Plus Environment


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perature, our δ-HMX model returns a vapor pressure of Pδ = 5.119 × 10−7 bar and ∆hvap,δ = 164.73 kJ/mole. Table 2 lists experimental results for ∆h0trans of the β-δ transition. With the exception of Karpowicz and Brill 8 (we discuss their results in the next section), the studies report a value between 8 – 10 kJ/mole. For consistency with our choice of Ttrans,n , we let ∆h0trans = 9.16 kJ/mole. Tables S1–S4 in the Supporting Information present the same set of information (Pδ and ∆hvap,δ from our δ-HMX model and appropriate values for Ttrans,n and ∆h0trans ) needed to construct the vapor pressure profiles of α-HMX and γ-HMX with (2). Table 2: Experimental results for the enthalpy change ∆h0trans of the β-δ phase transition at atmospheric pressure. For consistency with our choice of the transition temperature, we choose ∆h0trans to be the mean value obtained by McKenney and Krawietz 24 at a 1 K/minute heating rate. This value is substituted into (2) to represent the vapor pressure profile for β-HMX and ultimately, to determine the parameters a and b for this compound. Reference Hall 28 Krien et al. 29 Rylance and Stubley 30 Kishore 31 Landers and Brill 7 Karpowicz and Brill 8 Brill and Karpowicz 21 Quintana et al. 33 McKenney and Krawietz 24 McKenney and Krawietz 24

∆h0trans (kJ/mole) 9.83 ± 0.8 9.79 ± 0.13 9.3 8.8 10.1 13 9 8.1 8.56 9.16

Notes

100 – 690 bar, 3 µm samples 460 – 2080 bar, 3 µm samples 10 K/min Mean value at 5 K/min heating rate Mean value at 1 K/min heating rate

In addition to the vapor pressure, we need another property — a good choice being the density — to determine the two parameters a and b. Figure 2 shows the density of β- and δHMX, while Figure S1 in the Supporting Information presents the density of α- and γ-HMX. For both compounds depicted in Figure 2, the density vs. temperature correlations reported by Saw 12 and Engel et al. 42 roughly represent the lower and upper limits, respectively, of the values in the literature. The results from several other studies, 39,43–49 including Xue et al., 39 reside somewhere in the shaded regions. The density from the Peng-Robinson EOS is particularly sensitive to the value of b. We have obtained the solid curves in Figures 2 10 ACS Paragon Plus Environment

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where the coefficients a0 and a1 are listed along with the corresponding value of b in Table 3. Just like for the α, β, and γ phases, we have applied (2) to infer the vapor pressure of the liquid from phase transition data and the vapor pressure of the δ polymorph. At atmospheric pressure, δ-HMX is the thermodynamically stable polymorph at higher temperatures, and a number of authors have determined the melting temperature Tmelt,n (Table S6). We choose Tmelt,n = 553.65 K. At this temperature, our δ-HMX model predicts a vapor pressure of Pδ = 5.582 × 10−4 bar and an enthalpy of vaporization of ∆hvap,δ = 161.50 kJ/mole. For the liquid phase, the ∆h0trans in Equation (3) refers to the enthalpy of melting ∆h0melt . This quantity must be subtracted from ∆hvap,δ , rather than added to ∆hvap,δ , to obtain ∆hvap . Table 3: Value of b and coefficients for a (units of L2 ·bar/mole2 ) used in (4) and (5). Compound α-HMX β-HMX γ-HMX δ-HMX Liquid HMX

a1 −0.479287 −0.474522 −0.483766 −0.455847 −0.001424

a0 b (L/mole) 440.832443 0.155219 433.157727 0.151399 456.408907 0.165756 438.938747 0.162123 6.025981 0.163711

Pure liquid HMX is very unstable (rapidly decomposes), but its stability can be enhanced by mixing it with another compound, such as RDX. For this reason, earlier studies have not been able to measure ∆h0melt directly, but they have been able to deduce it from the slope of the liquidus curve in HMX/RDX phase diagrams. An important relation for this purpose is the Gibbs-Helmholtz equation 50 ∂(µi /T ) ∂T

!

P,z

=−

¯i H , T2

¯ i is where µi is the chemical potential of component i in a mixture of composition z, and H the partial molar enthalpy of i. Combining this with the following relation for ideal mixtures


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µi (T, P, z) = µi (T, P ) + RT ln zi , where zi is the mole fraction of i and µi (T, P ) is the chemical potential of pure i, we get

R ln zi =

∆h0melt

1 Tmelt,n

1 − T

!

.

(6)

This equation is also (approximately) valid for non-ideal mixtures that are rich in component i (i.e., mixtures where zi ≈ 1). From liquidus curve data for the HMX/RDX system, one can therefore apply (6) to obtain ∆h0melt of both pure components. Based on the liquidus curves presented in a patent application by Wright and Chute, 51 Maksimov 52 has determined ∆h0melt of δ-HMX to be 69.9 kJ/mole. However, his result is nearly twice that of McKenney and Krawietz, 24 who find from their HMX/RDX experiments that ∆h0melt = 37.8 kJ/mole. Their value is comparable to experimental and solubility modeling results presented at a recent conference by Bhattacharia et al. 53 These authors report the ∆h0melt from DSC measurements and an accompanying activity coefficient model to be 31.9 ± 3.9 kJ/mole and 31.5 ± 9.5 kJ/mole, respectively. Since McKenney and Krawietz is one of the two studies that we focus on for HMX/RDX mixtures, we use their result for ∆h0melt . Available data on the density of liquid HMX is also quite limited. As input into their 0 theoretical kinetic model, Henson et al. 13 set the volume change of melting ∆vmelt to be

0.019 L/mole. Bedrov et al. 54 have performed molecular simulations to predict the density of the liquid between 550 – 800 K in 50 K increments. Their result at 550 K is 1650.9 kg/m3 . The only way for Henson et al. to be compatible with Bedrov et al. is if the density of δ-HMX at the melting temperature Tmelt,n is 1846.6 kg/m3 . One can see from Figure 2(b) that this requirement is clearly unrealistic. Therefore, only one of these two values can be satisfied. 0 Since Henson et al. do not cite the source from where they obtained their ∆vmelt , we have

elected to adjust the value of b to get a liquid density that matches that of Bedrov et al. Our 13 ACS Paragon Plus Environment


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0 δ-HMX density at Tmelt,n is 1667.3 kg/m3 , which corresponds to ∆vmelt = 0.00175 L/mole. 0 If we were to use the ∆vmelt from Henson et al. (0.019 L/mole), the liquid density would be

1506.4 kg/m3 instead. Our liquid density at 600 K is 1623.2 kg/m3 , in fairly good agreement with the value of 1614.4 kg/m3 predicted by Bedrov et al.

