A molecular dynamics study of the early-time mechanical heating in shock-loadedoctahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine-based explosivesYao Long and Jun Chen Citation: Journal of Applied Physics 116, 033516 (2014); doi: 10.1063/1.4890715 View online: http://dx.doi.org/10.1063/1.4890715 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Anisotropic shock sensitivity for β-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine energetic material undercompressive-shear loading from ReaxFF- lg reactive dynamics simulations J. Appl. Phys. 111, 124904 (2012); 10.1063/1.4729114 Initial chemical events in shocked octahydro-1,3,5,7-tetranitro-1,3,5,7- tetrazocine: A new initiationdecomposition mechanism J. Chem. Phys. 136, 044516 (2012); 10.1063/1.3679384 On the low pressure shock initiation of octahydro-1,3,5,7–tetranitro-1,3,5,7-tetrazocine based plastic bondedexplosives J. Appl. Phys. 107, 094906 (2010); 10.1063/1.3407570 Flame spread through cracks of PBX 9501 (a composite octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine-basedexplosive) J. Appl. Phys. 99, 114901 (2006); 10.1063/1.2196219 An estimate of the linear strain rate dependence of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine J. Appl. Phys. 86, 6717 (1999); 10.1063/1.371722

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A molecular dynamics study of the early-time mechanical heatingin shock-loaded octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine-basedexplosives

Yao Long1,a) and Jun Chen1,2,b)

1Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics,P. O. Box 8009, Beijing 100088, China2Center for Applied Physics and Technology, Peking University, Beijing 100871, China

(Received 8 April 2014; accepted 9 July 2014; published online 18 July 2014)

We study the shock-induced hot spot formation mechanism of octahydro-1,3,5,7-tetranitro-1,3,5,7-

tetrazocine-based explosives by molecular dynamics, compare different kinds of desensitizers and

different shock velocities. A set of programs is written to calculate the physical picture of shock

loading. Based on the simulations and analyses, the hot spots are found at the interface and are

heated by plastic work in three ways: the interface intrinsic dissipation, the pore collapse, and the

coating layer deformation. The work/heat transition rate is proved to be increasing with a loading

speed. VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4890715]

I. INTRODUCTION

The shock initiation of plastic bonded explosive (PBX) is

started from the hot spot ignition.1,2 For solid explosive, the

hot spot takes place in micron space scale and nanosecond

time scale, therefore, it is hard to be observed in experiment.

A theoretical investigation of the micro-mechanism about hot

spot formation is helpful to understand the shock initiation,

nonideal detonation, and branched-chain reaction phenomena.

Till now, there are a large amount of experimental and the-

oretical studies aim at the shock initiation behavior: Goddard’s

group reported the classical molecular dynamics (MD) simula-

tions about the hot spot formation process in pentaerythritol tet-

ranitrate via reactive force-field;3,4 Wei’s group reported the

quantum molecular dynamics study of the shock-induced

decomposition mechanism of nitromethane;5 Zhang employed

a friction analysis method to evaluate the desensitizing ability

of planar structural explosive and additives;6–8 Menikoff inves-

tigated the hot spot formation mechanism of solid explosive9,10

and reported a compaction wave profile;11,12 Reaugh et al.implemented a systematic study about the defect evolution,13

hot spot formation,14 and deflagration15 problems; Tarver’s

group reported a set of experiments about the shock initiation

processes;16–18 Cawkwell reported the dislocation nucleation

phenomenon of hexahydro-1,3,5-trinitro-1,3,5-triazine under

high loading;19 Jaramillo simulated the inelastic deformation of

solid explosive;20 and Sewell reported a set of mechanical

properties for energetic materials, such as the elastic constants21

and equation of state.22

In principle, the hot spot is heated by the conventional

plasticity in a molecular solid that just happens to be an ener-

getic material, and the exothermic reactions during a chemical

process. At the early-time under shock loading, the hot spot is

induced by a set of mechanical effects such as the creep dam-

age,23,24 crack friction,25,26 adiabatic shock compression, and

plastic work.27 To investigate them, we simulate the shock

loading processes of complex mixture explosives by molecular

dynamics and write a set of programs to calculate the tempera-

ture field, stress field, particle velocity field, density field, and

energy field. The mechanical heating behavior is analyzed.

