microscopic study of the equation of state of β-hmx using...
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Microscopic stud
aDepartment of Aerospace Engineering an
Cincinnati, Cincinnati, OH 45221-0070, USbDepartment of Mechanical, Aerospace
Polytechnic Institute, Troy, NY 12180, USA.
Cite this: RSC Adv., 2015, 5, 55892
Received 1st May 2015Accepted 19th June 2015
DOI: 10.1039/c5ra08062b
www.rsc.org/advances
55892 | RSC Adv., 2015, 5, 55892–559
y of the equation of state of b-HMX using reactive molecular dynamicssimulations
Guang Yu Wang,*a Qing Peng,*b Gui Rong Liua and Suvranu Deb
The equation of state (EoS) is the relation between physical quantities describing thermodynamic states of
materials under a given set of conditions such as pressure, temperature, and volume. The EoS plays a
significant role in determining the characteristics of energetic materials, including Chapman–Jouguet
point and detonation velocity. Furthermore, the EoS is the key to connect microscopic and
macroscopic phenomena in the study of energetic materials. For instance, the foundation of the
ignition and growth model for high explosives is two Jones–Wilkins–Lee (JWL) EoSs, one for
unreacted explosive and the other for reacted explosive. Thus, an accurate calculation of the EoS is
required for the study of energetic materials. In this paper, the EoSs for both unreacted and reacted
b-HMX are calculated using molecular dynamics simulations with the ReaxFF-lg potential. The
microscopic simulation results are compared with experiments, first-principles calculations, and a
continuum ignition and growth model. Good agreement is observed. Our result indicates that
molecular dynamics simulations are a reliable alternative method to experiments to generate the EoS
for energetic materials.
1. Introduction
Due to the extreme conditions (very short time, high pressure,high temperature, phase transformation, and chemical reac-tion) that are hard to achieve,1,2 it still remains difficult toinvestigate the details of the detonation of high explosivesthrough experiments today. As a result, the simulation of highexplosives has long been an interest of researchers and variousnumerical models have been proposed to simulate the deto-nation process.3–7 The equation of state (EoS), which describesthe relation of the state variables in detonation, is a signicanttool to determine the characteristics (Chapman–Jouguet point,detonation velocity, etc.) of high explosives. The EoS also plays acritical role in many numerical models for high explosives,3,4,6,7
for instance, the foundation of ignition and growth models forhigh explosives is two Jones–Wilkins–Lee (JWL) EoSs, one forunreacted explosive and one for reacted explosive.3,4 Generally,to obtain the data needed for tting parameters of reacted JWLEoSs, a cylinder test needs to be conducted. The cylinder testconsists of detonating a cylinder of explosive conned bycopper and measuring the velocity of the expanding copperwall. Thus, it is complicated and also costly to conduct cylindertests for various high explosives.
d Engineering Mechanics, University of
A. E-mail: [email protected]
and Nuclear Engineering, Rensselaer
E-mail: [email protected]
00
The EoS for unreacted high explosives is obtained usingRankine–Hugoniot conditions, which are a set of conservationequations of mass, momentum, and energy. Much research hasbeen done on the Hugoniot curve of solids. Walsh and Christianinitiated the idea in 1955 to use planar shock waves to deter-mine the EoS of condensed materials.8 Aer that, shock waveexperiments have been widely applied to calculate the EoS for awide range of materials, including elements, compounds,alloys, porous materials, etc.9–13 Some researchers havesuccessfully applied planar shock waves in the study of Hugo-niot curve of explosives.14,15
Molecular dynamics is a powerful tool that can be used toinvestigate many microscopic processes. In recent years, thedetonation of high explosives has been explored much moreprofoundly with the introduction of reactive force elds(ReaxFF) in molecular dynamics. Van Duin and S. Dasguptarst proposed ReaxFF for hydrocarbons,16 which enablesconvenient and fast molecular dynamics simulations of large-scale reactive chemical systems. Furthermore, Liu et al. intro-duced an additional vdW-like interaction to ReaxFF, whichenables ReaxFF to obtain the correct density of molecularcrystals.17 The reactive force eld with van der Waals interac-tion corrections at low gradient regions, the so-called ReaxFF-lg, is used in this study.
