Moulson A.J., Herbert J.M. Electroceramics: Materials, Properties, Applications
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Many development problems have been encountered and overcome, including
the risk of de-poling during processing (cutting, bonding, welding, sterilization)
and of fractures due to stresses imposed on the elements during bending the
array to conform to the catheter surface. Because both types of fault can degrade
the image, the achievement of optimum mechanical strength is of the utmost
importance demanding a very high quality microstructure free from any stress-
raising defects. Typically the grain sizes of the various candidate piezoceramics
lie in the range 1 to 5 mm. E.L. Nix et al. [22] have carried out a detailed
analysis of the microstructures and mechanical and piezoelectric properties of a
range of candidate piezoceramics, including a single crystal material.
402 PIEZOELECTRIC CERAMICS
Fig. 6.36 The intravascular catheter: (a) the ‘phased array’ principle; (b) a schematic of the
catheter head (the guide-wire tube is 1.47 mm o.d.). The catheter as a whole is passed through
the artery along a pre-inserted guide-wire ((b) reproduced with permission of Jomed Inc.).
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6.5.4 Piezoceramic^polymer composites
Because of the special relationship between the structure of piezocomposites,
especially the 1–3 variety, and the major application – for underwater
technologies – the topic is dealt with in Section 6.4.6.
6.5.5 Summary
The discussion has highlighted a selection of applications chosen to illustrate the
two major functions of piezoceramics namely to generate voltages or
displacements. The choice, necessarily limited, also spans the displacement and
power levels commonly encountered.
Other applications include the following, listed according to the principal
functional modes.
. Small voltage generators (sensors): car crash sensors to activate air-bags, noise
cancellation, active suspension for cars, anti-‘knock sensor’ for i.c. engines,
liquid level sensing, etc.
. Displacement actuators: deformable mirrors, computer-controlled knitting
machine actuators, fuel-injection, ink-jet printer, cooling fans, electrical
relays, etc.
. High frequency generators at both ‘power’ and ‘signal’ levels: ultrasonic welding
of polymers, lithotripters, humidifiers, ‘buzzers’ (‘sounders’) of all types, non-
destructive testing devices, depth gauges for boats, under water and various
medical imaging technologies, etc.
The fascinating and instructive article, ‘Smart Electroceramics’ by R.E.
Newnham and G.R. Ruschau [23] is strongly recommended.
Appendix: Piezoelectric Relations for
Ceramicspoledinthe3Direction
Strains as functions of stresses in terms of compliances:
x
1
¼ s
11
X
1
þ s
12
X
2
þ s
13
X
3
ð6A:1Þ
x
2
¼ s
12
X
1
þ s
11
X
2
þ s
13
X
3
ð6A:2Þ
x
3
¼ s
13
X
1
þ s
13
X
2
þ s
33
X
3
ð6A:3Þ
APPENDIX: PIEZOELECTRIC RELATIONS FOR CERAMICS POLED IN THE 3 DIRECTION 403
x
4
¼ s
44
X
4
ð6A:4Þ
x
5
¼ s
44
X
5
ð6A:5Þ
x
6
¼ 2ðs
11
s
12
ÞX
6
ð6A:6Þ
Stresses as functions of strains in terms of stiffness coefficients:
X
1
¼ c
11
x
1
þ c
12
x
2
þ c
13
x
3
ð6A:7Þ
X
2
¼ c
12
x
1
þ c
11
x
2
þ c
13
x
3
ð6A:8Þ
X
3
¼ c
13
x
1
þ c
13
x
3
þ c
33
x
3
ð6A:9Þ
X
4
¼ c
44
x
4
ð6A:10Þ
X
5
¼ c
44
x
5
ð6A:11Þ
X
6
¼ 2ðc
11
c
12
Þx
6
ð6A:12Þ
Relations between the c and s coefficients:
c
11
¼
s
11
s
33
s
2
13
f ðsÞ
ð6A:13Þ
c
12
¼
s
12
s
33
s
2
13
f ðsÞ
ðA:14Þ
c
13
¼
s
13
ðs
11
s
12
)
f ðsÞ
ð6A:15Þ
c
33
¼
s
2
11
s
2
12
f ðsÞ
ð6A:16Þ
c
44
¼
1
s
44
ð6A:17Þ
f ðsÞ¼ðs
11
s
12
Þfs
33
ðs
11
þ s
12
Þ2s
2
13
gð6A:18Þ
The Poisson ratios:
n
12
¼s
E
12
=s
E
11
ð6A:19Þ
n
13
¼s
E
13
=s
E
33
ð6A:20Þ
404 PIEZOELECTRIC CERAMICS
Relationships between clamped and free permittivities:
e
x
1
¼ð1 k
2
15
Þe
X
1
ð6A:21Þ
e
x
3
¼ð1 k
2
p
Þð1 k
2
t
Þe
X
3
ð1 k
2
33
Þe
X
3
ð6A:22Þ
In Eqs (6A.23)–(6A.34) e represents the permittivity e
X
at constant stress and s
represents the compliance s
E
under short-circuit conditions.
