MODELING
AND
CHARACTERIZATION
PERFORMANCE
OF GEOMETRIC
OF RAINBOW
Final Report
AND
for Contract
Submitted
Dr. Zoubeida Office NASA
THUNDER
EFFECTS ACTUATORS
NCC-1-283
to:
Ounaies
of the Director
Langley Research Center 3 West Reid Street MS 132C
Hampton,
VA 23681-2199
Prepared
Robert The Gilbert
C. Robinson
by:
W. Schwartz
Department of Ceramic and Materials Clemson University Clemson, SC 29634-0907
April
11, 2000
Engineering
ON THE
MODELING AND CHARACTERIZATION PERFORMANCE OF RAINBOW
Table Table
of Contents
I. Summary
OF GEOMETRIC EFFECTS AND THUNDER ACTUATORS
ON THE
of Contents
........................................................................................................................
p. ii
....................................................................................................................................
II Discussion
of Results
p. 1
................................................................................................................
A. Background
..................................................................................................................
B. Application
of unimorph
theory
to Rainbow
and Thunder
actuators
p.2 p.2 ...........................
p.5
C. Equivalent Circuit Modeling ........................................................................................ D. Finite Element Modeling ...........................................................................................
p.9 p. 11
E. Summary
p.20
III. Suggestions
for Future
IV. List of Publications V. References
of Results
...................................................................................................
Work
...............................................................................................
and Presentations
Resulting
from NASA
Funding
..............................................................................................................................
...............................
p.20 p.25 p.26
MODELING
AND
CHARACTERIZATION
PERFORMANCE
OF GEOMETRIC
OF RAINBOW
AND
EFFECTS
THUNDER
ON THE
ACTUATORS
Robert W. Schwartz, J. Ballato, W. D. Nothwang, and P. Laoratanakul Department of Ceramic and Materials Engineering, Clemson University; Clemson, SC 29634-0907 Contact Information: Phone: 864-656-7880; Fax: 864-656-1453; e-mail:
[email protected]
I.
Summary Dome
mismatch fabrication
formation
stress between temperatures.
profile
within
region
of the piezoelectric.
domains),
and improve
direct extensional the non-linearity
and Thunder
the metallic Accompanying
the devices
the piezoelectric.
seems
in Rainbow
These
tensile
counterintuitive;
a stress
Use the results
Goal
aimed
1 outlined
simulations
impact
electromechanical
of the research
were
of c-to-a
in this region
compared
of
to standard
of the contributions this improvement
of the applied
device stress
the surface
(ratio
of the device
performance
to the direction
within
configuration
response
expansion
cooling from of an intemal
stresses
switching. Further consideration stress thus appears warranted.
(displacement
of stress to in response
electric
field should
of the lower
region
of the
to:
and load-bearing
of the modeling
and thickness performance. element
the domain
thermal
Conduct finite element and equivalent circuit simulation-based investigations to understand the effects of actuator geometry on internal stress distribution and actuator performance
2.
affect
movement
perpendicular
objectives
to relieve
tensile
devices, presumably because d-coefficients.1 Interestingly,
impede, not contribute to 90 ° domain piezoelectric that is under compressive 1.
wall
in improved
and flextensional of the piezoelectric
of significant
stresses
the 90 ° domain
occurs
piezoelectric layers during this process is the generation
and the development
This results
The specified
and
actuators
ratio) above
required
was met during
of actuators
Goal
fabricated
1 and other
to predict
for the fabrication
of geometry and shape on stress Goal 2, verification of modeling at meeting
studies
capabilities). the processing of Rainbow
the time period with
different
proposed
activities
in equivalent
These additional studies also yielded interesting suggest that in addition to stress effects, mass
mechanics
effects
and
of the contract. geometries
were
geometry
effects
results loading
circuit
(geometry
with optimized A number
carried
of finite
out and the
level was successfully modeled. results was not completed due to the extent
performance. and which
(i.e., modulus
conditions
ceramics
of activities
modeling
of device
that are summarized below and more straightforward
on performance
that are independent
of
stress)
also contribute to the enhanced performance of these devices. The results of both the finite element and equivalent circuit modeling that have been obtained to date were initially summarized in an interim report 2 to Dr. Wallace Vaughn (Advanced
Materials
and
Processing
Branch).
Results
in both
of these
areas
obtained
submission of the interim report this past March are summarized below. In previous research carried out on this contract, we used 3-dimensional modeling
to demonstrate
that the tensile
stress
levels
in the surface
of Rainbow
finite
since
the
element
actuators
were
dependent on the geometry of the device and were significantly greater than the stresses present in analogous Thunder devices. 2 Further, the predicted stress levels in the devices are in agreement with the observed displacement performance of the two families of actuator devices. 3
Higher stress levels were predicted by finite compared
to Thunder
devices generate effects of internal
devices.
greater stress
The influence
displacements, again emphasizing the importance on the d-coefficients of these materials.
of a variety
of specimen
rectangular geometry, and device within the material have also been thicknesses of approximately studies. 4 Initial results indicate are key
parameters
element
analysis
ratio)
has
different
aspect
presented
below. In addition
good
predicted
on stress
are under
to the finite
(reduced
layer
ratio,
circular,
More
profile, The
which
obtained
agreement
with
published
exceed
of cantilever
and
suggests these
equivalent
by non-linear
vs. rectangle that
devices
actuators
predictions.
circuit
the
aspect
will
display
prepared
Modeling
modeling
geometry
results 6 for resonance
reported
values.
effects
indicate
excellent
studies
but aimed
agreement
with
results
of Rainbow
frequencies,
Recent
finite
and rectangle
the
of rectangular
to verify
studies,
results
(circle
properties
investigation element
recent
geometry
are
devices
out to predict the resonance and strain behavior of these actuators. were modeled as a black box equivalent circuit. 5 Results of these
significantly
characterization
level.
actuator
differences.
ratios
has also been carried approach, the devices displayed
that
effect
performance
effects
the
thickness) on the internal stress profile and dome formation modeled. 2 Maximum stresses are predicted for reduced layer
stress
indicate
a significant
significant
geometry
in understanding
35%, in agreement with results of previous 2D finite element that reduced layer/piezoelectric layer ratio and device thickness
in defining also
element modeling for the Rainbow devices of both of these devices 3 indeed shows that the Rainbow
Testing
In this studies
the
strains
at
further
between
predicted
and measured resonance frequencies for actuators with both different length and thickness ratios. In these studies, the highest predicted strains were observed at - 26% reduced thickness/total thickness ratio, close to the experimentally observed value of 30-35%, 4 suggesting that not in addition
to stress,
II. Discussion A.
mass
loading
and other
mechanics
effects
must also be considered.
of Results
Background The
few tenths although
intrinsic of 1%.
this
strain
that can typically
Unimorph
and bimorph
performance
improvement
be obtained
from
technologies was
a piezoelectric
were
accompanied
by
capabilities of the device. A thorough review of applications is given by Smits et al. 7 For applications requiring still larger technologies (THin layer Wafer)
have been UNimorph
ceramics.
(100-300% demonstrated
devised DrivER)
In particular,
Rainbows
a decrease
in the
(PLZT)
manufacturing
carbon
block
in a furnace
975°C
is employed.
process
or other involves
at temperatures
Oxygen
from
the following between
the air reacts
steps.
600 and
2
piezoelectric A ceramic
1200°C;
with the carbon
load-bearing benders actuator
very high displacements
of Rainbow devices.1 controlled partial reduction containing,
a
actuators include THUNDER g And INternally Biased Oxide
are not only able to generate
high-Pb
is about this strain,
and bimorph two recent
strain), but they do so with reasonable load bearing capabilities. that Rainbow actuators can sustain point loads of about 10 kg. I°
(Pb,La)(Zr,Ti)O3
The typical
material to amplify
of unimorph displacements,
to amplify the strain. These and RAINBOW 9 (Reduced
previously reviewed the performance characteristics The Rainbow device is obtained by the (PZT),
developed
been has
of Pb(Zr,Ti)O3 oxide
materialsJ
disc is placed
typically,
to produce
It has Haertling
onto a
a temperature CO,
which
l of
in turn
reactswith the ceramicthat is in contactwith the block. The ceramicin reduced, reduction
forming metallic step, the carbon
During
cooling,
thermal
domed,
or saddle-like,
thermal
expansion
transition
Pb distributed in a multiphase ceramic block and composite are removed from
expansion
mismatch
of the piezoelectric
thus occurs in this rectangular Rainbows Rainbow presence
of
and
an
of these
and
unimorph
between
depending
between layer;
upon
the
two
the two
the
Thunder
layers
is most
most of the doming
actuators
stress
devices
are
profile.
that
referred This
is observed
and bimorph
and
causes
stress
devices.
