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Journal of the European Ceramic Society 36 (2016) 189–202
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Journal of the European Ceramic Society
journal homepage: www.elsevier.com/locate/jeurceramsoc
Theoretical and experimental determination of the major
thermo-mechanical properties of RE2 SiO5 (RE = Tb, Dy, Ho, Er, Tm, Yb,
Lu, and Y) for environmental and thermal barrier coating applications
Zhilin Tian a,b , Liya Zheng a,b , Jiemin Wang a , Peng Wan a,b , Jialin Li a , Jingyang Wang a,∗
a
High-performance Ceramics Division, Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences,
110016 Shenyang, China
b
University of Chinese Academy of Sciences, Beijing 100049, China
a r t i c l e
i n f o
Article history:
Received 29 June 2015
Received in revised form 7 September 2015
Accepted 12 September 2015
Available online 26 September 2015
Keywords:
Rare-earth silicates
Environmental barrier coating
Mechanical property
Thermal property
a b s t r a c t
X2-RE2 SiO5 orthosilicates are promising candidate environmental/thermal barrier coating (ETBC) materials for silicon-based ceramics because of their excellent durability in high-temperature environments
and potential low thermal conductivities. We herein present the mechanical and thermal properties of
X2-RE2 SiO5 orthosilicates based on theoretical explorations of their elastic stiffness and thermal conductivity, and experimental evaluations of the macroscopic performances of dense specimens from room to
high temperatures. Mechanical and thermal properties may be grouped into two: those that are sensitive
to the rare-earth (RE) species, including flexural strength, elastic modulus, and thermal shock resistance,
and those that are less sensitive to the RE species, including thermal conductivity, thermal expansion
coefficient, and brittle-to-ductile transition temperature (BDTT). The orthosilicates show excellent elastic
stiffness at high temperatures, high BDTTs, very low experimental thermal conductivities, and compatible thermal expansion coefficients. The reported information provides important material selection and
optimization guidelines for X2-RE2 SiO5 as ETBC candidates.
© 2015 Elsevier Ltd. All rights reserved.
1. Introduction
Silicon-based ceramics such as SiC, Si3 N4 , and SiC-matrix composites, exhibit excellent high-temperature mechanical properties,
good stability in dry air, and potential applicability in gas turbines
[1]. However, when exposed to combustion environments containing water vapor, silicon-based ceramics fail promptly because of
volatilization of gaseous silicon hydroxide [2]. The unsatisfactory
environmental durability is the bottleneck to their technological
applications in combustion environments [3]. Two chemical degradation mechanisms are commonly observed: recession, which
involves loss of the silicon-based ceramics because of repeated
oxidation and volatilization in the presence of high-velocity water
vapor moving through the turbine [4,5], and hot corrosion, which
involves formation of pits in the silicon-based ceramics because of
liquid silicate formation in the presence of alkali salts commonly
present in combustion atmospheres [4]. To address these issue and
∗ Corresponding author.
E-mail address: [email protected] (J. Wang).
http://dx.doi.org/10.1016/j.jeurceramsoc.2015.09.013
0955-2219/© 2015 Elsevier Ltd. All rights reserved.
protect silicon-based ceramics and composites, environmental barrier coating (EBC) must be developed.
Great progress has been achieved in developing functionally graded thermal barrier coating/environmental barrier coating
(TBC/EBC) systems [4,6]; these technologies are hybrid coating systems wherein the multifunctional EBC layer prevents recession and
hot corrosion and the TBC layer reduces thermal conduction. However, considering the mechanical and thermal mismatches between
TBCs and EBCs as well as the chemical incompatibility of some
multifunctional coatings, trial-and-error is necessary to determine
proper coating components. Therefore, developing advanced ETBC
coating candidates possessing the integrated functions of EBCs and
TBCs is crucial to protect silicon-based ceramics and composites
from chemical attack and suppress thermal conduction simultaneously; these properties are believed to enable significant and
efficient increases in the performance of protective components
in gas turbines.
Rare-earth (RE) silicates are acknowledged as third generation
of EBC materials, and a number of silicates also have been identified
as low-thermal conductivity ceramics [7,8]. RE2 SiO5 , for example,
has attracted extensive research attention because of its hightemperature stability and water vapor corrosion resistance [5]. This
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Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
2. Theoretical calculation methods and experimental
procedure
2.1. Theoretical calculation methods
Fig. 1. Crystal structure of X2-RE2 SiO5 .
orthosilicate can be screened as a low-thermal conductivity material according to Clarke’s criteria for thermal insulators [9], which
suggests that RE2 SiO5 is an excellent and promising ETBC candidate.