The equations of state for the direct effect:
D
1
¼ e
1
E
1
þ d
15
X
5
ð6A:23Þ
D
2
¼ e
1
E
2
þ d
15
X
4
ð6A:24Þ
D
3
¼ e
3
E
3
þ d
31
ðX
1
þ X
2
Þþd
33
X
3
ð6A:25Þ
The equations of state for the converse effect:
x
1
¼ s
11
X
1
þ s
12
X
2
þ s
13
X
3
þ d
31
E
3
ð6A:26Þ
x
2
¼ s
11
X
2
þ s
12
X
1
þ s
13
X
3
þ d
31
E
3
ð6A:27Þ
x
3
¼ s
13
ðX
1
þ X
2
Þþs
33
X
3
þ d
33
E
3
ð6A:28Þ
x
4
¼ s
44
X
4
þ d
15
E
2
ð6A:29Þ
x
6
¼ 2ðs
11
s
12
ÞX
6
ð6A:30Þ
The coupling coefficients are related to the d coefficients by
k
33
¼
d
33
ðs
33
e
3
Þ
1=2
ð6A:31Þ
k
31
¼
d
31
ðs
11
e
3
Þ
1=2
ð6A:32Þ
k
p
¼
d
31
fðs
11
þ s
12
Þe
3
=2g
1=2
¼ k
31
2
1 n
12
1=2
ð6A:33Þ
k
15
¼
d
15
ðs
44
e
1
Þ
1=2
ð6A:34Þ
k
t
¼
e
33
ðe
x
3
c
D
33
Þ
1=2
ð6A:35Þ
APPENDIX: PIEZOELECTRIC RELATIONS FOR CERAMICS POLED IN THE 3 DIRECTION 405
The open- and short-circuit compliances are related by
s
D
11
¼ s
E
11
ð1 k
2
31
Þð6A:36Þ
s
D
12
¼ s
E
12
k
2
31
s
E
11
ð6A:37Þ
s
D
33
¼ s
E
33
ð1 k
2
33
Þð6A:38Þ
s
D
44
¼ s
E
44
ð1 k
2
15
Þð6A:39Þ
The piezoelectric coefficients are related by
g
31
¼ d
31
=e
X
3
ð6A:40Þ
g
33
¼ d
33
=e
X
3
ð6A:41Þ
g
15
¼ d
15
=e
X
1
ð6A:42Þ
e
31
¼ e
x
3
h
31
¼ d
31
ðc
E
11
þ c
E
12
Þþd
33
c
E
13
ð6A:43Þ
e
33
¼ e
x
3
h
33
¼ 2d
31
c
E
13
þ d
33
c
E
33
ð6A:44Þ
e
15
¼ e
x
1
h
15
¼ d
15
c
E
44
ð6A:45Þ
h
31
¼ g
31
(c
D
11
þ c
D
12
Þþg
33
c
D
13
ð6A:46Þ
h
33
¼ 2g
31
c
D
13
þ g
33
c
D
33
ð6A:47Þ
h
15
¼ g
15
c
D
44
ð6A:48Þ
d
31
¼ e
31
ðs
E
11
þ s
E
12
Þþe
33
s
E
13
ð6A:49Þ
d
33
¼ 2e
31
s
E
13
þ e
33
s
E
33
ð6A:50Þ
d
15
¼ e
15
s
E
44
ð6A:51Þ
Problems
1. Quote from p. 341
This suggests that the large d-coefficients shown by some ferroelectric materials are due
to a combination of large electrostriction coefficients and large spontaneous
polarization and permittivity values.
Explain the reasonableness (or otherwise) of the statement in terms of the elementary
physics of the processes involved.
406 PIEZOELECTRIC CERAMICS
2. (a) Show that the two ways of expressing the units of the piezoelectric d-coefficient are
dimensionally equivalent.
(b) From the electrostriction data given in Fig. 6.15 estimate an effective maximum d
33
coefficient. [Answer: 200 pC N
1
]
(c) Suggest an explanation for the dependence of the electrostriction coefficient on
grain size shown in Fig. 6.15. [Answer: 200 pm V
71
]
3. A rectangular plate of piezoceramic of dimensions 10 mm63mm61mm is
electroded over the 10 mm63 mm faces. Calculate the new dimensions when a
potential difference of 100 V is applied between the electrodes and the field and
poling directions coincide (d
33
¼ 525 pC N
71
; d
31
¼220 pC N
71
). [Answer:
9.99978 mm62.999934 mm61.0000525 mm]
4. A piezoceramic disc, thickness t and diameter D, is electroded over its faces and poled.
A constant voltage is applied in the sense that t increases in an unrestrained way.