A lot of research has been conducted on high explosives(such as TNT, HMX, and RDX) since the introduction of ReaxFF.Strachan, Kober, and Van Duin et al. studied thermal decom-position of RDX under various temperature and densities.18
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Table 1 Parameters of unreacted JWL EoS of PBX 9501 (ref. 25)
A (GPa) B (GPa) r1 r2 u CV (J kg�1 K�1) T0 (K)
Unreacted PBX 9501 732 000 �5.2654 14.1 1.41 0.8867 2.7806 � 106 298
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Chenoweth, Cheung et al. studied thermal decomposition of apoly polymer with ReaxFF.19 Zhang et al. studied the thermaldecomposition of 1,3,5-trinitrohexahydro-s-triazine (RDX)bonded with polyurethane (Estane) using molecular dynamicssimulation equipped with ReaxFF.20 Zhou et al. studied thethermal and chemical response of shock induced HMX withReaxFF.21 However, these papers mainly focus on the thermaldecomposition of high explosives. We have studied the EoS22
and elastic properties23 of unreacted b-HMX using rst-principles calculations at zero temperature. To the authors'best knowledge there is no report on molecular dynamicssimulations of the reacted EoS of high explosives yet. Thus, inthis paper, we propose a method of molecular dynamics withReaxFF-lg to calculate the EoSs for both unreacted and reactedHMX.
2. Method2.1 Hugoniot EoS
The Hugoniot EoS describes the locus of thermodynamic statesof unreacted high explosives aer a shock passing through. Itcan be derived from the conservation of mass, momentum, andenergy, known as Rankine–Hugoniot conditions:24
r1us ¼ r2(us � u2) conservation of mass (1)
p2 � p1 ¼ r2u2(us � u2) ¼ r1usu2 conservation of momentum (2)
Fig. 1 Configuration of cylinder test.
Table 2 Parameters of the reacted JWL EoS of PBX 9501
A (GPa) B (GPa) r1
Reacted PBX 9501 1668.9 59.69 5.9
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p2u2 ¼ r1us
�1
2u2
2 þ E2 � E1
�conservation of energy (3)
where us is the shock wave speed, r2 and r1 are the densities ofmaterial behind and in front of the shock, u2 is the particlevelocity of the material behind the shock, p2 and p1 are thepressures of thematerial behind and in front of the shock, and E2and E1 are the internal energies per unit mass in the two regions.
From eqn (1) and (2), the following relations can be obtained:
u2
us¼ 1� n2
n1(4)
us ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 � p1
n1 � n2
rn1 (5)
u2 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðp2 � p1Þðn1 � n2Þ
p(6)
E2 � E1 ¼ 1
2ðp2 þ p1Þ
�1
r1� 1
r2
�¼ 1
2ðp2 þ p1Þðn1 � n2Þ (7)
where n2 and n1 are the specic volumes of the material behindand in front of the shock, and can be expressed as:
n ¼ 1
r(8)
In the ignition and growth model of high explosives, theequation of state of the unreacted explosive has a simple form,which can be expressed as follow:4
Punreacted ¼ Ae�r1n þ Be�r2n þ uCVT
n(9)
where A, B, r1, r2 and u are parameters tted based on experi-ment data, CV is the specic heat of the unreacted high explo-sive, T is temperature, n is specic volume as in eqn (8), andpunreacted is the pressure of the compressed solid explosive. Itcan be seen from eqn (9) that temperature is introduced in theexpression, which differs from traditional equations of state forsolid materials.
The parameters of the unreacted JWL EoS of PBX 9501 arelisted in Table 1.25 PBX 9501 is chosen because it consists of 95weight% HMX, 2.5 weight% estane binder, and 2.5 weight%BDNPA/F.
r2 u CV (J kg�1 K�1) T0 (K)
2.1 0.45 1.0 � 106 298
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Fig. 2 Simulation cells and geometry of b-HMX. (a) The geometry ofthe primitive unit cell of b-HMX (b) MD simulation box of b-HMX with64 HMX molecules (1792 atoms). The C, H, O, and N atoms are rep-resented by yellow, aqua, red, and grey balls, respectively.
