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
NUMERICAL SIMULATION OF THE VACANCY DIFFUSION IN
SHOCKED CRYSTALS
Yuri Skryla) and Maija M. Kukljab)
a
' Institute of Mathematics and Computer Science, University of Latvia, Latvia, [email protected]
'Michigan Technological University, MI, 49931, USA, [email protected]
Abstract. A time-dependant lattice response for the vacancy diffusion in a strain field induced in a
solid by the strong impact/shock wave is simulated by means of the Fickian diffusion model for a
one-dimensional case. A thermodynamic analysis for the probability of hot-spot formations in the
regions of the vacancy super-saturation is performed. An analytical solution for the structure of the
impact wave progressing across the solid is obtained.
impact wave. In other words, we consider a
dynamical process of interaction between external
excitation of the crystal (shock or impact wave) and
pre-existed lattice defects (vacancies). A new
method is to be developed to simulate and predict
the time-dependent crystal response to the impact
wave excitation and interplay between lattice
defects and sensitivity of high explosive solids to
detonation initiation.
INTRODUCTION
Although there are many significant theoretical
and experimental advances in shock wave physics
and chemistry for last two decades, a reliable
mechanism and clear understanding of the
mechanical impact to detonation energy transfer in
high explosive solids is not yet established [1],
There is a variety of different models for hot-spot
formations and many different views on the
development of the detonation process. However,
the complete theory of detonation initiation in a
generally agreed upon form does not exists.
This investigation is an attempt to describe and
analyze a high explosive (HE) crystal excited by
the mechanical impact leading to detonation. As
was shown in a recent study, the energy barrier for
the critical chemical bond decomposition is reduced
for any RDX (C3H6N6O6) molecule near a free
surface [2], and the electronic structure of defects
plays a crucial role in initiation of chemistry in
explosive materials [3]. We will show how simplest
defects, such as molecular vacancies and their
diffusion, are affected by the impact wave
propagating across the crystal. A simple and
demonstrative method used here displays a
possibility of hot-spot formation in the propagating
THE STRUCTURE OF AN IMPACT WAVE
All necessary parameters of the impact wave for
solids can be derived from conservation laws as for
gases and liquids [4]. The only difference is in the
equation of state and in an appearance of as, stress
component, in equations for the impulse and the
energy. Material strength as describes a solid and its
form depends on a specific model of the solid
chosen. An elastic-plastic model is often used to
simulate impact/detonation waves [5,6]. In this
study, we will use only an elastic model because
only that part of the impact wave structure where
the model of plastic flow may not be working is
considered.
599
The conservation laws of mass, impulse, and
energy are valid for the impact wave front. Let us
write these equations for one-dimensional problem
in Euler coordinates [7]:
,
dt
AN ANALYTICAL SOLUTION FOR AN
IMPACT WAVE STRUCTURE
(1)
dx
dpv
dt
form for the HMX crystal and/or for another HE
solid with the same EOS.
Let us assume that the impact wave moves with
the constant speed D through a crystal with known
parameters, pQ, Po, and v0 in the positive direction.
If the profile of the crystalline parameter changes is
established, one can obtain the following
distribution function:
(2)
dx '
1 pv 2}|+__|pe
9 ( v +1
_|pe + _
_p v3} | =d<7V
__
(3)
where p is the density; v is the velocity, e is the
internal energy; <7 is the Cauchy stress; jc is the
spatial coordinate; t is the time. It should be added
here that there is a well-known relation between a
strain rate£ and the velocity v for solid states:
£=|.
v',
(11)
(12)
PI
(4)
(14)
The Cauchy stress, a is
where p is the hydrostatic pressure. Material
strength crs and the viscous stress av can be
presented in the following form [5]:
as = 2Ge
(6)
crv = 2r]£
(7)
where G is the shear modulus and 77 is shear
viscosity.
The equation of state (EOS) for the pressure as a
function of the density and the internal energy has
been determined for the HMX (C6H6N6O6)
where H is the front's width:
16T7P!
H=-
The density of the compressed matter p\ behind
the wave front can be derived either by fixing the
pressure p\ or the speed of the impact wave D. In
this study, we will use an amplitude of the impact
wave pi as a measure of the external perturbation
of the crystal. In this case, a square of the wave
speed is to be written as:
crystal as [8]:
(8)
Po(
(9)
where
-1
F(/i)=-
(18)
(10)
where p0 is an initial density; C is the speed of
sound at ambient conditions; % is the Gruneisen
coefficient and 5 is some EOS parameter. Function
/(|LI) is valid for the compression only. Once these
equations are solved, one can find all the parameters
for the impact wave structure, such as a speed and
(19)
In order to get the density of the compressed
matter behind the impact wave front for the
moderate pressures, one needs to solve a quadratic
equation the root of which is represented as:
width of the wave front, a density distribution, and
also pressure and temperature maps. We will show
below a solution of these equations in an analytical
where
(20)
(21)
600
(22)
NUMERICAL RESULTS AND DISCUSSION
c = -p~.
(23)
In eqs. (17)-(23) variables p+= p\ + pQ and
p~= pi - PQ have been used. This solution can be
used if pi - pQ<2G(l +S ~*).
