Physica A 188 (1992) 357-366
North-Holland
PflYSICA
Dissociative phase transitions from hypervelocity
impacts
C . T . W h i t e , D . H . R o b e r t s o n a n d D.W. B r e n n e r
Naval Research Laboratory, Washington, DC 20375-5000, USA
Molecular dynamics simulations are used to study hypervelocity impacts of an ultrathin
flyer plate with a semi-infinite two-dimensional model diatomic molecular solid. These
hypervelocity impacts are shown to produce a dissociative phase transition from a molecular
to a close-packed solid in the target material. Although this close-packed phase persists for
less than 10 picoseconds and is confined to a domain less than 10 nanometers wide it
nevertheless behaves in a manner consistent with continuum theory.
1. Introduction
A wide variety of laboratory methods have been developed to launch flyer
plates, microprojectiles, pellets, and fragments at hypervelocities. These methods include the use of: fiber-coupled optical pulses to accelerate 5 ~m thick A1
targets to velocities in excess of 3 k m / s [1]; high-explosives that can launch
0 . 1 5 m m thick stainless-steel plates at velocities between 6 to 7 k m / s [2];
plasma accelerators that launch 300 micron diameter glass beads at velocities
up to 10 k m / s [3]; multistage gas-dynamic launchers that can accelerate 1.0 m m
thick Ti alloy plates to velocities over 10 k m / s [4]; and electric guns that launch
0.3 m m thick plastic plates at velocities in excess of 15 k m / s [5].
These hypervelocity p r o j e c t i l e s - w h e n impacted on stationary t a r g e t s - can
be used to study a broad range of basic and applied problems. Examples
e n c o m p a s s studies of the behavior of matter under extreme conditions[6],
simulations of m e t e o r impact - including m a n - m a d e space debris - on satellite
shields (for recent work, see e.g. papers, presented in ref. [7]), investigations
of very high pressure phase transitions [8], studies of the conditions necessary
to initiate detonations [9], and the exploration of potential high pressure
processing routes to novel materials such as "hexagonal d i a m o n d " [10] (lonsdaleite) originally discovered in granular form in meteorites.
Because of the short times involved, molecular dynamics ( M D ) simulations
provide a potentially powerful probe of processes occurring during the very
early stages of hypervelocity impacts. In addition, these methods can be used
0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
358
C.T. White et al. / Phase transitions from hypervelocity impacts
to directly simulate the hypervelocity impact of nanometer particles at surfaces
and hence, for example, to study the pitting and corrosive effects of collisions
of such particles with orbital vehicles. However, with the exception of Holian's
studies of a sphere of 683 particles striking a rectangular plate containing 8000
atoms [11], little attention has been given to using MD simulations to study
hypervelocity impacts.
In this paper we report results of a series of MD simulations of the
hypervelocity impact of a nanometer thick plate with a model two-dimensional
diatomic molecular solid. For the range of hypervelocity impacts studied
(10-20 km/s) even this ultrathin plate is found to have sufficient momentum to
induce a polymorphic (dissociative) phase transition from a molecular to a
close-packed solid in the target material. Although the close-packed phase
persists for less than 10 picoseconds and is confined to a domain less than 10
nanometers wide, it behaves in a manner consistent with continuum theory.
We describe the model used in these studies in the next section. Then in
section 3 the results are reported and discussed. Finally, section 4 provides a
summary and concluding remarks.
2. Model
The 2D model used to obtain the results reported in the next section has
been introduced in an earlier work [12]. This model is based on empirical bond
order potentials patterned after those used by Tersoff [13] to describe Si but
tailored here to treat a diatomic molecular solid. Within this approach the total
potential energy of a collection of N identical atoms is represented as
N
N
V= ~'~ ~', {fc(rij) [VR(rij ) - BiYA(rO)] + V,,dw(rij)},
i
(1)
j>i
where the parameters and functions used in this expression are given in table I.