RE2 SiO5 silicates present two monoclinic polymorphs. Compounds containing RE elements from La to Gd crystallize with a
space group of P21 /c, and are called X1 phases, while those with
RE elements from Dy to Lu feature a space group of C2/c, and
are called X2 phases. The X1 and X2 phases represent the lowtemperature and high-temperature phases, respectively [10], and
RE2 SiO5 specimens with Tb and Y have been reported to appear in
both phases. As shown in Fig. 1, the unit cell of X2-RE2 SiO5 contains
32 atoms, including two different RE3+ sites, RE1 and RE2, one Si
site, and five O sites labeled O1–O5. Four O (O1–O4) atoms form a
Si-centered distorted tetrahedron, while atom O5, with no Si atom
as the nearest neighbor, is loosely bonded to four RE ions, thereby
forming a distorted polyhedron. X2-RE2 SiO5 consists of rigid [SiO4 ]
tetrahedra and relatively soft [REO6 ] and [REO7 ] polyhedra [11].
X2-RE2 SiO5 orthosilicates possess smaller thermal expansion coefficients and lower thermal conductivities, making them suitable
for use as coating materials for silicon-based ceramics. Considering their beneficial properties, X2-RE2 SiO5 silicates were selected
as the representative material in the present investigations.
Mechanical and thermal properties, as well as property matching among substrates, are of great importance when selecting ETBC
materials with high reliability in harsh environments. Two main
obstacles prohibited the efficient evaluation of X2-RE2 SiO5 silicates
as ETBC. First, the mechanical and thermal properties of X2-RE2 SiO5
change with the RE elements incorporated into it; thus, guidelines
require extensive information of mechanical and thermal properties. Second, integrated ETBC candidates must satisfy important
criteria, such as low thermal conductivity, no phase transformation,
good thermal shock resistance (TSR), and excellent mechanical stability in the harsh service environments. Unfortunately, obtaining
accurate intrinsic properties through investigations of X2-RE2 SiO5
coating samples is difficult. As a convenient and reliable alternative, evaluating the mechanical and thermal properties of bulk and
dense X2-RE2 SiO5 samples is recommended.
We herein present theoretical predictions and experimental evaluations of the mechanical and thermal properties of
X2-RE2 SiO5 . The room-temperature mechanical properties of
X2-RE2 SiO5 samples were studied, and their dynamic Young’s
moduli and brittle-to-ductile transition temperatures (BDTT) were
explored at different temperatures. Thermal conductivities were
systemically investigated through analyses of crystal structures,
theoretical predictions, and experimental measurements. The thermal expansion coefficients of X2-RE2 SiO5 were obtained from 500
to 1473 K, and their TSR parameters were derived from measured
properties. The results obtained provide important guidelines for
the design and evaluation of X2-RE2 SiO5 as potential ETBC materials.
For X2-RE2 SiO5 (RE = Tb, Dy, Ho, Er, Tm, Yb, Lu and Y), firstprinciples calculations were performed using the VASP (Vienna
Ab-initio Simulation Package) code based on density functional
theory (DFT) [12]. Electron–ion interactions were represented by
using the projector augmented wave method [13], and the electronic exchange correlation energy was treated as a generalized
gradient approximation [14]. The plane wave basis cut off was set
to 450 eV, and special ␬-point sampling integration was used over
the Brillouin zone by applying the Monkhorst–Pack method with
a 3 × 3 × 3 ␬-point mesh [15]. The lattice parameters and internal freedom of the unit cell were fully optimized until the total
energy difference was smaller than 1 × 10−6 eV. Elastic constants
were determined from the linear fits of stress as functions of applied
homogeneous elastic strains [16]. The criterion of convergences
for total energy was selected as 1 × 10−6 eV in the elastic constant
calculations.
X2-RE2 SiO5 crystallizes in the C2/c space group and features 13
independent elastic constants. The polycrystalline bulk modulus
B, shear modulus G, and Young’s modulus E of the orthosilicate
were determined based on Voigt, Reuss, and Hill approximations
[17–19]. According to Voigt approximation, the bulk and shear
moduli can be calculated from elastic constants as:
BV =
2
1
(c11 + c22 + c33 ) + (c12 + c13 + c23 )
9
9
GV =
1
1
(c11 + c22 + c33 − c12 − c13 − c23 ) + (c44 + c55 + c66 ) (2)
15
5
(1)
According to Reuss approximation, the bulk and shear moduli
are calculated from compliance matrix components as:
BR =
1
(s11 + s22 + s33 ) + 2 (s12 + s13 + s23 )
GR =
1
(4)
4 (s11 + s22 + s33 ) − 4 (s12 + s13 + s23 ) + 3 (s44 + s55 + s66 )
(3)
The Voigt and Reuss averaging methods respectively assume
that strains and stresses are continuous in polycrystal and present
the upper and lower bounds of the effective bulk and shear moduli
for polycrystals. The Voigt–Reuss–Hill (VRH) approach combines
the upper and lower bounds by assuming the average Voigt and
Reuss elastic moduli to be a good approximation of the macroscopic
elastic moduli:
BVRH =
1
(BR + BV )
2
(5)
GVRH =
1
(GR + GV )
2
(6)
Young’s modulus E can be calculated using VRH’s shear and bulk
modui:
EVRH =
9GVRH BVRH
3BVRH + GVRH
(7)
Longitudinal and transversal sound velocities, l and s , are
derived from the shear modulus GVRH and bulk modulus BVRH :
l =
s =
BVRH +
4GVRH
3
G
VRH
(8)
1/2
(9)
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
Average sound velocity m is calculated as follows:
m =
3(s l )3
2l 3 + s 3
1/3
(10)
Using the average sound velocity m , the Debye temperature D
is obtained as follows [9]:
D =
1/3
h 3n NA M
kB 4
m
(11)
where h is Planck’s constant, kB is the Boltzmann’s constant, n is the
number of atoms in the primitive cell, NA is Avogadro’s number, is the density, and M is the molecular weight.
