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
SIMULATION OF THE EFFECTS OF SHOCK STRESS AND
ELECTRICAL FIELD STRENGTH ON SHOCK-INDUCED DEPOLING
OF NORMALLY POLED PZT 95/5
S. T. Montgomery, R. M. Brannon, J. Robbins, R. E. Setchell, and D. H. Zeuch
Sandia National Laboratories, Albuquerque, NM 87185
Abstract. Shock-induced depoling of the ferroelectric ceramic PZT 95/5 is utilized in pulsed power
applications. Experiments to examine the shock response of normally poled PZT 95/5 under uniaxial
strain conditions show that depoling kinetics, as reflected in current generation through an external
circuit, is inhibited by both decreasing the shock pressure and increasing the electric field within the
ceramic. A model to describe the response of the ferroelectric ceramic has been developed and
implemented into simulation codes. Measured currents in the external circuit and transmitted
waveforms at a window interface have been compared with results from simulations for experiments
with shock pressures varying from 0.6 to 4.6 GPa and electric fields varying from 0.3 to 37 kV/cm.
INTRODUCTION
constant speed, allowing the charge previously
bound by the remanent polarization to flow through
an electric load connected to the ceramic.
A more detailed description of the FE-AFE
phase transformation, based on a generalized free
energy, dependent on order parameters describing
the spontaneous polarization and its ordering in the
crystal lattice, was given by Montgomery (2). This
description of the transformation was incorporated
into a response model for unpoled ceramic that
included the change in specific volume experienced
during the phase transformation and the kinetics of
the transformation through a rate equation for the
change of polarization ordering in the lattice. The
model was incorporated into the one-dimensional
wave propagation code WONDY (3) to study the
material responses under uniaxial strain loading.
Interest in establishing an improved capability
for numerically simulating pulsed power devices
and piezoelectric gauges motivated development of
three-dimensional wave propagation codes that can
treat electromechanical responses of materials. The
finite element Lagrangian code SUBWAY (4) and
arbitrary Lagrangian-Eulerian code ALEGRA (5)
currently provide capability to simulate complex
The lead-zirconate-titanate ceramic having a
Zr:Ti ratio of 95:5 and modified with 2% niobium,
subsequently referred to as PZT 95/5, has been
used for a number of years as a charge source for
shock activated pulsed power applications. The
nominal state of this ceramic is ferroelectric (FE),
but it is near an antiferroelectric (AFE) phase
boundary. Applying a voltage across electrodes on
this ceramic can produce a remanent polarization
due to a restructuring of ferroelectric domains
within the material. In pulsed power applications,
the bound charge on the ceramic is released into a
circuit connected to the electrodes when the
remanent polarization is destroyed by shock
compression of the material into the AFE phase.
A simple model for the response of PZT 95/5
was described by Lysne and Percival (1) for bars of
ceramic loaded in the normal mode configuration
(i.e. the direction of shock propagation is normal to
the direction of remanent polarization in the
ceramic). They assumed that the shocked material
immediately transformed to the AFE phase as the
shock front propagated through the material at
201
combination of these two stress measures (which
we will reckon as positive in compression), i.e.,
three-dimensional transient electromechanical
situations, and have driven renewed interest in
improving the capability for numerically simulating
the material responses of PZT 95/5. Extensive
experimental work has been performed to improve
our understanding of the response of PZT 95/5 for
multi-axial stress quasi-static loading (6). A
program of shock impact experiments (7) was
undertaken to develop our understanding of the
dynamic material response including the kinetics of
the phase transformation for various combinations
of stress and electric field in the ceramic.
In the current study, results from an improved
simulation model for the electromechanical
response of the ceramic are compared with a series
of nearly uniaxial strain experiments (7) on
normally poled PZT 95/5 ceramic with
approximately 9% void space.
vM+Q(vN-vM} = PH,
and we conclude that the phase stability of a
domain is dependent on its orientation within
ceramic.
The parameter Q reflects
electrostrictive properties of the crystal while
parameter PH is related to the term governing
stability of spontaneous polarization. When
(1)
FE
the
the
the
the
the
stress on the crystal is hydrostatic <JM = crM = P,
hence the transformation occurs when PH = P and
so PH is called the hydrostatic transformation
pressure. Zeuch et al. (6) have examined the FE to
AFE transformation of a number of unpoled and
poled PZT 95/5 ceramics under conditions of
hydrostatic compression and fixed difference in
principal stress and confirmed that the
transformation is dependent on FE domain
orientation.