The PBX is a composite energetic material consists of

base explosive, desensitizer, plasticizer, and binder. It is

widely used in civil and military engineering28–30 instead of

the conventional explosives. The octahydro-1,3,5,7-tetrani-

tro-1,3,5,7-tetrazocine (HMX) based PBXs are coated by

two kinds of desensitizers: 1,3,5-triamino-2,4,6-trinitroben-

zene (TATB) and graphite, where the HMX has four phases:

a,31 b,32 c,33 d,34 and TATB has just one phase. The b-HMX

is the most stable phase with best performance so is inter-

ested in present work. The crystalline structures of HMX and

TATB are demonstrated in Fig. 1.

The following work consists of four parts: in Sec. II, the

force-field for molecular dynamics simulation is introduced;

in Sec. III, the shock dynamics simulation is implemented;

in Sec. IV, the physical picture of hot spot formation is ana-

lyzed; the conclusion is presented in Sec. V.

II. FORCE-FIELD

The force-field of PBX consists of three sets of poten-

tials to describe the energies of particles, coatings, and inter-

faces. The total energy is written as

FIG. 1. The crystalline structures of (a) b-HMX and (b) TATB.

a)Electronic mail: long_yao@iapcm.ac.cnb)Author to whom correspondence should be addressed. Electronic mail:

jun_chen@iapcm.ac.cn

0021-8979/2014/116(3)/033516/8/$30.00 VC 2014 AIP Publishing LLC116, 033516-1

JOURNAL OF APPLIED PHYSICS 116, 033516 (2014)

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Etotal ¼ Eparticle þ Ecoat þ Einterface: (1)

The force-field of HMX was reported by Smith,35,36 con-

sisted of pair potentials, Coulomb interactions, bond, angle,

dihedral, and improper energies. The particle energy is

Eparticle ¼ Epair þ ECoulomb þ Ebond þ Eangle þ Edihedral

þ Eimproper: (2)

The force-field of TATB was reported by Gee,37 consisted of

pair potentials, Coulomb interactions, bond, angle and dihe-

dral energies. The coating energy is

Ecoat ¼ Epair þ ECoulomb þ Ebond þ Eangle þ Edihedral: (3)

The force-field of graphite was reported by Tersoff38,39 and

Kolmogorov.40 The coating energy is

Ecoat ¼1

2

Xi6¼j

f rijð Þ Aije�kijrij � bijBije

�lijrij

� �; (4)

where ij denotes the atom pair in graphite, rij denotes the

pair distance, f(rij) is a cut-off function, bij is the three-body

interaction function, and Aij, Bij, kij, lij are coefficients. The

force-field of interface was reported in our previous

works41,42 in the form of pair potential. The interfacial inter-

action energy is

Einterface ¼X

ij

uijðrijÞ; (5)

where ij denotes the atom pair across interface and uij

denotes the pair interaction. As a result, a complete force-

field to describe the complex PBX system is obtained, can be

invoked by the LAMMPS program43 for molecular dynamics

simulation.

III. SIMULATION

The simulation models are polycrystal structures in

four types: the HMX ideal model, the HMX pore model,

the TATB-coated HMX, and the graphite-coated HMX.

The pores are placed at the particle boundaries and the coat-

ings are around the particles. The model size is about

400� 60� 1000 A, where the length in y dimension is very

narrow because we are just interested in the physical pic-

ture in xz plane. The periodic boundary condition is applied

in xy dimensions and a vacuum space of 1000 A is placed in

z dimension. Each model consists of eight particles and has

five extra coating layers if necessary. The particle size is

about 200 A, the particle orientations are random chosen,

and the total atom number of one model is about 1 900 000.

The initial models are optimized by molecular

mechanics (MM) and MD via the LAMMPS program.43 The

MM searches the lowest energy configuration of the atomic

system with a force convergence tolerance of 10�8 kcal/mol/

A, and the MD relaxes the atomic system at 300 K by Nose-

Hoover thermostat44–46 with a damping factor of 50 fs and a

time step of 0.5 fs. We run 20 000 MD steps at NPT ensem-

ble and NVT ensemble orderly for relaxation and present the

resulting structures in Fig. 2, where the pores are marked by

circles.