Table 3 Lattice constants of b-HMX from experiment,27 DFT-D2calculations22
a (A) b (A) c (A) a b g
Experiment27 6.54 11.05 8.70 90 124.30 90DFT-D2 (ref. 22) 6.542 10.842 8.745 90 124.41 90
Fig. 3 Variation of volume (a), total energy (b), temperature (c), and press300 K in MD simulations. Three ensembles (NPT, NVT, NVE) are used se
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2.2 JWL EoS
The expansion of the detonation products of high explosives isconsidered to be an isentropic process. It consists of compli-cated chemical reactions, occurring over very small timeinterval. The cylinder test has long been the principal method toobtain the data needed for tting parameters of the JWL EoS ofexplosive detonation products.26 It consists of detonating acylinder of explosive conned by copper and measuring thevelocity of the expanding copper wall. The standard tube has a1-inch internal diameter, 0.1-inch copper wall, and is 12 incheslong. The conguration of cylinder test is shown in Fig. 1.
In the ignition and growth model, the reacted JWL EoS ofhigh explosives has a similar form as the unreacted JWL EoS:
preacted ¼ Ae�r1n þ Be�r2n þ uCVT
n(10)
preacted is the pressure of the detonation products. The otherparameters have a similar meaning as those in eqn (9).
The parameters of the reacted JWL EoS of PBX 9501 are listedin Table 2.25
ure (d) of b-HMX during the geometry relaxation at the temperature ofquentially to relax the system. Each ensemble takes 50 ps.
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3. Molecular dynamics simulation3.1 Structure and relaxation of b-HMX
We focus on the b-HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine), a typical secondary explosive with chemicalformula of C4H8N8O8. The geometry of the primitive unit cell ofthe b-HMX is shown in Fig. 2(a) with the cell box, which wasobtained from our previous DFT-D2 study.22 The b-HMX is aP21/c monoclinic crystal with space group number 14. Thelattice parameters are listed in Table 3. Replication of theprimitive unit cell of HMX, is shown in Fig. 2(b). We tested threedifferent systems with 4 � 2 � 4, 5 � 3 � 5, and 8 � 5 � 7primitive unit cells, with 1792, 4200, and 15 680 atoms,respectively. We found that the results are converged at thesystem with 1792 atoms. Therefore, we report our results fromthe simulation cells with 1792 atoms as depicted in Fig. 2(b).
The structure of b-HMX at zero temperature was obtainedfrom our previous DFT-D2 (ref. 22) calculations. The system isrstly relaxed using the NPT, NVT, and NVE ensemblessuccessively with ReaxFF-lg. Due to the inadequate consider-ation of van der Waals (vdW) attractions, when using originalReaxFF to relax the solid molecule, the equilibrium volumewill be about 10–15% higher than experiment.17 This disad-vantage is overcome by ReaxFF-lg, which gives sufficient
Fig. 4 Variation of volume (a), total energy (b), temperature (c), and pre
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consideration of vdW interactions in its description of reactiveforce elds. In the structural of relaxation, the time step is setto be 0.25 fs while the simulation time is set to be 50 ps foreach ensemble.
The lattice constants for the unit cell of b-HMX from exper-iment,27 DFT-D2 (ref. 22) studies are listed in Table 3.
The supercell of b-HMX consists of 64 b-HMX molecules,which contains 1792 atoms in total. The density of b-HMXbefore relaxation is 1893.4 kg m�3.
The variation of volume, pressure, temperature, and totalenergy is shown in Fig. 3.
It is well known that b-HMX is very sensitive to uctuationsin temperature. It will transform from b-HMX to d-HMX above436 K.28 Therefore, it is critical to control these temperatureuctuations. It can be seen from Fig. 3(c) that, during therelaxation, the temperature is limited to around 300 K, whichensures no phase transformations during relaxation. Accord-ing to Fig. 3(a), the volume of the supercell uctuates signi-cantly in the NPT stage, and the maximum volume of thesupercell reaches 17 558.66 A3. In the successive NVT and NVEstages, the volume stabilizes at 16 924.73 A3 (or 528.9 A3 perprimitive unit cell). Aer relaxation, the density of b-HMX is1893.4 kg m�3, which is very close to the experimental value(1.838 kg m�3) at ambient conditions. According to Fig. 3(b),
ssure (d) of b-HMX under compression at T ¼ 1000 K,VV0
¼ 0:5.