From their observations they
conjectured that a FE domain is stable so long as aN
is less than or equal to some critical value which
corresponds to the case Q = 1 in Equation (1).
PHASE TRANSFORMATION
An improved electromechanical response model
for PZT 95/5, which treats the material as a porous
solid with a matrix comprised of a mixture of
material in the FE and AFE phases, is under
development. The response model is structured
into three major parts: a decomposition of the FE
phase into a mixture of regions, called FE domains,
having uniform spontaneous polarization, a module
that establishes the phase, rate of transformation
between phases, spontaneous polarization, and
permittivity in each domain, and a calculation for
the mechanical state of the material involving pore
collapse, material phase, and transformation strain.
A more detailed description of these aspects of the
response model is given by Brannon et al. (8).
Calculation of the transformation from the FE
to AFE phase for a given FE domain direction is
achieved using a transformation criterion derived
from a generalized free energy for the FE phase.
Examination of a free energy expansion from a
high temperature cubic lattice for the rhombohedral
FE phase exhibited by PZT 95/5 shows a linear
dependence on the mean stress, OM, and the normal
stress component, <JN acting on a plane
perpendicular to the direction of spontaneous
polarization. Consequently, the boundary between
the FE and AFE phase depends on a linear
NORMALLY POLED SIMULATIONS
Results from simulations, using the ALEGRA
wave propagation code, of a number of experiments
described by Setchell et al. (7) on normally poled
PZT 95/5, performed to examine the applicability
of the new PZT 95/5 model are presented here. In
these experiments a planar-impact of a 12.75-mm
Z-cut sapphire disc onto a ~3.2-mm Z-cut sapphire
disc was used to generate a long duration shock
pulse through a 4-mm thick bar of PZT 95/5
ceramic located behind the thinner disc. The
ceramic bar was backed by a 1.5-mm thick Z-cut
sapphire buffer with a diffusely reflective film that
enabled the particle speed in the sapphire to be
deduced from VISAR signals measured through a
12.7-mm thick Z-cut sapphire window adjacent to
the buffer. Setchell et al. (7) supported the ceramic
bar with identical electrically shorted support
blocks so that during the first wave transit only
-7% of the instrumented PZT 95/5 bar deviated
from a uniaxial strain loading.
The simulations reported here were performed
with boundary constraints consistent with uniaxial
202
strain and are only valid for the first wave transit
through the ceramic: approximately the first 1-^isec
of the calculated response curve. The bar had
lateral dimensions of 10-mm by 29-mm with the
4x29=mm faces having electrodes separated by 10mm. A load resistor separated one of the electrodes
from electric ground and current generated through
the load was monitored using a current-viewing
resistor connected in series to the load resistor. All
of the ceramic bars used in the experiments had a
nominal remanent polarization of 0.3 C/m2 as
determined by applying a looping voltage and
measuring the charge on the electrodes.
Simulations were performed for impact speeds
of 58.3, 152.7, and 336 m/s, corresponding,
respectively, to 9, 24, and 47 kilobars normal stress
into the ceramic, for a resistive load of 10-£1 An
additional simulation was conducted for an impact
speed of 154.1 m/s and a resistive load of 1150-Q.
The material properties used in the simulations
were estimated as much as possible from other
unrelated experiments (1,2,6,7). The parameter
governing the onset of pore collapse was selected
so that the calculated particle speed of the 152.7
m/s impact agreed roughly with the measured
particle speed.
2.0
2.5
Time (nsec)
FIGURE 1. Comparisons of measured (solid lines) and
calculated (symbols) particle velocity histories for three
impact speeds and a resistive load of 10 Q.
It is not surprising that the best agreement between
the calculations and experiments occurs for the
intermediate impact speed since the model
parameter for the onset of pore crushing was
calibrated to that experiment.
Clearly, the details of pore collapse require
improvement, as reflected in Figure 1, for which
RESULTS & DISCUSSION
36.0
336 m/s
A comparison of the calculated particle velocity
at the sapphire window buffer is given in Figure 1
for three impact speeds. Time is measured from the
instant of impact for all figures. Reasonable
agreement between the measured and calculated
profiles was achieved for all but the high-speed
impact.
Figure 2 shows a comparison between the
calculated and measured currents for the high and
low speed impacts with a 10-Q resistive load.
Figure 3 shows a comparison between the
calculated and measured currents for the 152.7 m/s
and 154.1 m/s impacts with load resistors of 10 and
1150-Q, respectively. The electric field within the
ceramic varied from 0.3 to 37 kV/mm over this
range of resistive load.