The molecular bonding portion of the potential consists of a repulsive, VR(r ),
and attractive, VA(r), term, both modeled by exponentials, while a LennardJones potential, Vvdw, is used to describe the weak long-ranged van der Waals
interaction. This Lennard-Jones potential is truncated not only at large distances but also at bonding separations to allow for covalent bonding. The bond
order function, /}ij ~ l(B# + Bii), entering eq. (1) introduces many-body effects into the potential by modifying Va(r ) according to the local bonding
environment. These many-body effects arise from the hidden electronic degrees of freedom that are not treated explicitly in the model. For an isolated
diatomic molecule/?~ is unity and the potential reduces to a generalized Morse
function familiar in the description of diatomic bonding (see e.g. ref. [14]). For
359
C.T. White et al. / Phase transitions from hypervelocity impacts
Table I
T h e c o m p o n e n t s and p a r a m e t e r s used in eq. (1). In the simulations each atom was a s s u m e d to
have the m a s s of N.
VR(r ) =
exp[--a~
Do = 5.0 eV
S : 1.8
(r -- re) ]
a =2.7A
VA(F ) = S D ~ 1 e x p [ - a 2X/27S (r - re) ]
1
r~ = 1
G=5.0
Z fc(%)exp[m(%-r,k)] ) "
B#=(I+G
f~(r)=
/1,
k~.j
~{l+cos[~r(r-2)]}
[0,
m=2.25,~-1
n = 0.5
,
2~<r<3
3 ~< r
~ =5.0× 10-3eV
cr = 2.988 A
0,
r < 1.75
P0 = 0.4727 eV ~
Po+r[P~+r(P2+rP3)],
1.75 ~< r < 2 . 9 1
Pl = - 0 . 6 9 9 6 e V A
2.91 ~< r < 7.32
P2 = 0.3364 e V A
7 . 3 2 ~< r
P3 = - 0 . 0 5 2 0 eV ,~
Vvdw(r) = / 4~[(tr/r)~2 - ( g / r ) 6 ] '
~0,
'
more highly coordinated structures/~ij is no longer unity, but rather decreases
with the increasing number and strength of competing bonds the atoms i and j
form. This decrease in/~ij reflects the finite number of valence electrons these
atoms have available for bonding. If the coupling G entering /~ij defined in
table I is small, then the strength of the bond between these two atoms is not
rapidly reduced by these other competing bonds and the potential will favor
highly coordinated metallic systems. However, if G is large, then the potential
will favor low-coordinated structures with a few strong bonds as in a molecular
solid.
Herein G is chosen to favor a valence of one, insuring that without the van
der Waals bonding term, Vvdw, the ground state at zero temperature and
pressure is a collection of 1N independent diatomic molecules. For the
parameters given in table I, each of these diatomic molecules has a binding
energy of 5 eV and a vibrational frequency of 1682 cm -1 at an equilibrium bond
distance of 1.0 ,~, all similar to molecular oxygen. Inclusion of Vvdw causes this
system to condense into a diatomic molecular solid which in two dimensions
has a crystalline binding energy of 0.04 eV per molecule, a distance of closest
approach between atoms in nearest-neighbor molecules of 3.3 ,~, and a solidstate speed of sound of 1 . 9 k m / s , all well within physical norms. Further
properties of this m o d e l - i n c l u d i n g its shock H u g o n i o t - a r e presented elsewhere [12].
360
C.T. White et al. / Phase transitions from hypervelocity impacts
3. Results and discussion
Hypervelocity impact simulations are initiated by slamming a thin flyer plate
into the edge of the semi-infinite model diatomic molecular crystal initially at
rest. Both the flyer plate and the molecular crystal are initially taken to have
near-zero temperature and pressure. The dynamics of the system resulting
from this hypervelocity impact is then studied by integrating Hamiltonian's
equations of motion using a Nordsieck predictor-corrector method [15].
During these simulations periodic boundary conditions are imposed perpendicular to the direction of shock propagation. A series of studies are carried
out corresponding to different flyer plate velocities ranging from 10 to 20 km/s.
In all these simulations the flyer plate is formed from 16 layers of the molecular
crystal.