The shock loading is driven by a piston runs from down-

side to upside in four different kinds of speeds: 500 m/s,

1000 m/s, 1500 m/s, and 2000 m/s. It is simulated by MD at

NVE ensemble via the LAMMPS program,43 where the initial

temperature is 300 K and the time step is 0.5 fs. The method

to calculate a set of physical quantities is introduced in

Appendix. The physical pictures of graphite-coated HMX

under shock loading are displayed by the ATOMEYE pro-

gram,47 as shown in Fig. 3. The high energy, high stress, and

high temperature zones at the interfaces represent hot spots,

require detailed analysis.

IV. DISCUSSION

Because the chemical reaction heat is ignored in the

present work, the hot spot is heated by adiabatic shock com-

pression and plastic work. They are discussed separately.

A. Hugoniot curve

The hot spot temperature induced by adiabatic shock

compression is derived from the Hugoniot curve of solid ma-

terial. The Hugoniot equation is

E� E0 ¼1

2Pþ P0ð Þ V0 � Vð Þ; (6)

where E0, P0, and V0 are the initial energy, pressure, and vol-

ume of the system. To solve this equation, the energy/

FIG. 2. The optimized structures of (a)

HMX ideal model, (b) HMX pore

model, (c) TATB-coated HMX, and

(d) graphite-coated HMX. The vacuum

spaces at upside are not displayed.

033516-2 Y. Long and J. Chen J. Appl. Phys. 116, 033516 (2014)

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volume vs. temperature/pressure functions of HMX, TATB,

and graphite are calculated by MD at NPT ensemble for the

temperatures from 300 K to 1000 K and the pressures from

0 GPa to 30 GPa. The results are fitted as

EðP; TÞ ¼X4

i¼0

X4

j¼0

eijPiTj; VðP; TÞ ¼

X4

i¼0

X4

j¼0

�ijPiTj: (7)

The Hugoniot equation of each component is solved numeri-

cally for T0¼ 300 K and P0¼ 0 GPa. The shock velocity (D)

and particle velocity (u) are calculated by the formulas48

D ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P� P0

q0 1� V=V0ð Þ

s; u ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

q0

P� P0ð Þ 1� V

V0

� �s; (8)

where q0 is the initial density. The results are plotted in Fig. 4.

Fig. 4(b) shows that the D-u curve satisfies a linear rela-

tionship for u� 500 m/s; therefore, it can be fitted as

D ¼ cþ ku; (9)

where c is equivalent to the isentropic sound speed and k is a

non-unit coefficient. The fitted parameters are presented in

Table I and are compared with the available experiments.

Next, we obtain the D-u curves of mixture explosives

from the shock dynamics simulation results and fit the c and

k parameters. Substitute Eq. (8) into Eq. (9), the Hugoniot

curve is derived as

P ¼ P0 þq0c2 1� V=V0ð Þ

1� k 1� V=V0ð Þ½ �2(10)

as plotted in Fig. 5. Because the pore density of HMX pore

model is quite larger than actual explosive, therefore, the rel-

evant Hugoniot curve is lower than the experimental result.

However, the other Hugoniot curves are consistent with the

experiments from LASL.49,50

B. Hot spot

Fig. 6 shows the temperature fields of the four simula-

tion models under shock loading, where the high temperature

zones represent hot spots. And then, the maximum tempera-

ture of a simulation model at arbitrary time is denoted as a Tvs. t function, as shown in Fig. 7. For HMX ideal model and

TATB-coated HMX, the temperature is unchanged with time

approximately; but for HMX pore model and graphite-

coated HMX, the T vs. t curve has two peaks, corresponding

FIG. 3. The (a) temperature field, (b) pressure field, (c) particle velocity field, (d) density field, and (e) energy field of graphite-coated HMX at 8.5 ps under

shock loading, where the piston speed is 2000 m/s.

FIG. 4. The Hugoniot curves of HMX,

TATB, and graphite single crystals in

(a) P-V axes and (b) D-u axes, where

the experiments are from LASL shock

Hugoniot data, Ref. 49.

033516-3 Y. Long and J. Chen J. Appl. Phys. 116, 033516 (2014)

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to the times that the shock wave runs across the pores or the

intersections of coatings. Therefore, the hot spot temperature

is defined as the maximum value of the T vs. t curve. It is

increasing with a piston speed, as shown in Fig. 8(a).