Based on the elastic constants obtained, the directional dependence of Young’s modulus (for monoclinic crystals) in 3D
representation can be calculated by Nye [20]:
t/L. The internal friction Q−1 corresponding to the flexural vibration mode was calculated as Q−1 = k/ff , where k is the exponential
decay parameter of the amplitude of the flexural vibration component. The bulk moduli of X2-RE2 SiO5 were evaluated from the
relationship between E and G,
B=
GE
3 (3G − E)
2.2. Material preparation
Bulk X2-RE2 SiO5 samples were prepared by using a two-step
processing method. First, pure X2-RE2 SiO5 powders were synthesized from a 1:1 molar ratio of commercially available powders
of RE2 O3 (Rear-Chem. Hi-Tech. Co., Ltd., Huizhou, China) and SiO2
(Sinopharm Chemical Reagent Co., Ltd., Shanghai, China). The powders were mixed by wet ball milling for 24 h in a Si3 N4 jar, and
the slip obtained was dried at 60 ◦ C for 24 h, and then passed
through a 120 mesh sieve to obtain the desired powders. The
powder mixture was thermally treated at 1550 ◦ C for 1 h to synthesize pure X2-RE2 SiO5 powders. Dense X2-RE2 SiO5 ceramics
were obtained through hot pressing using pure X2-RE2 SiO5 powders. Hot pressing was performed at 1600 ◦ C at a heating rate
of 10 ◦ C/min for 1 h under 30 MPa in a flowing Ar atmosphere.
The densities of the as-synthesized samples were determined by
using the Archimedes method, and the phase compositions of the
samples were identified using an X-ray diffractometer (XRD) with
CuK␣ radiation (D/max-2400, Rigaku, Tokyo, Japan). Microstructures were observed with a SUPRA 55 scanning electron microscope
(LEO, Oberkochen, Germany) and grain size distributions were analyzed using Image-Pro Plus software.
2.3. Mechanical properties
The dynamic Young’s modulus and internal friction of the samples were evaluated through an impulse excitation technique using
samples with dimensions of 3 mm × 15 mm × 40 mm. The samples
were measured in a graphite furnace RFDA-HTVP-1750-C (IMCE,
Diepenbeek, Belgium) at a heating rate of 4 ◦ C/min in an Ar atmosphere. Vibration signals captured by a laser vibrometer were
analyzed by a resonance frequency and damping analyzer, and
Young’s modulus was calculated from the flexural resonant frequency, ff , according to ASTME 1876-97:
E = 0.9465
mff
w
2
L3
t3
T1
(13)
where E is the dynamic elastic modulus in Pa, m is the mass of
the specimen in kg, ff is the fundamental flexural resonant frequency in Hz, and w, L and t are the width, length and thickness,
respectively, of the specimen in meters. T1 is a correction factor,
that depends on the Poisson ratio v and the thickness/length ratio
(14)
Bending strength was measured in samples with dimensions of
3 mm × 4 mm × 36 mm, and all tests were performed in a universal testing machine (CMT4204, SANS, Shenzhen, China). The
three-point bending method with a crosshead speed of 0.5 mm/min
was applied, and three samples for each X2-RE2 SiO5 specimen
were measured. Prior to testing, the surfaces of the samples were
polished with diamond paste of 1.5 ␮m to minimize machining
flaws. Fracture toughness was measured using the single-edge
1
3
2 2
2
2 2
2 2
4
4
2 2
2 2
3
4
2 2
2
= l1 s11 + 2l1 l2 s12 + 2l1 l3 s13 + 2l1 l3 s15 + l2 s22 + 2l2 l3 s23 + 2l1 l2 l3 s25 + l3 s33 + 2l1 l3 s35 + l2 l3 s44 + 2l1 l2 l3 s46 + l1 l3 s55 + l1 l2 s66
E
where sij are the elastic compliance coefficients that can be
obtained through the inversion of the elastic constants (sij = cij −1 )
and l1 , l2 and l3 are the directional cosines to the x-, y- and z-axes,
respectively.
191
(12)
notched beam method from specimens with dimensions of
4 mm × 8 mm × 36 mm. Notches were introduced by diamondcoated wheel slotting. The thickness of the blade was 0.15 mm and
the width of the notches was 0.2 mm. The saw depth was nearly half
of the specimen height (8 mm). The specimens were fractured in a
three-point bending test with a crosshead speed of 0.05 mm/min.