A sequence of snapshots for the simulation begun with the 15 km/s velocity
plate is shown in fig. 1. At 0.8 ps (fig. la) only a single compressional shock
wave is visible. Across this shock front the diatomic molecular solid (DMS)
transforms directly to a close-packed solid (CPS). Although the density of this
CPS, Pces, is approximately 2.4 times larger than the density of the DMS,
PDMS, the average nearest-neighbor interatomic separation in this phase, dNN,
is actually 20% larger due to an increase in the interatomic coordination. These
properties of the CPS make this DMS to CPS transition similar to the
dissociative transition reported in diatomic molecular solids of iodine [16] and
bromine [17]. This transition occurs at 21 GPa in I 2 and 81 GPa in Br 2 leading
to an increase in the overall relative density, PCes/PDMs, of 1.7 for 12 and 2.3
for Br2, and an increase in KNNof around 15% for both materials.
At 1.6 ps (fig. lb) a second leading compressional shock wave has appeared
in the 15 km/s simulation. Across this new front the diatomic molecules are
compressed and rotated from their initial positions in the undisturbed crystal
but retain their molecular identity. Once this second shock front is clearly
visible it rapidly separates from the dissosciative front as can be seen by
comparing fig. lb. to fig. lc. Although both the compressional and dissociative
shock fronts propagate at velocities in excess of the speed of sound in the
undisturbed crystal, the velocity of the dissociative front rapidly decreases from
more than 1 2 k m / s at 0.5ps to close to 3 km/s at 2.4ps. In contrast, the
velocity of the leading compressional shock remains pinned close to 12 km/s so
long as the dissociative front exists, only beginning to gradually slow after the
CPS disappears. As the dissociative front slows the accompanying dissociative
zone also narrows as this region is consumed by the rarefaction wave from
behind (fig. 1a-d). Figs. 1 and 2 show that as the particle flow velocity, Upces,
behind the dissociative front approaches 3 km/s (fig. 2) this front also begins to
smear with the material behind beginning to enter the mixed phase region by
C.T. White et al. / Phase transitions from hypervelocity impacts
~
Q
~e~mQ~
, @ t,~.,.,~¢.~.o.,.l'~o,..oo...o,..........o.
* ~.~.*...
(a)
~*.-o~O=
%,,.'~.~~~l~J~.~~',~o,~?,,,
" • ...'~'...".
~ . ' ~ . . . n L , ' ~ ' ~ ' ~ ~
. , , "~.~ ".• " s~4e
. ~.
¢ ~~ -~ "•~4~
u l ~• •:~.¢
. . ~. ~~
~w~cmm4
~ v ~ - - ~
,..,
361
~ ,.
~"
'_~.'-.--,~~..
"
"~
~
~
-
~~
"~'.
L~;~'~t~
l T ~
~
~•
. ~ ¢ ~ t . ~ ~ - " , ~
Fig. 1. Snapshots of the simulations begun with the 15 k m / s flyer plate at: (a) 0.8; (b) 1.6; (c) 2.4;
(d) 3.2; and (e) 4.0 ps after the impact. The shock waves are propagating from left to right.
2.4 ps (fig. lc), which is clearly present at 3.2 ps (fig. ld) just prior to the
complete disappearance of the dissociative zone.
A dissociative phase together with an accompanying split shock wave such as
shown in fig. lb is also present in all the other simulations begun with flyer
plate velocities between 10 and 20 km/s. However, the induction time for the
appearance of the leading compressional wave varies, with this wave appearing
almost immediately in the 10 km/s simulation but taking progressively longer
to appear as the initial impact plate velocity is increased. Although the
appearance of the leading compressional wave is delayed for higher impact
plate velocities, once formed this front always exhibits a velocity close to but
slightly less than 12km/s, provided the dissociative region remains well
362
C.T. White et al. / Phase transitions from hypervelocity impacts
,a
°,-,
"'."~.~....:
•
"::2.:.
•
2
....
0
I ....
0.5
I ....
I ....
1
1.5
..,.,.*.
i ....