There are two mechanisms to heat the hot spot at the

early-time under mechanical loading: the adiabatic shock

compression and the plastic work. The hot spot temperature

induced by the adiabatic shock compression can be derived

from the Hugoniot curve, so is named as the Hugoniot tem-

perature; and the hot spot temperature induced by plastic

work is named as the plastic temperature. Based on the

Hugoniot curves of HMX, TATB, and graphite, the Hugoniot

temperature of each component is plotted in Fig. 8(a).

The calculated Hugoniot temperature is quite lower than

the actual hot spot temperature; therefore, the hot spot is

mainly heated by plastic work. By subtracting the Hugoniot

temperature from the total hot spot temperature, the plastic

temperature is obtained, as shown in Fig. 8(b).

There are three mechanisms to induce the plastic work

at interface under shock loading: the interface intrinsic dissi-

pation, the pore collapse, and the coating layer deformation,

where the interface intrinsic dissipation includes interface

friction effect and wave dispersion effect. The HMX ideal

model just has the interface intrinsic dissipation effect, the

HMX pore model has both the interface intrinsic dissipation

effect and the pore collapse effect, and the TATB/graphite-

TABLE I. The c and k parameters of HMX, TATB, graphite, and mixture

explosives.

c (m/s) k

HMX single crystal 3035 1.7146

TATB single crystal 2277 2.2106

Graphite single crystal 3674 2.5280

This work HMX ideal model 3024 1.6820

HMX pore model 2593 1.7160

TATB-coated HMX 2826 1.7588

Graphite-coated HMX 2447 1.8968

HMX 2901 6 407a, 3070b 2.058 6 0.490a, 1.79b

Expt. TATB 2037a 2.497a

HMX-TATB 2670b 1.83b

aFrom LASL explosive data, Ref. 50.bFrom LASL shock Hugoniot data, Ref. 49.

FIG. 5. The Hugoniot curves of poly-

crystal models in (a) P-V axes and (b)

D-u axes, where the experiments

are from LASL shock Hugoniot data,

Ref. 49.

FIG. 6. The temperature fields of (a)

HMX ideal model, (b) HMX pore

model, (c) TATB-coated HMX, and

(d) graphite-coated HMX at 8.5 ps

under shock loading, where the piston

speed is 2000 m/s.

033516-4 Y. Long and J. Chen J. Appl. Phys. 116, 033516 (2014)

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coated HMX have both the interface intrinsic dissipation

effect and the coating layer deformation effect. A strongly

deformed graphite layer is demonstrated in Fig. 6(d).

Fig. 8(b) shows that the maximum plastic temperature

of HMX ideal model is about 500 K, which represents an

interface intrinsic dissipation effect only. Although the fric-

tion coefficient and wave dispersion rate vary with the inter-

face type, the resulting plastic temperature induced by this

effect should be around 500 K. Therefore, the maximum

plastic temperature of HMX pore model (about 2400 K) is

mainly induced by the pore collapse effect, and the maxi-

mum plastic temperature of graphite-coated HMX (about

1200 K) is mainly induced by the coating layer deformation

effect. The order of the three effects to induce plastic work is

“pore collapse”> “coating layer deformation”> “interface

intrinsic dissipation.”

At last, the desensitizing ability of TATB and graphite

is analyzed. Since the hot spot is strongly heated by the pore

collapse effect, the TATB and graphite coatings can decrease

the intergranular pores for desensitization, while TATB has

better desensitizing ability because it consists of small mole-

cules that can fill a large amount of nano-scale pores.

C. Plastic work

Based on the Hugoniot curves, we obtain the temperature

field and energy field induced by adiabatic shock compression,

denote them as T0ðuðxÞ;TypeðxÞÞ and E0ðuðxÞ;TypeðxÞÞ,

FIG. 7. The T vs. t curves of (a) HMX ideal model, (b) HMX pore model,

(c) TATB-coated HMX, and (d) graphite-coated HMX, where the piston

speed is 2000 m/s.

FIG. 8. (a) The hot spot temperature

vs. piston speed functions and (b) the

hot spot temperature induced by plastic

work only.

FIG. 9. The (a) plastic work, plastic

heat and (b) work/heat transition rate

of HMX pore model and graphite-

coated HMX, where the piston speed is

2000 m/s.