Phase transitions Figure 3 illustrates the effect of pressure on several different phase transitions that HMX may undergo. In all of the figures, the instantaneous slope of the solid curves are given by dT Ttrans ∆vtrans = , dP ∆htrans

(7)

where the three quantities on the right-hand side (these are defined in the previous section) are to be evaluated at the T and P of interest. The dashed lines in Figures 3(a) and 3(b), as well as the three interrupted lines in 3(c) and 3(d), show what the transition temperature would be if the slope at all T and P were equal to the constant 0 dT Ttrans,n ∆vtrans = . dP ∆h0trans

(8)

Unlike in (7), the right-hand side in (8) refers to quantities at atmospheric pressure. These quantities are important for two reasons. First, they have been either measured experimentally or calculated by molecular simulations. The previous section describes how we have fit a and b to match values in the literature for Ttrans,n , ∆h0trans , and the density (which in 0 turn determines ∆vtrans ). Second, and perhaps more importantly, from common experience

with substances such as water, we do not expect there to be much difference between the slopes computed from (7) and (8) over the pressure range shown in the figure. For example, the dashed line and solid curve in Figure 3(a) almost completely overlap each other. This is useful because it means that information about the phase transition at atmospheric pressure 14 ACS Paragon Plus Environment

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can give a good indication of what the transition temperature will be at higher pressures. The only HMX phase transition for which we have found experimental data above atmospheric pressure is the β-δ transition. Figure 3(a) includes data from two studies 7,8 mentioned earlier in Table 1. The best-fit line to these two data sets intersects the y-axis at 457.0 K. This is the atmospheric pressure transition temperature Ttrans,n suggested by the data. There is a large discrepancy between our predictions and the experiments. One reason is because our choice of Ttrans,n (462.5 K) is higher. But even if we were to shift it down to 457.0 K, our solid curve would still lie significantly above the dotted line. This implies that, 0 compared to the data, our ∆vtrans is too high and/or our ∆h0trans is too low. By definition

0 ∆vtrans = vδ − vβ =

1 1 − , ̺δ ̺ β

0 where ̺β (̺δ ) is the molar density of β-HMX (δ-HMX). Figure 2 indicates that our ∆vtrans 0 of 0.0114 L/mole lies near the middle of the experimental range. This ∆vtrans corresponds

to a 7.1% volume change, which is similar to the 6.7% change reported by Herrmann et al. 11 Likewise, our ∆h0trans of 9.16 kJ/mole is an intermediate value in the accepted range (Table 2). 0 Thus, the ∆vtrans and ∆h0trans implied by the two studies in Figure 3(a) are not compatible

with results reported by many of the studies discussed in the previous section. In order to determine the smallest possible slope dT /dP , it is necessary to place physically 0 0 realistic lower and upper bounds on ∆vtrans and ∆h0trans , respectively. Small values of ∆vtrans

correspond to a small β-HMX density and a large δ-HMX density. Figure 2 suggests that 0 the smallest ∆vtrans that one might reasonably expect could be achieved by extrapolating

Gump and Peiris’ results for the β polymorph to 457.0 K, and combining that with the 0 results of Engel et al. for the δ polymorph. 42,45 This would yield ∆vtrans = 0.00808 L/mole.

The largest ∆h0trans in Table 2 (besides that from Karpowicz and Brill) is 10.6 kJ/mole. The dashed-dotted line in Figure 3(a) is the transition temperature profile that would be obtained



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Figure 3: Phase transition temperatures of HMX vs. pressure. All the solid curves represent our predictions. Their slopes are given by (7), where Ttrans , ∆vtrans , ∆htrans refer to instantaneous values of the temperature, volume change, and enthalpy change of the transition, respectively. The slopes of the interrupted (dashed, dotted, and dashed-dotted) lines are calculated from (8). This equation involves the same three quantities as in (7), but they are evaluated at atmospheric pressure instead. The text discusses in detail how (7) and (8) can explain many of the results, including why the two β-δ experimental data sets 7,8 in (a) do not seem to be consistent with many of the studies mentioned in the previous section, why the γ-δ transition temperature in (b) decreases with pressure, and the behavior of the solid-solid transition curves in (c) and melting curves in (d).


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0 with Ttrans,n = 457.0 K and with these extremal values for ∆vtrans and ∆h0trans . Although

this line approaches some of the data points, it still does not intersect any of them, despite the fact that it has the smallest conceivable slope. Measuring the phase transition temperature is a very challenging prospect due to some complicating factors that appear to be rooted in the highly metastable nature of the polymorphs. Landers and Brill 7 have obtained their results by applying Raman spectroscopy to a nitrogen-pressurized capillary tube immersed in a heat bath. This setup is an improved version of one used in an earlier study. 32 They have found that increasing the rate of heating does not affect the slope dT /dP , but it does shift the value of Ttrans,n to a higher temperature. The particle size also influences the results. In fact, for all but their most fine (3 µm) samples, they observe a sharp slope break at around 690 bar. Above this pressure, dT /dP suddenly becomes much smaller. In a follow-up study by Karpowicz and Brill, 8 the slope break is attributed to an experimental artifact, specifically to decomposition products that become trapped in the crystal lattice. The Landers and Brill results that we have presented in Figure 3(a) are for their 3 µm samples. These are the closest results they have at higher pressures to the true β-δ phase equilibrium curve. Landers and Brill state that if 0 Ttrans,n = 431 K (see Teetsov and McCrone 20 in Table 1) and ∆vtrans = 0.00866 L/mole, 9

the ∆h0trans of the best-fit line through the points up to 690 bar would be 10.1 kJ/mole. Karpowicz and Brill have extended the pressure range of Landers and Brill by measuring the transition temperature from 460 – 6200 bar. However, they have found that their results above 2080 bar (corresponding to a temperature of about 513 K) are marred by significant decomposition and exhibit a dramatic slope break. The slope break is evident in our figure. We have not included the two points above 2080 bar in our determination of the best-fit line. By setting Ttrans,n to be 431 K in (8), Karpowicz and Brill obtain a ∆h0trans of 13 kJ/mole. This value is significantly higher than the others listed in Table 2. It suggests that the true β-δ equilibrium curve should reside above the best-fit dotted line in Figure 3(a). 17 ACS Paragon Plus Environment