The
The
the
Curie
in the structure
circular,
square,
actuators
contributes
to more
to
traditional
due
the
direct
fabrication
of a
ratio.
below
development
to as "stress-biased"
internal
the formation
pronounced
stress
is chemically
dimension
have prepared ratios. 12
compared
flextensional
layers
lateral/thickness
temperature range. Investigators with a variety of sizes and thickness
intemal
performance standard
mismatch
geometry
this region
matrix. Following the the furnace and cooled.
and to the
enhanced extensional
process
for both
of
these devices involves an elevated temperature processing step, and the stress profile that is generated is primarily the result of the thermal expansion mismatch between the piezoelectric and the underlying reduced (Rainbow) or metallic (Thunder) layers. Details of the fabrication process
and the performance
characteristics
of the devices
may be found
In order to model the performance of Rainbow actuators, mechanical properties of the reduced layer and have reported expansion behavior. _3'_4 It is interesting detailed studies of the thermal expansion ferroelectric
character.
temperature
of-
A distinct
330°C,
indicative
presence of this ferroelectric for, 16 and it is also neglected nature
In the development of the stress profile
we may
change
in the thermal
of a phase
expansion
transformation
phase in studies of device in the analysis below.
Using are
this
performance
has
is observed transition).
not been
at a 15 The
accounted
of these devices, some attention has been given previously to the by Li, Furman, and Haertling. 4 Interest in the stress profile of these
state
domain configuration the electromechanical
that the d-coefficients
approach,
considered.
behavior
(i.e., the Curie
of the device
the stress state of the device is altered, as might occur through for example, differences in device performance are anticipated. response
1 and 11.
to note that depending on the reduction conditions, behavior _5 of this phase suggest that it retains some
devices arises because of its effects on phenomena, which concurrently impacts Alternatively,
in References
several groups have studied the on the modulus and thermal
the effects It has
been
of stress suggested
and
domain response
are dependent
changes
on the non-linearity that
wall movement of the devices. on stress,
in the fabrication in the
it is this significant
and as process
electromechanical stress-induced
non-
linearity that results in the enhanced performance of the Rainbow and Thunder devices compared to more traditional devices. This explains our interest in understanding the nature of the internal stress profile within these devices. By defining which process techniques and device aspects may be manipulated to optimize this stress profile, we may develop devices greater performance compared to current Thunder and Rainbow devices. These issues discussed in greater detail previously. 2 ceramic
The Thunder actuator is fabricated using a curable polyimide polymer,
geometry with still have been
by bonding a thin metal layer to a piezoelectric s This three layer structure is then heated to elevated
temperatures, typically around 350°C, which results in curing of the polymeric layer. During subsequent cooling, in an analogous manner to the Rainbow ceramic, the thermal expansion mismatch between the metallic and ceramic layers results in a domed or saddle-like configuration
for the actuator.
The strain
characteristics
of the Thunder
devices
are, in general,
similar to thoseof the Rainbow_ but the load-bearingcapabilitiesof the Thunderdevicesappear to havenot beeninvestigatedasthoroughly. Also, the attainabledisplacementsof the Thunder devicesappearto besomewhatlessthanthoseof Rainbowceramics) Thus, standard
the
general
unimorph
configurations
devices,
only
of
they
are
Rainbow
domed
and
and
can
Thunder possess
across both the reduced and the unreduced piezoelectric layer. modeled the performance of Rainbow actuators by assuming
ceramics
are
a significant
Since
the d-coefficients understanding
understanding performance. planar
regarding
be significantly
in these
devices
through
Before considering the performance of these
as well as standard unimorph by the effective d31 coefficient
for single ceramics,
may
devices
of the stress,
potentially
may
devices and strategies to further improve actuator our studies of the effects of actuator geometry on
finite
contribute
a
lead to a better
element
simulation.
A variety
of circular
in the piezoelectric analysis approaches.
layer
to the enhanced
response
actuators, the observed of the piezoelectric layer.
that
and were
comments additional
is observed.
For
stress-
displacement response will be This coefficient is well defined
crystal materials such as BaTiO3, and is, of course, also known for polycrystalline including lead zirconate titanate materials of a variety of compositions. While
crystal
materials,
the observed
d31 response
is truly
an "intrinsic"
effect,
may also affect
what
is typically
referred
and c-crystallographic directions in these materials extrinsic response is affected by material properties least the past
10 years,
response to improve single crystal based c- and
was obtained
investigators
have
the performance bimorphs, which
a-domain
Upon
switching
have different such as grain
response
inherent size and
to take advantage
from both the domain since
the a-
polarizabilities. The stress state, and for at
of an increased
extrinsic
of piezoelectric actuators. 17 Zhang and Cross 17 prepared due to the presence of an internal stress, demonstrated a
configuration.
due to domain
attempted
to as the "intrinsic"
in
in polycrystalline
even in materials that are poled, the observed response has contributions and extrinsic effects, i.e., domain wall motion effects. In addition,
configuration
mixed
by the magnitude
devices
have in the
pC/N compared x-ray diffraction
these results, it is appropriate to make a few other characteristics of Rainbow and Thunder devices and which
biased, dictated
ceramics, intrinsic
impacted
in these
devices were modeled and the stress levels present with device geometry using both linear and non-linear
aspects
single
profiles
of the performance of these In this report, we summarize
stresses
rectangular correlated
may
of the stress
to
profile
Li, Furman and Haertling that the d31 coefficient
surface region of Rainbow ceramics, which is under tension, is as high as -508 to a value of-271 pC/N for a standard ceramic. Studies of the (200) and (002) peak intensities tend to support these conclusions. 4 quantitative
similar
stress
application
with the metallic
of the layer
field,
a greater
providing
strain
a restoring
response
elastic
force
to regenerate the original domain pattern after the removal of the electric field. Analysis of the results of these investigators indicates that a performance enhancement of a factor of 30 compared to a typical single crystal device is possible. Is For a more direct comparison, the response
of poled
polycrystalline
actuators
was considered,
and again,
the bimorph device was noted. Other approaches to predict
the performance
of Rainbow
been
and Cross
modified
pursued.
for unimorph acknowledged variations incorporate
For example, actuators that stress
in d31 with influences
Wang
to describe effects may
stress
are,
of this type
have
the performance impact the effective
in fact,
pointed
into the model
4
out
and Thunder
the
standard
paper,
used
no
performance devices
constituent
of Rainbow d31 coefficient.
in this
that was
improved
have
to describe
was
also
equations
ceramics and 16 However, attempt
for
have while
made
performance.
to The
presenceof the internalstressprofile was,however,highlightedasthe majordifferencebetween Rainbowandstandardunimorph-typeactuators._6 While thereare a few reportsin the literatureon the effects of stresson the ferroelastic propertiesof ceramicssuchas PZTt9'2°andBaTiO3,thereare few, if any, reportson the effects of both electric and mechanicalstresseson the electromechanicalresponseof thesematerials. Therefore,in the singlecrystal domainactuator,16andin stress-biaseddevicessuchasRainbow andThunderactuators,it becomesquitedifficult to predictthe performanceof the devicesdueto the unknownmagnitudeof the extrinsiccontributionsto the response,the presenceof a radically different domain configuration comparedto a standardpoled piezoelectricunimorph, andthe presenceof a restoring stress field that tends to bias the structure toward an a-domain configurationwhile the electricfield tendsto enforcethe formationof a c-domainconfiguration. Due to the ferroelastic natureof the material, the presenceof both mechanicaland electrical fields can also be expectedto promotesubstantiallymore complex aging andfatiguebehavior thatcanimpactthe utilization of thedevicesin a variety of applications. During the courseof this program,due to the complexity of thesedevices,we initiated efforts to understanda variety of effectsthat we felt could contributeto the performanceof these actuators. Specifically, we havebegunto look at massloading effects on the generatedstrain (displacement)of thesedevicesusing an equivalentcircuit approach. This methodallows for predictionof the macroscopicbehaviorof the piezoelectricdevicewithout requiring anin-depth knowledge of the microscopic characteristics. 21 The equivalent circuit model views the piezoelectric device as a "black box" having some combination of resistance,capacitance, inductance,etc.22 Basedupon thesebulk propertiesand material propertiessuch as Young's modulus,a macroscopicmodel is developedto predict the responseof the device to an input electrical signal.5'23 The effects of massloading appearto have been neglectedin previous analysesof bothRainbowandThunderdevices,aswell as in the study of unimorphandbimorph devices. The possible importanceof this device "geometry" parameteris further discussed below. In the presentationof programresultsthat follows, the enhancedperformanceof stressbiasedpiezoelectricactuatorssuchasRainbow andThunderdevicesis further discussed.While a quantitativemodel thatpredictsdeviceperformancewasnot developed,nor wasthis a goal of this program,a number of aspectsof the devices that may contribute to their responseare investigated. First, results that were obtainedusing the standardconstituentequationsfor unimorphswill be discussedto suggestthat at least part of the observedresponseof these devicesis due to simple mechanicaleffects. Second,results of equivalent circuit modeling studieswill bepresentedthat imply that massloadingeffectsin thesedevicesalsoplay a role in their performance. Third, we examinethe resultsof a study of the effect of stresson dielectric constant. And finally, we presentthe outcomefor finite elementanalysesof the stressprofile andsurfacestressesin thesedevices. These results will illustrate that the geometry of the device has a significant effect piezoelectric contribution B. Application Prior
of unimorph to discussing
biased actuators, tip displacement
on the mechanical stress field, to the performance of these devices. theory
to Rainbow
the extension
and Thunder
of unimorph
some background on the derivation and charge generation is discussed.
