Lecture 8-2 - University of Pittsburghqiw4/Academic/MEMS1082/Lecture 8-2.pdf · Piezoelectricity...
Transcript of Lecture 8-2 - University of Pittsburghqiw4/Academic/MEMS1082/Lecture 8-2.pdf · Piezoelectricity...
Department of Mechanical Engineering
Piezoelectricity and Pyroelectricity
Lecture 8-2
Department of Mechanical Engineering
Direct Effect
Piezoelectricity
(Electric polarization)=(Piezoelectric coefficients) (Mechanical stress)
σdP =
[C/m2]First rank vector
[C/N]Third rank tensor
[N/m2]Second rank tensor
Piezoelectric coefficients constitute a third rank polar tensor
(Mechanical strain)=(Piezoelectric coefficients) (Electric field)
[ ]
First rank vector
[m/V]
Third rank tensor
[V/m]
Second rank tensor
Converse Effect dE=ε
Department of Mechanical Engineering
Direct Effect
Piezoelectricity Converse Effect
( ) ( ) ( )Ed t=ε( ) ( )( )σdP =3x1 3x6 6x1 6x1 3x16x3
Matrix form
jkijki dP σ= iijkjk Ed=εTensor form
Units:Vm
VCmN
NC
JJ
NC
NC
=
⋅⋅
=
=
Thermodynamic Proof
TdSdPEddU kkijij ++= εσFirst Law
Electric work inside materials
Department of Mechanical Engineering
PiezoelectricityFree Energy
ijijkk PETSUG εσ+−−=
SdTdEPddG kkijij −−−= σε
G is function of state because system completely specified by σ, E, and T
SdTTGdE
EGdGdG
Eijk
Tkij
TEij ,,, σσ
σσ
∂∂
+
∂∂
+
∂∂
=
ijij
G εσ
−=∂∂
kk
PEG
−=∂∂ S
TG
−=∂∂
and
Department of Mechanical Engineering
Various Transduction effects
Piezoelectricity
k
ij
kij EEG
∂
∂−=
∂∂∂ ε
σ
2
ij
k
ijk
PE
Gσσ ∂
∂−=
∂∂∂
=2
Converse piezoelectric effect
Direct piezoelectric effect
TTG ij
ij ∂
∂−=
∂∂∂ εσ
2
ijij
ST
Gσσ ∂∂
−=∂∂
∂=
2
Thermal expansion coefficient
Piezo-caloric effect
TP
TEG k
k ∂∂
−=∂∂
∂ 2
kk ES
ETG
∂∂
−=∂∂
∂=
2
Pyroelectric Coefficient
Electro-caloric effect
Department of Mechanical Engineering
3-subscript Tensor Notation
Tensor and Matrix Coefficients
Converse Effect
jkijki dP σ=
iijkjk Ed=ε
Direct effect
331333213231131
231232212221121
1311312112111111
σσσσσσσσσ
dddddddddP
++++++=
For example:
2-subscript Matrix Notation
jiji dP σ= i=1,2,3j=1,2,3,4,5,6
6365354343332321313
6265254243232221212
6165154143132121111
σσσσσσσσσσσσ
σσσσσσ
ddddddPddddddP
ddddddP
+++++=+++++=
+++++=
iijj Ed=ε
3362261166
3352251155
3342241144
3332231313
3322221122
3312211111
EdEdEdEdEdEdEdEdEdEdEdEdEdEdEdEdEdEd
++=++=++=++=++=++=
εεεεεε
i, j, k=1, 2, 3
Department of Mechanical Engineering
Tensor and Matrix Coefficients
Identify piezoelectric coefficients
13313
12212
11111
dddddd
===
11212111216
11311313115
12313212314
222
dddddddddddd
=+==+==+=
Identify equivalent stresses
432235311362112
333222111
,,,,
σσσσσσσσσσσσσσσ
=========
………………………
………………………………