2.4. Thermal properties
Thermal diffusivity was obtained using a laser flash analyzer
(Netzsch LFA 457, Bavaria, Germany) in an Ar atmosphere from
room temperature to 1273 K. Prior to the thermal diffusion test,
the two sides of the samples were coated with a Ti adhesion
layer through multi-arc ion plating to minimize thermal irradiation through the sample and achieve a thin colloidal graphite layer
to ensure complete and uniform absorption of the laser pulses.
Thermal expansion coefficients were obtained from temperaturedependent changes in the length of the specimens from room
temperature to 1473 K in air as determined using a vertical hightemperature optical dilatometer (ODHT, Modena, Italy). Sample for
this test measured 3 mm × 4 mm × 14 mm.
3. Results and discussion
3.1. Phase composition and microstructure
The two-step synthesis method yielded X2-RE2 SiO5 samples
with relative densities higher than 97%. Fig. 2 compares the standard XRD spectrum of Dy2 SiO5 (ICDD PDF No. 40-0289) with the
experimental XRD patterns of X2-RE2 SiO5 and shows that single X2-RE2 SiO5 phases are generally formed. Minute amounts of
impurities (RE2 O3 or RE2 Si2 O7 ) are detected in Dy2 SiO5 , Ho2 SiO5 ,
Tm2 SiO5 and Yb2 SiO5 because of the narrow X2-RE2 SiO5 phase
region in the RE2 O3 -SiO2 system, and the diffraction peaks shift
toward higher angles as the atomic radius of the RE element
decreases. Fig. 3 shows the morphologies of X2-RE2 SiO5 samples
obtained after thermal etching at 1300 ◦ C for 30 min; here, most
grains are equiaxial. The average grain size of the X2-RE2 SiO5 samples is approximately 1 ␮m, and their grain size distributions are
listed in Table 1.
3.2. Room temperature mechanical properties
The mechanical properties of the X2-RE2 SiO5 samples, including
its elastic constant, Young’s modulus, shear modulus, bulk modulus, Poisson ratio, and flexural strength, are listed in Tables 2 and 3.
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Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
icant anisotropy in the elastic moduli of the X2-RE2 SiO5 silicates is
observed. The maximum and minimum Young’s moduli of the X2RE2 SiO5 samples are summarized in Table 4, and the ratio of the
maximum to the minimum values is approximately 2. Planar mappings of the Young’s moduli of the samples on the (0 0 1), (0 1 0),
and (1 0 0) atomic planes are also presented in Fig. 4 to provide
more details on the variation in Young’s modulus. Mappings on the
(0 1 0) and (0 0 1) atomic planes show strong anisotropic properties,
whereas those on the (1 0 0) atomic plane display relatively weak
anisotropy.
To quantify the anisotropic elastic stiffness, we used the
anisotropy index AU [24], which is defined by:
AU = 5
GV
BV
+
−6
GR
BR
(15)
For an isotropic system, AU is identically equal to zero and the
deviation from zero reflects the degree of anisotropy. In addition,
the anisotropy of the bulk and shear moduli can be obtained from
Voigt and Reuss approximations. The indices are defined as follows
[25]:
AB =
BV − BR
BV + BR
(16)
AG =
GV − GR
GV + GR
(17)
Fig. 2. XRD patterns of prepared X2-RE2 SiO5 .
Elastic constants (cij ) were initially obtained by DFT calculations,
and the results are shown in Table 2. The theoretical elastic moduli of the X2-RE2 SiO5 polycrystals are derived from the elastic
constants obtained and compared with the experimental data in
Table 3, thereby demonstrating that the theoretical Young’s and
shear moduli match the experimental values very well. The calculated Young’s moduli of all of the X2-RE2 SiO5 polycrystals are
less than 170 GPa; Lu2 SiO5 presented the highest Young’s modulus
(169 GPa) while Tb2 SiO5 presented the lowest (146 GPa). A similar
trend may be observed in terms of shear moduli.
Except for the Young’s modulus of Yb2 SiO5 and the flexural
strength of Ho2 SiO5 , the elastic moduli and flexural strengths of the
samples typically increase with decreasing atomic radii of the RE
elements. Yb-containing materials have been reported to exhibit
abnormal properties [21]; for example, the density, melting and
boiling points of Yb differ from those of other lanthanides. This
finding may be attributed to the closed-shell electron configuration
of Yb ([Xe] 4f14 6s2 ), which provides only two 6s valance electrons
available for chemical bonding. By contrast, three valence electrons
are available in most of the other lanthanides. The abnormal flexural strength of Ho2 SiO5 may be attributed to the strong anisotropy
of elastic stiffness, as discussed in the following section. The elastic
moduli and flexural strengths of the X2-RE2 SiO5 samples demonstrate a clear dependence on the atomic number of the RE element.
This finding may be explained by enhancements in RE O bonding
strength with decreasing ionic radius of the RE species. Wu et al. [8]
found electronegativity and RE O bonding strength enhancements
with increasing RE3+ radius in RE9.33 (SiO4 )6 O2 silicate oxyapatites.