5
theory
which
may
impact
the
extrinsic
actuators to predict
the performance
of standard unimorph Briefly, the standard
of stress-
theory for predicting unimorph equations
are derived from the generalconstitutive equationsthat describeelectromechanicalresponse shown in Figure 1. These modified equations, shown below, indicate the dependence of unimorph
performance
on material
as w, 1, and thickness ratio. Cross 16 to more directly relate piezoelectric piezoelectric also
layer to the metallic layer thicknesses.
Other approaches, been used recently
actuators. actuators
properties
such as s and d, and geometrical
These equations have since been further actuator performance to device parameters layer
compliance
and
the
of the
metal
based on the use of unimorph theory and standard beam to predict the performance of Rainbow and Thunder
16'24 These methods have focused on the prediction when fabricated in rectangular cantilever geometries.
contributions been carried
ratio
parameters
to piezoelectric response of the devices out with the inclusion of these effects. I ELECTRICAL
such
modified by Wang and such as the ratio of the layer
to the
theory, have stress-biased
of the tip displacement of the The need to include non-linear
has also cited, 16 although
no modeling
has
1
CONSTITUTIVEEQUATIONS
I MECHANICAL
Figure
1.
I
[
we are electric
D=eS+_SE
T=cDS-h'D
E = -h S + 13 sD
S=seT+d'E
D=dT
S=s oT + g' D
E = -g T + i3TD
equations
tip deflection was predicted et al. 25 For study of device
to describe
the electromechanical
response
of
using the constituent equations for unimorphs of performance in applications such as positioning,
primarily interested in the displacement performance of the devices field, and the following equations from Wang et al. were utilized: 25 8=aF+bV
b = 3d3jL2 tp 2
+_TE
THERMAL
Heckmann diagram and constitutive piezoelectric ceramics.
Cantilever Smits 7 and Wang
T=cES-e'E
under
an applied
(1)
AB(B+ 1) 1 +4AB+6AB 2 +4AB 3 +A2B 4
(2)
Em
a
(3)
=-
Ep till
B
=-
(4)
tp
6
where the coefficient,
term b describes the tip deflection, 8, per applied volt V, d3t is the piezoelectric L is the length of the cantilever, tp is the thickness of the piezoelectric layer, and tm is
the thickness
of the metallic,
or for the case of Rainbows,
the reduced
layer.
The
term A shows
the effects of the ratio of the Young's modulus of the metallic (Era) to the piezoelectric on tip deflection, and the term B describes the effects of the ratio of the piezoelectric thickness
(tm/tp)
the
on
tip deflection
of the
cantilever.
values used are presented below in Table I; for the modulus of 6.9 x 10 I° N/m E was used for the underlying
TABLE Material
properties
utilized
in equivalent
circuit
materials
Using Eqn. employed
1-4, the relative in Rainbow and
Rainbow
Thunder aluminum
actuators,
actuator layer.
that
the
was
modulus
modeled,
a
I. and unimorph
Equivalent
Motional time constant Permittivity Conductivity Coupling Coefficient Rel. Permittivity (Piezo. Constant) Density - Piezoelectric layer Density - Reduced layer Elastic Constant - Piezo layer Elastic Constant - Reduced layer
For
layer (Ep) to metal
Circuit
modeling
of Rainbow
Model
Unimorph
1 x 10"12sec 1.53 x 104 F/m 0 0.60 1730 7.95 g/cm 3 6.90 g/cm 3 1.11 x 10 n N/m 2
ceramics.
Model
7.42 x 10 I° N/m 2 6.26 x 10 I° N/m 2
displacement of unimorph Thunder devices (PZT 5A
actuators fabricated with the and reduced PZT 5A for the
Rainbow simulation; and PZT 5A and aluminum for the Thunder simulation) were predicted. The results of these analyses are shown below in Figure 2. Interesting, these predictions indicate that both devices should display similar effects of the tm/tp (metal/piezoelectric thickness) ratio, despite the slight differences in the moduli of the reduced layer (in the Rainbow) and the metal (aluminum) often used in the fabrication of Thunder devices. The use of stainless steel as the metal
layer
displacements
shifts
the
curve
are predicted
to the
left.
for a metal
We
note
(or reduced)
that,
in this
layer
thickness
figure,
the
of the total device thickness. Interestingly, this value is close to that observed the Rainbow devices that the generate maximum displacement (see Figure that while both above maximized stresses devices
surface
tensile
and below), performance
stresses
may
be maximized
maximum
of approximately
for this device
device
30 - 35 %
experimentally for 3a). This suggests
geometry
(see
discussion
other simpler geometrical or mechanical factors also play a role in the of these devices. Stated otherwise, the maximization of the tensile
for this geometry may simply prepared with this configuration.
serve
to further
7
enhance
the performance
of "unimorph"
0.9
8 0.6 F= 07 0.8 _.
- -
!
0.5
i5 0.4
_
0.2 0.1 0 0
0.2
0.4 Percent
Figure 2.
0.6
Metal
0.8
(%)
Unimorph predictions of Rainbow and Thunder geometry effects on relative displacement the cantilever tip.
of
These predictions are somewhat different from those of Wang and Cross who reported that maximum tip displacements are obtained for a tm/tp ratio of 1, i.e., a metal thickness that is 50% of the total device thickness. However, as mentioned above, the value of 30 - 35 % is close to that observed experimentally response. For example, Haertling least for circular thickness:reduced
for Rainbow and coworkers
domed devices) layer thickness.
devices that generate maximum displacement have reported that maximum displacements (at
are obtained for a thickness 4 Similar observations have
ratio of approximately 2:1 Piezo been noted for the performance of
Thunder devices, with devices that are constructed with a metal layer that is 50% of the thickness of the piezoelectric layer displaying the greatest displacement response. 26 The difference between
these
(rectangle investigation 1.2
results
vs.
and those
circle),
or
tip
of Wang deflection
and Cross vs.
,--0.--$
-I--I
to device
geometry
measurement,
but
effects further
1.2
v ......... $ kV/cm
kVlem
W/cm
--'11-" I kVlcm
I
l[ .-i-mvl,,,
0.8
0.6 1
0.6f
io4 I i
be related deflection
is required.
......... i
'"
16 may
mid-point
0.8
0,4
.-._- I_
!O oll , , _______ 0
10
20
30
40
50
60
z
0.2 0 0
70
10
20
30
Percent Aluminum Percent Reduced
Figure 3.
Measured displacement
performance of (a) Rainbow and (b) Thunder actuators. 3
8
40
50
The modeling response
of
unimorph
some of the general vs. actuator geometry
perhaps biased
application
related
While displacement
to these
devices
therefore
appears
aspects of device performance, although are really not observed for the Thunder
to the polymeric
devices
theory
layer
are driven
at higher
unimorph performance,
theory this
that
is present
in the devices,
attractive
in
similar predictions of data of Wise. 3 This is
or the fact that the stress-
fields. qualitatively "mechanical"
predicts aspect
the alone
effects of device does not explain
geometry on the enhanced
response of Rainbow and Thunder actuators. An additional contribution to the performance of these actuators is still believed to originate from the stress profile that exists in these devices.14'lT'27 To illustrate this point, Wise has already discussed the performance of Rainbow and Thunder devices and has response. 3 The modeling results internal
stress
is important
reported presented
that Rainbows below suggest
in the performance
of these
shown that the Rainbow devices possess significantly Thunder devices (60 MPa). X-ray diffraction studies fabricated
with
different
stress
levels
support
the
demonstrate that, in fact, devices.
a greater displacement as has been speculated,
Finite
element
argument
that enhanced
domain
Again, this result is contrary to expectations for but the studies by Moon 28 and other investigators
conclusion.
of actual
studies
performance are required to "quantitatively" predict C. Equivalent
Circuit
Another
devices,
has
higher maximum stress (400 MPa) than by Moon (summarized below) of actuators
occurs at higher stress. 28 response of PZT materials, Further
modeling
as well
to determine the possibility the performance of Rainbow
as analysis
of published
of extending and Thunder
wall
motion
the ferroelastic 4 support this results
"standard" ceramics.
of device
unimorph
theory
Modeling
approach
used
to approximate
the
performance
of Rainbow
ceramics
was
an
equivalent circuit approach. In this phase of the program, a simple, resonator model was used to predict the changes in resonance frequency different device configurations. TM The focus of the investigation was
one-dimensional plate and strain response for to study the effects of
reduced
different
layer
thicknesses, while
thickness the
mass
unimorph
on device of the top
modeling
performance. electrode
accounts
In order
was
varied
principally
approach utilized below permits the study of simple mass loading effects
for
to simulate
in the
equivalent
geometrical
effects,
reduced
circuit and
layer
model.
the
finite
Thus, element
investigation of stress effects, this method allows on the displacement performance of the device.
for the Recent
investigations have addressed the development of more accurate equivalent circuit models for the analysis of multilayer structures, 29 but as a first step, in this study, the simplest one-dimensional plate
resonator
model
was utilized.