Department of Mechanical Engineering
For first-, second-, third- and fourth-rank polar tensors, the well-known equations of transformation from an orthogonal x1, x2, x3 system to another similarly orthogonal x’1, x’2, x’3 system are according to their definition
Transformation of Axes
jiji TaT =′
klklikij TaaT =′
lmnknjmilijk TaaaT =′
mnoplpkojnimijkl TaaaaT =′
where the aij direction cosines are the elements of the (aij) matrix. The (aij) matrix connects the original and the `new' co-ordinates according to the matrix equation
Department of Mechanical Engineering
Transformation of Axes
=
′′′
3
2
1
333231
232221
131211
3
2
1
xxx
aaaaaaaaa
xxx
Tensor form ∑ ==′j
jijjiji xaxaxEinstein convention
From new coordinate to old
=
′′′
3
2
1
333231
232221
131211
3
2
1
xxx
aaaaaaaaa
xxx
( ) ( ) ( )xax t ′=or
( ) ( )( )xax =′
or
transpose
( )a : Direction cosine matrix
Tensor form ∑ ′=′=j
jijjjii xaxax
Department of Mechanical Engineering
Transformation of Axes
Rotated Axes and angles
θφθφθϕθ
ϕφθϕφϕφϕφθ
ϕθϕφϕφθϕφϕφθ
cossinsinsinsinsinsin
sinsincoscoscoscossinsincoscos
cossinsincoscossincossinsincoscoscos
33
32
31
23
22
21
13
12
11
====
−=−−=
−=+=−=
aaaaaaaaa
Department of Mechanical Engineering
Transformation of Axes
First rotation φ +π/2 about X3
( )
−=
1000sincos0cossin
φφφφ
Ia
Second rotation θ about X’1
( )
−=
θθθθ
cossin0sincos0001
IIa
Third rotation π/2−θ about X”3
( )
−=
1000coscos0cossin
ϕϕϕϕ
IIIa
Department of Mechanical Engineering
Transformation of Axes
General rotation is product of three individual rotation
( ) ( ) ( ) ( )IIIIII aaaa =
( )
−
−
−=
1000sincos0cossin
cossin0sincos0001
1000coscos0cossin
φφφφ
θθθθϕϕ
ϕϕa
( )
−−−
−+−=
θφθφθϕθϕθφφθϕφϕφθ
ϕθϕφϕφθϕφϕφθ
cossinsincossinsinsinsincossincoscoscossinsincoscos
cossinsincoscossincossinsinsincoscosa
Department of Mechanical Engineering
Symmetry Axes Parallel to X3
2-fold φ=180o Orthorhombic(Tetragonal& cubic)(Hexagonal)
( )
+−
−=
100010001
a
3-fold φ=120o
( )
+−
−−
=10002/123
0232/1
a
Trigonal(Hexagonal)
4-fold φ=90oTetragonal& cubic
( )
++
−=
100001010
a
6-fold φ=60o( )
+
−
=100
02/123
0232/1
aHexagonal
Department of Mechanical Engineering
Mirror plane perpendicular to X2, and Center of symmetry=inversion center Monoclinic, orthorhombic,
tetragonal, hexagonal, cubic
( )
+−
+=
100010001
a
Triclinic, monoclinic, orthorhombic, trigonal, tetragonal, hexagonal, cubic
Mirror plane
( )
−−
−=
100010001
a
Center of symmetry
Department of Mechanical Engineering
Pyroelectric tensor (Change in polarization)=(Pyroelectric coefficients)(Change in temperature)
A vectorA scalar
TpP i ∆=∆pi is pyroelectric coefficient
jiji PaP ∆=′∆ TT ∆=′∆
How does p transform?