The cation field strength (CFS) can be used to represent bond
strengths in RE silicates [22,23]. Hampshire proposed that the CFS
can be calculated using the equation: CFS = Zc /rc 2 , where Zc is the
cationic charge and rc is the cationic radius. The ionic radius of
lanthanides decreases with increasing of atomic number because
of lanthanide contraction. Using the ionic radius of RE3+ , the calculated CFS values of the X2-RE2 SiO5 samples are also listed in Table 3;
the values obtained suggest strengthening of RE O bonding in X2RE2 SiO5 with the decrease of ionic radii of RE elements.
The directional dependence of the Young’s moduli of the X2RE2 SiO5 samples was calculated using Eq. (12) and the results
are plotted in Fig. 4. This figure demonstrates the variation of
elastic modulus along different crystallographic directions. Signif-
For isotropic structures, the Voigt and Reuss approximations must
have the same values of B or G. The AB and AG must be zero, and
deviations from zero indicate anisotropy. The anisotropy indices
AU , AB , and AG of X2-RE2 SiO5 are calculated and shown in Table 5.
For AU , Ho2 SiO5 and Y2 SiO5 respectively possess the greatest and
least anisotropy indices. The values of AB of Ho2 SiO5 and Y2 SiO5
are similar in magnitude, whereas the AG of Ho2 SiO5 is more than
twice that of Y2 SiO5 . Therefore, the large anisotropy of the elastic
stiffness of Ho2 SiO5 mainly comes from the significant anisotropy
of its shear modulus.
3.3. Temperature-dependent Young’s modulus and internal
friction
During the start-up or shut-down operation of turbine followed
by oxidation or corrosion of the substrates, cracks may form and
coating may spall off as a result of severe thermal cycling. Thus, the
temperature dependence of elastic stiffness and the BDTT of coating materials need to be determined [26]. Temperature-dependent
Young’s modulus and internal friction of the samples were measured by using the resonant frequency method, and the BDTT
and Debye temperature were derived from the experimental data
obtained. Fig. 5 shows the temperature-dependent Young’s moduli and internal friction of the X2-RE2 SiO5 samples. The Young’s
moduli decrease nearly linearly from room temperature to relatively high temperatures and then rapidly dropped at certain
critical temperatures; these phenomena are accompanied by exponential increases in internal friction. The X2-RE2 SiO5 samples also
maintain over 80% of their room-temperature Young’s moduli at
1500–1800 K, thereby demonstrating excellent retention of the
high-temperature Young’s moduli. The Young’s modulus could not
be measured at higher temperatures, because the noise observed
under this condition is too high to resolve the resonance frequency
of the samples.
Internal friction, which is also referred to as the mechanical
loss spectrum, can be used to determine the BDTT. Internal friction of X2-RE2 SiO5 orthosilicates maintain low magnitudes from
room temperature to at least 1400 K and abruptly increase thereafter. Exponential increases in internal friction are accompanied by
decreases in Young’s moduli at high temperatures, thus accounting
for the so-called “brittle-to-ductile” transition in ceramic materials
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
193
Fig. 3. Surface morphologies of X2-RE2 SiO5 after thermal etching at 1300 ◦ C for 30 min.
Table 1
Average grain size and grain size distributions of X2-RE2 SiO5 (in ␮m).
Average grain size
2–1.5
1.5–1
1–0.5
0.5–0
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
0.73
6.5%
19.4%
41.9%
32.2%
0.77
6.7%
16.7%
56.6%
20.0%
0.76
6.9%
13.8%
51.7%
27.6%
0.98
10.5%
36.8%
31.6%
21.1%
0.84
4.0%
20.0%
56.0%
20.0%
1.02
16.7%
27.8%
44.4%
11.1%
0.51
1.6%
3.3%
42.6%
52.5%
0.74
3.1%
18.8%
40.6%
37.5%
Table 2
Elastic constants of X2-RE2 SiO5 (in GPa).
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
C11
C12
C13
211
215
220
225
225
223
240
224
51
54
58
61
66
55
67
92
80
81
83
84
78
102
86
55
C15
17
16
15
14
8
16
9
5
C22
C23
C25
C33
C35
C44
C46
C55
C66
169
179
190
201
214
185
221
208
37
40
44
47
49
49
49
29
−16
−16
−30
−17
−20
−18
−20
0
181
184
185
188
179
202
194
154
−18
−20
−24
−26
−20
−21
−35
0
43
45
47
49
52
46
52
47
10
10
11
11
10
15
11
10
75
76
76
77
72
83
81
64
57
62
63
65
67
62
71
65
194
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
Fig. 4. Directional Young’s moduli of X2-RE2 SiO5 in: 3-D scenario, (1 0 0), (0 1 0), and (0 0 1) atomic plane.
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
195
Table 3
Young’s modulus E (GPa), bulk modulus B (GPa), shear modulus G (GPa), Poisson’s ratio , flexural strength f (MPa), Debye temperature D (K) and CFS (Å−2 ) of X2-RE2 SiO5 .
Method
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
Calc.
Expt.
Calc.