033516-5 Y. Long and J. Chen J. Appl. Phys. 116, 033516 (2014)

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where x is a position vector, u(x) denotes the particle velocity,

and Type(x) denotes the component type. By subtracting the

contribution of adiabatic shock compression from the total

temperature/energy field, the contribution of plastic work to

temperature and energy is

DTðxÞ ¼ TðxÞ � T0ðuðxÞ;TypeðxÞÞ;DEðxÞ ¼ EðxÞ � E0ðuðxÞ;TypeðxÞÞ:

(11)

Therefore, the total plastic work (W) and plastic heat (Q) are

calculated by

W ¼ð

DE xð Þd3x; Q ¼ð

3

2kBDT xð Þn xð Þd3x; (12)

where n(x) denotes the atom count per volume, kB is the

Boltzmann constant, and the integral is over the simulation

model. The rate of work/heat transition is defined as

r ¼ Q

W: (13)

Fig. 9(a) shows the total plastic work/heat vs. time

curves of HMX pore model and graphite-coated HMX.

Although the plastic work and plastic heat increase with the

simulation time, the rate between them is unchanged approx-

imately, as demonstrated in Fig. 9(b). The mean rate over the

whole simulation time (t0) is defined as

hri ¼ 1

t0

ðt0

0

r tð Þdt: (14)

Because the high speed loading leads to higher kinetic

energy vs. potential energy rate, and the plastic heat is trans-

formed from the kinetic energy of material; therefore, the

work/heat transition rate is increasing with the piston speed,

as shown in Fig. 10.

V. CONCLUSION

We simulate the shock dynamics of four explosive mod-

els: the HMX ideal model, the HMX pore model, the TATB-

coated HMX and the graphite-coated HMX, and write a set

of programs to calculate the temperature field, pressure field,

particle velocity field, density field, and energy field. The

Hugoniot curves of complex mixture explosives are obtained

and the hot spot formation mechanism is analyzed. Some

interesting results are presented:

First, the hot spots are found at the interface and are

heated by plastic work. Second, there are three mechanisms to

induce the plastic work: the interface intrinsic dissipation, the

pore collapse, and the coating layer deformation. The order of

them to heat the hot spot is “pore collapse”> “coating layer

deformation”> “interface intrinsic dissipation.” Third, the

plastic work and plastic heat are calculated. The work/heat

transition rate is proved to be increasing with a piston speed.

ACKNOWLEDGMENTS

The authors gratefully acknowledge Professor D. Q.

Wei and Dr S. Liu for supporting HMX models. This work

was supported by the National Natural Science Foundation

of China (Nos. 11004011 and 11172048), Development

Foundation of China Academy of Engineering Physics

(No. 2011A0101001), Defence Industrial Technology

Development Program (No. B1520132013) and the

Foundation of National Key Laboratory.

APPENDIX: THE METHOD TO CALCULATE THEPHYSICAL PICTURE UNDER SHOCK LOADING

To calculate the temperature field, the conventional cell-

averaging method results in a rough picture because it requires a

large cell volume to include a number of atoms so that the aver-

aging is statistically reasonable; therefore, we use an interpola-

tion formula to map the physical quantity from atom to mesh. A

small cell volume is used and a high quality picture is obtained.

To build a mesh, the model size is denoted as

Lx� Ly� Lz and the node number is

Nx ¼Lx

d; Ny ¼

Ly

d; Nz ¼

Lz

d; (A1)

where d is the mesh size, 2 A. The coordinate of a node is

xi ¼ id; yj ¼ jd; zk ¼ kd (A2)

and the volume of one cell is

DV ¼ ðxiþ1 � xiÞðyjþ1 � yjÞðzkþ1 � zkÞ ¼ d3: (A3)

Then, we denote the position of the nth atom as [xn, yn,

zn], assume it is in the ijk cell that satisfies

xi � xn < xiþ1; yj � yn < yjþ1; zk � zn < zkþ1 (A4)

and denote the extensive quantity and intensive quantity per

atom as an and bn. Based on the inversion of interpolation

formula, a set of weights is defined as

FIG. 10. The mean work/heat transition rate vs. piston speed functions.