From a consideration of the energies involves in the definition of k
33
derive the
relationship:
k
33
¼ d
33
/ðs
E
33
e
X
33
Þ
1=2
in which k
33
is the electromechanical coupling coefficient, s
E
33
the elastic compliance
measured at constant applied electric field and e
X
33
the permittivity at constant
mechanical stress.
For a p.d. of 1.5 kV applied between the electrodes calculate the extension under no
restraining force, and estimate the force necessary to block the extension, the ‘blocking
force’ ðd
33
¼500 pC N
1
, t ¼1 mm, diam:¼10 mm, s
E
33
¼20 10
12
m
2
N
1
). [Answer
300 kgf]
5. A uniform tensile stress of 50 MPa acts along the axis of a piezoceramic cylindrical rod
of length 12 mm and diameter 6 mm. Calculate the potential difference developed
between the ends of the rod. Given that the coupling coefficient is 0.5, calculate the
total stored energy in the rod (e
X
33
¼ 700e
0
; d
33
¼ 350 pC N
71
). [Answer: 33.9 kV;
25 mJ]
6. A cylindrical rod of ceramic BaTiO
3
is poled along an axis perpendicular to its
electroded end-faces. The cylinder stands on one flat end and carries a load of 10 kg
uniformly distributed over the upper face. Calculate the open-circuit voltage difference
developed between the end-faces, the mechanical strain and the length change.
In a separate experiment, a potential difference is applied between the ends of the
unloaded cylinder. Calculate the voltage necessary to obtain the same strain as
developed previously. Explain why the developed and applied voltages differ.
Cylinder dimensions: height, 10 mm; diameter, 10 mm. Electromechanical proper-
ties of ceramic: s
D
33
¼ 6.8 10
12
m
2
N
1
; h
33
¼ 1.5 GN C
71
; d
33
¼ 190 pC N
71
.
[Answer: 130 V, 8.6610
76
and 0.086 mm; 453 V]
PROBLEMS 407
7. Using the data given in Problem 6 analyse the energetics, assuming a coupling
coefficient of 0.5 and e
r
1600.
8. A piezoceramic bimorph is loaded in a symmetrical ‘three-point’ arrangement, in
which the loads are applied via small steel rods of circular section parallel to each
other and perpendicular to the length of the bimorph. Neglecting the effects of the
metal shim, calculate the voltage developed when the centre is displaced 2 mm from
the plane containing the outer loading lines (distance between outer loading points,
20 mm; bimorph width, 3 mm; bimorph depth, 1 mm; g
31
¼4:7 10
3
VmN
1
;
g
33
¼11:4 10
3
VmN
1
;c
D
11
¼ 168 10
9
Pa; c
D
12
¼ 78:2 10
9
Pa; c
D
13
¼ 71:0
10
9
Pa). [Answer: 2:6 V].
9. The following electroded pieces fabricated from the same piezoceramic are available:
(i) a disc – diameter 21 mm, thickness 0.8 mm, fully poled through its thickness, (ii) a
square section rod 2033 mm fully poled along its length, and (iii) a rod of the
same dimensions as in (ii) but fully poled perpendicular to its length.
An impedance analyser is also available.
Outline a strategy for determining e
X
3
, e
x
3
, s
D
33
, s
E
33
, k
33
, k
3
, d
33
, g
33
, k
p
, and
calculate the various parameters given the following.
Mass of disc is 2.15 g and the planar resonance and antiresonance frequencies are
102 kHz and 120 kHz.
The longitudinal resonance (the lengthwise poled rod supported at its centre
(node)) and antiresonance frequencies are 71.8 kHz and 88.1 kHz.
The resonance and antiresonance frequencies for the bar poled across its thickness
are 72.6 kHz and 76.9 kHz.
The capacitance of the disc measured at 1 kHz is 6.1 nF.
From the data make an estimate of the Poisson ratio and comment on its realism.
[Answers: e
x
¼ 1592e
0
Fm
1
; s
D
33
¼ 10.4 10
12
m
2
N
1
; k
33
¼ 0.62; k
31
¼ 0.37;
s
E
33
¼ 16.9 10
12
m
2
N
1
; d
33
¼ 303 pC N
1
; g
33
¼ 21.5 10
3
VmN
1
; k
p
¼
0:60]
10. Discuss the design of the bimorph for a hi-fi ‘tweeter’ and describe a fabrication
route.
11. Regarding the components of a 0–3 composite as characterised by their resistivity
values rather than permittivities, suggest strategies for poling the composite.
12. An experimental 1–3 piezoceramic/polymer composite disc transducer operating at
the fundamental thickness mode resonance frequency is intended to detect objects in
water having dimensions of approximately 10 mm. It is firmly fixed to a backing.