2
2.5
Time (ps)
Fig. 2. Scatter plot of UpcPs 3 . 0 A b e h i n d the dissociative front for the hypervelocity impact
s i m u l a t i o n b e g u n with the 15 k m / s flyer plate.
defined. In addition, in all simulations, only after the particle flow velocity
behind the dissociative front slows to near 3 km/s, does the dissociative zone
begin to lose its identity. These general properties of these hypervelocity
impact simulations as well as the explicit behavior shown in fig. 1 are all
consistent with continuum theory as we now show.
More than 35 years ago the first shock-induced polymorphic phase transition
was observed [18]. An important property of these phase transitions- predicted from continuum theory [8, 19, 20] and observed experimentally [19] - is the
associated compressional shock wave splitting that may occur. If split compressional shock waves are present, then continuum theory predicts [20] that the
leading shock front starts the material flowing, bringing it to the point of
transition, while the transition occurs across the second compressional shock
front. A split shock wave with the transition occurring across the second front
is just what is observed in figs. lb and lc.
The continuum theory of planar shock waves also predicts [20] that a shock
wave propagates into a medium (assumed at rest and characterized by a
pressure P0 and a specific volume V0) with a velocity D given by
D=(1-V/Vo)
lUp,
(2)
where the particle flow velocity behind this shock front, Up, is given by
Up = V ( v o -
v)(p - po),
(3)
C.T. White et al. / Phase transitions from hypervelocity impacts
363
with V the specific volume and P the pressure of the shocked material. These
results are derived from the R a n k i n e - H u g o n i o t relations (see e.g. ref. [21]) by
assuming that these relations can be applied as jump boundary conditions
across the shock front. In view of the sharp shock fronts depicted in fig. 1, this
assumption should be quite good. Therefore, because the leading shock front
carries the pressure, PT, and specific volume, VT, of the phase transition, its
velocity from eqs. (2) and (3) is given by D , = V o ~ / ( P T - P o ) / ( V o - V x ) , and
hence is predicted as pinned at a constant value determined by the properties
of the phase transition and the undisturbed molecular crystal. In addition, the
second front can only exist as a shock front so long as the particle flow velocity
behind this front exceeds the particle flow velocity in the leading compressional
zone, Up1. But from eq. (3), u m = ~/(V~ - V T ) ( P T - Po) and hence Up, is also a
constant determined by the properties of the phase transition and the initial
state. Continuum theory therefore explains why when two shock waves are
present in the system - regardless of the impact plate velocity - the first shock
is always observed to propagate at a near constant velocity D,, while the
second always begins to fragment as the particle velocity behind this front
approaches a fixed value u . Elsewhere we have shown [12] that for the
present model: VT = 4.5 Ag/atom; P1 -- 0.813 eV/Ag; V, -- 6.14 A ' / a t o m ; and
P0 ~0.00eV/A2; results which imply that D l = 11.5 km/s and Up, ~ 3 . 0 k m / s .
These predictions differ no more than 5% from the corresponding nearconstant values of D 1 and Up, observed in all these flyer plate simulations
confirming the consistency of the continuum interpretation.
These results are also consistent with the increasingly delayed appearance of
the leading compressional wave observed with increasing flyer impact plate
velocities. Specifically, so long as the velocity of the close-packed front Dcp s
exceeds D,, only a single shock wave will be present with the phase transition
occurring across this front. However, eq. (2) implies that as the particle flow
velocity in the close-packed region, Upc~s decreases, so too will D(.ps, to the
point that Dcp s = D~. The leading compressional front will then be emitted
and the shock waves will split as depicted in fig. lb. Therefore, consistent with
the data, the time it takes for the shock waves to split should lengthen with
increasing initial flyer plate velocities, because these higher initial velocities will
result in higher earlier values of Upcps.
