Deduced property matrices of domain-engineered relaxor single crystals of
[110]L×[001]T cut: Effects of domain wall contributions and domain-domain
interactions
Rahul Shukla, Kalidindi Kotam Rajan, M. Shanthi, Jing Jin, Leong-Chew Lim et al.
Citation: J. Appl. Phys. 107, 014102 (2010); doi: 10.1063/1.3270426
View online: http://dx.doi.org/10.1063/1.3270426
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JOURNAL OF APPLIED PHYSICS 107, 014102 共2010兲
Deduced property matrices of domain-engineered relaxor single crystals
of †110‡L Ã †001‡T cut: Effects of domain wall contributions and
domain-domain interactions
Rahul Shukla,1,2 Kalidindi Kotam Rajan,3 M. Shanthi,2 Jing Jin,3 Leong-Chew Lim,2,3,a兲
and Prasanna Gandhi1,a兲
1
Suman Mashruwala Advanced Microengineering Laboratory, Department of Mechanical Engineering,
Indian Institute of Technology Bombay, Mumbai 400076, India
2
Department of Mechanical Engineering, National University of Singapore, Singapore 119260, Singapore
3
Microfine Materials Technologies Pte. Ltd., 10 Bukit Batok Crescent, No. 06-02, The Spire 658079,
Singapore
共Received 1 May 2009; accepted 8 November 2009; published online 4 January 2010兲
Multidomain single crystal properties can be more accurately deduced from measured multidomain
data instead of single-domain data, especially for the shear properties and clamped dielectric
permittivities, which are strongly influenced by domain wall contributions and domain-domain
interactions in the material. Using the above finding, the property matrices of Pb共Zn1/3Nb2/3兲O3
− 共6 – 7兲%PbTiO3, Pb共Zn1/3Nb2/3兲O3 − 4.5%PbTiO3, and Pb共Mg1/3Nb2/3兲O3 − 28%PbTiO3 single
crystals of 关110兴L ⫻ 关001兴T 共P兲 cut were deduced from the measured 关100兴L ⫻ 关001兴T 共P兲
multidomain data. The results show that relaxor single crystals of 关110兴L ⫻ 关001兴T 共P兲 cut have
2
2
E
E
/ s11
values, especially for Pb共Zn1/3Nb2/3兲O3 − 共6 – 7兲%PbTiO3 of which d31
/ s11
extremely high d31
2
⬇ 52 nN/ V , making such crystal cuts candidate materials for unimorph actuator applications.
© 2010 American Institute of Physics. 关doi:10.1063/1.3270426兴
Over the past decade, it has been shown that relaxor
ferroelectric single crystals, notably Pb共Zn1/3Nb2/3兲O3 –
PbTiO3 共PZN-PT兲 and Pb共Mg1/3Nb2/3兲O3 – PbTiO3 共PMNPT兲, exhibit exceptionally high dielectric and electromechanical properties compared to state-of-the-art lead zirconate titanate 共PZT兲 ceramics with minimum hysteresis even
when driven at relatively high voltages.1,2 With improved
growth techniques, large-size high-uniformity PZN-PT and
PMN-PT single crystals have been successfully grown by
several groups.3–10 High longitudinal and transverse piezoelectric properties were obtained when the crystals were
poled in the 关001兴 crystal direction instead of the 关111兴 spontaneous polarization direction.11–13 More recently, Shukla
and co-workers14,15 have shown that 关001兴-poled PZN-共6–
E
7兲%PT crystals of 关110兴L cut exhibit high d31 / s11
ratio and
electromechanical coupling factors compared to the 关100兴L
cut. In contrast, 关111兴-poled single-domain PZN-PT and
PMN-PT single crystals have extremely high shear
properties.11,16–19 These single crystal materials are potential
new-generation materials for high performance sensors and
actuators.
Device design with anisotropic materials is relatively
complicated involving the use of property matrices and computer simulation technique. The anisotropic nature of these
single crystals thus needs to be properly understood before
one incorporates them into devices. These property matrices
can be measured experimentally by the resonance and ultrasonic techniques or their combination.20–22
a兲
Authors to whom correspondence should be addressed. Electronic addresses: [email protected] and [email protected].
