CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
STEADY FLOW DETONATIONS FROM MOLECULAR DYNAMICS
SIMULATIONS
David R. Swanson and Carter T. White*
University of Nebraska—Lincoln 68508
*Naval Research Laboratory, Washington, DC 20375
Abstract: Results from over-driven piston supported simulations as well as freely detonating simulations
provide strong evidence that the continuum theory concepts of steady flow and self-similarity hold for
atomistic simulations. The latter provides a means to estimate the CJ point without a Hugoniot
determination.
INTRODUCTION
TABLE I, Potential Efcetgy ffettn atod PafratoefceiB
This paper investigates the relationship of
continuum theory to atomistic simulations of
detonating solids. The atomic model of an
energetic diatomic molecular solid used in this
work is based on reactive empirical bond order
potentials, and is capable of supporting a
detonation in two and three dimensions with
properties intrinsic to the material.
Molecular
dynamics
simulations
of
chemically-sustained
shock
waves
require
potentials capable of simultaneously following the
dynamics of thousands of atoms in a rapidly
changing environment, while including the
possibility of exothermic chemical reactions
proceeding along chemically reasonable reaction
paths from cold solid-state reactants to hot gasphase molecular products. In addition, for a
molecular solid (which is typical of many energetic
materials), these potentials must incorporate both
the strong intramolecular forces that bind atoms
into molecules and the weak intermolecular forces
that bind molecules into solids.
The AB model used is detailed in Table 1.
This model reacts according to the chemical
reaction
J { 1 + «4r(r -
jr. -hi))]}
T, -M < T < T, -M -h 1-OA
=1-1.0 atoU: D** = ?.0 eV: jD^* = Df* = 5.0 eV:
1.8: n = ?.T A"1 : r, = 1.0 A: G= 50: in = ?.?5 A~3 :
0.5: <5 = 0.4 A at 6 = 1.0 A: e= 5.0 > 10-3 eV: a = ?.988 A:
= ?.6568eV: J^ = -3.3581 eV A-1: -B>= 13?30eV A--:
= -0.177 ? eV A-3: Qo = -2.510? > 10-° eV.
= 5.79U > 10-3 eV A-1 : Q^ =
= -1 .1031 > 10-4 eV A-5.
2AB -> A2 + B2 + 6 eV
395
and exhibits physically realistic equilibrium and
reaction properties [1,2], although it does not
model any specific real system.
Continuum theory predicts the existence of a
CJ point behind the detonation front. Between the
front and the CJ point is a region in a state of
nonequilibrium steady flow in which the average
values of physical properties do not change with
time. Furthermore, behind the CJ point the length
and time scales associated with dissipative
processes have been lost and the system becomes
self-similar.
(Ic)
These equations relate the undisturbed explosive
lying at rest with pressure P0 = 0 and specific
volume F 0 = l / p 0 to the state behind the
detonation front characterized by a pressure Ph a
specific volume Vh and a particle flow velocity v
with both v and the detonation velocity D
measured in the reference frame of the undisturbed
material. Because P0, V0 are known, the RankineHugoniot relations are a set of three equations with
four unknowns, v, D, P/, and Vj. The first of these
relations determine v in terms of D, PI and F/
leaving two equations with three unknowns. The
first of these remaining equations (Equation Ib)
defines the Rayleigh line while the second
(Equation Ic) defines the Hugoniot curve. The
problem is then formally determined by selecting
the solution to Equations Ib and Ic that
corresponds to the minimum D for an unsupported
detonation. This additional condition is known as
the Chapman-Jouguet hypothesis that was put on a
firmer foundation by Zel'dovich [4].
Because Equation Ic does not involve D, the
Hugoniot curve can be represented by a function of
the form P{ = H(V19VQ,P0) defined by the set
{P}, Fj}, which satisfies Equation 1 c assuming P0
and V0 are known. The final state Hugoniot is one
of the central constructs of the ZND theory of
detonations and can be determined provided the
equation of state, E(P,V) is known. However this
function can be determined to good approximation
from detonation simulations without direct
recourse to the equation of state. This is done by
performing a series of supported simulations with
each corresponding to a different piston velocity.
In these simulations the flow immediately in front
of the piston face soon settles down to a constant
profile moving at the piston velocity and
characterized by Pj and Vj that can be measured
accurately from the simulations by averaging the
results near the piston face in both space and time.
The state of the system in front of the piston is
fully reacted and in thermodynamic equilibrium. In
addition, if the shock profile is steady from the
piston face to the detonation front, then this state
STEADY OVERDRIVEN FLOW
These simulations were performed in two
dimensions, and were carried out with periodic
boundary conditions enforced perpendicular to the
direction of shock propagation to model the infinite
crystal. The model material was taken as semiinfinite in the direction of the shock front
propagation. This was achieved by adding material
in front of the propagating shock wave as
necessary. Also, in all the detonation simulations
the initial temperature of the system was fixed at a
low but nonzero value to allow the formation of a
stable molecular crystal while preventing spurious
soliton solutions. Shock waves were introduced
into the simulations by impacting the crystal's free
edge with a flyer plate. In the piston-supported
simulations, a piston proceeding in the direction of
the shock was inserted into the system at a velocity
that matched the local particle flow velocity. The
piston was modeled by a quadratic potential. Once
the boundary and initial conditions were
established, individual atomic trajectories were
followed by integrating Hamilton's equations of
motion using a Nordsiek predictor-corrector
method [3].