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The γ-δ transition portrayed in Figure 3(b) is an interesting case where the highertemperature phase (δ) has a larger density. In this sense, it is analogous to the familiar liquid water-ice I transition. Engel et al. 42 is the only study we have found that has measured the 0 density of γ-HMX vs. temperature (Figure S1). Our ∆vtrans = −2.2%, which is smaller in 0 magnitude than the −3.3% reported by Herrmann et al. 11 The negative ∆vtrans implies that

the transition temperature decreases with pressure. It would be quite a rare occurrence, however, to observe this phase transition in practice. As we discuss in more detail later, the γ phase is not thermodynamically favorable (does not have the lowest Gibbs energy) compared to the α and β polymorphs at temperatures below the γ-δ transition curve where it might exist. It will tend to transform to α-HMX or β-HMX unless effort is made to inhibit the transition. The elusive nature of γ-HMX compared to the other polymorphs is documented in past studies (see Cady and Smith 9 and references cited therein). Figure 3(c) presents three solid-solid phase transitions that play an important role in determining the phase diagram of HMX. The α-β curve has by far the largest slope because its ∆h0trans of 1.26 kJ/mole is multiple times smaller than the ∆h0trans of the α-δ transition (7.89 kJ/mole) and the ∆h0trans of the β-δ transition (9.16 kJ/mole). The densities of the 0 polymorphs in descending order are β > α > δ. The β-δ curve (∆vtrans = 0.0114 L/mole) 0 overtakes the α-δ curve (∆vtrans = 0.0074 L/mole) at higher pressures because of the greater

density disparity. The liquid phase has a much higher enthalpy than any of the solid polymorphs. This has two noticeable effects on the three melting transitions shown in Figure 3(d). First, the melting curves have significantly smaller slopes than the solid-solid transition curves in Figure 3(c). Second, the large enthalpy difference between the liquid and solid phases implies that the differences in ∆h0trans among the melting curves are relatively insignificant. The slope of the melting curves are therefore mainly determined the density 0 0 0 order of the solid phases (β > α > δ =⇒ ∆vtrans,β−liquid > ∆vtrans,α−liquid > ∆vtrans,δ−liquid ).

The α-β transition curve intersects the y-axis at 420.4 K. This transition temperature 18 ACS Paragon Plus Environment

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Ttrans,n is 30 – 40 K higher than the experimental Ttrans,n listed in Table S5. Nevertheless, this result is a natural and unavoidable consequence of how we have modeled the vapor pressure profiles of α-HMX and β-HMX with the Clausius-Clapeyron equation (2). If we substitute the stated values for Ttrans,n , Pδ , etc. into (2), the Clausius-Clapeyron curves of the two polymorphs would intersect at 426.2 K. Our Ttrans,n is a little different than this because (2) is only an approximation to the true vapor pressure. We know, for instance, that the enthalpy of vaporization is not a constant, as implied by (2). For this reason, we have fit a and b to (2) in only a very narrow temperature range (1 or 2 degrees) around the temperatures Ttrans,n of 470.0 K for α-HMX and 462.5 K for β-HMX. Away from these temperatures, there will inevitably be some deviation. Nevertheless, it is clear that the deviation must be limited to a few degrees at most, certainly much less than 30 K. This means that given the ∆h0trans data in Tables 2 and S2, it is impossible to satisfy the α-β transition temperatures in Table S5 if we choose Ttrans,n of α-δ to be 470.0 K and Ttrans,n of β-δ to be 462.5 K. A careful examination of Tables 1 and S1 reveals, however, that the Ttrans,n obtained in experiments where the new HMX phase is crystallized in a heated solution tends to be significantly lower (30 – 40 K lower in most cases) than the Ttrans,n from experiments where HMX is heated in a surrounding gas phase. The latter situation (the absence of a solvent) is more relevant for the applications mentioned in the Introduction. Molecular simulations have predicted that dissolution of β-HMX in acetone alters the morphology of the crystals. 55 In addition, the acetone molecules may also serve as nucleation sites, similar to the role played by nitroplasticizer in the experiments of Smilowitz et al. 15 described in the Introduction. It is therefore possible that the α-β experiments in Table S5, all of which involve crystallization in solution (two of the studies 9,19 use acetone), could have underestimated the temperature of this phase transition by up to 40 K compared to the case where solvent is absent. Our result for Ttrans,n of α-β is reasonable if we account for this factor. More studies are necessary to improve our understanding of the effect of solvation on the transition temperature. 19 ACS Paragon Plus Environment


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The HMX phase diagrams in Figure 4 indicate the most thermodynamically stable phase for a given temperature and pressure. Since only a pure species is considered here, this is equivalent to finding the HMX phase that has the lowest chemical potential or fugacity coefficient. At room temperature, the most stable form is the β polymorph. As HMX is heated from room temperature at a fixed pressure, Figure 4(a) indicates that β-HMX may undergo one, two, or three phase transitions, depending on the pressure. It is important to note that this figure demonstrates what would happen if the heating occurs quasi-statically, meaning that enough time is given for the HMX to fully explore the energy landscape and assume the state that is a global minimum in the Gibbs energy. Figure 4(b), which we discuss shortly, depicts the “phase diagram” that may be observed in reality if the heating is not done in an idealized, quasi-static manner. Both figures correctly indicate that the γ phase is not the most stable phase for any set of conditions. They also agree with past studies which have found that out of the four polymorphs considered in this study, only β-HMX is relevant at sufficiently high pressures. 56


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for α-γ and 470.9 K for β-γ, while it is 470.0 K for α-δ, 462.5 K for β-δ, and 444.0 K for γ-δ. This means that if α-HMX were heated quasi-statically at atmospheric pressure, it would transform to δ-HMX at a temperature where δ-HMX is more thermodynamically stable than γ-HMX. Furthermore, the δ polymorph would continue to be the more stable form as the HMX is further heated. A similar analysis at any other pressure would yield the same outcome. Later, we show that a necessary and sufficient condition for the absence of γ-HMX in the phase diagram is that Ttrans,n of β-δ must be higher than Ttrans,n of γ-δ. Although the existence and properties of α-HMX are well-documented in the literature (we have cited several such studies in the paper as well as in the Supporting Information), there are also many studies that have observed only the β-δ transition. One can see, for instance, that there are more entries in Table 1 for β-δ than there are in Tables S1 (α-δ) or S5 (α-β). This raises the question of why the α-β or α-δ transitions are not observed in some studies, even though there is a region in the phase diagram where α-HMX is the most thermodynamically stable phase. The answer may again lie in the fact that the HMX polymorphs are characterized by large nucleation and growth barriers that inhibit phase transitions. It is these barriers that have allowed the density of γ-HMX to be measurable over a wide temperature range [Figure S1(b)] even though this polymorph is not thermodynamically favorable at any set of conditions. It is beyond the scope of our thermodynamic model to explicitly include kinetic effects, but we can suggest the following. When β-HMX is heated up to the α-β transition curve, one possibility is that the driving force for α-HMX nucleation is not strong enough for significant transformation to occur in the short time frame of the experiments. For practical reasons (most notably the production of decomposition gases), it is virtually impossible to heat energetic materials in an idealized, quasi-static way. Consequently, the heating is often done at rates of at least 1 K/minute even though metastable HMX states have been known to last for days and even weeks. 10–12 The driving force may become strong enough for a phase transition to be observable only when β-HMX 22 ACS Paragon Plus Environment