0021-8979/2010/107共1兲/014102/5/$30.00
Using the resonance technique, the measured property
matrix of 关111兴-poled single-domain PZN-共6–7兲%PT single
crystal of 关11̄0兴L ⫻ 关112̄兴W ⫻ 关111兴T cut has been obtained by
Jin and co-workers.19 Their results are reproduced here in
Table I. The measured property matrices for 关100兴L
⫻ 关001兴T 共P兲 cut of PZN-4.5%PT,20 PZN-共6–7兲%PT,23 and
PMN-28%PT 共Ref. 24兲 single crystals have also been determined by earlier researchers and are listed in Table II for
easy reference. More recently, the measured property matrices of 关001兴-poled PZN-共6–7兲%PT single crystal but of
关110兴-length cut has been constructed and refined by Shukla
and co-workers.14,15 Their results are listed in row 1 of Table
III for easy reference. All the above property values are hereafter referred to as the experimentally deduced 共ED兲 properties.
Instead of experimental measurements, the coordinate
transformation approach can be used to deduce the property
matrices provided that at least one complete set of property
matrix of a known set of crystal axes is available.25 The
relation between the property values with respect to any two
different sets of axes is defined by the transformation matrix
关a兴. In Cartesian coordinates 共x , y , z兲 system, this is done by
means of three successive anticlockwise rotations in a sequence. These three successive angles of rotation are called
the Euler angles 共␸, ␪, ␺兲. Several conventions are in use
depending on the axes about which rotations are carried
out.26 Here we shall use the zxz convention; i.e., the initial
set of axes is first rotated by an angle ␸ about the z-axis 共via
rotation matrix 关a1兴兲, then by an angle ␪ about the x-axis 共via
rotation matrix 关a2兴兲, and finally by an angle ␺ about the
z-axis 共via rotation matrix 关a3兴兲. The resultant rotation matrix
关a兴 is a 3 ⫻ 3 matrix given by
107, 014102-1
© 2010 American Institute of Physics
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014102-2
J. Appl. Phys. 107, 014102 共2010兲
Shukla et al.
TABLE I. Measured properties of 关111兴-poled single-domain PZN-共6–7兲%PT single crystal of 关11̄0兴L ⫻ 关112̄兴W ⫻ 关111兴T cut 共Ref. 19兲.
Elastic stiffness constants: cijE and cijD 共1010 N / m2兲
cE11
PZN-共6–7兲%PT
18.0
cE12
cE13
8.0
4.8
cE14
cE33
cE44
cE66
cD11
cD12
cD13
cD14
cD33
cD44
cD66
⫺2.6
17.1
1.6a
5.0a
19.4
8.0
6.4
⫺2.2
20.2a
4.0
5.7
Elastic compliance constants: sijE and sijD 共10−12 m2 / N兲
PZN-共6–7兲%PT
sE11
sE12
sE13
sE14
sE33
sE44
sE66
sD11
sD12
sD13
sD14
sD33
sD44
sD66
32.8a
⫺28.5a
⫺1.2
100.1
6.5
390.4
122.6
7.7a
⫺3.4
⫺1.4
6.1
5.8a
31.7
22.3
Piezoelectric constants: eij 共C / m2兲, dij 共10−12 C / N兲; dielectric constants: ␧ij 共␧0兲 and coupling coefficients 共kij兲
PZN-共6–7兲%PT
e15
e22
e31
e33
d15
d22
d31
d33
␧S11
␧S33
␧T11
␧T33
k15
k31
k33
kt
31.7
30.9
⫺4.6
12.6
6000a
⫺1280a
⫺35a
93a
4222
567
11000a
700a
0.79a
0.18a
0.33a
0.39a
a
Measured properties.
冤
冥
ayy = − sin ␺ sin ␾ + cos ␪ cos ␾ cos ␺ ,
axx axy axz
关a兴 = 关a3兴 · 关a2兴 · 关a1兴 = ayx ayy ayz ,
azx azy azz
共1兲
where aij are functions of the direction cosines of respective
rotation matrices and are given by
ayz = cos ␺ sin ␪ ,
azx = sin ␪ sin ␾ ,
axx = cos ␺ cos ␾ − cos ␪ sin ␾ sin ␺ ,
azy = − sin ␪ cos ␾ ,
axy = cos ␺ sin ␾ + cos ␪ cos ␾ sin ␺ ,
azz = cos ␪ .
