Isothermal equations of state of beta octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine athigh temperaturesJared C. Gump and Suhithi M. Peiris Citation: Journal of Applied Physics 97, 053513 (2005); doi: 10.1063/1.1856227 View online: http://dx.doi.org/10.1063/1.1856227 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in First-principles high-pressure unreacted equation of state and heat of formation of crystal 2,6-diamino-3, 5-dinitropyrazine-1-oxide (LLM-105) J. Chem. Phys. 141, 064702 (2014); 10.1063/1.4891933 Thermal equations of state and phase relation of PbTiO3: A high P-T synchrotron x-ray diffraction study J. Appl. Phys. 110, 084103 (2011); 10.1063/1.3651377 Equations of state of 2,6-diamino-3,5-dinitropyrazine-1-oxide J. Appl. Phys. 110, 073523 (2011); 10.1063/1.3646492 Equation of state, phase transition, decomposition of β-HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine) athigh pressures J. Chem. Phys. 111, 10229 (1999); 10.1063/1.480341 Monte Carlo calculations of the hydrostatic compression of hexahydro-1,3,5-trinitro-1,3,5-triazine and β-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine J. Appl. Phys. 83, 4142 (1998); 10.1063/1.367168

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Isothermal equations of state of beta octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine at high temperatures

Jared C. Gumpa! and Suhithi M. PeirisIndian Head Division, Naval Surface Warfare Center, Indian Head, Maryland 20640

sReceived 29 September 2004; accepted 9 December 2004; published online 14 February 2005d

Isothermal pressure-volume equations of state of beta HMXsoctahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocined at temperatures of 30, 100, and 140 °C under both hydrostatic andnonhydrostatic compressions have been obtained using synchrotron angle-dispersive x-raydiffraction experiments. The samples were heated to the isotherm temperature and compressed up to5.8 GPa. At all temperatures HMX remained in the beta phase up to 5.8 GPa. However, at 140 °Cupon decompression to ambient from nonhydrostatic pressures above 4 GPa, HMX underwent aphase transition to the delta phase. The same transition was seen upon decompression to ambientfrom hydrostatic compression; however, parts of the sample remain in theb phase, resulting in amixed-phase sample. The diffraction data were analyzed to yield unit-cell dimensions at eachpressure, and further analyzed to yield thermal expansion, bulk modulus, and the pressure derivativeof the bulk modulus. ©2005 American Institute of Physics. fDOI: 10.1063/1.1856227g

I. INTRODUCTION

HMX or octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocineis a very popular high-explosive ingredient. Its wide use inmany formulations creates a need to understand its materialproperties. These properties need to be investigated not justunder ambient conditions, but also under the high pressureand temperature conditions leading to detonation. In an at-tempt to simulate the high-pressure and high-temperatureconditions of detonation, diamond-anvil cells with heatingcoils were used.

HMX is known to have four phases:a, b, g, andd.1 Themost stable form at room temperature isb-HMX, which isthe form investigated in these experiments. Botha- andd-HMX are stable at high temperatures, andg-HMX is ahydrate. The transition from theb to thed phase is known tooccur at about 165 °C at ambient pressure.1–4 There is also aspeculation that it is thed phase of HMX that is sensitive toinitiation resulting in detonation.2–4 The effect of compres-sion on the phase transition and the complete phase diagramof HMX are not yet fully understood.

The isotherm ofb-HMX at room temperature was pre-viously studied by Olingeret al.5 and by Yoo and Cynn.6 Adetailed comparison of the results from these two studies wasreported by Menikoff and Sewell.7 Their report highlightedthe difficulty in extracting consistent bulk modulus informa-tion from the isothermal data due to sensitivity to the fittingform and data domain. They suggested that some of the dis-crepancy between bulk modulus values might be due to alack of low-pressures,1 GPad data. We report here 15 ormore data points at each temperature within a moderate pres-sure ranges,6 GPad, which should prove useful in elucidat-ing previous discrepancies. These results also expand therange of isothermal equation of statesEOSd data ofb-HMX

to temperatures above room temperature. We report here thestudy of the compression ofb-HMX at a temperature ap-proaching theb−d phase change.