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Fig. 5 Comparison of the EoS calculated by MD with those obtained from experiment, DFT-D2, and unreacted JWL EoS of PBX 9501. It shouldbe noted that Yoo's experiment was conducted at room temperature and the DFT-D2 simulation was conducted at 0 K in (a–c). The unreactedJWL EoS and EoS obtained by MD are calculated at 300 K, 1000 K, 2000 K in (a–c).
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aer the initial uctuation, the pressure of the system uctu-ates around 0 GPa.
3.2 Unreacted equation of state
The unreacted EoS of b-HMX is obtained through a series ofNVT simulations. NVT is preferred because as mentioned inSection 2, the effect of temperature is introduced in theunreacted JWL EoS of high explosives. Furthermore, accordingto Yoo,29 the pressure–volume relation calculated underhydrostatic conditions should be in good agreement withHugoniot curve. The simulation scheme is as follows: compressthe b-HMX along a-direction of the supercell to a predenedvolume and temperature, thus, for each set of volumes andtemperatures, there is a corresponding pressure. In the imple-mentation of each NVT simulation, the time step is set to be 0.1fs and the total simulation time is set to be 15 ps. The variationof volume, pressure, temperature, and total energy is shown inFig. 4, where the compression direction is a-direction of thesupercell and the predened volume ratio is 0.5 (compress thevolume to 0.5 its original volume) while the predenedtemperature is 1000 K.
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The comparison of the EoS calculated by MD with thoseobtained from experiment,29 DFT-D2,22 and unreacted JWL EoSof PBX 9501 is shown in Fig. 5. It should be noted that theexperiment was performed at room temperature, and DFT-D2was performed at 0 K.
According to Fig. 5(a), the EoS calculated by DFT-D2 is veryclose to experiment29 when volume ratio is greater than 0.72. Itcan also be observed from Fig. 5(a) that when the volume ratio isgreater than 0.8, all three calculated EoSs are close to experi-ment. However, substantial deviation from experiment is foundfor both unreacted JWL EoS and MD EoS when the volume ratiois less than 0.8, which implies that both numerical predictionsfail at high compression. A possible reason for the failingprediction made by MD simulation is that Reaxff-lg may be notsuitable for simulating unreacted process.
According to Fig. 5(a–c), when the volume ratio is greaterthan 0.7, the EoS calculated by MD is greater than the unreactedJWL EoS at all predened temperatures (300 K, 1000 K, 2000 K).However, when the volume ratio is less than 0.7, the pressurecalculated by the unreacted JWL EoS becomes much larger thanthat obtained from MD simulations and others.
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Fig. 6 Influence of temperature on unreacted JWL EoS and EoS calculated by MD.
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The inuence of temperature on unreacted JWL EoS and EoScalculated by MD is shown in Fig. 6. It can be observed fromFig. 6(a) that unreacted JWL EoSs under different temperatureare close to each other, especially when volume ratio is small,which indicates that for unreacted JWL EoS in ignition andgrowth model, the pressure is not sensitive to variation of
Fig. 7 Variation of volume (a), total energy (b), temperature (c), and pre
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temperature. However, the situation is quite different for EoScalculated by MD. According to Fig. 6(b), the pressure of thesupercell is signicantly inuenced by temperature. Whenvolume ratio is greater than 0.65, the pressure at high temper-ature is larger than that at low temperature, while in low volumeratio zone, the situation is opposite.
ssure (d) of b-HMX in NVT expansion, T ¼ 1000 K,VV0
¼ 4:0.
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3.3 JWL equation of state
Similar to unreacted EoS, the JWL equation of state of b-HMXwas also obtained through NVT simulations. Prior to the NVTsimulation, NPHUG is used to compress the system to around64 GPa. NPHUG performs a time integration of the Hugoniot-stat equations of motion developed by Ravelo.30 Aerwards, thesystem expands from the compressed state. The simulationscheme is similar: expand the b-HMX along a specic direction(a-direction of the supercell) to reach a predened volume andtemperature, thus, for each set of volume and temperature,there is a corresponding pressure. The time step is set to be 0.1fs and the total simulation time is set to be 15 ps. The variationof volume, pressure, temperature, and total energy is shown inFig. 7, where the expansion direction is a-direction of thesupercell, the predened volume ratio is 2.621 (expand thevolume to 2.621 its original relaxed volume), and the predenedtemperature is 1000 K.