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is heated beyond the α-δ transition curve. Above this curve, δ-HMX is the most thermodynamically stable phase, so the transition that is observed is β-δ. Thus, the apparent phase diagram may look something like Figure 4(b), where the β phase region has been extended up to the α-δ curve so that it occupies what would have been the α phase region. To conclude this section on phase transitions, we note that it is possible to have an equation of state where the α polymorph is not thermodynamically favorable at any temperature and pressure (i.e., an EOS where the equilibrium phase diagram does not have an α phase region). This would occur if Ttrans,n of α-δ is less than Ttrans,n of β-δ. Because the α-δ curve has a smaller slope than the β-δ curve (for reasons we have already mentioned), the former would never intersect the latter in this case. It would lie below the β-δ curve for all pressures. In order for the resulting phase diagram to make sense, the α-β curve gets shifted upward so that it is always above the other two curves. Otherwise, if this shift did not occur, one would obtain a contradiction where in a certain region of the phase diagram, α-HMX must be both more stable and less stable than δ-HMX. We have explored this scenario, and indeed we have found that if we keep Ttrans,n of β-δ as 462.5 K, but decrease Ttrans,n of α-δ from 470.0 K to 460.0 K, Ttrans,n of α-β increases dramatically from 420.4 K to 479.0 K. This transition temperature for α-β is more than 90 K greater than the temperatures reported by the experimental studies in Table S5, all of which involve crystallization in solution. As we have discussed earlier, Tables 1 and S1 suggest that dissolution in a solvent may reduce the transition temperature by up to 40 K compared to the case where solvent is not present, so a 90 K difference seems unlikely. For this reason, we have elected to keep Ttrans,n of α-δ as 470.0 K so that our Ttrans,n of α-β remains within 40 K of the results in Table S5. Consequently, our model predicts that the equilibrium phase diagram [Figure 4(a)] does have an α phase region. We return to these arguments from a more general perspective in the last major section of this paper.



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Heat capacity Figures 5(a) and 5(b) show the temperature dependence of the constant-pressure heat capacity CP of β- and δ-HMX, respectively. Our results are non-dimensionalized by R. The CP of α- and γ-HMX are presented in Figure S2. Several experimental 10,29,30,57–59 and a few theoretical studies 60–62 are included for comparison. Some of these studies have fit their discrete data points to a temperature correlation. For all four polymorphs, we obtain good agreement with the experimental results of Rylance and Stubley. 30 The AAD is 3.5% for β-HMX and 2.2% for δ-HMX. If HMX were heated quasi-statically at atmospheric pressure, its heat capacity would change as shown in Figure 5(c). If only the β-δ transition is observed, as in Figure 4(b), the change in the heat capacity would be more similar to that depicted in Figure 5(d). Discontinuities in CP due to solid-solid transitions are very small compared to the sharp increase that occurs at the δ-liquid melting temperature. Our previous work 23 explains how we calculate the heat capacity departure (residual) from the first and second temperature derivatives of the fugacity coefficient. We obtain CP by adding the ideal gas heat capacity CPig to the departure. As before, we get CPig by fitting a third-order polynomial in temperature to the quantum simulation results of Osmont et al.: 63

CPig = c3 T 3 + c2 T 2 + c1 T + c0 . For HMX, c3 = 2.184101 × 10−10 , c2 = −8.547314 × 10−7 , c1 = 1.194553 × 10−3 , and c0 = −7.49962 × 10−3 if CPig is in units of kJ/mole/K. Lyman et al. 60 have also performed quantum (Hartree-Fock) simulations of the gas phase. They have combined their gas-phase calculations with experimental phase transition data to predict CP of the β, δ, and liquid phases. Theirs is the only study we have found that has attempted to quantify the heat capacity of liquid HMX. Their liquid CP is not shown in Figure 5, but at the melting temperature, it is slightly smaller than their prediction for δ-HMX. Our liquid CP is more


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physically realistic because it is (noticeably) larger than our δ-HMX heat capacity. 55

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Figure 5: Heat capacity CP of HMX vs. temperature at atmospheric pressure. Compared to Rylance and Stubley, 30 the AAD is 3.5% for β-HMX (a) and is 2.2% for δ-HMX (b). The plots in (c) and (d) correspond to the phase diagrams in Figures 4(a) and 4(b), respectively.

Bulk modulus and sound speed The isothermal bulk modulus KT is defined as ∂P KT = −v ∂v

!

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T



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It is the inverse of the isothermal coefficient of compressibility βT . Figure 6(a) compares our predictions for KT of β-HMX with experimental and theoretical results in the literature. 5,43,45,61,64–69 The bulk modulus of the other polymorphs follow similar trends, so we focus on β-HMX to illustrate the behavior. There is a lot of variation in past studies, especially around room temperature where most of the results have been obtained. Menikoff and Sewell 65 have done a critical examination of two earlier studies 5,64 to analyze possible causes of the discrepancy. They have found that KT depends significantly on the particular fitting form (e.g., the various flavors of the Birch-Murnaghan equation) applied to pressure vs. volume data. The results are also sensitive to the fitting domain. Data points at higher pressures (those above 1 GPa) tend to skew the results towards smaller values of KT . This is further complicated by non-hydrostatic conditions that tend to emerge in the surrounding medium at higher pressures. Due to these uncertainties, Menikoff and Sewell conclude that the best one can say is that KT = 14 ± 3.5 GPa at room temperature. Our prediction at lower temperatures lies above this range, but it is in within Gump and Peiris’ margin of error at 303 K. Unlike many other experimental studies in the figure, they have included several data points below 1 GPa in their third-order Birch-Murnaghan fit. This may explain why their KT is relatively large. Since our EOS is oriented towards pressures below 1 GPa, it is reasonable that our results would agree most closely with those of Gump and Peiris. The bulk modulus KT is related to the sound speed u through

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!1/2

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where ρ is the mass density and CV is the constant-volume heat capacity. We calculate CV from the relation

CV = CP − T αP2 KT v,


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Figure 6: (a) Isothermal bulk modulus KT of β-HMX and (b) sound speed in β-HMX vs. temperature at atmospheric pressure. Zaug 43 has calculated the bulk moduli in (a) from his sound speed measurements (taken at two different temperatures: 297 and 380 K) in (b). The vertical lines in (a) indicate the error bars in Gump and Peiris 45 or Menikoff and Sewell. 65 where the thermal coefficient of expansion αP is defined as 1 αP = v

∂v ∂T

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The equations indicate that smaller values of KT correspond to smaller sound speeds. Zaug 43 reports sound speed measurements at two temperatures — 297 K and 380 K — obtained via impulsive stimulated light scattering (ISLS) on single crystals of β-HMX heated in diamond anvil cells. Figure 6(b) shows his ISLS results for the variation of the sound speed u as the acoustic wave vector is rotated within the crystal at these two temperatures. Our result for u at 297 K is near the upper end of Zaug’s range of variation, but it is below his range at 380 K. This is a reflection of the behavior illustrated in Figure 6(a). The figure shows that although our results for KT are reasonable, they decrease more strongly with temperature compared to the KT from other studies. The Peng-Robinson EOS was originally developed to model pure fluids and fluid mixtures. The compressibility βT of fluids increases more with temperature than does βT of solids. This may explain the tendency of our KT = 1/βT to



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decrease relatively strongly with temperature. It suggests a possible drawback of applying the Peng-Robinson EOS to solids. However, given the still-unresolved uncertainties in the measurements of KT and u, it would be helpful to have more results, particularly at higher temperatures, before more conclusive statements can be made.