Expt.
Calc.
Expt.
Calc.
Expt.
Calc.
Expt.
Calc.
Expt.
Calc.
Expt.
Calc.
Expt.
B
G
E
BV
BR
BH
GV
GR
GH
100
86
103
92
107
86
111
93
112
99
114
94
118
105
104
86
96
98
61
59
63
62
65
61
66
64
67
67
66
65
71
70
62
65
55
58
99
101
107
108
109
101
104
109
110
111.5
111
114.5
97
100.5
58
146
144
152
151
153
148
159
157
161
163
158
158
169
172
152
155
60.5
58
61.5
60
63
62
64.5
58
62
64
67.5
59
60.5
f
155 ± 5
153 ± 3
236 ± 10
156 ± 4
200 ± 5
207 ± 20
214 ± 19
177 ± 3
D
CFS
0.251
0.229
0.250
0.225
0.254
0.214
0.256
0.219
0.255
0.225
0.264
0.221
0.253
0.226
0.248
0.200
416
416
419
422
418
416
423
425
424
430
413
418
426
433
513
528
3.52
3.61
3.70
3.79
3.87
3.98
4.047
3.704
Table 4
Maximum Young’s modulus (Emax ), minimum Young’s modulus (Emin ) and the ratio of Emax /Emin of X2-RE2 SiO5 .
Emax (GPa)
Emin (GPa)
Emax /Emin
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
216
108
2.00
217
113
1.92
221
114
1.94
220
121
1.82
207
127
1.63
221
111
1.99
229
123
1.86
183
113
1.62
Table 5
Anisotropy indices AU , AG and AB of X2-RE2 SiO5 .
U
A
AB
AG
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
0.587
0.020
0.052
0.558
0.020
0.049
0.663
0.029
0.057
0.537
0.018
0.048
0.360
0.018
0.031
0.639
0.022
0.056
0.610
0.031
0.052
0.326
0.035
0.025
Fig. 5. Temperature-dependent Young’s moduli and internal friction of X2-RE2 SiO5 .
[27]. Kardashev [28] suggested that exponential increases in internal friction correspond to the BDTT. The BDTT can be determined
by linear fitting of the internal friction data at various temperature ranges. As indicated in Fig. 5, the BDTTs of X2-RE2 SiO5 are
determined using the method proposed by Kardashev [28], and
the results are shown in Fig. 6. The BDTTs of X2-RE2 SiO5 generally decrease with decreasing ionic radius of the RE elements. The
BDTTs of Tb2 SiO5 , Dy2 SiO5 , Ho2 SiO5 and Er2 SiO5 , for example, are
between 1600 and 1700 K. By contrast, other X2-RE2 SiO5 samples
with smaller RE elements possess lower BDTTs ranging from 1480 K
to 1570 K. The largest difference between the values obtained is
approximately 180 K. Damping mechanisms are positive factors for
Fig. 6. Brittle-to-ductile transition temperatures of X2-RE2 SiO5 as a function of RE
ionic radius.
improving toughness in hard and brittle materials and illustrate
the ability of a material to dissipate part of the vibrational energy
locally, thereby improving toughness by crack blunting. Sufficient
strain tolerance, which can extend the life of ETBC coatings, can be
obtained at temperatures above the BDTT.
Brittle-to-ductile transition is usually associated with the
changing of fracture modes. In brittle fractures, cracks typically
propagate by cleavage along specific crystallographic atomic planes
or along weak grain boundaries that are embrittled by impurity
196
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
Fig. 7. Fracture morphologies of Yb2 SiO5 after flexural strength tests at (a) room temperature, (b) 1473 K, and (c) 1573 K.
segregation. For example, Fig. 7 shows the fracture surfaces of
Yb2 SiO5 (BDTT = 1519 K) after testing of its flexural strength at room
temperature, 1473 K, and 1573 K. While, transgranular fracture
occurred at room temperature, as shown in Fig. 7(a), the fractures at 1473 K reflect a mixture of transgranular and intergranular
modes, as shown in Fig. 7(b). At 1573 K, intergranular fractures are
clearly observed, because the morphology of grains can be easily
distinguished on fracture surface in Fig. 7(c). Schaller [27] suggested that brittle-to-ductile transition can be decomposed into
two stages namely, brittle-to-tough transition and tough-to-ductile
transition. As the cracks propagating along the grain boundary can
dissipate more energy, fracture toughness must be enhanced at elevated temperatures. The fracture toughness of Yb2 SiO5 at 1573 K
was measured to be 3.5 ± 0.3 MPa m1/2 , much higher than that at
room temperature 2.0 ± 0.1 MPa m1/2 . Combining the current comprehensive information, the BDTT of Yb2 SiO5 was determined by
the mechanical spectrum to be 1519 K. BDTT can provide reliable
knowledge on the ductile performance of ductility of X2-RE2 SiO5
silicates and is helpful for aiding the selection of optimal ETBC
candidates.
Experimental Young’s moduli were used to deduce the experimental Debye temperatures of X2-RE2 SiO5 . Wachtman et al.