033516-6 Y. Long and J. Chen J. Appl. Phys. 116, 033516 (2014)

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kijk;n ¼xn � xið Þ yn � yjð Þ zn � zkð Þ

xiþ1 � xið Þ yjþ1 � yjð Þ zkþ1 � zkð Þ

kiþ1j;n ¼xn � xiþ1ð Þ yn � yjð Þ zn � zkð Þ

xi � xiþ1ð Þ yjþ1 � yjð Þ zkþ1 � zkð Þ

kijþ1k;n ¼xn � xið Þ yn � yjþ1ð Þ zn � zkð Þ

xiþ1 � xið Þ yj � yjþ1ð Þ zkþ1 � zkð Þ

kijkþ1;n ¼xn � xið Þ yn � yjð Þ zn � zkþ1ð Þ

xiþ1 � xið Þ yjþ1 � yjð Þ zk � zkþ1ð Þ

kijþ1kþ1;n ¼xn � xið Þ yn � yjþ1ð Þ zn � zkþ1ð Þ

xiþ1 � xið Þ yj � yjþ1ð Þ zk � zkþ1ð Þ

kiþ1jkþ1;n ¼xn � xiþ1ð Þ yn � yjð Þ zn � zkþ1ð Þ

xi � xiþ1ð Þ yjþ1 � yjð Þ zk � zkþ1ð Þ

kiþ1jþ1k;n ¼xn � xiþ1ð Þ yn � yjþ1ð Þ zn � zkð Þ

xi � xiþ1ð Þ yj � yjþ1ð Þ zkþ1 � zkð Þ

kiþ1jþ1kþ1;n ¼xn � xiþ1ð Þ yn � yjþ1ð Þ zn � zkþ1ð Þxi � xiþ1ð Þ yj � yjþ1ð Þ zk � zkþ1ð Þ

: (A5)

The physical quantity mapped from atom to mesh is

aijk ¼X

n

kijk;nan; bijk ¼X

n

kijk;nbn

� �� Xn

kijk;n

� �;

(A6)

where the summation is over all the atoms of the simulation

model.

The method to calculate the temperature, pressure, and

energy is introduced. Because the shock wave is running

along z dimension, the temperature is calculated by using the

atom velocities in x and y dimensions. The formula is

Tn ¼mn

2kB

v2x;n þ v2

y;n

� �; (A7)

where kB is the Boltzmann constant, mn is the mass of the

nth atom, and vx,n and vy,n are the atom velocities in xydimensions. The formula of total stress per atom is51

rn ¼ mnv2z;n þ

1

2

XNp

l¼1

z1Fz;1 þ z2Fz;2ð Þ þ1

2

XNb

l¼1

z1Fz;1 þ z2Fz;2ð Þ

þ 1

3

XNa

l¼1

z1Fz;1 þ z2Fz;2 þ z3Fz;3ð Þ

þ 1

4

XNd

l¼1

z1Fz;1 þ z2Fz;2 þ z3Fz;3 þ z4Fz;4ð Þ

þ 1

4

XNi

l¼1

z1Fz;1 þ z2Fz;2 þ z3Fz;3 þ z4Fz;4ð Þ; (A8)

where the first term in right-hand side denotes the pressure

induced by atom motion, the second to fifth terms denote the

pressures induced by pairwise, bond, angle, dihedral and

improper interactions, and Np, Nb, Na, Nd, and Ni denote the

number of many-body terms. The formula of total energy per

atom is51

en ¼1

2mn v2

x;n þ v2y;n þ v2

z;n

� �þ 1

2

XNp

l¼1

Ep;l þ1

2

XNb

l¼1

Eb;l

þ 1

3

XNa

l¼1

Ea;l þ1

4

XNd

l¼1

Ed;l þ1

4

XNi

l¼1

Ei;l; (A9)

where the second to fifth terms in right-hand side denote the

energies of pairwise, bond, angle, dihedral and improper

interactions.

As a result, the temperature field at the mesh is

Tijk ¼X

n

kijk;nTn

� �� Xn

kijk;n

� �(A10)

the pressure field is

Pijk ¼1

DV

Xn

kijk;nrn

� �� Xn

kijk;n

� �(A11)

the particle velocity field is

uijk ¼X

n

kijk;nvz;n

� �� Xn

kijk;n

� �(A12)

the density field is

qijk ¼1

DV

Xn

kijk;nmn (A13)

and the internal energy field is

Eijk ¼1

DV

Xn

kijk;nen �1

2qijku2

ijk: (A14)

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