共2兲
Equation 共1兲 works conveniently for properties of lower
rank tensors in nature such as polarization 共p兲, dielectric permittivity 共␧兲, stress 共T兲, and strain 共S兲. For properties of
higher order tensors in nature, it is more convenient to use
axz = sin ␺ sin ␪ ,
ayx = − sin ␺ cos ␾ − cos ␪ sin ␾ cos ␺ ,
TABLE II. Measured properties of 关001兴-poled multidomain PZN-共6–7兲%PT, PZN-4.5%PT, and PMN-28%PT single crystals of 关100兴 length cut, i.e.,
关100兴L ⫻ 关001兴T 共P兲 cut single crystal.
Elastic stiffness constants: cijE and cijD 共1010 N / m2兲
cE11
PZN-共6–7兲%PTa
PZN-4.5%PTc
PMN-28%PTd
10.7
11.1b
12.47
cE12
cE13
9.7
10.2
11.06
10.1
10.1
10.91
cE33
cE44
cE66
cD11
10.8
10.5
11.47
5.5b
6.4b
7.04b
6.1
6.3b
8.92
10.8
11.3
12.77
cD12
cD13
9.9
10.4
11.36
9.5
9.5
9.61
cD33
cD44
cD66
14.4b
13.5b
17.15b
6.1
6.7b
7.42
6.1
6.3
8.92
sD66
Elastic compliance constants: sijE and sijD 共10−12 m2 / N兲
PZN-共6–7兲%PTa
PZN-4.5%PTc
PMN-28%PTd
sE11
sE12
sE13
sE33
90b
82b
54.87b
⫺13.5
⫺28.5
⫺14.51
⫺72
⫺51
⫺37.81
145
108
80.3
sE44
18.11
15.6
14.2
sE66
sD11
sD12
sD13
sD33
sD44
16.5
15.9
13.07
54.4
61.5
36.71
⫺49
⫺49
⫺32.6
⫺5.7
⫺9.0
⫺2.47
16.8b
20.6b
11.6b
16.4
14.9
13.5
16.5
15.9
13.07
Piezoelectric constants: eij 共C / m2兲, dij 共10−12 C / N兲; dielectric constants: ␧ij 共␧0兲 and coupling coefficients 共kij兲
PZN-共6–7兲%PTa
PZN-4.5%PTc
PMN-28%PTd
e15
e31
11.04
8.9
9.28
⫺3.23
⫺3.7
⫺4.69
e33
d15
d31
d33
␧S11
16.1
15
20.47
200
140
132
⫺1500b
⫺970
⫺1025b
2800b
2000
2000b
2030
3000
2583
␧S33
814
1000
835
␧T11
␧T33
k15
k31
k33
kt
2250b
3100b
2750b
7000b
5200b
6550b
0.31b
0.23
0.22b
0.54b
0.50b
0.55b
0.94b
0.91b
0.92b
0.50b
0.50b
0.58b
a
Reference 23.
Measured properties.
c
Reference 20.
d
Reference 24.
b
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014102-3
J. Appl. Phys. 107, 014102 共2010兲
Shukla et al.
TABLE III. Measured 共ED兲 and deduced 共S → M and M → M兲 properties of 关001兴-poled multidomain PZN-共6–7兲%PT single crystal of 关110兴 length cut, i.e.,
关110兴L ⫻ 关001兴T 共P兲 cut.