TpaTpaPaP jijjijjiji ′∆=∆=∆=′∆new old old new
All new TpTpaP ijiji ′∆′=′∆=′∆
jiji pap =′Pyroelectricity is a first rank polar tensor (vector) property
Department of Mechanical Engineering
Effect of symmetry on p
3x1 3x3 3x1
Matrix form
First rankTransformation matrix of symmetryElement of group
( ) ( )( ) ( )ppap ==′3x1
Neumann’s Principle:Property unchanged aftertransformation
Centro-symmetric crystals have inversion symmetry
( )
≡
−−−
=
−−
−=
′′′
=′
3
2
1
3
2
1
3
2
1
3
2
1
100010001
ppp
ppp
ppp
ppp
p
0321 === ppp (No pyroelectricity)
Pyroelectricity is absent in all centro-symmetric crystals (a null property)
Department of Mechanical Engineering
Poled ferroelectric ceramic
Symmetry group=m//X3
-fold axis//X3 ( )
−=
1000cossin0sincos
θθθθ
a
For all θ( )
≡
+−
=
−=
′′′
=′
3
2
1
3
21
21
3
2
1
3
2
1
cossinsincos
1000cossin0sincos
ppp
ppppp
ppp
ppp
p
θθθθ
θθθθ
If θ=90o 021 == pp
03 ≠p
( )
=
3
00
pp
Department of Mechanical Engineering
Direct Effect
Piezoelectricity
(Electric polarization)=(Piezoelectric coefficients) (Mechanical stress)σdP =
[C/m2]First rank vector
[C/N]
Third rank tensor
[N/m2]
Second rank tensor
kljklijjij'
i daPaP σ=='mnnlmkjklij
'i aadaP σ=
'mn
'imn
'i dP σ= jklnlmkij
'imn daaad =
Piezoelectric coefficients constitute a third rank polar tensor
Tensor form transformation
Department of Mechanical Engineering
Piezoelectric-Matrix Transformations
Stress-symmetric second rank tensors
kljlikij aa σσ =′ Equivalent tensor forms
How does 6x1 matrix transform?( ) ( )( )σασ =
3x3 3x33x3 3x3( ) ( )( )( )taa σσ =′
What is (α)?
To determine (α), write out tensor and matrix forms and equate term by term
.......................222122111121212111
21111 ++++=′ σσσσσ aaaaaa
6165154143132121111 σασασασασασασ +++++=′
21111 a=α
21212 a=α 121116 2 aa=α
Other α coefficients can be determined in similar way
Department of Mechanical Engineering
α Transformation
( )
++++++
+++=
211222112311211322132312231322122111
311232113311311332133312133312321131
322131223123332132233322332332223121
323131333332233
232
231
222121232322223
222
221
121111131312213
212
211
222
222
222
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
α
( )
+++++++++
=
122122113211123122313221323122211211
132123113311133132313321313321231113
231213221332331223323322333223221312
231313333323233
223
213
221212323222232
222
212
211111313121231
221
211
222
222
222
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
aaaaaaaaaaaaaaaaaaaaaaaaaaa
tα
Department of Mechanical Engineering
Stress
Piezoelectric-Matrix Transformations
( ) ( )( )σασ =′6x1 6x6 6x1
new old
( ) ( ) ( ) ( )( ) ( )σσαασα ==′ −− 11
Polarization
new old( ) ( )( )PaP =′ ( ) ( ) ( ) ( )( ) ( )PPaaPa ==′ −− 113x1 3x3 3x1
Piezoelectricity ( ) ( )( ) ( )( )( ) ( )( )( ) ( )( )( )σ
σασ′′=
′===′ −
ddadaPaP 1
( ) ( )( )( ) 1−=′ αdad3x6 3x3 3x6 6x6
new old
( ) ( )( ) ( ) ( )( )( ) ( ) ( )ddaada ==′ −−− ααα 111
( ) ( ) ( )( )αdad ′= −1
old new
Department of Mechanical Engineering
Matrix Method: Point Group 2
Sucrose, 2//X2( )
−+
−=
100010001
a
( ) ( )( )( ) 1−=′ αdad
( )
−
−
−+
−=′
100000010000001000000100000010000001
100010001
363534333231
262524232221
161514131211
dddddddddddd
ddddddd
( ) ( )ddddddddddddddddddd
d =
+−+−−−−+−+++
−−−−=′
363534333231
262524232221
161514131211
( )
=
3634
25232221
1614
000000
0000
dddddd
ddd
8 independent piezoelectric coefficients
Department of Mechanical Engineering
Piezoelectric ceramics
Point group =m Direct inspection method
Many mirror planes//X3
Mirror X1 1 -1 2 2 3 3
111 111 112 +122 Eliminate all coefficients with odd No. of 1’s
0331321313312221
231213212133132123122111
=============
ddddddddddddd
Mirror X2 2 -2 1 1 3 3Eliminate all coefficients with odd No. of 2’s
0332323121112233211222 ======= ddddddd
Mirror at 45o to X1 and X2 1 2 2 1 3 3
322311 dd = 223113232131 dddd ===
Department of Mechanical Engineering
Transformation of Axes
d matrix for group =m
( )
=
0000000000000
333131
15
15
dddd
dd
For PZT, BaTiO3, PbTiO3, etc