II. EXPERIMENTAL METHODS

An important issue to consider for this experiment issample purity. Impurities inb-HMX can cause expansion ofthe unit-cell volume. The purity of the samples used in thisstudy, as determined by high performance liquid chromotog-raphysHPLCd, was insured by thrice recrystallizingb-HMXin acetone.

Angle-dispersive x-ray diffraction measurements wereperformed at the Cornell High Energy Synchrotron SourcesCHESSd with a monochromatedsl=0.49 Åd x-ray beam.Diamond-anvil cells with heating coilsfhydrothermaldiamond-anvil cellssHDACdg developed by Bassettet al.8

were prepared with 127-mm-thick Inconel or stainless-steelgaskets drilled with 250–360-mm holes and loaded withpowderedb-HMX. A ruby piece was added to monitor thepressure and the frequency shift of the rubyR1 line was usedas a pressure gauge.9 NaCl sin a HMX:NaCl volume ratio of8:1d was included in the room-temperature samples and thevolume compression of salt with Birch’s EOS for salt wasused to calculatein situ pressure.10 Hexane was used as thepressure medium for hydrostatic room-temperature experi-ments. Flourochemical FC-75 was used as the pressure me-dium for hydrostatic high-temperature experiments.

Samples were heatedsat a rate of about 4°C/mind to thedesired temperaturesTd and then maintained at that tempera-ture during compression to high pressuressPd. At eachT andP, the sample was exposed to x rays for 10–30 min and anx-ray sensitive image plate collected the diffraction patterns.The plate was then scanned by computer and analyzed withSIMPA software.11 The diffraction peaks were fitted to Gauss-ian wave forms to determined spacing, then indexed usingthe b-phase monoclinic structure with aP2s1d /n spacegroup.

adAuthor to whom correspondence should be addressed; electronic mail:gumpjc@ih.navy.mil

JOURNAL OF APPLIED PHYSICS97, 053513s2005d

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III. EXPERIMENTAL RESULTS

To obtain the isothermal equation of state ofb-HMXdifferent samples were maintained at the desired temperatureand compressed. The compression up to 5.8 GPa at bothroom temperature and 100 °C was reversible, and decom-pression at those temperatures continued to show thebphase. However, at 140 °C, when the samples were com-pressed above 4 GPa under nonhydrostatic conditions, de-compression resulted in transformation to thed phase as de-termined from x-ray diffraction data, while samplescompressed above 4 GPa under hydrostatic conditions re-sulted in only a part of the sample being converted to thedphase. That is, upon decompression, the nonhydrostaticallycompressed samples only showed thed phase, while the hy-drostatically compressed samples showed both theb and dphases. Thed phase thus formed upon decompression at140 °C remained stable for a few hours after the sampleswere cooled back to room temperature. Figure 1 shows thex-ray diffraction patterns upon compression and decompres-sion under nonhydrostatic conditions at 140 °C. The decom-pression patterns for the hydrostatic run as well as ad-phasepattern calculated from single-crystal data are also includedfor comparison.12

The x-ray diffraction patterns obtained were indexed us-ing the hkl assignments from the ambient-pressure mono-clinic b-HMX structure. On average, 15 diffraction peaksvisible in the ambient pressure diffraction pattern weretracked throughout the compression until the peak intensitywas too weak to allow a peak position to be determined. The

hkl indices of these 15 peaks are 011, 020, 11̄0, 1̄02, 1̄12,

1̄20, 1̄22, 022, 1̄32, 1̄41, 2̄32, 1̄33, 042, 3̄12, and 3̄13. The

1̄32 peak was not used for the measurements with salt as apressure medium due to the fact that the NaCl 200 line over-lapped with it. Unit-cell volumes were calculated from theindexed diffraction patterns. Tables I–IV list the unit-cell pa-rameters at each pressure for the various isotherms studied.