PZT
zero acoustic wave transmission from simulations were carried out by running material
parameters
are shown
above
in Table
In the initial simulations, a device layer/total thickness ratio) identical
results
of the
could
was
used
as the basis
of the model
with
the piezoelectric layer to the electrode. A variety of the equivalent circuit model using MATHCAD and the
reduced
simulations
5A ceramic
be compared
I. geometry (actuator thickness, width, length, and to that reported by Elissalde 6 was utilized, so that to experimentally
reported
results.
The
initial
output of the model indicated that the resonance frequencies predicted were a function of several competing variables. It was interesting to note, however, that as shown in Figure 4, the predictions frequency
of the equivalent results
of Elissalde.
circuit model were in good agreement 6 The primary variables that defined
9
with the published the resonance
resonance
frequency
were
the dimensionsof the device, but additional variables,suchas load (in this instance,electrode mass)andthe ratio of reducedthicknessto the unreducedthickness,alsoimpactedthe resonance behaviorof the actuator. In comparingthe resultsof the equivalentcircuit model with thosein the literature,it is interestingto notethat evenfor the simple onedimensionalmodel that was developed,the predictionsof the modelcorrelatedfairly closelywith the observedexperimental results. 1.E+08 •
TN_ Excitation
_
F_xpednm-Cat (Bissaldeet.al.,
1.E+07
1.E+06 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t.er_ (mm) Figure 4.
Comparison of equivalent circuit actuator length with experimentally
Additional layer
simulations
thickness.
electrode the ratio
have
also
In this first scenario,
modeling results of Rainbow resonance measured values of Elissalde. s
been
the simple
is utilized. Specifically, the primary of oxide thickness to reduced layer
thickness,
and
off
resonance
carried
performance
out accounting
approach
for the
of approximating
device parameters that thickness, i.e., electrode
for
different
frequencies
effects
vs.
of reduced
this layer
as a massy
were examined included mass at constant wafer
tp/tm thicknesses
(different
electrode
masses). simulation dependent
The general trends observed in the literature for these effects were duplicated by our efforts, with one striking difference. The maximum strain at resonance was strongly on mass loading. Therefore, a "resonance" behavior for strain was observed as shown
in Figure
5.
surface
Keeping
in mind
of the piezoelectric
that the reduced
wafer,
layer
it was possible
wafer, to change the amount of loading level (which corresponds to a reduced
was
being
to simply
treated
adjust
simply
this mass,
as a mass
on one
or thickness
of the
on the surface. It is important to note layer/total thickness ratio of 26.2%)
that this loading again correlates
surprisingly parameters
well with the value of 30 - 35 % reported by Haertling for actuator fabrication that gave optimized strain responses.l'4 As internal stress is not a component within
the model,
it leads
tensile
stress
one
to believe
for this reduction
that
ratio,
additional
exist
reasons,
for the enhanced
been reduced 30 - 35%. The equivalent circuit modeling effects contribute to the enhanced performance of devices noted
that because
magnitude qualitatively
of the
nature
beyond
of the assumptions
the maximization
performance
of the devices
that have
results suggest that even simple mass with this geometry. It should also be
invoked
in the derivation
of the model,
of the strain cannot be taken at its absolute value, but rather must relative to other predicted displacement values. However, it is still
I0
of surface
the
be examined interesting to
note thatmaximum displacementsare again that display the highest strains. strain confirm these predications,
predicted
for the Rainbow
geometries
(-30
- 35%)
Recent experimental studies 3° of the effects of mass loading although the enhancement in response is not three orders
on of
magnitude
as suggested
Figure 5.
Effect of mass loading (relative thickness of the reduced layer) on the displacement performance of Rainbow actuators. The experimental data is indicated as the line in the figure.
D. Finite
Element
by Fig. 5.
Modeling
Approach This
section
describes
the key results
obtained
to date from
the finite
element
simulations. A few of these results presented below were documented previously Report. 2 Together with these results, significant new results are also discussed. In the Interim of the thermal
Report,
expansion
rectangular Rainbow used a 3-dimensional
surface
coefficient geometries. approach
stresses, mismatch)
stress
profiles
and device
were
studied
for circular,
deformations square,
modeling
in the Interim (as a result
and a variety
Compared to pre ious studies by other investigators,' to model deformation and internal stress. We also V
•
•
4
of
14
used
we a
different thermal coefficient of expansion (TCE) for the piezoelectric layer compared to that used previously by other investigators. 2 Here, we used two TCEs for the thermal response of the material
above
and below
the Curie
of the analysis. Briefly,
the model
that was
constructed different formation
as a two-layer
plate,
point.
We anticipate
developed with
each
thermal coefficient of expansion. and internal stress result during
that this should
for simulation layer
having
improve
of the Rainbow's the
same
lateral
the accuracy
dome
shape
dimensions
was but
a
Further, the basis of the model, i.e., that dome cool down from the reduction temperature, is in
agreement with experimental results that indicate that most of the dome formation, and therefore, most of the stress development, occurs during cooling below the Curie transition. 28 The thermal expansion values were assigned based on a review of the literature 31 and previous University.
experimental characterization 28 The temperature of the
of expansion behavior two plate structure was
11
carried then set
out at to the
Clemson reduction
temperature, the
1250 K, followed
cooling
process,
by cooling
a dimensional
of the structure
mismatch
identified above, leading to doming of accompanying internal stress distribution. The
finite
commercially 8-node brick and y axes, model
model
was
the
Rainbow
constructed
and
two
temperature, layers
device
occurs
and
analyzed
the
using
it was only necessary symmetry
plane
independent
to model
conditions
be constrained
isotropic The values
one quarter
require
that
to be zero normal
of the structure.
displacements
of
to that plane.
material
properties
for these
properties
were
used
are reported
Table Material
properties
of the reduced
Young's
for the
reasons
The
(version
nodal
layer and oxide
Modulus
Poisson's
Layer
Oxide Layer
for
both
in Table
the
5.4),
a
aspect
points
of the
lying
on
a
axis of the structure
where it was completely to 300 K, temperaturereduced
and
oxide
layer
II.
layer required
Ratio
for finite
Density
6.26x 10 l0
0.342
6900
7.42x10
0.390
7700
element
Thermal
(kg/m 3)
I°
an
dimensional, about the x
Another
central
of
II.
(N/m 2) Reduced
During
development
ANSYS
was allowed to move vertically, except for the point at the lower edge, fixed. To simulate the dome formation that occurred during cooling elements.
300 K.
available software package. The models were constructed using three elements.14 Because the structures that were modeled were symmetric
is that
symmetry
element
between
the
to room
modeling.
Expansion
Coefficient 1x 10 .5 0.25x10
"s(300-643
K)
0.67x 10 .5 (643- ! 250 K)
Linear
3-D models
In our studies of circular geometries, which element methods, 4 using a linear analysis, we
can also be studied by obtained similar results
reported
for the stress
6a. 2 This figure
function circular
of position Rainbow.
thickness
ratio
profile,
as shown
(thickness) The results
(reduced
in Figure
for different directions presented are for the
layer/total
sample
thickness)
shows
2-dimensional finite to those previously the internal
stress
as a
in a 30 mm diameter, 0.5 mm thick center of the circular sample, and the for this
particular
sample
was
0.3.
The
region of tensile stress on the left of the figure is associated with the reduced region while that on the right is associated with the surface of the piezoelectric oxide region. The x and y components the thickness while
of internal stress, normally called the planar stresses, are identical of sample. It is evident that the entire reduced layer is subjected
the lower
part of the oxide
layer
is under
There is an abrupt change in the planar stress intersection between the stress lines and x-axis the Rainbow
oxide
layer.
In general,
compression
and the surface
and vary through to a tensile stress region
is in tension.
at the reduced layer/oxide layer interface. The signifies the location of the neutral plane within
the z-component
12
of the internal
stress
is negligible.
400 300 200 D..
100 0 co -100 E -200 -300 -400 -500 Thickness
(mm)
600
+0.1 "-m-- 0.2
4O0
---,ik-- 0.3
=
200
--0--0.5 --)1(---0.7
0 -200 n
-40O -600 Thickness (mm)
(a; top) The distribution of internal stress when a circular Rainbow is cooled down from reduction temperature to room temperature. Positive values on the vertical scale represent tension. Results presented are for the center of the circular specimen; and (b; bottom) the distribution of planar stress through the thickness of a circular Rainbow with different thickness ratios. Results are for the center of the circular specimens.
Figure 6.
These
results
two dimensional the surface
are qualitatively
finite
region
element
similar
analyses
of the ceramic,
to those
of circular
which
of Li and
Rainbow
approaches
coworkers
ceramics.
4']4 who
The tensile
300 MPa for this sample
carried stress
geometry,
out
within results
in a different domain configuration (higher a-domain percentage) compared to typical monolithic PZT piezoelectrics and a higher effective d3] piezoelectric coefficient in the surface region. 4 To further
study
the effects
of actuator
with different reduced layer-to-piezoelectric these simulations are shown in Figure 6b. actuators
are obtained
when
geometry
layer thicknesses Maximum tensile
approximately
optimal
for maximizing
Rainbow
the stress
for actuators
is reduced
during
the
in Figure 7 which plots the stress at the of reduced layer ratio. Given the effects thickness ratios of approximately 30 -
electromechanical
13
profiles
were also studied. The results of stresses in the surface region of the
30 - 35 % of the piezoelectric
fabrication process. These results are further illustrated surface of the center of a circular Rainbow as a function of tensile stress on enhancement of the d-coefficients, 35% appear
on stress,
response.