Elastic stiffness constants: cijE and cijD 共1010 N / m2兲
cE11
PZN-共6–7兲%PT 共ED兲a
PZN-共6–7兲%PT 共S → M兲
PZN-共6–7兲%PT 共M → M兲
15.73
16.3
16.3
cE12
cE13
cE33
cE44
cE66
cD11
2.55
3.4
4.1
9.14
9.6
10.1
9.9
10.2
10.8
5.72b
6.5
5.5
0.46
0.3
0.5
16.1
19.2
16.4
cD12
2.92
4.86
4.25
cD13
8.07
9.5
9.5
cD33
cD44
cD66
13b
14.5
14.4
7.14
7.2
6.1
0.46
2.5
0.45
Elastic compliance constants: sijE and sijD 共10−12 m2 / N兲
PZN-共6–7兲%PT 共ED兲a
PZN-共6–7兲%PT 共S → M兲
PZN-共6–7兲%PT 共M → M兲
sE11
sE12
sE13
sE33
sE44
sE66
39.1b
46.4
42.4
31.5b
36.4
34.1
⫺65.2
⫺79.4
⫺72
130.4
162
145
17.5
19.9
18.1
218.8
483
207
sD11
10.8
7.82
6.8
sD12
3.4
0.81
⫺1.4
sD13
sD33
sD44
sD66
⫺4.8
⫺5.7
⫺5.7
16.1b
14.3
16.8
14
14.1
16.4
218.8
40.1
207
Piezoelectric constants: eij 共C / m2兲, dij 共10−12 C / N兲; dielectric constants: ␧ij 共␧0兲 and coupling coefficients 共kij兲
PZN-共6–7兲%PT 共ED兲a
PZN-共6–7兲%PT 共S → M兲
PZN-共6–7兲%PT 共M → M兲
e15
e31
17.67
33.2
11.0
⫺5.29
⫺7.53
⫺3.23
e33
15.31
18.9
16.1
d15
309
511
200
d31
d33
⫺1425a
⫺1498
⫺1500
2721a
3010
2800
␧S11
␧S33
␧T11
1413a
3004
2030
850a
3001
814
3106a
7570
2250
␧T33
k15
k31
7256a
7561
7000
0.45a
0.44
0.33
0.85a
0.85
0.92
k33
kt
0.94a
0.91
0.93
0.49a
0.30
0.50
a
Reference 15.
Measured properties.
b
the complete 6 ⫻ 6 transformation matrix directly.27 Let 关M兴
and 关N兴 be the Bond stress transformation matrix and the
Bond strain transformation matrix, respectively. The various
transformed properties, i.e., dielectric permittivity 共␧兲, piezoelectric strain coefficient 共d兲, piezoelectric stress coefficient
共e兲, compliance 共s兲, and stiffness 共c兲 can thus be obtained via
the following expressions:
or
关␧⬘兴 = 关a兴关␧兴关a兴
␧⬘ij = aika jl␧kl ,
关d⬘兴 = 关a兴关d兴关Ñ兴
or
d⬘ijk = aila jmakndlmn ,
关e⬘兴 = 关a兴关e兴关M̃兴
or
e⬘ijk = aila jmaknelmn ,
关s⬘兴 = 关N兴关s兴关Ñ兴
or
s⬘ijkl = aima jnakoalpsmnop ,
共3兲
and
关c⬘兴 = 关M兴关c兴关M̃兴
or
c⬘ijkl = aima jnakoalpcmnop ,
where i , j , . . . , o, p = x , y , z, and ,
关a兴 关M̃兴, and 关Ñ兴 are the
transposes of the matrices 关a兴, 关M兴, and 关N兴, respectively.
The unprimed property matrices are the measured properties
using the original set of axes and the primed ones are the
deduced property matrices with respect to the new set of
axes.
In the deduction of property matrices using the abovedescribed technique, the 关111兴-poled single-domain properties are important for the deduction of multidomain properties of any coordinate axes of interest, which would
otherwise require a fair number of test samples and time to
generate the property matrices. Such a technique has been
used by Liu and Lynch28 and Damjanovic and co-workers,29
who used the measured single-domain property values to deduce selective piezoelectric coefficients of multidomain re-
laxor ferroelectric PZN-PT and PMN-PT single crystals, respectively. The following two assumptions were invoked in
deducing the property values from single-domain data; i.e.,
共i兲 all the domain variants are equally probable and 共ii兲 domain wall contributions and domain-domain interactions are
negligible.