Figure 2 shows volume versus pressure data for theb-HMX phase at different temperatures under hydrostaticcompression. Usually isotherms are parallel in theP-V plane,however, these isotherms appear to merge. That is, HMX

becomes more compressible at higher temperatures ap-proaching theb-d phase transition temperature.

The P-V data were fitted directly in theP vs V/Vo planeusing the third-order Birch–Murnaghan formalism.10 Resultsfrom these fits were then used to obtain the following EOSparameters, whereKo,T is the isothermal zero-pressure bulkmodulus, andKo,T8 is the pressure derivative of the zero-pressure bulk modulus. The EOS fits at the three tempera-tures are shown as the solid, dashed, and dotted lines in Fig.2. The increase in compressibility as HMX is heated is alsoseen in the decreasing bulk modulus with increasing tem-perature.

s1d T=30 °C s303 Kd nonhydrostatic compression

Vo = 516.5 Å3 ± 3.20,

Ko,T = 14.8 ± 0.56 GPa, Ko,T8 = 20.7 ± 1.73.

s2d T=30 °C s303 Kd hydrostatic compression

Vo = 515.7 Å3 ± 1.47,

Ko,T = 21.0 ± 1.02 GPa, Ko,T8 = 7.45 ± 0.95.

s3d T=100 °C s373 Kd hydrostatic compression

Vo = 526.4 Å3 ± 1.24,

Ko,T = 14.1 ± 0.82 GPa, Ko,T8 = 11.6 ± 1.41.

s4d T=140 °C s413 Kd hydrostatic compression

Vo = 533.4 Å3 ± 3.76,

Ko,T = 13.5 ± 0.56 GPa, Ko,T8 = 9.0 ± 0.85.

Volume thermal expansion was calculated assuming lin-earity in theV,T relationship. The ambient pressure volumessV0d obtained from samples at room temperature, 100 °C,and 140 °C were plotted as a function of temperature. Theslope of theV vs T plot was determined by a linear least-squares fit. Thermal expansionsad was then calculated bydividing the slope by the ambient pressure volumes becausea=s1/Vds]V/]TdP. The average volume thermal expansion

FIG. 1. X-ray diffraction patterns ofHMX at 140 °C compressed nonhy-drostatically to above 5.0 GPa andthen decompressed. The decompres-sion patterns for a hydrostatic data setand a calculatedd-HMX spectrum areincluded for comparison.

053513-2 J. C. Gump and S. M. Peiris J. Appl. Phys. 97, 053513 ~2005!

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obtained from our data in this temperature range at ambientpressure is 0.000 27 K−1. Because of the merging isothermalEOS, at higher pressure thermal expansion is so reduced thatabove 3 GPa there is hardly any volume expansion from30 to 140 °C.

IV. DISCUSSION

The wide use of HMX in many explosive formulationscreates a need to investigate its material properties at thehigh-pressure and temperature conditions achieved duringdetonation. As seen in this study the compression at differenttemperatures is sufficiently different to result in an EOSmerging above 3 GPa. Such isotherm systems are thermody-namically allowed and have been seen in other materialswhen temperatures are close to a phase-transitiontemperature.13 These results also mean that HMX becomesmore compressible at higher temperatures, to the extent ofnot expanding when heated from 30 to 140 °C at pressuresgreater than 3 GPa. This observation has now been sup-ported by molecular-dynamic simulations performed bySewell from Los Alamos National Laboratory. His theoreti-cal curves calculated at 50, 100, and 150 °C are plotted withthe hydrostatic experimental isotherms from this work in Fig.3.