It may
also be noted
in this figure
that other actuator layer, which would
piezoelectric Compared
to standard
unimorph
geometries can result in compressive stresses throughout suppress the electromechanical response of the device,
the even
actuators.
300 250 200
0.
150 09
100
e-
t_
5O
O.
0 -50 Thickness Ratio
Figure
7.
Linear
The planar stress at the surface of the oxide layer in the center of the sample reduced layer/total thickness ratio.
3D modeling
A similar ratio of stainless were
predicted
While
this
devices, would
might
be expected
60 MPa,
be expected
magnitudes
performance
and Rainbow
compared
based
on the
of predicted
of the two families
to result
in a higher
While
linear
analyses
of the
dome
occurs
Rainbow
profile center
to - 350 MPa for the Rainbow
stresses
effective
processing are
temperature
in agreement
devices.
piezoelectric
with
3 Because
coefficient,
actuators,
the
higher higher
in agreement
device.
2
of Thunder observed
stress
levels
displacements
with
experimental
analyses
characteristics during
Modeling
devices
lower
of actuator
are predicted for Rainbow compared to Thunder studies of the two families of devices. 3 Non-linear
of
was carried out on Thunder actuators using a 0.3 thickness thickness. Tensile stress levels in the surface of the device
to be approximately relative
displacement
of Thunder
modeling approach steel to total device
effect
the
- Comparison
as a function
approaches
does indeed vs. the edge
such
as
formation
processing currently
those
and
above
intemal
is generally
under
so large
investigation
display a dependence of the device varies).
are
stress utilize
useful
profile, that
in
understanding
the bending nonlinear
a non-linear
analysis method,
on radial position as expected For circular Rainbows, higher
general
deformation
that
is required. and
the stress
(i.e., the stress at the stresses are noted near
the outer 1/3 of the devices, in agreement with observed x-ray diffraction results for (200) and (002) peak intensities compared to the central region of the Rainbow. 4 These results are shown in Figure predicted.
8. Also in this figure, Hoop
stresses
it is interesting
represent
to note that both hoop
the x-component
of stress
and radial
in the y-direction
stresses (and
may be
y-stress
in
x-direction), while the radial stresses are the x-stress in the x-direction and the y-stress in the ydirection. Both show a trend of increasing stress with distance from the center of the surface of
14
3.00E+08
/1
2.50E+08 2.00E+08 O..
1.50E+08
to to
_- STR
X-X
-.- STR
X-Y
- STR
X-Z
-.- STR
Y-X
-*- STR
Y-Y
--- STR
Y-Z
1.00E+08
o3
5.00E+07
E k..
t"-
0.00E+00 -5.00E+07
f
5
lo
\
-1.00E+08 Radius
Stress across the surface of a circular Rainbow ceramic with a 0.3 thickness ratio, 15 mm radius and 0.5 mm thickness; center (0 mm) and outer edge (15 mm). Definition of nomenclature: l't coefficient: direction; 2 na coefficient: stress direction.
Figure 8.
the piezoelectric, stress
is higher
expected
(mm)
although toward
to contribute
the variation the
edge
in radial
stress
of the actuator.
to the non-linear
is greater,
and the magnitude
It is important
piezoelectric
to note
coefficient,
that
which
of the radial
both
stresses
will be defined
are
by the
planar stress. Considering the variation in surface stress from the center to the edge of the device, this figure also demonstrates the need to model stress profiles in these devices using .nonlinear modeling approaches. Previous linear analysis studies 2 of the stress across the surface indicated
that the stress Results
methods
was relatively
for thickness
are shown
that the highest
ratio
in Figure
tensile
stresses
agreement with experimental linear analysis of the stress predicted
for higher reduced Similar studies have
the results analyses ratio
are presented of the stress
of the actuator
9.
effects
on surface
The results
are again
stress
are similar
predicted
determined
to those
for a reduced
by non-linear
of the linear layer
thickness
layer thickness ratios. been carried out on rectangular 10 through
at the center
has a significant
12.
Figure
of the cantilever
effect
on stress
cantilevers
ratio
of different
I 0 illustrates actuators.
profile,
analysis
approximation As with stresses geometry
the results aspect
the are and
of non-linear
As may be seen,
with higher
in
of 30 %, in
observations of actuator displacement performance. at the center of the actuator, compressive surface
in Figures profile
uniform.
ratios
the aspect generating
higher tensile surface stresses at the center of the actuator. However, the results also indicate that the compressive stresses in the lower part of the piezoelectric are greater for this actuator, which may reduce the effective piezoelectric coefficient of this region of the device, 4 and hence, reduce
the displacement
performance
of the devices.
Actuators
that
(length: width) may offer the optimum combination of high tensile of the device and lower compressive stresses in the lower portion aspect
ratios
of 2:1, while
have reduced
tensile
stresses
displacement
performance.
having
lower
in the upper
compressive
stresses
part of the device,
15
have
an aspect
ratio
of 3:1
stresses in the upper portion of the device. Devices with
in the lower and may,
part of the device
as a result,
display
also
reduced
4.E+08 3.E+08 -.-0.1
2.E+08
----0.2 n
1 .E+08
,-0.3
Ct)
-_- 0.5 f,.,.
0.E+00
!
-.-0.7
O3 t-,
10
-1 .E+08
O_
-2.E+08 -3.E+08 Radius
(mm)
-4.E+08
Figure 9.
Effect of thickness reduction ratio on the stress at the surface of the piezoelectric layer, from the center (0 mm) to the edge (15 mm); circular Rainbow ceramics with 15 mm radius and 0.5 mm thickness.
6.00E+08 3xl .0
4.00E+08
3xl .5 3x0.5 3x3
_2.00E+08 0.00E+00 -2.00E+08 -4.00E+08
d
-6.00E+08 Thickness
Figure 10.
Effect of Rainbow actuator aspect ratio on the surface profile at the center of rectangular actuators with a 0.3 (30%) thickness ratio. Dimensions shown represent width and length in cm.
]6
3.50E+08 3.00E+08
a
3xl.0
.....
b
2.50E+08
b
IS Z 3x0.51 33_-5
a c d
"" 2.00E+08 13.
1.50E+08 ¢n 1.00E+08 5.00E+07 0.00E+00 -5.00E+07 Radius Figure 11.
Planar stress along the length direction in rectangular Rainbow actuators with different aspect ratios; actuator length: 30 mm; center of rectangular actuator: 0 mm, edge of rectangular actuator: 15 mrn.
Taken
together,
these
results
cantilever displacement performance, standard unimorph theory, 7'32 which width
or internal
(square
devices)
stress have
levels
thus
suggest
of the devices
minimal
Stresses
along
ratio
have
stresses
the surface also
been
Again,
as with stress
profile,
aspect
ratio;
aspect
higher
in the surface
of the actuators
stress
along
region
higher
aspect
lower
effects
similar
and width
of the actuators
on
However, as a function
in Figures
with
the
of 1
to predictions
directions
is a function
stresses,
ratio
displacements.
are illustrated
tensile
ratio
with an aspect
of the device,
in the length
the surface
be
Cantilevers
and the results
produced
will
contradiction to the predictions of in the descriptive equations for the
to have generate capabilities.
investigated
ratios
there
(see below).
for circular devices, and would be predicted they probably display enhanced load-bearing of aspect
that
which is in apparent does not include terms
11 and
12.
of the actuator
magnitude
of the
stress approaching 280 MPa for the 3.0 x 0.5 cm actuators in the length direction. As actuator squareness increases (decreased aspect ratio), the tensile stress toward the center of the device decreased dramatically and even becomes compressive in the center. These results for the square actuator (Figure
strongly
resemble
correlation
between
devices.
A
of the circular
the magnitude
strong
correlation
with
the
devices. above
of the surface is expected
stress here
Surface
stress
geometries
are
along
in progress
and the displacement
because
of
the
the width
effects
direction
to verify
performance of
stress
these
four
the
of the on
the
dmik coefficients.
piezoelectric
Modeling
those
12) displayed similar trends. Tests of actuators fabricated
deformation Finally,
Figure
of Rainbow 13 presents
actuators the
3-dimensional
geometries to show the induced deformation that occurs the simulations are in good agreement with experimental
17
linear
models
of
actuator
during device fabrication. The results observations for rectangular actuators
of
3.00E+08 D3x0"5 E 3xl.0
a
b
D
c
c
_3x3
a 2.00E+08
b
3xl.5
1.00E+08 t_ (3.
d
v
Or) t_
0.00E+00 d
&,..
09 -I.00E+08
-2.00E+08 -3.00E+08 Radius Figure 12.
Planar stress along the width direction in rectangular Rainbow actuators with different aspect ratios; actuator length: 30 mm; center of rectangular actuator: 0 mm, edge of rectangular actuator: 15 ram.
fabricated
at Clemson
along
the x-axis
longer
x-axis
curvatures Deformation
University.