The piezoelectric ceramic rods have a density of 7500 kg m
3
, the polymer
1200 kg m
3
and the volume fraction of the ceramic is 0.25. The elastic compliance of
the ceramic (s
D
33
)is10 10
12
m
2
N
1
and that of the polymer 115 10
12
m
2
N
1
.
Sketch out the form the composite should take defining the disc thickness and the
dimensions and spacing of the rods, briefly justifying steps in the argument.
408 PIEZOELECTRIC CERAMICS
Given that the appropriate relative permittivities of the ceramic and the polymer
are respectively 1500 and 3.5, and that the d
33
coefficient for the ceramic is
375 pC N
1
, calculate a value for the hydrophone figure of merit and show that the
units are m
2
N
1
. For this estimate it may be assumed that the composite has been
structurally modified so that the d
31
contribution is negligible. Comment on the
realism or otherwise of the calculated value.
Calculate the capacitance per unit area of the composite, and its acoustic
impedance.
Describe a production route suited to manufacturing the composite in large
numbers.
[Answer: 40 000 (units of 10
15
m
2
N
1
); 1 mFm
2
; 9 Mrayl]
Bibliography
1. Jaffe, B., Cook, W.R. and Jaffe, H. (1971) Piezoelectric Ceramics, Academic Press,
London.
2. Herbert, J.M. (1982) Ferroelectric Transducers and Sensors, Gordon and Breach,
London.
3. Burfoot, J.C. and Taylor, G.W. (1979) Polar Dielectrics and their Applications,
Macmillan, London.
4. Levinson, L.M. (ed.) (1988) Electronic Ceramics . Marcel Dekker, New York.
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perovskite Pb Zr
1x
Ti
x
O
3
, Phys. Rev. B, 63, 14103–1 to 9.
8. Guo, R. et al. (2000) Origin of the high piezoelectric response in Pb Zr
1x
Ti
x
O
3
,
Phys. Rev. Lett., 84, 5423–6.
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and Sintering of PZT Ceramics: II, Effect of PbO content on densification kinetics,
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10. Eitel, R.E. et al. (2001) New high temperature morphotropic phase boundary
piezoelectrics based on Bi(Me)O
3
–PbTiO
3
ceramics, Jpn. J. Appl. Phys., 40, 5999–
6002.
11. Park, S.-E. and Shrout, T.R. (1997) Characteristics of relaxor-based piezoelectric
single crystals for ultrasonic transducers, IEEE Trans. Ultrasound, Ferroelectrics and
Frequency Control, 44, 1140–7.
12. Kondo, M. et al. (1997) Piezoelectric properties of PbNi
1/3
Nb
2/3
O
3
–PbTiO
3
–
PbZrO
3
ceramics, Jpn. J. Appl. Phys., 36, 6043–5.
13. Roberts, G. et al. (2001) Synthesis of lead nickel niobate–lead zirconate titanate solid
solutions by a B-site precursor method, J. Am. Ceram. Soc., 84, 2869–72.
BIBLIOGRAPHY 409
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1/3
Nb
2/3
)O
3
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1/3
Nb
2/3
)O
3
, Jpn. J. Appl. Phys., 36, 6141–4.
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410 PIEZOELECTRIC CERAMICS
7
PYROELECTRIC MATERIALS
The discussion draws on the review articles by R.W. Whatmore [1] and by R.W.
Whatmore and R. Watton [2]. The now classical text by R.A. Smith, F.E. Jones
and R.P. Chasmar [3] is recommended for supplementary reading.
7.1 Background
True pyroelectricity results from the temperature dependence of the spontaneous
polarization P
s
of polar materials and is therefore shown by ferroelectric
materials whether they are single-domain single crystals or poled ceramics.
Because a change in polarization in a solid is accompanied by a change in surface
charges (see Section 2.7.1) it can be detected by an induced current in an external
circuit. If the pyroelectric material is perfectly electrically insulated from its
surroundings, the surface charges are eventually neutralized by charge flow
occurring because of the intrinsic electrical conductivity of the material. Effective
neutralization occurs in a time approximately equal to the electrical time
constant re of the material.
When an electric field E is applied to a polar material the total electric
displacement D is given by
D ¼ e
0
E þ P
total
¼ e
0
E þðP
s
þ P
induced
Þ
ð7:1Þ
which, from Eqs (2.76) and (2.81), is
D ¼ eE þ P
s
ð7:2Þ
Therefore
@D
@T
¼
@P
s
@T
þ E
@e
@T
Electroceramics: Materials, Properties, Applications. 2nd Edition. Edited by A. J. Moulson and J. M. Herbert.
& 2003 John Wiley & Sons, Ltd: ISBN 0 471 49747 9 (hardback) 0 471 49748 7 (paperback)