In addition, our studies show that decompression ofb-HMX sfrom above 4 GPad to ambient pressure at 140 °Cresults in thed phase. Our results have recently been con-firmed by Raman spectroscopy performed under the guid-

ance of Smilowitz at Los Alamos National Laboratory.14 Thisis an unusual effect considering thed phase is the higher-temperature phase, herein accessed by decompression. How-ever, in comparison to theb phase, thed phase is also thelower-density phase. Because diamond-anvil cell decompres-sion can be considered “quick” in terms of phase-transitionkinetics, the fast lowering in density may make thed phasekinetically more accessible upon decompression at 140 °C.The kinetic argument is further substantiated when consider-ing that after several days at ambient pressure thed phasedoes slowly convert to theb phase, as seen by Hermannetal.15

A visual comparison of our room-temperature compres-sion data under hydrostatic and nonhydrostatic conditionscan be seen in Fig. 4. The nonhydrostatic room-temperaturedata show less compression at high pressure than the hydro-static room-temperature data. This difference is mainly dueto a reduction in compression along theb axis, as evidencedin Tables I and II. Nonhydrostatic loading, which producesnonuniform pressure gradientssmostly shear stressd, seemsto allow for lattice distortion with little accompanying vol-ume change as opposed to the uniform compression of hy-drostatic loading. When working at 100 and 140 °C, there isvery little difference between the compression curves underhydrostatic versus nonhydrostatic conditions. This suggeststhat the elevated temperature of the material sufficientlychanges the material properties such that the nonuniformpressure gradients are reduced.

TABLE I. Unit-cell parameters at each nonhydrostatic pressure at room temperature. Conventional beta is 180deg. minus reported value.

PressuresGPad

Cell asÅd

Cell bsÅd

Cell csÅd

bs°d

VolumesÅ3d

0±0.2 6.53±0.01 11.02±0.02 8.73±0.02 55.39±0.0018 517.28±1.510±0.02 6.52±0.01 11±0.05 8.71±0.05 55.77±0.0044 516±4.020±0.02 6.52±0.01 11.01±0.05 8.7±0.05 55.77±0.0044 516.49±4.06

0.21±0.05 6.44±0.02 11.07±0.05 8.6±0.06 56.52±0.0048 511.33±4.950.36±0.02 6.48±0.02 11.02±0.06 8.61±0.06 55.72±0.0054 508.18±4.920.45±0.02 6.47±0.01 10.92±0.04 8.67±0.03 55.57±0.0028 505.41±2.630.75±0.04 6.44±0.02 10.94±0.05 8.59±0.05 55.66±0.0049 499.93±4.420.76±0.04 6.44±0.01 10.78±0.04 8.72±0.03 55.82±0.0025 501.11±2.520.86±0.04 6.46±0.01 10.76±0.03 8.71±0.02 55.88±0.0022 501.58±2.211.04±0.05 6.44±0.02 10.92±0.05 8.53±0.05 54.8±0.0046 490.48±4.051.11±0.06 6.45±0.01 10.78±0.03 8.66±0.02 55.37±0.0022 495.13±2.11.21±0.06 6.41±0.02 10.78±0.09 8.64±0.09 55.72±0.0058 492.81±6.861.66±0.08 6.44±0.06 10.95±0.18 8.48±0.19 54.89±0.0149 489.15±15.071.73±0.09 6.44±0.02 10.62±0.05 8.69±0.06 55.4±0.0043 489.17±4.431.85±0.09 6.45±0.02 10.84±0.04 8.54±0.05 54.14±0.0041 484.41±3.871.89±0.1 6.47±0.01 10.71±0.03 8.67±0.04 53.87±0.0033 485.56±3.182.01±0.1 6.37±0.04 10.72±0.14 8.66±0.14 55.49±0.0093 486.53±10.992.21±0.11 6.4±0.01 10.74±0.03 8.57±0.03 54.73±0.0031 480.47±2.72.51±0.13 6.36±0.02 10.62±0.04 8.59±0.05 54.96±0.0046 475.07±3.92.57±0.13 6.27±0.03 10.56±0.08 8.61±0.11 56.28±0.0075 474.42±7.532.59±0.13 6.21±0.03 10.58±0.11 8.59±0.09 56.84±0.0093 472.64±7.922.63±0.13 6.32±0.04 10.62±0.17 8.53±0.14 55.6±0.0111 472.09±12.043.19±0.16 6.37±0.07 10.62±0.1 8.58±0.07 54.53±0.0152 472.21±9.343.52±0.18 6.37±0.02 10.61±0.07 8.57±0.07 54.5±0.0049 471.27±5.234.11±0.21 6.36±0.1 10.49±0.12 8.57±0.11 54.3±0.0212 464.72±12.885.21±0.26 6.36±0.05 10.66±0.19 8.4±0.18 53.64±0.0134 458.51±14.03