The comparisons shown in Figures 1-3 illustrate
promise for the applicability of the response model
over a wide range of input stress and electric field.
30.0 -
~ 24.0
<C,
¥ 18.0
58.3 m/s
5 12.06.00.0
0.0
0.5
1.0
1.5
2.0
Time (^sec)
FIGURE 2. Comparison of the measured (solid lines) and
calculated (symbols) currents from the specimen connected to
a 10-Q resistive load for the highest and lowest impact speeds.
the calculated second compression wave arrives
very late in comparison to the experiment. We
expect that the calculated current, seen in Figure 2,
203
pressure on the projection of the local electric field
along the direction of spontaneous polarization,
thus adjusting the normal compressive stress
required for transformation.
35.0
30.0
io-n,
ACKNOWLEDGEMENTS
10.0
This work was performed at Sandia National
Laboratories. Sandia is a multiprogram laboratory
operated by Sandia Corporation, a Lockheed
Martin Company, for the United States Department
of Energy, under Contract DE-AC04-94AL85000.
5.0
0.0
0.0
0.5
1.0
1.5
2.0
Time (^sec)
FIGURE 3. Comparison of the measured (solid lines) and
calculated (symbols) currents from the ceramic specimen for
the -153 m/s impact experiments.
REFERENCES
1.
for the high-speed impact experiment would match
the experimentally observed profile much better as
the treatment of pore collapse in the response
model is improved.
According to estimates of PH ~ 200-300 MPa,
as provided by quasistatic measurement (6). We
see additional confirmation of the applicability that
the FE to AFE phase transformation boundary is
given by Equation (1) with Q = 1. The lowest
mean stress easily exceeds 500 MPa for all impact
speeds, however since the ceramic is normally
poled a better estimate of the stress required to
force the phase transformation is given by the
lateral confining normal stress corresponding to the
direction of remanent polarization. The reduced
current for the slowest-speed impact experiment
reinforces this observation.
We expect that the agreement between
experiment and calculation can be improved by
modifying the FE domain distribution used in the
model. One aspect of depoling response found by
Setchell et al. (7) that was not explored here
involved the observed reduction in current output
for the slowest impact experiment with a resistive
load o f - 1150-Q. The transformation condition
given by Equation (1) does not include any
influence on the phase boundary due to the local
electric field. A simple modification to Equation
(1) that allows electric field stabilization, or
destabilization, of the FE phase is being studied.
This can be achieved by including a linear
dependence of the hydrostatic transformation
2.
3.
4.
5.
6.
7.
8.
204
Lysne, P. C., and Percival, C. M., J. Appl. Phys. 46,
1519-1525(1975).
Montgomery, S. T., "Analysis of Transitions
Between Ferroelectric and Antiferroelectric States
Under Conditions of Uniaxial Strain," in Shock
Waves in Condensed Matter -1985, edited by Y. M.
Gupta, Plenum Press, New York, 1986, pp. 179-183.
Kipp, M. E., and Lawrence, R. I, "WONDY V - A
One-Dimensional
Finite-Difference
Wave
Propagation Code," SANDS 1-0939, Sandia National
Laboratories, Albuquerque, NM (June 1982).
Montgomery, S. T., Graham, R. A., and Anderson,
M. U., "Return to the Shorted and Shunted Quartz
Gauge Problem: Analysis with the SUBWAY
Code," in Shock Waves in Condensed Matter 1995. edited by S. C. Schmidt and W. C. Tao, AIP
Conference Proceedings 370 Part 2, New York,
1996. pp. 1025-1028.
Summers, R. M., Peery, J. S., Wong, M. K., Hertel,
E. S., Trucano, T. G., and Chhabildas, L. C., Int. J.
Impact Engng. 20, 779-788, (1997).
Zeuch, D. H., Montgomery, S. T., and Holcomb, D.
J., J. Mater. Res. 14, 1814-1827 (1999).
Setchell, R. E., Montgomery, S. T., Chhabildas, L.
C., and Furnish, M. D., "The Effects of Shock Stress
and Field Strength on Shock-Induced Depoling of
Normally Poled PZT 95/5," in Shock Compression
of Condensed Matter - 1999, edited by M. D.
Furnish, L. C. Chhabildas, and R. S. Hixson, AIP
Conference Proceedings 505, New York, 2000, pp.
979-982.
Brannon, R. M., Montgomery, S. T., Aidun, J. B.,
and Robinson, A. C., "Macro- and Meso-scale
Modeling of PZT Ferroelectric Ceramics," in these
proceedings.