Although the time it takes for the shock waves to split will increase with
increasing flyer plate velocities, the condition Dcp s = D1 together with eqs. (2)
and (3) imply that when the splitting first occurs Upc~s will always be given by
u 2 = (1 - V c e s / V o ) D , ,
(4)
where Vcp s is the volume in the close-packed region. To arrive at this result the
364
C.T. White et al. / Phase transitions from hypervelocity impacts
close-packed region was assumed effectively incompressible, which is consistent with the model Hugoniot [12]. This Hugoniot also shows that
V c p s ~ 2 . 3 5 A2/atom and hence eq. (4) implies that u 2 ~ 7 . 1 km/s.
To test eq. (4), we have calculated Upcps as a function of time at ~3.0 A
behind the close-packed front averaged over a 3 . 0 ~ wide strip, for the
simulation begun with the 15 km/s flyer plate impact. The results are shown in
fig. 2 as a scatter plot. This data suggest that from 0.5 to 1.5 p s , Upcps(t ) =
a t + b , where the coefficients a and b are determined from a least squares fit of
this data and are given by: a = - 2 . 8 km/(ps s); and b = 8.2 km/s. Next, using
this result and assuming eq. (4) is valid, shows that if the shock waves split at
time to, then their separation Xs(t ) at a later time t is given by: Xs(t ) = [a(Vcps/
V0 - 1) ~][(t- t0)2/2], which implies that t o = t - [ 2 ( X s / a ) ( V c p s / V o - 1)] 1/2.
Using this expression for t 0 and the observed Xs(t ) from 0.5 to 1.5 ps, we then
calculate a range of to's and average to obtain {o ~ 0 . 5 1 ps #1 with a 95%
confidence interval of -+0.04 ps. Finally, substituting i 0 into the least squares fit
relation for Upcps(t ) yields u 2 ~ 6 . 8 k m / s , which is close to the 7 . 1 k m / s
obtained before. This level of agreement further confirms the consistency of
the simulated results with the continuum theory.
4. Summary
In conclusion, in this paper we have reported the results of a series of MD
simulations of hypervelocity impacts of an ultra-thin plate with a model
diatomic molecular solid. At these high impact velocities even this ultra-thin
plate is found able to induce a dissociative phase transition to a close-packed
solid with accompanying complex behavior such as shock wave splitting.
Although this close-packed phase persists for less than 10 ps and is confined to
a domain less than 10 nm wide it still behaves in a manner consistent with
continuum theory.
These simulations also show that hypervelocity impacts sufficient to induce a
local dissociative transition should also be capable of causing a good deal of
chemistry through rearrangement of the atoms in the dissociative zone. Although not readily apparent in fig. 2, because only one type of atom is present
and hence only transfer reactions are allowed, this rearrangement leads to the
formation of product molecules in the rarefaction region behind the dissocia~ A l t h o u g h this result implies that the leading compressional wave is actually already present in
fig. la, it also shows at 0.8 ps this front is still too near to the dissociative front to be seen with the
resolution of fig. la.
c.T. White et al. / Phase transitions from hypervelocity impacts
365
tive f r o n t that are c o m p o s e d of a t o m s which were n o t m a t e s in e i t h e r the target
m a t e r i a l or the flyer plate. T h e q u e s t i o n of c h e m i s t r y i n d u c e d by h y p e r v e l o c i t y
i m p a c t s - i n c l u d i n g the r e l a t i o n s h i p of this issue to q u e s t i o n s i n v o l v i n g the
b e g i n n i n g of d e t o n a t i o n - i s c u r r e n t l y u n d e r f u r t h e r i n v e s t i g a t i o n u s i n g m o r e
c o m p l e x m o d e l s d e s c r i b i n g a m a t e r i a l with m o r e t h a n o n e type of a t o m .
Acknowledgments
This w o r k was s u p p o r t in part by the Office of Naval R e s e a r c h ( O N R )
t h r o u g h the N a v a l R e s e a r c h L a b o r a t o r y a n d t h r o u g h the O N R Physics Division. C o m p u t a t i o n a l s u p p o r t was p r o v i d e d in part by a g r a n t of c o m p u t e r
r e s o u r c e s from the N a v a l R e s e a r c h L a b o r a t o r y . O n e of us ( D H R ) acknowledges a N R C / N R L P o s t d o c t o r a l R e s e a r c h Associateship.
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