053513-3 J. C. Gump and S. M. Peiris J. Appl. Phys. 97, 053513 ~2005!

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In addition to our data, in Fig. 4, we show the data ofOlinger et al.5 and Yoo and Cynn.6 Yoo and Cynn’s data,denoted by open squares, show their ambient-pressure unit-cell volume to be much higher than the values obtained by us

and other groups at ambient pressure.15,16 This indicates apossibility that their HMX material perhaps contained impu-rity, which is known to expand the unit cell.1 Our data wereobtained with a very pure batch of HMX thrice recrystallized

TABLE II. Unit-cell parameters at each hydrostatic pressure at room temperature. Conventional beta is 180 deg.minus reported value.

PressuresGPad

Cell asÅd

Cell bsÅd

Cell csÅd

bs°d

VolumesÅ3d

0±0.02 6.52±0.01 11.06±0.02 8.7±0.01 55.24±0.0028 515.5±1.240±0.02 6.51±0.01 11.06±0.02 8.69±0.02 55.52±0.0033 516.04±1.890±0.02 6.53±0.01 11.06±0.02 8.69±0.02 55.36±0.0036 515.79±1.61

0.1±0.08 6.51±0.01 11.06±0.02 8.68±0.02 55.49±0.0046 514.62±2.450.17±0.19 6.5±0.01 11.02±0.02 8.66±0.02 55.39±0.0047 510.52±2.110.39±0.11 6.51±0.03 10.84±0.06 8.66±0.02 55.38±0.0037 502.97±4.050.49±0.11 6.48±0.01 11.02±0.03 8.66±0.02 55.08±0.0046 507.37±1.950.83±0.27 6.47±0.01 10.78±0.01 8.65±0.01 55.06±0.0017 494.06±1.070.95±0.15 6.44±0.01 10.93±0.02 8.59±0.02 55.25±0.0043 496.62±1.850.96±0.1 6.48±0.04 10.72±0.07 8.61±0.03 55.34±0.0046 492.06±5.121.36±0.25 6.42±0.01 10.68±0.02 8.61±0.02 54.97±0.0026 483.19±1.581.38±0.13 6.43±0.02 10.82±0.07 8.57±0.05 55.53±0.0107 491.82±5.752.08±0.24 6.41±0.03 10.7±0.09 8.5±0.06 55.4±0.013 480.25±6.332.13±0.08 6.41±0.02 10.65±0.07 8.57±0.09 55.04±0.0127 479.77±6.282.16±0.17 6.36±0.03 10.77±0.1 8.47±0.07 55.48±0.0151 477.97±7.682.49±0.32 6.38±0.03 10.56±0.04 8.57±0.04 54.94±0.006 472.17±3.572.68±0.18 6.32±0.03 10.66±0.1 8.5±0.07 55.5±0.0148 472.1±7.642.85±0.13 6.39±0.02 10.51±0.07 8.55±0.1 55.1±0.0134 470.56±6.52.99±0.26 6.34±0.05 10.54±0.06 8.54±0.05 54.95±0.0089 466.87±5.283.49±0.2 6.26±0.04 10.54±0.16 8.54±0.12 54.34±0.0223 457.65±12.383.5±0.09 6.33±0.07 10.47±0.09 8.55±0.1 54.73±0.0152 462.58±8.68