The Rankine-Hugonoit relations may be
written in the form [4]:
D
D
(la)
(Ib)
396
as Pj and Vj exceed their corresponding values at
the CJ point. This is illustrated in Figure 2 where a
series of velocity profiles at different times from a
detonation simulation supported by a piston are
depicted. The fact that these somewhat averaged
profiles when aligned all coincide provides good
evidence that the shock profile is indeed
everywhere steady in this case. On the other hand,
for supported underdriven simulations this is no
longer true. Although in this case the detonation
moves at a steady velocity and the flow in front of
the piston is also steady and in equilibrium, there is
a region behind the front that relaxes to satisfy the
rear boundary conditions and this region changes
with time. Within this region mass, energy, and
momentum accumulate so that the RankineHugoniot relations are no longer satisfied in front
of the piston face. In practice this failure of the
piston method poses no problem because
unsupported simulations can never reach these
states.
Steady flow from the CJ point to the front in
the simulations provides an important verification
of the assumptions of ZND theory. These results
also indicate that behind the CJ point the
simulations should become self-similar because the
scales associated with the dissipative processes and
chemistry have been lost.
will satisfy the Rankine-Hugoniot so that Pj and F/
will lie on the Hugoniot curve. The set {P^^}
defining the Hugoniot curve is then determined by
performing a series of such simulations, each
corresponding to a different piston velocity.
The Hugoniot curve determined in this way is
shown in Figure 1. Calculated points on this curve
are represented as diamonds connected with solid
lines as a guide to the eye. Also shown in Figure 1
is the Rayleigh line (dotted line) determined by the
CJ hypothesis. This hypothesis selects the Rayleigh
line (dotted line) with minimum D and hence
minimum slope originating at the initial state. As
can be seen from Figure 1 the requirement of
minimum slope makes this line tangent to the
Hugoniot curve. The point of tangency is known as
the CJ point. We find that D determined from the
slope of this Rayleigh line agrees with D measured
directly from the simulations within less than a
percent providing strong confirmation of the ZND
theory in general and the CJ hypothesis in
particular for the AB model.
Determining the Hugoniot from supported
piston simulations requires that the detonation
profile from the piston face to the front is steady
for a given piston velocity for only then will the
Rankine-Hugoniot relations be satisfied near the
piston face. We have shown directly from the
simulations that this is an excellent assumption so
long as the detonation is overdriven, that is so long
1.0
0.8
•§ 0.6
^
PL,
0.4
0.2
0.0
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
-300
V/Vo
-250
-200
-150
-100
-50
Distance (Ang.)
Figure 1.
Hugoniot curve calculated from supported
detonation simulations (diamonds). The solid lines connecting
the diamonds are drawn as a guide to the eye. Also shown is
the Rayleigh line (dotted line) determined according to the CJ
hypothesis.
Figure 2: Density profiles from an over-driven simulation
driven by a piston moving at 2.75 km/sec in the reference
frame of the material. Shown are results averaged over 10 ps
in 0.02 ps intervals from 10-20 (solid line), 20 -30 (dashed
line), 30 - 40 (dash-dotted line) and 40 - 50 ps (dotted line).
397
CONCLUSIONS
-40
The method used to determine the system
properties at the CJ point presented in the previous
paragraph relies on the self-similar behavior of the
relaxation following this point while the methods
presented earlier rely on the steady flow of the
detonation prior to this point. The fact that these
complementary approaches yield consistent results
for the CJ point provides strong evidence linking
the simulations to the continuum theory of
detonations.
80
Figure 3. The density (amu/A2) versus x/t (A/ps) at 12
(diamonds) and 20 (dashed line only) ps after initiation.
The horizontal line at 3.15 amu/A2 corresponds to the
density behind the peak where the simulation first
becomes self-similar while the horizontal line at 2.25
amu/A2 corresponds to the density of the unshocked
crystal.
ACKNOWLEDGEMENTS
This work was supported by ONR. DRS
acknowledges the use of the Research Computing
Facility at UNL.
SELF-SIMILARITY
REFERENCES
In Figure 3 we plot the density versus x/t from
an unsupported simulation at 4 (diamonds) and 20
(crosses) ps after initiation. Although there is
considerable scatter in these results, because the
data was collected in 0.1 nm bins without any
smoothing or time averaging, the results at 4 ps do
coincide to good approximation with those at 20 ps
provided the density behind the peak density is less
than approximately 3.15 amu/sq.Ang. However,
the density where the detonation first becomes selfsimilar should be the CJ density, pc/. Thus, we
have directly from Figure 3 that pc/ ~ 3.15
amu/sq.Ang. which yields a normalized specific
volume at the CJ point of
^o
Po
2.25
Pc,
3.15
1.
2.
3.
4.
-0.71.
This corresponds to a distance of about 3 nm
behind the detonation front. Therefore, Figure 3
not only establishes the self-similar character of the
detonation behind the CJ point, but also can be
used to determine its properties without recourse to
the Hugoniot.
398
D. W. Brenner, D. H. Robertson, M. L. Elert, and
C. T. White, Phys. Rev. Lett. 70, 2174 (1993); ibid
76,2202(1996).
C. T. White, D. H. Robertson, M. L. Elert, and D.
W. Brenner, in Macroscopic Simulations of
Complex Hydrodynamic Phenomena, (Eds. M.
Mareschal and B. L. Holian), Plenum Press, New
York (1992) p. 111.
C. W. Gear, Numerical Initial Value Problems in
Ordinary Differential Equations, Prentice-Hall,
Englewood Cliffs (1971).
Ya. B. Zel'dovich and Yu. P. Raizer, Physics of
Shock Waves and High-Temperature Hydrodynamic
Phenomena, Vols. 1 and 2, (Academic Press, New
York, 1966, 1967).