The comparison of the reacted JWL EoS of PBX 9501 withthat obtained through MD is shown in Fig. 8.
It can be observed from Fig. 8(c) that when the predenedtemperature is set to be 3000 K, the MD result agrees very well
Fig. 8 Comparison of the reacted JWL EoS with the EoS obtained fromrespectively.
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with the reacted JWL EoS. However, when temperature isdecreased, the MD result gradually deviates from the reactedJWL EoS. The lower the temperature is, the greater the MDresult deviates from reacted JWL EoS. It indicates that ReaxFF-lgexcels in describing chemical reactions at high temperature (forexample, at 3000 K, which is close to the temperature of deto-nation products of PBX 9501 right aer explosion) wherechemical reactions occur violently.
Similarly, the inuence of temperature on reacted JWL EoSand EoS calculated by MD is shown in Fig. 9.
It can also be observed from Fig. 9(a) that the three reactedJWL EoSs almost coincide, which indicates that in the ignitionand growth model, temperature has little inuence on theshape of reacted JWL EoS. However, as shown in Fig. 9(b), inMDsimulation, the inuence of temperature on EoS is signicant.When temperature increases, the pressure of the systemincreases correspondingly. Larger amount of increments areobserved at low volume ratio zone than at high volume ratiozone. It seems that in ignition and growth model, the effect oftemperature is not thoroughly considered, and that is probablywhy the reacted JWL EoSs under three different temperaturenearly coincide with each other.
MD. Temperature is set to be 1000 K (a), 2000 K (b), and 3000 K (c)
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Fig. 9 Influence of temperature on reacted JWL EoS and EoS calculated by MD.
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4. Conclusion
In summary, we have studied the unreacted JWL EoS of b-HMXand reacted JWL EoS of PBX 9501 using reactive moleculardynamics simulations with ReaxFF-lg potentials. For theunreacted case, the pressure–volume relation obtained fromMD simulations is compared with experiment, DFT-D2, andunreacted JWL EoS of PBX 9501 in the ignition and growthmodel. It is found that when temperature is low (300 K), the MDresult agrees relatively well with unreacted JWL EoS in ignitionand growth model. However, when temperature is increased,the MD result gradually deviates from unreacted JWL EoS. Inaddition, the pressure of compressed supercell calculated byboth MD and unreacted JWL EoS are higher than that obtainedfrom experiment. It suggests that the unreacted JWL EoS inignition and growth model is not capable to describe the pres-sure of the compressed system well at high compression ratio.Furthermore, the inuence of temperature on EoS is investi-gated. It is found that unreacted JWL EoS is not sensitive to thevariation of temperature, in other words, when temperaturechanges substantially, the unreacted JWL EoS only changes alittle. The situation is quite different for MD simulation. Theunreacted EoS calculated by MD is signicantly inuenced bytemperature.
For the reacted case, it is found that when temperature is ashigh as 3000 K where the chemical reaction is violent (3000 K isclose to the temperature of detonation products of PBX 9501right aer explosion), the EoS calculated by MD agrees very wellwith the reacted JWL EoS in ignition and growth model.However, when temperature becomes lower, the EoS calculatedby MD gradually deviates from the reacted JWL EoS. Further-more, the inuence of temperature on reacted JWL EoS isstudied. It is found that reacted JWL EoS is not sensitive totemperature. When temperature increases from 1000 K to 3000K, the reacted JWL EoS only have insignicant change, oppositeto the situation happening to MD simulations. Therefore, weconclude that in current ignition and growth model, tempera-ture only plays trivial role in EoS.
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Acknowledgements
The authors appreciate the nancial support from the UnitedStates DoD (DTRA): Grant # HDTRA1-13-1-0025. This work wassupported in part by an allocation of computing time from theOhio Supercomputer Center.
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