3.74±0.28 6.34±0.06 10.36±0.08 8.56±0.07 54.91±0.0123 459.76±7.14.29±0.26 6.28±0.03 10.27±0.09 8.55±0.07 54.94±0.014 451.3±5.834.32±0.12 6.3±0.06 10.31±0.08 8.56±0.09 54.54±0.0137 452.43±8.024.45±0.1 6.34±0.07 10.2±0.09 8.56±0.11 54.6±0.0164 451.83±9.374.47±0.28 6.27±0.03 10.4±0.09 8.51±0.07 54.46±0.0137 450.96±6.854.81±0.32 6.23±0.03 10.38±0.1 8.49±0.08 54.51±0.0144 447.13±6.935.02±0.32 6.29±0.12 10.2±0.19 8.56±0.1 54.69±0.0157 447.91±13.155.22±0.18 6.4±0.09 10.16±0.14 8.45±0.07 54.29±0.0095 446.12±9.965.42±0.31 6.21±0.03 10.35±0.11 8.48±0.09 54.83±0.0164 445.39±6.97

TABLE III. Unit-cell parameters at each hydrostatic pressure at 100 °C. Conventional beta is 180 deg. minusreported value.

PressuresGPad

Cell asÅd

Cell bsÅd

Cell csÅd

bs°d

VolumesÅ3d

1.12±0.06 6.42±0.01 10.9±0.04 8.59±0.04 55.12±0.007 493.35±2.931.21±0.06 6.42±0.02 10.9±0.05 8.59±0.04 55.16±0.0083 493.39±3.491.62±0.08 6.44±0.01 10.76±0.03 8.61±0.03 55.43±0.0048 490.79±2.71.94±0.1 6.39±0.02 10.86±0.05 8.54±0.05 55.3±0.0102 487.05±4.252.03±0.1 6.45±0.01 10.61±0.04 8.66±0.04 55.23±0.0064 486.89±3.262.6±0.13 6.36±0.02 10.54±0.05 8.54±0.04 55.41±0.0079 471.36±3.882.68±0.13 6.37±0.07 10.42±0.08 8.67±0.08 55.43±0.012 473.32±7.862.88±0.14 6.37±0.06 10.46±0.07 8.51±0.05 55.57±0.0098 467.53±6.673.1±0.16 6.39±0.05 10.36±0.05 8.69±0.05 54.64±0.0083 469.14±5.263.62±0.18 6.32±0.11 10.31±0.11 8.63±0.11 55.15±0.0203 461.56±11.273.94±0.2 6.34±0.06 10.32±0.08 8.5±0.06 55.53±0.011 458.76±7.34.28±0.21 6.3±0.18 10.22±0.16 8.56±0.16 55.97±0.0316 456.58±284.74±0.24 6.32±0.06 10.22±0.07 8.52±0.06 55.34±0.0104 452.54±6.435.02±0.25 6.34±0.06 10.11±0.06 8.52±0.06 55.2±0.0096 448.32±5.755.96±0.3 6.29±0.05 10.07±0.06 8.5±0.05 55.18±0.0091 442.2±5.32

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in acetone. Therefore, our data reflect the compression of thepure material. However, Yoo and Cynn’s data are useful tomodel the behavior of commercially available HMX such asthe materials that are used in weapons production.

A numerical comparison of the room-temperature hydro-static bulk modulus and its first pressure derivative can befound in Table V. Olingeret al. used a pseudovelocity for-malism to extract the thermodynamic parameters from theirdata.5 Yoo and Cynn used a third-order Birch–Murnaghan

TABLE IV. Unit-cell parameters at each hydrostatic pressure at 140 °C. Conventional beta is 180 deg. minusreported value.