HMX/RDX mixtures Phase diagrams Some of the most essential information concerning a mixture is contained in its temperaturecomposition phase diagram. The HMX/RDX phase diagram in Figure 7 presents our results along with data from two experimental studies conducted at atmospheric pressure. We have generated the phase diagram by combining the HMX models in this work with the RDX models in our earlier paper. 23 Before we analyze our results in more detail, we first focus on the experimental studies. Our discussion will cover the various regions and features of the HMX/RDX phase diagram, including a feature — a peritectic point — that is not seen in the RDX/TNT phase diagram that we have previously modeled. The peritectic point arises because HMX, unlike RDX and TNT, has prominent solid polymorphs. To date, Quintana et al. 33 and McKenney and Krawietz 24 are the only two studies that present detailed experimental data on HMX/RDX mixtures. A patent application by Wright and Chute 51 includes a rough sketch of the HMX liquidus curve, but not enough details for us to be able to use it. A conference abstract by Bhattacharia et al. 53 mentions some experimental and solubility modeling results, but their data have not yet been published. The experimental studies in Figure 7 present data along three different parts of the phase diagram. This information is sufficient to develop a model for the rest of the phase diagram. One part they have examined is the RDX liquidus curve. This curve represents the composition of the liquid phase, which is a mixture that contains both liquid HMX and 28 ACS Paragon Plus Environment

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Figure 7: Temperature-composition phase diagram of the HMX/RDX binary system at atmospheric pressure. The solid curves represent the phase boundaries from our model. An overall view is shown in (a), while magnified views near the melting temperature of pure RDX, eutectic point, and peritectic point are presented in (b), (c), and (d), respectively. Four single-phase regions are present: β-HMX-rich solid (labeled as β), δ-HMX-rich solid (δ), RDX-rich solid (RDX), and liquid. These four phases combine to form five distinct twophase regions, each of which is represented by addition of the two constituent phases (e.g., the two-phase region between β-HMX-rich solid and RDX-rich solid is denoted as β + RDX). Three-phase equilibria are represented by points which denote the composition of the three phases that are in equilibrium. As a visual aid, we have connected the points together with imaginary tie lines. The eutectic tie line connects the eutectic point, which represents the liquid, with points for the β-HMX-rich and RDX-rich solid phases. The peritectic tie line connects the peritectic point, which represents the β-HMX-rich solid, with points for the δ-HMX-rich solid phase and the liquid phase. Just like in Figure 4, kinetic effects have influenced the experimental data and the subsequent appearance of the phase diagrams, particularly the parts concerning the β-δ transition and the peritectic tie line. 29 ACS Paragon Plus Environment


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liquid RDX, that is in equilibrium with the RDX-rich solid phase. The latter is primarily composed of solid RDX, but also contains a small amount of β-HMX. Another data set is along the eutectic tie line. The studies have deduced the existence of this tie line by heating samples of different HMX:RDX ratios and finding that all of the samples partially melt (form a liquid phase) at around the same temperature, called the eutectic temperature. McKenney and Krawietz have visualized the melting process in one of their samples through hot stage microscopy. The eutectic point represents the composition of the liquid that forms at the eutectic temperature. McKenney and Krawietz report a eutectic point of (0.749 – 0.755 mole fraction RDX, 461.65 ± 0.3 K). This is similar to the eutectic point from Quintana et al., who report a composition of 0.757 mole fraction RDX (70 mass% RDX) and a temperature of about 463 K. The third set of data from the experiments pertains to the β-δ phase transition temperature as a function of the mixture composition. McKenney and Krawietz have performed X-ray diffraction on pure HMX to confirm that the two polymorphs involved in the solid-solid transition are indeed β and δ. It is assumed that the same transition occurs in HMX/RDX mixtures. The experiments reveal that the transition temperature increases with RDX concentration up until a certain composition, beyond which it plateaus. This asymptotic value for Ttrans,n represents a peritectic temperature. 70 The three phases in equilibrium at this temperature are connected by the peritectic tie line. A model for the rest of the phase diagram can be constructed from the experimental data. Below the eutectic temperature, there is a two-phase region with β-HMX-rich solid and RDX-rich solid (labeled as β + RDX in the figure). Above the peritectic temperature, there is a two-phase region with δ-HMX-rich solid and liquid (δ + liquid). Between the two temperatures, there are 2 two-phase regions: β + δ to the left of the peritectic point, and β + liquid to the right of the peritectic point. The phase boundaries that envelope the twophase regions are calculated from the conditions for phase equilibrium. These conditions require the chemical potential of each component to be the same in both phases. The 30 ACS Paragon Plus Environment

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chemical potentials (or more practically speaking, the fugacities) in a phase can be calculated if its a and b parameters are known. As in our earlier paper, 23 we calculate a and b of a mixture from the van der Waals mixing rules, which state that for a mixture of c components

a=

c c X X

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where zi is the mole fraction of component i, ai is the a parameter of i, bi is the b parameter of i, and kij is the binary interaction coefficient between components i and j. These coefficients are defined so that kii = 0 and kij = kji is generally non-zero for all i, j. Due to their “symmetric, off-diagonal” nature, only three different interaction coefficients are needed to characterize all possible binary mixtures in the phase diagram. We need to determine 1) kβ-HMX/RDX for the β-HMX-rich and RDX-rich solid phases, 2) kδ-HMX/RDX for the δ-HMX-rich solid phase, and 3) kliquid for the liquid phase. In general, these coefficients may depend on temperature, but we treat them as constants in this work. We determine the three binary interaction coefficients through eutectic point and peritectic temperature data. Three phases are in equilibrium at the eutectic temperature: βHMX-rich solid, RDX-rich solid, and liquid. This equilibrium is therefore described by four constraints on the chemical potentials, two for HMX and two for RDX. Since the eutectic point has been established experimentally (i.e., the eutectic temperature and the composition of the liquid are known), the four equations are solved to find the composition (the RDX mole fraction) of the two solid phases and the interaction coefficients kβ-HMX/RDX and kliquid . The three phases in equilibrium at the peritectic temperature are β-HMX-rich solid, δ-HMX-rich solid, and liquid. The four conditions for equilibrium at this temperature are solved to obtain the composition of the three phases and the value of kδ-HMX/RDX . The experimental studies in Figure 7 have determined this temperature to be about 476.0 K, but 31 ACS Paragon Plus Environment