Knowing the information regarding a strain
distribution in the impact wave structure, one can
determine an effect of the impact wave on the
redistribution of the vacancies by numerical
solution of the diffusion equation. An initial
vacancy distribution was considered as a uniform
distribution and was equal to 1016 cm"3 [11]. The
amplitude of the impact wave was chosen out of the
range from 8 to 12 GPa. Data used for calculations
of the wave structure in HMX are presented in
Table 1 [5]. The diffusion constants of vacancies
corresponding to a typical organic crystal (£"0=1.85
eV and D0=7 1011 m2 c"1 for antracene) were taken
from [11].
A typical response of the vacancy system to
impact wave progressing across the solid is shown
in Fig. 1. The concentration profiles C along with
the strains (CTS,<7V) and pressure/? are shown by solid
lines at 12 GPa at the time of 10~7 c. For
comparison, concentration profiles at 8 and 10 GPa
are shown at the same moment in time. Parameters
of the impact wave are listed in Table 2.
It is clearly seen that the impact wave entering
the crystal gathers the vacancies and pulls them
along, forming a super-saturated zone, where the
vacancy concentration is high and can exceed the
initial concentration by an order of magnitude
(distribution 3). The super-saturated zone is not
large and is a small fraction of the wave width. The
level of the vacancy saturation is a strong function
of pressure. It is seen from the Table 2 that the
temperature is strongly dependent on the pressure
and reaches about 1181Katl2 GPa. Therefore, the
temperature is the major factor, which governs
redistribution of vacancies.
THE VACANCY DIFFUSION IN A STRAIN
FIELD
An impact wave progressing across a crystal
creates a region of elevated pressure and
temperature. Let us assume that changes of the
pressure and the temperature occur only in the wave
region of H width and the gradients of these values
are large enough. Naturally, these gradients will
affect different defects of the crystal causing their
redistribution. We consider redistribution of
molecular vacancies, which initially was assumed to
be uniform. The similar effect of the impact wave
on the dislocation redistribution will be reported in
future communications.
By using the thermodynamic approximation the
system of equations for the vacancy motions in the
external strain field can be written as following [9]:
dx[
dx
kB\T
(25)
kbT
is the diffusion coefficient; C is the vacancy
concentration; CD is the elastic energy per molecule;
kB is Boltzman's constant and Tis the temperature.
The elastic energy per molecule, or per vacancy,
co can be recalculated from the formula for the
elastic energy density (assuming the molecular
density Nm of the crystal is known)
where
TABLE 1. Material's parameters used in calculations.
(26)
2YNn
Here, Y is Young module and <JS is to be found from
the general solution of the eqs. (l)-(3) system.
The method of exponential difference has been
used here [10]. This method allows us to obtain a
rather good solution on relatively rough grids in
cases of high gradients of the strain field. By the
co = -
Properties
Density, p0
Specific heat, CD
Shear modulus, G
Young module, Y
Shear viscosity, r|
Yo
sweep method this equation has been solved for the
1-D case.
S
C
601
HMX
1.9
1.28
10
26
300
1.1
2.058
2.9
Units
gm cm"3
Jgm- 1 ^ 1
Gpa
Gpa
Poise
kmc"1
The essential conclusion here is that the
described model illustrates a possibility of the
dynamical chemical active zone (hot-spot)
formation due to the impact wave progressing
across the crystal. The impact wave interacts with
the lattice defects (vacancies) creating the vacancy
super-saturation zone, which is moving further with
the impact wave. It was recently found by means of
quantum-chemical calculations that this region is
energetically more favorable for the initiation of
chemistry [2] than other parts of the solid. Once the
exothermic chemical reaction in the hot-spot region
is triggered, this will result in an appearance of the
self-sustained regime, and therefore, an impact
wave may be transferred to detonation.
TABLE 2. Parameters of the impact wave.
N
1
2
3
Pi
GPa
8
10
12
Pi
gem"3
2.40
2.48
2.55
Vi\ D
kmc 1
1.16; 5.54
1.34; 5.77
1.50; 5.99
T!
K
826
1001
1181
H
|im
196
170
150
The width of the wave front turned out to be
rather wide, several tenth of mm. The possible
reason for this is a high viscosity characteristic for
solids. The super-saturated zone is about 100 um,
which is large enough to estimate the void
formation probability based on thermodynamics.
Therefore, it is interesting enough to find
characteristic sizes of hot-spots formed by vacancy
nucleation, which can appear in such supersaturated vacancy solid solutions.
As is well known from physics of crystal
growth from gas phase, the speed of nucleus
formation is an exponential function of the order of
super-saturation, C/C0, and is of an explosive
character when it approaches 4.2 [13]. Considering
a magnitude of super-saturation, C/Co greater then
4, we found that a radius of vacancy aggregates
falls into the range l-4xlO"6cm depending on the
super-saturation magnitude. This is consistent (or
even somewhat larger) with the critical radius for
the irreversible chemical reaction to occur, 8x10~7
cm [14]. Voids of this size may be formed with
pressures exceeding 10 GPa (see Fig. 1).
6.8
ACKNOWLEDGEMENT
This work has been supported in part by the US
Office of Naval Research. MMK is very grateful to
D.R.Wolff for his encouragement and support.
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1.0
FIGURE 1. Vacancy concentration (right axis) and Cauchy
stress components (left axis) distributions at 10~7 c after shock
wave loading.
602