PressuresGPad

Cell asÅd

Cell bsÅd

Cell csÅd

bs°d

VolumesÅ3d

0.79±0.04 6.55±0.01 10.96±0.03 8.68±0.02 55.32±0.0039 512.36±3.291.17±0.06 6.54±0.02 10.8±0.04 8.65±0.03 55.21±0.0069 501.63±3.221.2±0.06 6.51±0.01 10.89±0.03 8.65±0.03 55.03±0.0047 502.42±2.2

1.37±0.07 6.42±0.03 10.72±0.03 8.61±0.03 55.26±0.0046 486.67±3.041.58±0.08 6.46±0.03 10.76±0.04 8.61±0.05 54.55±0.0088 487.63±4.311.58±0.08 6.52±0.02 10.74±0.05 8.64±0.04 55.2±0.0078 496.7±4.351.7±0.09 6.41±0.03 10.68±0.04 8.62±0.03 55.18±0.0052 484.14±3.4

1.99±0.1 6.43±0.06 10.71±0.05 8.6±0.05 54.46±0.0101 481.42±6.22±0.1 6.39±0.02 10.76±0.02 8.57±0.02 55.27±0.0033 484.62±4.36

2.3±0.12 6.36±0.04 10.59±0.06 8.55±0.05 55.48±0.0081 474.8±5.482.41±0.12 6.41±0.05 10.56±0.06 8.55±0.04 55.81±0.0087 478.46±6.812.52±0.13 6.36±0.03 10.68±0.04 8.54±0.03 55.42±0.0049 477.83±8.682.79±0.14 6.34±0.06 10.55±0.07 8.55±0.07 55.5±0.0103 470.74±6.972.93±0.15 6.4±0.02 10.58±0.03 8.55±0.02 55.07±0.0043 474.08±2.563.23±0.16 6.38±0.03 10.41±0.04 8.54±0.03 55.05±0.0056 464.37±3.923.34±0.17 6.37±0.03 10.57±0.03 8.52±0.02 55.09±0.0046 470.62±2.723.51±0.18 6.35±0.05 10.33±0.06 8.58±0.06 55.02±0.0089 461.68±5.43.72±0.19 6.36±0.04 10.45±0.05 8.53±0.03 55.39±0.0071 467.03±4.333.87±0.19 6.34±0.04 10.36±0.04 8.49±0.03 55.21±0.0066 457.68±3.874.2±0.21 6.31±0.06 10.3±0.07 8.53±0.07 55.26±0.0107 455.76±6.53

4.72±0.24 6.3±0.07 10.13±0.08 8.56±0.08 55.18±0.0124 448.03±7.484.86±0.24 6.27±0.05 10.32±0.06 8.43±0.05 55.41±0.0093 448.9±5.635.08±0.25 6.27±0.05 10.3±0.06 8.44±0.05 55.41±0.0087 448.95±5.745.38±0.27 6.3±0.07 10.1±0.08 8.52±0.08 55.21±0.012 445.55±7.225.6±0.28 6.25±0.06 10.29±0.07 8.41±0.05 55.51±0.0103 445.95±6.38

FIG. 2. IsothermalP-V data at 30, 100, and 140 °C shown symbols, andEOS obtained by fitting each set of data to the Birch–Murnaghan equationof state formalism.

FIG. 3. Comparison of 50, 100, and 150 °C isotherms calculated by Sewellwith the experimental data.

053513-5 J. C. Gump and S. M. Peiris J. Appl. Phys. 97, 053513 ~2005!