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for reasons that we discuss below, our model is not capable of reaching such high peritectic temperatures. The highest peritectic temperature we can achieve is 469.1 K. As we approach this temperature, the RDX mole fraction in the δ-HMX-rich solid phase goes to zero. Since it is unrealistic for the solubility of RDX to be so low in this phase, we choose a slightly lower peritectic temperature (468.7 K) and find the kδ-HMX/RDX and phase compositions that are consistent with this temperature. Our results for the binary interaction coefficients and the compositions at the eutectic and peritectic temperatures are listed in Tables S23 – S25 of the Supporting Information. All of the phase boundaries are determined from this information by solving the appropriate conditions for equilibrium. We are able to closely reproduce the experimental RDX liquidus curve. If HMX liquidus curve data becomes available in the future, it may be necessary to treat kliquid as being temperature-dependent to fit the data. This information is not currently available because both studies have reported that significant decomposition occurs at temperatures slightly above the peritectic temperature. There are a number of details regarding the β-δ transition and the peritectic tie line that warrant further discussion. Once again, the kinetics of the phase transitions have affected the results. McKenney and Krawietz have heated samples of pure HMX in a differential scanning calorimeter at two rates: 1 K/minute and 5 K/minute. They have found that the slower rate leads to smaller transition temperatures (462.5 K vs. 470.0 K; see Table 1) and larger ∆htrans,n (9.16 kJ/mole vs. 8.56 kJ/mole; see Table 2). The increase in Ttrans,n with the heating rate is consistent with our discussion concerning Figure 4. Quintana et al. have obtained their results using differential thermal analysis, which is similar to DSC, but they have applied an even faster heating rate of 10 K/minute. Since we are interested in a single equation of state that is applicable to both pure HMX and HMX/RDX mixtures, we have adopted the transition temperature of McKenney and Krawietz at 1 K/minute as Ttrans,n of β-δ. These are their closest results to the true equilibrium. However, due to time constraints, McKenney and Krawietz have applied a heating rate of 5 K/minute for their HMX/RDX 32 ACS Paragon Plus Environment

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mixture data in Figure 7. The mismatch in the heating rate explains why our model cannot reach the peritectic temperature observed in the experiments. Note that the difference between our Ttrans,n of pure HMX (462.5 K) and our peritectic temperature (468.7 K) is roughly the same as the difference between McKenney and Krawietz’ 5 K/minute values for Ttrans,n of pure HMX (470.0 K) and the peritectic temperature (476.0 K). The agreement would improve if the experiments were carried out a slower rate. In fact, if they were carried out at a sufficiently slow rate (one that admittedly may be impossible to achieve in practice), Figure 4(a) suggests that a single-phase region that is rich in α-HMX would appear at an intermediate temperature between the β and δ regions. Since the Ttrans,n of α-β (420.4 K) is significantly lower than the experimentally observed eutectic temperature (about 462 K), the α-β transition likely occurs at a temperature below which liquid can exist. Thus, this phase diagram would exhibit a peritectoid temperature where α-HMX-rich solid, β-HMXrich solid, and RDX-rich solid are in equilibrium. This is a peritectoid, rather than peritectic, temperature because all three phases are solids. Above this temperature, we would have a eutectic temperature with α-HMX-rich solid, RDX-rich solid, and liquid. Finally, above the eutectic there would be a peritectic temperature where α-HMX-rich solid, δ-HMX-rich solid, and liquid are in equilibrium. McKenney and Krawietz have performed remelting operations where they thermally cycle several samples of different HMX:RDX ratios. Their results suggest that during the thermal cycling, liquid HMX in the initial eutectic melt recrystallizes completely to δ-HMX by the second or third remelting operation. The phase diagram that describes these results is similar to Figure 7, except that the single-phase β-HMX-rich solid region is completely replaced by δ-HMX-rich solid so that the peritectic point, the β + δ region, and the β + liquid region no longer appear. The RDX-rich solid is still composed of RDX and β-HMX. The eutectic temperature in the remelting phase diagram represents equilibrium between δ-HMXrich solid, RDX-rich solid, and liquid. We have not been able to reproduce this phase 33 ACS Paragon Plus Environment


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diagram using the same values for the binary interaction coefficients. This is not surprising because a particular set of interaction coefficients should yield a unique phase diagram. It may be possible to achieve agreement with both initial melting and remelting results using temperature-dependent coefficients, but we have not explored this possibility. More importantly, however, we choose not to model the remelting data because they clearly do not represent true equilibrium states. For instance, the remelting data suggests that δHMX-rich solid is present at ambient conditions, but this phase is certainly metastable at these conditions. In a recently published conference paper, Jaansalu 71 presents an activity coefficient model for two HMX/RDX phase diagrams: one for the remelting data and another for the hypothetical situation where only β-HMX is present. Our earlier paper 23 discusses his model in greater detail. Up until now, we have considered HMX/RDX mixtures only at atmospheric pressure. Figure 8 shows phase diagrams at higher pressures up to 2500 bar. All of the parameters in the equation of state (a and b of the pure components, plus the binary interaction coefficients) do not depend on pressure. As a result, they are already known from the fits at atmospheric pressure. For a given pressure, we solve the equilibrium conditions to find the corresponding eutectic temperature and the composition of the three phases that coexist in equilibrium at this temperature and pressure. We do similar calculations for the corresponding peritectic temperature. Both of these landmark temperatures increase with pressure, but as can be seen in Figure 8, pressure has a stronger effect on the peritectic temperature. As a result, the area of the β + liquid region increases with pressure, but the area of the δ + liquid region decreases with pressure. This behavior can be understood from Figures 3 and 4. The β-δ curve rises more with pressure than the β-liquid curve does, and the former intersects the latter at about 1653 bar. At pressures higher than this, the first transition curve encountered by β-HMX as it is heated from room temperature is the β-liquid curve. In other words, above 1653 bar, β-HMX melts before it has a chance to transition to δ-HMX. Therefore, δ-HMX 34 ACS Paragon Plus Environment

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is absent from the pure HMX phase diagram at these higher pressures (Figure 4). These results are reflected in the HMX/RDX phase diagram. Above 1653 bar, the δ-HMX-rich solid phase does not appear at all, leaving β-HMX, RDX, and the respective liquid phases to form a simple, binary eutectic system that is similar to the RDX/TNT system.



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Figure 8: Temperature-composition phase diagram of the HMX/RDX binary system at pressures of (a) 500 bar, (b) 1000 bar, (c) 1500 bar, and (d) 2500 bar. The β-δ transition temperature increases with pressure more than the melting temperatures do (Figure 3). Consequently, the area of the β + liquid region increases with pressure, but the area of the δ + liquid region decreases with pressure. Above 1653 bar, the melting point of βHMX is lower than the β-δ transition temperature so that δ-HMX no longer appears in the pure HMX phase diagram (Figure 4). The HMX/RDX system at these higher pressures becomes a simple, binary eutectic system where β-HMX is the only polymorph present. The peritectic temperature that signifies the transition of β-HMX-rich solid to δ-HMX-rich solid (and liquid) is absent above 1653 bar.