These
of approximately
length
compared
4 cm long,
2.5 mm and 0.3 mm along to the model
(which
are expected compared to the results for similar Thunder devices
A 3;=0.5 cm rectangular
Rainbow
1 cm wide
with 0 3 thickness
studied
simulations, are reported
actuators
the shorter
display
heights
y-axis.
Because
of the
3 cm long actuators),
slightly
higher
in agreement below.
ratio
dome
A 3xl
cm rectengLqar
with
R_nbow
observation.
with 0.3 thickness
ratio
415
_._
-1 -1 g
•
-2
-t5
I
1 2_/_1_
5
-15
-
15
A 3xl
•
-2
5 cm rectangular
y_O
Rainbow
with 0,3 thickness
ratio
x
A 3x3
cm rectangular
Rainbow
_th
-o
0.3
thickness
ratio
5 -15"_
-15 15
Figure 13.
Equilibrium shapes of rectangular Rainbow with different dimensions. have different x, y, and z scales.
18
-ln
15
Note that the boxes
Measured
stress
effects
on domain
wall movement
Recent results in other x-ray diffraction studies of (002) and (200) peak intensities University support arguments for the stress enhancement of the electromechanical
Clemson response
of these
devices.
28 Specifically,
Rainbow ceramics appears to enhance electric field, thus improving the strain greater stress
detail levels
movement layer
by preparing in the surface
for an applied
ratios.
region
electromechanical
wall
tensile
response
More
wall motion complete
response extrinsic
of the
is anticipated
material,
which
enhanced
for the
with
5 times
region
of circular wall
with
higher
surface
to more
for a more
three accurately
accurate
switching
than
the
high
field
times
and greater
stresses.
This
that at lower
describe
assessment
of a The stress.
the ferroelastic
of the
domain
d-coefficient
Rainbows.
movement
to the presence of the devices.
devices. 4 also supports
with simple
wall
reduced
response,
domain wall motion due to the observed response
will allow
larger
of
to different
in domain
intrinsic
and
this hypothesis.
In
switching
electric fields in standard, poled polycrystalline ferroelectric ceramics of tensile stress in the surface region of the Rainbow actuator results
that is a_sproximately ceramic.
region
with different
electromechanical
devices
to begin
lead
changes
90 ° domain
is, in fact, approximately
domain
which
prepared
are for the surface
is required
of stress-enhanced
in the surface
14 illustrates
for samples
promote
enhanced contribute
high stresses
of this kind
stress
ratios, 2s
contributions to the performance of these stress-biased Analysis of the earlier data of Li, Furman and Haertling
fact, a comparison by large presence
under
data
Figure
is associated
result again supports the notion that lateral tensile stress can significantly domain
stresses
stresses
movement
thickness
of 10 kV/cm
the reported
higher
90 ° domain
with different
field
of a tensile
the a-domain to c-domain switching that occurre under response of the device.S2 Moon investigated this effect in
of the actuator.
electric
In this figure,
As may be observed, since
Rainbows
the presence
at
of the
promoted
suggests that the in a d-coefficient standard
poled
Internal stress 3xl 0 7 N/m 2 6
--
A
-A
Internal stress 2x10 7 N/m 2 4--
_-
_-
_
;
a_
Internal stress lx10 r N/m 2 2--
0 3
m v
I
I
i
1
4
5
6
7
Grain size (l_n)
Figure 14.
Variations in domain wall movement as a function of the tensile stress in the surface region of the piezoelectric. Results obtained by in-situ x-ray diffraction measurements. 28
19
E. Summary
of Results
Based modeling,
on the
and
Thunder
equivalent
devices,
performance, devices
results
circuit
while
similar
different
devices,
mechanical
accurately
ratios
predict
issues
it seems to those
not
be
It should
also be noted
to applied
of the stress-biased
that
voltage
yield that
a term
that
similar
in Rainbow enhance
we need be able
unimorph
(reduced
in the
layer)
-
devices.
However,
to know
the magnitude
to quantify
to
the effects
coefficient
for the width
and device
of the layers
metal
the
unimorph
performance
stress-biased
theory,
motion,
significantly
optimum
devices,
unimorph
not contain
wall
characteristics
layer, and must d-coefficients.
in standard
does
may
mechanical
performance
of the stresses present within the piezoelectric stress on the non-linearity in the piezoelectric displacement
which
domain to conclude
stresses
surprising
in the highest
the performance
on
reasonable
tensile
the similar
perhaps
result
measurements
of the
Considering
it should
layer
effect
analysis,
the presence
are also critical.
piezoelectric
of stress
that relates
of the device.
Since
of tip we
have already shown that the stress profile in these devices is dependent on the aspect ratio, some differences in the performance of devices fabricated with these geometries would be predicted. Studies III.
of actuators
with varying
Suggestions
for Future
geometries
are currently
in progress
and are discussed
below.
Work
Summa_ A suggested the importance
general
of stress,
goal of future
geometry,
research
and loading
Rainbow and Thunder stress-biased actuators. work with finite element and equivalent circuit using fiber optic and LVDT sensors, different field and stress conditions. Research (1) The
in the following use
of finite
areas
element
in this area
effects
is to further
our understanding
on the displacement
performance
This goal would be accomplished modeling, characterization of device
and critical
experiments
to study
domain
of
of both
with further performance
switching
under
is recommended: modeling
in a dynamic
under load and applied electric field. different geometries and stress profiles
fashion
by studying
This will be carried to assist in quantifying
the devices
out for devices with effects of stress on
performance. of the equivalent (2) The extension five-layer model 29 that should performance. characterized. of
the
devices
be
to quantify
improved d33 under
determine tensile
will
for
studied
into the equivalent
We will begin for the measure
model
allow
Both geometry and mass In addition, aging effects
incorporated
(3)
circuit
the effects
stress
effects
and
the effects
accurate
loading effects of the dielectric the
circuit
by incorporating more
results
of
a recently predictions
developed of
actuator
on actuator strain will be and piezoelectric constants these
investigations
will
be
models. of stress
on the d-coefficients
in these
materials
prediction of device performance by using a tensile tester to different loads. While extensive research has been carried out to of compressive
stresses
has been reported.
20
on performance,
relatively
little work
on
(4) We will usein-situ x-ray diffraction techniquesto characterizedthe (002)/(200)peak intensity ratio of devicessubjectedto various electric fields. This research will be carded
out at Sandia
National
Laboratories
using
a micro-spot
beam to - 30 Ism and the devices will be characterized so that domain wall movement characteristics under stress conditions can be determined.
(5)
As studies
in the finite element
compare the results allows for the better
(6)
We will explore
(7) To
the extension
performance
use
of
of
a
to
of Suedlested
that would
of the devices. grating
on imprint
be expected
to
the
more
performance,
fully
identify
and
testing
This
technique,
investigations
driver
in the
the performance
of
the
throughout
dome
little
will involve
the application allow
will
better
for
the
throughout the piezoelectric layer, and on the non-linearity in the piezoelectric
that there
device
performance.
are significant
aging
This work
and
relaxation
is the first to begin
to
potentially, of aging
useful
lifetime.
by determining
The
thrust
the effects
of the
continued
of prolonged
work
stress
will
states
of the materials used in the fabrication studies the following experimental
on of and
are planned.
large
is known
and cantilever
is required
the
device
stress effects on d-coefficients enhancements to the piezoelectric displacements
of Rainbow
and
d-coefficients
Thunder
devices,
are
about
geometries
predict
the
effects
exhibit
the performance
of tensile
both stress of these
21
stresses
states. devices.
on these
properties,
An understanding For example,
considered
a more
assessment of dij vs. c will be undertaken. Significant previous studies have understanding the effects of compressive stress on dielectric and piezoelectric relatively
which
in displacement performance under dc field, dielectric constant, constant vs. time, despite the obvious importance of such effects
the causes
2. Quantification of tensile Since stress-induced prime
we will
to determine
deformation
which
cross-section
the piezoelectric, dielectric, and domain configuration Rainbow and Thunder devices. To support these modeling
approaches
local
indicate
to impact
quantify the temporal changes and stress effects on dielectric on device
areas progress,
Activities
1. Aging Effects As our results effects
the
is required.
monitor
microscopic characterization of d-coefficients thereby, quantification of the effects of stress coefficient. Details
research
theory to predict
unimorph
actuators
interferometry
layer
diffraction
of standard
to focus
and Thunder actuators. Two reports of this approach have 24'25 but further exploration in the use of this approach for
of stress-biased
Moire
piezoelectric
circuit
from the two different modeling prediction of device performance.
of stress-biased Rainbow appeared in the literature modeling
and equivalent
system
in a cross-sectional geometry, different fields and different
quantitative
been aimed at response, but even
of these
it should
a
though
properties
be possible
to
further developstandardunimorphtheoryto incorporatethe stresstermsin the dij coefficient(see below). To begin to quantify the effectsof tensilestresson the d-coefficientsof thesematerials, the centralregionof poled rectangularceramicplateswill be electrodedand the sampleswill be mountedin a tensile tester. A rangeof tensilestresses,up to mechanicalfailure will be applied to the piezoelectric.While holding the sampleat a constantknown stressstate,the low field and high field dielectric constantswill be measured. To characterizethe non-linearity of the piezoelectric coefficient, a fiber optic displacementsensor will be used to monitor the displacementin the 3-directionundervarioustensilestressesappliedin the 1-direction. This test situationclosely resemblesthe field-stressconditionsfor both Rainbow andThunder ceramics, and as such, should provide quantificationof the non-linearpiezoelectric higher order terms neededfor the developmentof moredetailedmodels. This will lead to better predictionof the performanceof thesestress-biased devices. The key experimentscarried out in this task will employ PZT with the material compositionand grain sizes typically used in Rainbow and Thunderfabrication. Investigationsinto the time-dependence of theserelationshipswill also be carriedout. 3. In-situ x-ray diffraction characterization of stress andfield effects The optimal method to fully characterize the interrelationships applied
stress,
a Moire
interferometry
and the domain
points
within
below
as an optional
significant
the internally
funding
wall movement
experiment stressed
research increase
that leads
(see below) piezoelectric
task.
layer.
