a molecular dynamics study of the early-time mechanical heating in shock-loaded...
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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: [email protected])Author to whom correspondence should be addressed. Electronic mail:
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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