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Phase transition enthalpies Figure 9 compares our predictions for the enthalpy change ∆h associated with a few different kinds of phase transitions to experimental data from Quintana et al. 33 Figure 9(a) shows the ∆h of melting along the eutectic tie line. The vertices of the triangle are set by the composition of the three phases in Table S23 of the Supporting Information. For HMX/RDX mixtures whose overall RDX mole fraction is less than that of the eutectic point (0.749 mole fraction RDX), ∆h represents the enthalpy change that occurs in the transition from a twophase β + RDX mixture to a two-phase β + liquid mixture. In contrast, for HMX/RDX mixtures whose overall RDX mole fraction is greater than that of the eutectic point, ∆h represents the enthalpy change that occurs in the transition from a two-phase β + RDX mixture to a two-phase RDX + liquid mixture. For all two-phase mixtures, the relative proportion of each phase is given by the lever rule. It is apparent from this rule that ∆h is greatest if the overall composition of the mixture matches that of the eutectic point, since such a mixture would melt completely to a single-phase liquid at the eutectic temperature. Figure 9(b) shows the ∆h associated with the β-δ transition. For HMX/RDX mixtures whose overall RDX mole fraction is less than that of the peritectic point (0.037 mole fraction RDX; see Table S24), ∆h is the enthalpy change exhibited by β-HMX-rich solid on the β side of the two-phase β + δ region. However, for HMX/RDX mixtures that reside to the right of the peritectic point, ∆h is the enthalpy change of transition along the peritectic tie line. A two-phase β + liquid mixture along this line transitions to a two-phase δ + liquid mixture if its temperature is increased by an infinitesimal amount. We can improve the agreement with Quintana et al. if we account for the differences in the enthalpy change ∆h0trans of transition from pure β-HMX to pure δ-HMX. According to Table 2, our ∆h0trans of 9.16 kJ/mole is about 1 kJ/mole greater than the ∆h0trans of 8.1 kJ/mole from Quintana et al. The dashed lines in the figure indicate what our results would be if ∆h0trans were shifted by this amount. The agreement is not as good as in Figure 9(a), but it gives a fair representation of the trend 37 ACS Paragon Plus Environment


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Figure 9: Enthalpy change (a) along the eutectic tie line and (b) of the β-δ phase transition at atmospheric pressure. The vertices of the triangle in (a) are set by the composition of the three phases in Table S23 of the Supporting Information. The apex represents the eutectic point. To the left (right) of the apex, ∆h is the enthalpy change that occurs in the transition from a two-phase β + RDX mixture to a two-phase β + liquid (RDX + liquid) mixture. The descending solid line in (b) is the ∆h associated with the transition from a two-phase β + liquid mixture to a two-phase δ + liquid mixture along the peritectic tie line. The dashed lines represent what our results would be if we account for differences in the ∆h0trans of transition from pure β-HMX to pure δ-HMX. As discussed in the Table 2 caption, our ∆h0trans of 9.16 kJ/mole is chosen to match that of McKenney and Krawietz; this is greater than the ∆h0trans of 8.1 kJ/mole from Quintana et al. in the enthalpy change of the β-δ transition as a function of the RDX mole fraction.

Can α-HMX be thermodynamically stable? In this section, we address an important question that we have alluded to a few times: is it possible for α-HMX to be the most thermodynamically stable (lowest Gibbs energy) phase over a certain range of temperatures and pressures? In other words, should the HMX phase diagram have an α phase region? We obtain a clear answer if we consider the transition temperatures Ttrans,n of the six distinct transitions among the four polymorphs: α-β, α-γ, α-δ, β-γ, β-δ, and γ-δ. There are 6! = 720 possible ways to arrange these six transitions, but nearly all of these arrangements are not logically self-consistent. To show this, we analyze 38 ACS Paragon Plus Environment

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two sets of three transitions. One set is composed of α-β, α-δ, and β-δ, while the other is composed of β-γ, β-δ, and γ-δ. We have demonstrated earlier (see the end of the Phase Transitions Section) that there are only two possibilities for the relative order among the former set. If Ttrans,n of α-δ is higher than that of β-δ, Ttrans,n of α-β must be lower than the other two. Otherwise, if Ttrans,n of α-δ is lower than that of β-δ, Ttrans,n of α-β must be higher than the other two. Any other order would lead to a contradiction where in a certain region of the phase diagram, α-HMX must be both more thermodynamically stable and less thermodynamically stable than δ-HMX. Similarly, the other set in our analysis also has only two possibilities for its relative order. If Ttrans,n of β-δ is higher than that of γ-δ, Ttrans,n of β-γ must be higher than the other two. Otherwise, if Ttrans,n of β-δ is lower than that of γ-δ, Ttrans,n of β-γ must be lower than the other two. We can therefore divide the transition temperature arrangements into four groups, based on whether Ttrans,n of α-δ is higher or lower than that of β-δ, and whether Ttrans,n of β-δ is higher or lower than that of γ-δ. It can be shown that these constraints reduce the original set of 720 possible arrangements into a much smaller set of only 16 logically self-consistent ones, four of which belong in each of the four groups. An exhaustive list of these 16 arrangements, along with the corresponding trend in the Gibbs energies, is presented in Tables S7–S22 of the Supporting Information. Our equation of state follows the one listed in Table S7. We may use the experimental phase transition data presented in Table 1 and in the Supporting Information to assess the likelihood of the 16 arrangements. First of all, we may eliminate the arrangements in Tables S12–S18 because in all seven cases, γ-HMX is thermodynamically stable over a certain range of temperatures and pressures. However, at least since the work of Cady et al. 9 over 50 years ago, it is widely believed that γ-HMX is only metastable at all conditions. We have not found any study which claims otherwise. This leaves nine arrangements: the five in Tables S7–S11 that have an α phase region, and the four in Tables S19–S22 that do not. To choose between these two sets of possibilities, we 39 ACS Paragon Plus Environment


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note that nearly all of the experimental α-δ transition temperatures in Table S1 are higher than the corresponding β-δ entries in Table 1. This suggests that the set of five arrangements in Tables S7–S11 is more likely because Ttrans,n of α-δ is higher than that of β-δ in this set, while the opposite is true in Tables S19–S22. Moreover, because all four arrangements in Tables S19–S22 belong to the group where Ttrans,n of α-δ < Ttrans,n of β-δ and Ttrans,n of β-δ > Ttrans,n of γ-δ, the α-β transition temperature in this group is higher than that of the α-δ, β-δ, and γ-δ transitions. In fact, the α-β transition occurs at the highest temperature in three out of the four arrangements (Tables S19–S21). This is in stark contrast to Tables S7– S11 and to past experimental observations, all of which place α-β towards the lowest end (e.g., compare experimental results in Table S5 with results in any of the other tables). In summary, the arrangements where α-HMX is thermodynamically stable (Tables S7–S11) are clearly more consistent with experimental data than the arrangements where either α-HMX is not stable (Tables S19–S22) or γ-HMX is stable (Tables S12–S18). We therefore conclude that the HMX phase diagram must have an α phase region [Figure 4(a)] and the HMX/RDX phase diagram must have an α-HMX-rich solid region. Because the α-β transition occurs at a relatively low temperature, we have suggested earlier that the HMX/RDX diagram would be characterized by a peritectoid temperature, in addition to eutectic and peritectic temperatures. Due to the strong metastability, however, it may be difficult (perhaps even impossible) to observe the α-β transition on experimentally feasible timescales.