However,
of the
current
to piezoelectric
to characterize the
level
the local
The
scope
basis
electric
response
is to carry
displacement
for
experiment the
out
is outlined
would
student
field,
at different
of this experiment
of this
of support
between
require
working
a
on this
program. An
alternative,
information,
reasonably
effective,
is to use of micro-spot
x-ray
but
less
diffraction
expensive
method
to characterize
(200) and (002) peaks as a function of position within the piezoelectric electric fields. We also propose to study the peak intensity ratio of these applied the
electric
(200)
field.
and
Because
(002)
peak
stress profile domain wall
on domain movement.
piezoelectric
response,
of the relationship
intensities, population, Further, it
will
be
between
it will be possible
to
the relative
obtain
similar
intensities
of the
layer under different peaks as a function of
the a- and c-domain to characterize
the
population effects
and
of internal
as well as the interaction of stress and electric field since domain wall movement is strongly correlated possible
by
using
this
technique,
to
obtain
on to
important
information to quantify the effects of stress on the piezoelectric coefficients of these materials. This research will be carried out at Sandia National Laboratories using a micro-spot x-ray diffraction
system
with approximately peak intensity 4. Inclusion The capacitance
that
can focus
500 p.m thick
the x-ray
piezoelectric
ratio at approximately of property
aging
experimentally
effects
coefficients
the
model
time-dependent
which
locations
into equivalent
models. As we have seen from Figure resonance behavior of Rainbow structures By making
to - 30 _tm.
layers,
15 different
determined
and piezoelectric
source
with
actual
22
model being
and Thunder
actuators
for determination
the depth
of the
of the layer.
models
phenomenological 10, the (despite
will allow
across
circuit
will be added
Rainbow
dependencies
of stress
parametrically
and
time
to the equivalent
on
circuit
is very accurate in determining the only exact for 1 D plates and stacks).
material
values
for
the
various
relaxation
mechanisms,the equivalentcircuit approachwill becomean evenmorepowerful predictivetool to modeldeviceperformance. 5. Finite
element
modeling
In this task, devices studies
device
of circular characterize to assess
load and field
element
modeling
in a "dynamic"
and applied electric field predict device performance
geometries.
A variety
of loads
fashion
by studying
of variations
Subsequently,
will be applied
as point
sources
to the center
in device
different
monitor
stiffness
magnitude
displacement
and load-bearing.
electric
performance
fields
will be applied
under
different
to the actuators
field
conditions.
performance for the actuators and the results obtained by finite element modeling. Electric
fields
different loads electric fields.
will
in the
finite
calculations
element
different
of stress
geometries
on actuator With regard
characteristics
and
profiles, stress
these
profiles
performance. to the performance
of the reduced
modeling to assist
approach
of
efforts
to devices
interface
will be carded
devices,
subjected
polymeric
devices,
"Robon"
As studies
circuit accurate
the equations
will
be used
previous
in the finite
compare the results predictions of strain equivalent relatively
FEM layer;
to study
models
element
from the two and frequency
have
it
is
play a critical
and equivalent
different response
method is substantially with respect to resonance
required
to predict
the response
with more typical piezoelectric resonators, which the equivalent circuit model originally By accurate
extending
predictions
the
equivalent
of resonance
stress
circuit and
strain
profiles
not included
of stress
also
role
unclear
circuit
research
devices
areas
model
on actuator performance.
that
include
progress,
we plan to
to look
follows
quartz
at gradient
should
plate
We will model functionally graded our recently developed five-layer
approximated model takes lead
resonators,
structures,
be obtained.
even
The results
structures by extending the equivalent model] 9 In this model, the material
by a number (5, or fewer, in the present case) into account the effects of impedance mismatches
to acoustic
reflections.
The
output
remains
resonance
23
for the
that associated
analyses will also be compared to finite element modeling. The comparison of these allow for further insight into mechanics and stress effects on device performance. using
the
Both methods allow geometry and while
closely
as AT- or SC-cut developed.
response
the
profile
the predictions of this method appear predictions. This is because the form of
of bimorph such was
if
in the stress
for devices
and
this layer.
modeling approaches. as a function of actuator simpler, frequency
to
out for devices
effects
and device performance. Therefore, the influence of a non-planar interface properties will also be modeled with regard to both stress profile and displacement For Thunder
to results
for a range of applied by experimentation. As
in quantifying
Rainbow
layer/piezoelectric
to equivalent displacement
will be compared
and load/actuator displacement will be characterized Again, these results of these simulations will be verified
with the characterization geometry
be applied
of these
without
Resonance
frequencies will be identified for these conditions and the results will be compared circuit predictions. In addition, modified unimorph theory will be used to predict
with
the
conditions. Again, the goal of these under different operational conditions
and square actuators, and also to the tips of cantilever actuators. Initially, we will the resulting deformations/displacements of the actuators under these different loads the impact
to
under
we will use finite
under different load is to be able to better
for different
load
of devices
for more
of these
results
will
circuit model gradation is
of small uniform steps. The new between the respective layers that
frequency,
maximum
displacement,
andsurfaceaccelerationon the •characterizing
the influences
this had been
simply
structure.
of the reduced
modeled
as a thicker
is fine for relatively thin electrodes, of the Rainbow thickness (such displacement) more exactly model (with simplifies electrical.
This new layer (and
model
also will be useful
in the case of the Rainbow heavier)
electrode.
in more devices.
In theory,
but as the reduced layer becomes as the - 30 % reduced layer
to mimic the response of a cantilever This thrust will compare the five-layer
a more significant fraction thickness that maximizes
structure model
when again the only driving force is to the one layer model as well as the
to actual effort.
device
performance.
6. Develop extension of standard unimorph theory for stress-biased actuators Standard unimorph theory does not account for either internal stress in the
unimorph
prediction
theory
of device
to predict
displacement
Rainbow
performance.
performance,
In the
the principal
The
in their paper that non-linear and they begin to address
previous
modification
y-direction (length)
to depend effects
stresses stresses.
on device
of width
in these Further,
width,
direction
width
extensions
of was
that occurs during correctly suggest
stress effects will play a role in determining device performance this effect for strain and dielectric displacement, however, the
stress acts only along the length We suggest this because our finite
direction
of
to the theory
magnitude of these effects is not considered. Further, based on the results of previous finite element modeling studies described above, statements in the paper
induced
effects
or cantilever
the inclusion of a term for 0-field tip deflection, based on the TCE mismatch processing which results in a domed structure. 16 The authors of this approach
mechanical consideration.
Previously,
this approximation
the transmission and interaction of acoustic waves through these layers must be modeled. The five-layer model is an extension of our previous single layer plate mass loading of electrodes) that accounts fully for these effects. The model also
experimental results to realize better simulations aging also will be included in this newest modeling
effects
accurately
devices while
tip deflection
stresses
are
of the element
of approximately
blocking is not.
device" seem to simulations suggest
force Again,
on the piezoelectric
the
some such
demand further that fabrication-
same
magnitude
at the tip of the cantilever this analysis
coefficients.
seems
of our as "the
as x-
is predicted
to not consider
As we model
the
the performance
of wider and thicker devices, the approximations made under Euler-Bernoulli beam theory are not necessarily valid, and device width may become an issue. In this task, we will further develop standard unimorph theory by modifying the boundary conditions
used
terms
into the
above
studies,
of geometries,
to describe equations improved should
cantilever
performance
as possible. predictions
When of device
and by incorporating coupled
performance
However, one effective this method, the device is not change grating electric thickness
the
non-linear
experimental
for actuators
piezoelectric
results
fabricated
from
the
with a variety
be possible.