In this work, we first deduced the property matrix of
PZN-共6–7兲%PT single crystal of 关110兴L ⫻ 关001兴T 共P兲 cut using the single-domain data of Jin and co-workers.19 This was
achieved by rotating the property matrices of 关11̄0兴L
⫻ 关112̄兴W ⫻ 关111兴T cut single-domain PZN-共6–7兲%PT single
crystal first by 54.73° 共␪兲 about the x-axis and then by
90° 共␺1兲, 180° 共␺2兲, 270° 共␺3兲, and 360° 共␺4兲 about the
z-axis and averaging the four possible domains to obtain the
properties of the 关110兴L ⫻ 关11̄0兴W ⫻ 关001兴T pseudocubic system 关refer to Fig. 1共a兲兴. The electromechanical coupling coefficients k31, k33, k15, and kt, which cannot be deduced directly from the axis transformation technique, are determined
2
S D
using the relationships k2ij = d2ij / 共␧TiisEjj兲 and k2t = e33
/ 共␧33
c33兲.
The deduced property matrix is given in row 2 of Table III.
They are hereafter referred to as the S → M 共single-domain to
multidomain兲 deduced properties.
Compared with the ED data 共row 1 of Table III兲, signifiD
共or
cant differences are observed in the shear properties c66
D
E
E
s66兲, c66 共or s66兲, d15, and e15, as well as clamped dielectric
S
S
, ␧33
, and
permittivities and their derived properties 共i.e., ␧11
e31兲. In addition, significant deviation is also observed in free
T
. Since the S → M transformation techdielectric constant ␧11
nique ignores domain wall contributions and domain-domain
interactions, the present finding suggests that such contributions may be important in shear as well as clamped properties of relaxor ferroelectric single crystals. This finding is
supported by the result of Delaunay and co-workers,30 who
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014102-4
J. Appl. Phys. 107, 014102 共2010兲
Shukla et al.
[001]
[111]
IV
erty matrix of PZN-共6–7兲%PT of 关110兴L ⫻ 关001兴T cut using
the coordinate transformation technique but based on the
measured property matrix of 关100兴L ⫻ 关001兴T multidomain
single crystal instead 共row 1 of Table II兲. This was achieved
by rotating the latter property matrix by ␸ = 45°, ␪ = 0°, and
␺ = 0° with respect to its own coordinate axes 关refer to Fig.
1共b兲兴. The deduced property matrix is listed in row 3 of Table
III. They are hereafter referred to as the M → M
共multidomain-to-multidomain兲 deduced properties.
Comparison with row 1 of the same table shows that
D
D
, s12
共and
except for a few properties 关i.e., d15 family and s11
D
D
c12兲, and s13, which are about 30%–40% off兴, the properties
deduced from the M → M approach are within 15% of the
ED values. This result shows that the M → M approach,
which takes account of the contributions from domain walls
and domain-domain interaction in the initial set of measured
properties, gives better prediction of multidomain properties
of relaxor ferroelectric single crystals. The above results indicate that the deduction of unknown multidomain single
crystal properties from known multidomain single crystal
properties 共M → M兲 has an advantage over deduction from
the single-domain data 共i.e., via the S → M approach兲.
Although the ED property matrices for 关110兴L ⫻ 关001兴T
cut of PZN-共6–7兲%PT have been reported by Shukla and
co-workers14,15 the same but for PZN-4.5%PT and PMN28%PT remain not available to date. Inspired by the success
of the M → M technique, the elastic, piezoelectric, and dielectric coefficients for PZN-4.5%PT and PMN-28%PT of
关110兴L ⫻ 关001兴T cut are deduced from the properties of
关100兴L ⫻ 关001兴T cuts given in Table II. The deduced properties are listed in Table IV for easy reference.
2
E
/ s11
is of practical importance in the
The parameter d31
design of piezoelectric actuators involving an elastic mem2
E
/ s11
ratios of the three crystal comber. Table V lists the d31
positions studied, all of 关110兴L ⫻ 关001兴T cut. It is evident that
2
E
/ s11
ratio,
PZN-共6–7兲%PT single crystal has the highest d31
which makes this crystal the choice material for making
single crystal unimroph actuators involving an elastic member.