Conclusions We have presented models based on the Peng-Robinson equation of state 22 for five different phases of HMX: α, β, γ, δ, and liquid. The equation of state has two parameters, a and b, and each phase is described by a different set of a and b. We have fit a and b to experimental


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data on the density and the vapor pressure. In the case of δ-HMX, a and b are adjusted directly to vapor pressure measurements. The vapor pressure profile of the other phases are inferred from phase transition data, specifically the enthalpy of transition and the transition temperature. By fitting a and b in this way, our phase transition temperatures fall within the range of experimental values. This enables us to generate a phase diagram for HMX. In addition, we have demonstrated that our models yield good agreement with heat capacity measurements, and reasonable results for the bulk modulus and sound speed. We have combined our HMX models with our previous work 23 on RDX to develop a Peng-Robinson model for HMX/RDX mixtures. The a and b parameters of a mixture are calculated from a and b of the components in the mixture, as well as an additional set of parameters, the binary interaction coefficients. We have determined these coefficients by solving the conditions for phase equilibrium to match experimental data at the eutectic and peritectic temperatures. We have produced an HMX/RDX phase diagram based on this information. It closely reproduces experimental results along the RDX liquidus curve. We have also made predictions about the form of the HMX/RDX phase diagram at higher pressures (up to 2500 bar) that are consistent with our phase diagram on pure HMX. In addition, we obtain fair agreement with data concerning the enthalpy change associated with a few different types of phase transitions in HMX/RDX mixtures. Our models are targeted towards conditions relevant to both cookoff experiments and rocket propulsion. Accurate representations of HMX phase transitions are essential for both of these applications because different phases have significantly different volumes, energy densities, and sensitivities, all of which affect the performance and safety of HMX. The phase behavior of HMX/RDX mixtures is also important because much of the military-grade RDX is produced through a process that introduces HMX impurities that may be as high as 21 mole%. 24 Since the RDX melting temperature decreases with HMX concentration, the impurities may cause liquid RDX — a highly volatile substance — to form at lower 41 ACS Paragon Plus Environment


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temperatures than expected. This could have important safety ramifications. The β polymorph crystallizes in solution to α-HMX at a relatively low temperature, 9,19,20 which suggests that α-HMX is thermodynamically stable in a temperature regime that falls between the β and δ phase regions. However, no study has experimentally observed the α-β transition in the absence of a solvent. Instead, the predominant transition in many studies is β-δ. This has created ambiguity in the literature regarding the thermodynamic stability of α-HMX. By choosing appropriate values for the α-δ, β-δ, and γ-δ transition temperatures, we have developed the equation of state so that it predicts that α-HMX is stable over a certain range of temperatures and pressures. We have shown that if this polymorph were not stable for any set of temperatures and pressures, one of the following must be true: 1. The HMX phase diagram has β, δ, and liquid phase regions only. This condition implies the α-β transition temperature is higher than the α-δ, β-δ, and γ-δ temperatures. 2. The HMX phase diagram has a γ phase region, in addition to β, δ, and liquid regions. Both of these conditions contradict virtually all of the experimental results regarding the transition temperatures. The HMX phase diagram must therefore have an α phase region in the limit of quasi-static heating. Consequently, the HMX/RDX phase diagram in this limit must have an α-HMX-rich solid region that spans three landmark temperatures: peritectoid, eutectic, and peritectic. However, this region has not been observed in the two studies 24,33 currently available on HMX/RDX mixtures, most likely because metastable phases can remain present far beyond the timescales that can be realized experimentally. Nevertheless, we encourage future studies on pure HMX and HMX/RDX mixtures to apply as slow a heating rate as possible in an effort to observe the α-β transition. The metastability of the HMX polymorphs is clearly still an unresolved issue. Its complexity presents great challenges that have been studied through experiments and continuumscale models. 13–18 However, this topic has not yet been investigated with molecular simula42 ACS Paragon Plus Environment

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tions, even though much mechanistic insight could be gained from these simulations. The equation of state in this work may help such future studies because it can calculate the Gibbs energy of any phase even at conditions where the phase is not thermodynamically favorable. The driving force predicted by the equation of state may be compared with those from the molecular simulations. This synergistic approach will complement the existing work on HMX nucleation/growth and improve our understanding of this important and confounding topic.

Acknowledgement This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The research was partially funded by the Joint DoD/DOE Munitions Technology Development Program (JMP).

Supporting Information Available This material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Forbes, J. W.; Tarver, C. M.; Urtiew, P. A.; Garcia, F.; Greenwood, D. W.; Vandersall, K. Pressure Wave Measurements from Thermal Cook-Off of an HMX based High Explosive. JANAF CS/APS/PSHS Joint Meeting. 2000; 10 pp. (2) McClelland, M. A.; Glascoe, E. A.; Nichols III, A. L.; Schofield, S. P.; Springer, H. K. ALE3D simulation of incompressible flow, heat transfer, and chemical decomposition of Comp B in slow cookoff experiments. 15th International Detonation Symposium. 2014; 11 pp.



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(3) Nichols III, A. L.; Schofield, S. P. Modeling the response of fluid/melt explosives to slow cook-off. 15th International Detonation Symposium. 2014; 9 pp. (4) Beauregard,

R.

L.

http://www.insensitivemunitions.org/history/

the-uss-forrestal-cva-59-fire-and-munition-explosions/. (5) Yoo, C.-S.; Cynn, H. Equation of state, phase transition, decomposition of β-HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine) at high pressures. J. Chem. Phys. 1999, 111, 10229, DOI: 10.1063/1.480341. (6) Hare, D. E.; Forbes, J. W.; Reisman, D. B.; Dick, J. J. Isentropic compression loading of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) and the pressure-induced phase transition at 27 GPa. Appl. Phys. Lett. 2004, 85, 949 – 951, DOI: 10.1063/1.1771464. (7) Landers, A. G.; Brill, T. B. Pressure dependence of the β-δ polymorph interconversion in octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine. J. Phys. Chem. 1980, 84, 3573–3577, DOI: 10.1021/j100463a015. (8) Karpowicz, R. J.; Brill, T. B. The β → δ transformation of HMX – Its thermal analysis and relationship to propellants. AIAA J. 1982, 20, 1586–1591, DOI: 10.2514/3.7992. (9) Cady, H. H.; Smith, L. C. Studies on the polymorphs of HMX ; Technical Report LAMS2652, Los Alamos National Laboratory, 1962; 50 pp., http://babel.hathitrust.org/ cgi/pt?id=mdp.39015086497560. (10) Koshigoe, L. G.; Shoemaker, R. L.; Taylor, R. E. Specific heat of Octahydro-1,3,5,7Tetranitro-1,3,5,7-Tetrazocine (HMX); Technical Report TPRL 314, 1983. (11) Herrmann, M.; Engel, W.; Eisenreich, N. Thermal expansion, transitions, sensitivities and burning rates of HMX. Propell. Explos. Pyrot. 1992, 17, 190–195, DOI: 10.1002/prep.19920170409. 44 ACS Paragon Plus Environment

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