[29] proposed an empirical formula describing the temperaturedependent Young’s modulus:
E = E0 − F × T × exp
−T 0
T
l =
s =
min
= 0.87kB
E0
1
0 2 (1 + )
(20)
where E0 is the fitted Young’s modulus at 0 K, 0 is the theoretical
density, and is the experimental Poisson ratio. Thereafter, the
Debye temperature can be obtained using Eqs. (19) and (20). The
Debye temperatures of the X2-RE2 SiO5 samples (listed in Table 3)
match those determined through DFT calculations well. In general,
except for that of Y2 SiO5 , Debye temperatures change within 4% in
magnitude and increase slightly with increasing RE atomic number.
3.4. Minimum thermal conductivity
Thermal conductivity is one of the key indices used to determine
the feasibility of integrated ETBC materials. The thermal conductivity of insulation materials can generally be divided into two
(21)
1
2 + 2
−3/2
+
1
2
+
3 − 6
3 + 2
−3/2
2/3 1/6 E 1/2
2/3 n
M 2/3
× kB NA
−1/3
(22)
where v is Poisson ratio.
Cahill et al. [31] calculated the min using a model of random
walks among localized quantum mechanical oscillators and formulized min as follows:
min
(19)
nNA
1
2
3
=
(18)
E0
(1 − )
0 (1 + ) (1 − 2)
M −2/3 E 1/2
where kB is the Boltzmann constant, NA is Avogadro’s constant, n
is the number of atoms in the primitive cell, is the density, E is
the Young’s modulus, and M is the molecular weight. To lower the
min , large mass contrast, complicated structure, and small elastic
stiffness are necessary.
Liu et al. [30] modified Eq. (21) by introducing elastic anisotropy
and calculated min from the crystal structural information and
elastic parameters:
min
where E0 is the Young’s modulus at 0 K, T is absolute temperature,
and F and T0 are fitting parameters. The temperature-dependent
Young’s moduli of the X2-RE2 SiO5 samples are fitted by Eq. (18)
(dash-dotted line in Fig. 5), and the fitted parameters are listed in
Table 6. To calculate Debye temperatures, knowledge of the longitudinal l and transversal sound velocities s is necessary. These
parameters can be obtained through the following relationship:
regions from room temperature to high temperatures [9]: (1) thermal conductivity decreases with ∼1/T (T is temperature) because of
anharmonic Umklapp phonon–phonon scattering; and (2) at high
temperatures, thermal conductivity is independent of the temperature and approximates the minimum thermal conductivity ( min ).
In most cases, min can be used as a qualitative material selection
guideline for identifying low thermal conductivity ceramics.
As suggested by Clarke, the phonon mean free path converges
to the average interatomic distance at high temperature limits,
and the average sound velocity m could be approximated as
0.87 E/ [9]. Hence, the min is calculated as follows:
=
1/3
6
i
kB n2/3
i
T 2
i
i
0
x 3 ex
(ex
− 1)2
dx
(23)
where n is the number density of atoms, i are the three speeds
of sound (one longitudinal and two transverse), x is the distance
that an atom is displaced from equilibrium, and i is the cutoff
frequency for each polarization expressed in degrees using the following equation:
i = i
kB
1/3
62 n
(24)
where is the reduced Planck constant, kB is the Boltzmann constant, and n is the number density of atoms per unit volume.
At extremely high temperatures, Eq. (23) can be simplified to:
min
=
kB 2/3
n
(2s + l )
2.48
(25)
The min of X2-RE2 SiO5 calculated by Eqs. (21), (22), and (25)
are listed in Table 7. The min values of X2-RE2 SiO5 are very low in
the vicinity of 1 W m−1 K−1 , thereby indicating their potential low
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
197
Table 6
Fitted parameters presented in Eq. (18) for X2-RE2 SiO5 .
E0
F
T0
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
143.72
0.02191
1359.75
151.88
0.02242
1408.22
147.60
0.01902
1070.57
155.70
0.01974
1061.82
160.78
0.02641
1387.69
157.99
0.02632
1596.06
171.31
0.01889
694.90
154.40
0.02094
1127.42
Table 7
Minimum thermal conductivities (W m−1 K−1 ) of X2-RE2 SiO5 .
min (Clarke)
min (Liu)
min (Cahill)
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
1.00
0.80
0.86
1.01
0.82
0.88
1.01
0.82
0.88
1.03
0.83
0.90
1.03
0.83
0.90
1.01
0.81
0.89
1.05
0.85
0.93
1.26
1.01
1.11
thermal conductivity. The min calculated using the three models
demonstrates a dependence on the atomic number of the RE elements. In addition, the min values calculated using Clarke’s model
are relatively high and the values calculated using other methods
are close to each other. These phenomena may be explained by
that fact that in Clarke’s model, all sound velocities are assumed to
be pure modes even for systems with low symmetry. By contrast,
in Liu’s and Cahill’s methods, minimum thermal conductivities are
treated as the summation of three acoustic branches (one longitudinal mode and two transverse modes), which may make them
more suitable for predicting the min of anisotropic materials [32].