7. Moire Interferometric study of domain wall motion The accurate characterization of stress effects
fine line diffraction
with
experimental cross-section
grating
in the stress
method would
would
be applied
state
of the device,
vs. piezoelectric on piezoelectric
will occur.
is non-trivial.
is to utilize be polished
a Moire interferometry technique. In to a reasonable smoothness. Then, a
to the device
in a domed
there
is no initial
and the interferogram across the thickness of the device field is applied to the Rainbow (or Thunder) actuator, of the piezoelectric
profile coefficients
Because
24
of the fineness
configuration.
deformation is uniform. deformation
Since
there
of the diffraction However, as an throughout the
of the diffraction
grating,
the
local deformation, with micron scale resolution, can be determined. Since the sign and magnitudeof the stressvaries across the piezoelectric,the local deformation (strain = dcoefficientx field) will vary. Characterizationof the local deformationwill thereforeallow for anin-situquantitativedeterminationof theeffectsof stressonthe piezoelectriccoefficient. IV.
List of Publications
and Presentations
Publications 1.
2.
J. Ballato, R. Schwartz, and A. Ballato, "Network Formalism for Modeling Functionally Gradient Piezoelectric Plates and Stacks and Simulations of Rainbow Ceramic Actuators," IEEE
UFFC,
(1999)
R.W.
Schwartz,
Submitted.
P. Laoratanakul,
"Understanding Mechanics Proceedings, SPIE, Smart 3.
W.D.
Nothwang,
Circuit
W. D. Nothwang,
R. W. Schwartz,
Modeling
J. Ballato,
and Stress Effects in Rainbow Materials and Structures (2000)
for Rainbow
and J. Ballato,
Ceramics,"
Y. Moon,
"Experimental
(2000)
and A. Jackson,
and Thunder Submitted
Actuators,"
Validation
to be submitted
of Equivalent
to IEEE
UFFC.
Presentations 1.
R.
W.
Schwartz,
M. L. of Actuators
"Development Actuators,"
American
(November 2.
R.
W.
17-18,
5.
Actuators,"
Stress
3M Company,
Effects
University,"
R. W. Schwartz,
Minneapolis,
the
Nashville,
Performance
MN (March
W. Schwartz, "Modeling Performance of Rainbow
Sandia
National
P. Laoratanakul, Mechanics
of the
Actuators," American 2000. Scheduled
and
Effects
Ceramic
W. D., Schwartz,
to Describe Society
Meeting,
TN
of
Stress-Biased
18, 1999). Invited.
J. Ballato, "Equivalent Ceramic Society Annual
Circuit Meeting,
Annual
Aging
and
Meeting,
and Characterization Ceramics," American Piezoelectric
Laboratories,
Modeling of Indianapolis,
Effects
of Mass Society
Loading Annual
R. W., and Ballato,
Fatigue St. Louis,
Effects
J. Ballato,
in Rainbow
25
NM
Research (June
Y. Moon, Thunder
on
the
Meeting, J., "The 30 - May
Strain
and
Response
St. Louis,
MO,
3, 2000.
A. Jackson, SPIE, and
of Piezoelectric 30 - May Circuit
American
Scheduled
at 1999).
Modeling
April
Use of an Equivalent Ceramics,"
21,
Actuators,"
Beach (March 6 - 9, 2000). J., "Simple Plate Resonator
in Piezoelectric
MO, April
and
of Stress and Ceramic Society
Ceramics
Albuquerque,
W. D. Nothwang, Stress
Smart Materials and Structures Meeting, Newport Nothwang, W. D., Schwartz, R. W., and Ballato,
Nothwang,
Sectional on
Annual Meeting, Indianapolis, IN (April, 1999). R. W. Schwartz, "Electronic Materials and
Characterization
8.
Southeast
1999).
"Understanding 7.
"Understanding
P. Laoratanakul and R. Geometric Effects on the
Clemson Invited. 6.
Society
W. D. Nothwang, R. W. Schwartz, and Piezoelectric Rainbow Actuators," American IN (April,
4.
Ceramic
and K. Gass, in Stress-Biased
1998).
Schwartz,
Piezoelectric 3.
Richard, P. Laoratanakul, P. L. Robertson, for VCSEL Positioning and the Role of Stress
3,
Model Ceramic
V.
2
3 4 5 6 7 s 9 _0 _ _2 _3 _4 _ 16 17 _s t9 20
2_ 22 23
References G. H. Haertling, "Rainbow Cet'amics - A New Type of Ultra-High Displacement Actuator," Am. Ceram Soc. Bull., 73, 93 (1994). R.W. Schwartz, P. Laoratanakul, and W. D. Nothwang, "Modeling and Characterization of Geometric Effects on the Performance of Rainbow Actuators, NASA Interim Contract Report, March 16, 1999; submitted to Dr. Wallace Vaughn. S.A. Wise, "Displacement properties of RAINBOW and THUNDER piezoelectric actuators," Sensor and Actuators A - Physical, 69 (1), 33 (1998). G. Li, E. Furman, and G. H. Haertling, "Stress-Enhanced Displacements in PLZT Rainbow Actuators," J. Am. Ceram. Soc., 8016], 1382, (1997). A. Ballato and J. Ballato, "Accurate Electrical Measurements of Modern Ferroelectrics," Ferroelectrics, 182, 29 (1996). C. Elissaide and L. E. Cross, "Dynamic Characteristics of Rainbow Ceramics," J. Am. Ceram. Soc., 78 (8), 2233, 1995. J.G. Smits, S. I. Dalke, and T. K. Cooney, "The constituent equations of piezoelectric bimorphs," Sensors and Actuators, A, 28, pp. 41 -61 1991. R. G. Bryant, "THUNDER actuators," presented at the Enabling Technologies for Smart Aircraft Systems Workshop, Hampton, VA, May, 1996. G.H. Haertling, "Rainbow ceramics - a new type of ultra-high displacement actuator," Am. Ceram. Soc. Bull., 73, 93 (1994). G. H. Haertling, "Chemically reduced PLZT ceramics for ultra-high displacement actuators," Ferroelectrics, 154, 101 (1994). G.H. Haertling, U. S. Patent 5,471,721. G. H. Haertling, "Rainbows and Ferrofilms - Smart Materials for Hybrid Microelectronics," Ceramic Transactions, 68, pp. 71-96, (The American Ceramic Society, Westerville, OH, 1996). C. Elissalde, L. E. Cross and C. A. Randall, "Structural-property relations in a reduced and internally biased oxide wafer (RAINBOW) actuator material, "J. Am. Ceram. Soc., 79 (8), pp. 2041-48, 1996. G. Li, PhD. Thesis, Clemson University, 1995. W. Vaughn, NASA-Langley, private communication, 1999. Q.-M. Wang and L. E. Cross, "Tip deflection and blocking force of soft PZT-based cantilever RAINBOW actuators, J. Am. Ceram. Soc., 82 (1), pp. 103-110, 1999. Q.M. Zhang and L. E. Cross, "BaTiO3 domain bimorph actuator," Ferroelectrics, 98, pp. 137-153, 1989. R.W. Schwartz, unpublished results (1999). R.F. Brown, "Effect of two-dimensional mechanical stress on the dielectric properties of poled ceramic barium titanate and lead zirconate titanate," Can. J. Phys., 39, pp. 741-753, 1961. Q. M. Zhang, J. Zhao, K. Uchino, and J. Zheng, "Change of the weak-field properties of Pb(Zr,Ti)O3 piezoceramics with compressive uniaxial stresses and its links to the effect of dopants on the stability of the polarizaitons in the materials,",/. Mater. Res., 12 (1), pp. 226-234, 1997. A. Ballato and T. Lukasek, "Distributed Network Modeling of Bulk Waves in Crystal Plates and Stacks," ECOM, p.1 (May 1975). A.R. von Hippel, Dielectrics and W.a..ves, (John Wiley and Sons, New York, 1954). V. Giurgiutiu, Z. Chaudhry, and C. A. Rogers, "Energy-Based Comparisons of Solid-State Actuators,"_ Center
for Intelligent Materials Systems and Structures, Sept. 1995. 24 W.Y. Shih, W.-H. Shih, and I. A. Aksay, "Scaling Analysis for the Axial Displacement and Pressure of Flextensional Transducers," J. Am. Ceram. Soc., 80 (5), 1073 (1997). 25 Q.-M. Wang, X.-H. Du, B. Xu, and L. E. Cross, "Electromechanical coupling and output efficiency of piezoelectric bending actuators," IEEE Trans. UFFC, 46 (3), pp. 638-646, 1999. 26 j. East, PZR Technologies, Inc., private communication, 1998. 27 G.H. Haertling, private communication, 2000. 28 Y.-K. Moon, Ph.D. Thesis, Clemson University (1999). 29 j. Ballato, R. Schwartz, and A. Ballato, "Network Formalisn for Modeling Functionally Gradient Piezoelectric Plates and Stacks and Simulations of Rainbow Ceramic Actuators," IEEE Trans. UFFC, (1999)Submitted. 30 W.D. Nothwang, unpublished results (2000).
26
31 G. H. Haertling, "Piezoelectric and Electrooptic Ceramics," in Ceramic Materials for Electronic_ 2 "d Ed., R. C. Buchanan, Editor, (Marcel Dekker, Inc., New York, 199 l), p. 163. 32 Q. M. Wang, X.-H. Du, B. Xu, and L. E. Cross, "Electromechanical Coupling and Output Efficiency Piezoelectric Bending Actuators," IEEE Trans. UFFC, 46 (3), 638 (1999).
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