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equation of state formalism.6 Menikoff and Sewell fit thedata of Olingeret al. with a third-order Birch–Murnaghanequation of state formalism so that they could compare it toYoo and Cynn’s data using the same fitting technique.7 Meni-koff and Sewell also performed a third-order Birch–Murnaghan fit of only Yoo and Cynn’s data that were below12 GPa to avoid any possible phase transitions occurring athigher pressures. Upon comparison, our bulk modulus andits pressure derivative most closely match those of Yoo andCynn when only their data below 12 GPa are used. However,the bulk moduli between these two fits are not within experi-mental error, whereas the pressure derivatives are in goodagreement. This highlights the importance of low-pressuredata points when evaluating the isotherm bulk modulus atzero pressure.

The comparison of room-temperature EOS parametersfrom this work to previous data5,6 has highlighted the diffi-culty of unambiguously determining thermodynamic param-eters from isothermal data. Even though the three data sets

were analyzed using the same fitting form, there is still somediscrepancy as to the bulk modulus and its first pressure de-rivative. These differences may result from the variations inthe data domains as suggested by Menikoff and Sewel,7 orfrom the sensitivity of the physical parameters of HMX tothe quantity of impurity in the HMX lattice.

The volume data obtained at different temperatures alsolet us calculate an ambient-pressure volume thermal expan-sion of 0.000 27 K−1 in this temperature range. An older pub-lication by Hermannet al.15 reports a volume thermal expan-sion of 0.000 13 K−1 and a proceedings article by Saw16

shows the same parameter to be 0.000 20 K−1. It is possiblethat the thermal expansion of HMX is very sensitive to theamount of impurities within the lattice, such as cyclotrimeth-ylenetrinitramine sRDXd inclusions, explaining this widerange of figures obtained for thermal expansion ofb-HMX.

V. CONCLUSIONS

This study has expanded the available information on theisotherms ofb-HMX to temperatures above room tempera-ture. The effect of temperature on the isothermal equations ofstate shows that HMX gets more compressible with tempera-ture. At high pressuress.3 GPad the isotherms begin tomerge, and by 6 GPa the effect of changing temperaturefrom room temperature to 140 °C is negligible. The bulkmodulus and its pressure derivative were calculated from thecompression curves by fitting them to a third-order Birch–Murnaghan formalism. These values were determined atroom temperature, as well as at 100 and 140 °C. Thermalexpansion was determined to be 0.000 27 K−1. This studyalso discovered that decompression ofb-HMX after com-pression to above 4 GPa at 140 °C results in the transforma-tion of theb phase to thed phase. This is deemed to be dueto the quick drop in density as we decompress, because thedphase has been observed to revert to theb phase after sometime.

ACKNOWLEDGMENTS

This work is based upon research conducted at the Cor-nell High Energy Synchrotron SourcesCHESSd which issupported by the National Science Foundation and the Na-tional Institutes of Health/National Institute of GeneralMedical Sciences under Award No. DMR00-225180. The au-thors gratefully acknowledge funding from Dr. Judah Gold-wasser, Code 333, Office of Naval Research, and thank Dr.Thomas Sewell of Los Alamos National Laboratory for pro-viding us with the results of his molecular-dynamic calcula-tions included here.

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FIG. 4. Room-temperature isothermalP-V data. Both hydrostatic and non-hydrostatic data sets from this study, as well as hydrostatic data from bothOlinger, Roof, and Cady and Yoo and Cynn are shown.

TABLE V. Comparison of the bulk modulussKod and its first pressurederivative sKo8d for several hydrostatic room-temperature studies. pv=Pseudovelocity fitting function. BM=third-order Birch–Murnaghan fittingfunction, as calculated by Menikoff and SewellsSee Ref. 5d.

Ko sGPad Ko8

This studysBMd 21.0±1.0 7.4±1.7Olinger, Roof, and CadysBMda 10.6±1.7 18.1±3.4Olinger, Roof, and Cadyspvd 13.5 9.3Yoo and CynnsBMd 12.4 10.4Yoo and CynnsBMd ,12 GPaa 16.0±2.5 7.3±1.4

aSee Ref. 5.

053513-6 J. C. Gump and S. M. Peiris J. Appl. Phys. 97, 053513 ~2005!

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