III
I
II
SM
[1 1 0]
[11 2 ]
[110]
[1 1 0]
(a)
[001]
[001]
IV
I
IV
III
I
II
III
II
MM
[010]
[1 1 0]
[100]
[110]
(b)
FIG. 1. 共Color online兲 Relationships between the coordinate axes 共a兲 of
关11̄0兴L ⫻ 关112̄兴W ⫻ 关111兴T cut single-domain crystal and 关110兴L ⫻ 关001兴T cut
multidomain single crystal used in the S → M approach and 共b兲 of 关100兴L
⫻ 关001兴T cut and 关110兴L ⫻ 关001兴T cut multidomain single crystals used in the
M → M approach. The solid lines show possible domain orientation states
and the dashed lines show the directions of polarization in respective
domains.
concluded that the effective properties of the macroscopic
system with multidomain structures are a function of singledomain properties and extrinsic contribution of domain
walls.
To include the contributions of domain walls and
domain-domain interaction, we have also deduced the prop-
TABLE IV. Deduced properties 共M → M兲 of 关001兴-poled multidomain PZN-4.5%PT and PMN-28%PT single crystals of 关110兴 length cut, i.e., 关110兴L
⫻ 关001兴T 共P兲 cut.
Elastic stiffness constants: cijE and cijD 共1010 N / m2兲
cE11
PZN-4.5%PT 共M → M兲
PMN-28%PT 共M → M兲
17.0
20.7
cE12
cE13
cE33
cE44
cE66
cD11
cD12
4.35
2.85
10.1
10.1
10.5
11.5
6.4
7.0
0.45
0.70
17.2
21.0
4.55
3.15
cD13
9.5
9.6
cD33
cD44
cD66
13.5
17.2
6.7
7.4
0.45
0.70
sD44
sD66
14.9
13.5
221
139
Elastic compliance constants: sijE and sijD 共10−12 m2 / N兲
PZN-4.5%PT 共M → M兲
PMN-28%PT 共M → M兲
sE11
sE12
sE13
sE33
sE44
sE66
sD11
sD12
30.7
23.4
22.8
16.9
⫺51.0
37.8
108
80.3
15.6
14.2
221
139
10.2
5.3
2.3
⫺1.2
sD13
sD33
⫺9.0
⫺2.5
20.6
11.6
Piezoelectric constants: eij 共C / m2兲, dij 共10−12 C / N兲; dielectric constants: ␧ij 共␧0兲 and coupling coefficients 共kij兲
PZN-4.5%PT 共M → M兲
PMN-28%PT 共M → M兲
e15
e31
e33
d15
d31
d33
␧S11
␧S33
8.9
9.28
⫺3.7
⫺4.7
15.0
20.5
140
132
⫺970
⫺1025
2000
2000
3000
2583
1000
835
␧T11
␧T33
3100
2750
5200
6550
k15
k31
k33
kt
0.21
0.22
0.82
0.88
0.90
0.93
0.43
0.57
Downloaded 24 Feb 2012 to 14.139.97.78. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
014102-5
J. Appl. Phys. 107, 014102 共2010兲
Shukla et al.
TABLE V. Comparison for 关110兴L ⫻ 关001兴T cut multidomain single crystals
of PZN-共6–7兲%PT, PZN-4.5%PT, and PMN-28%PT.
Crystal composition
PZN-共6–7兲%PT
PZN-4.5%PT
PMN-28%PT
d231 / sE11
共nN/ V2兲
51.9
30.6
44.9
In conclusion, domain wall contributions and domaindomain interactions may be significant when deducing the
property matrices of multidomain relaxor ferroelectric single
crystals. In this regard, it is advisable to use the M → M
approach of axis transformation technique when deducing
the property matrices of multidomain crystal of a different
cut. Using the said technique, the property matrices of PZN4.5%PT and PMN-28%PT single crystals of 关110兴L
⫻ 关001兴T cut are deduced from that of the 关100兴L ⫻ 关001兴T
cut. The results show that PZN-共6–7兲%PT single crystals of
2
E
/ s11
value than PZN关110兴L ⫻ 关001兴T cut exhibit a higher d31
4.5%PT and PMN-28%PT single crystals and are candidate
materials for actuator application involving an elastic member.
This work is supported by research grants received from
the Ministry of Education 共Singapore兲 and National University of Singapore, via Research Grant Nos. R-265-000-221112, R-265-000-257-112, R-265-000-257-731, and R-265000-261-123/490. One of the authors 共R.S.兲 acknowledges
the research scholarship and technical support received from
IIT Bombay and NUS for him to pursue his Ph.D. research.
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