Horai and Simmons [33] found that the silicon-oxygen network
functions as the propagating path of heat in many silicate minerals.
Because they are located inside the silicon-oxygen network, metallic atoms act as traps that consume part of the vibration energy to
excite the vibrational motion of metallic atoms. Effective phonon
transportation varies inversely with the mass of the metallic atoms.
Because RE atoms possess large atomic masses and tend to capture
large amounts of vibration energy, X2-RE2 SiO5 orthosilicates have
very low min .
Fig. 8. Temperature-dependent thermal diffusivities of X2-RE2 SiO5 .
3.5. Temperature-dependent thermal conductivity
The above predictions of min indicate that X2-RE2 SiO5 orthosilicates are promising low-thermal conductivity materials. In this
section, experimental thermal conductivities are measured at various temperatures and compared with the calculated ones. The
temperature dependence of thermal conductivity provides crucial information for thermal transport throughout the specified
temperature range. Thermal conductivity is experimentally determined from the measurements of thermal diffusivity a, heat
capacity Cp , and density using the following equation:
= aCp (26)
Fig. 8 shows the temperature-dependent experimental thermal
diffusivities of the X2-RE2 SiO5 samples. The thermal diffusivities
of X2-RE2 SiO5 generally decrease with the increasing temperature, and the values are enhanced at high temperatures because
of thermal radiation contributions. In terms of the constituent
binary oxides (RE2 O3 and SiO2 ), the isobaric heat capacities CP
of X2-RE2 SiO5 are determined from literature data [34] through
the Neumann–Kopp rule, which may reliably reproduce the heat
capacity of complex ternary oxides, such as RE9.33 (SiO4 )6 O2 [8],
yttria-stabilized zirconia [35], and rare earth zirconates [36], with
errors typically within ±3% compared with precise experimental
measurements [36]. Fig. 9 plots the heat capacities of X2-RE2 SiO5
and shows that silicates with smaller molecular weights tend to
present lower heat capacities. The experimental densities of X2RE2 SiO5 where the RE is Tb, Dy, Ho, Er, Tm, Yb, Lu, and Y are 6.35,
6.49, 6.56, 6.73, 6.90, 7.10, 7.24, and 4.22 g/cm3 , respectively.
Fig. 9. Heat capacities of X2-RE2 SiO5 .
The experimental thermal conductivities of X2-RE2 SiO5 are
calculated using Eq. (26), and results are shown in Fig. 10. The
experimental values range from 1.87 W m−1 K−1 to 3.48 W m−1 K−1
at room temperature, which is roughly twofold difference among
the different RE elements. At 1273 K, differences among experimental thermal conductivities decrease, and values range from
1.1 W m−1 K−1 to 1.6 W m−1 K−1 , except for Lu2 SiO5 , which is evidently affected by thermal radiation [37]. Fig. 10 demonstrates that
X2-RE2 SiO5 silicates have extremely low thermal conductivities,
although the thermal conductivities vary irregularly with the RE
198
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
Fig. 10. Experimental, theoretical and minimum lattice thermal conductivities of X2-RE2 SiO5 .
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
199
Table 8
Theoretical parameters for calculating intrinsic lattice thermal conductivities of X2-RE2 SiO5 using Slack’s model. T is absolute temperature in Kelvin.
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
s (m/s)
l (m/s)
m (m/s)
3015
3028
3006
3027
3035
2941
3008
3641
5229
5248
5237
5287
5289
5190
5228
6290
3348
3362
3339
3363
3371
3270
3341
4041
1.50
1.50
1.52
1.53
1.52
1.57
1.50
1.49
species. Multiple phonon scattering mechanisms among the experimental samples may explain this phenomenon.
According to the Debye model, lattice thermal diffusivity a is
defined as [38,39]:
1
a = m
3
(27)
where m is the average sound velocity and is the phonon mean
free path. As m is nearly temperature independent, a is determined
by the variation of at different temperatures. The phonon mean
free path can be divided into several independent parts as follows
[39–41]:
1
1
=
+
phonon
1
defect
+
1
boundary
+
1
i
(28)
i
where phonon , defect , boundary , and i are the phonon mean free
paths corresponding to anharmonic Umklapp phonon scattering,
defect scattering, grain boundary scattering and other mechanisms,
respectively. The values of defect , boundary and i are nearly temperature independent [42–44]; thus, the temperature dependence
of phonon mean free path is dominated by the behavior of phonon
at elevated temperatures, and phonon can be used to describe the
intrinsic lattice thermal resistance throughout the whole temperature range. defect , boundary , and i relate to extrinsic effects and
may be influenced by different microstructures and defect concentrations at low temperatures. Thermal radiation effects at high
temperatures also contribute to the rise in thermal conductivity.
So it can be concluded that thermal conductivities of X2-RE2 SiO5
were affected by different phonon scattering mechanisms in different temperature ranges which blocks the insight of intrinsic
lattice thermal conductivities.With the help of reliable theoretical models, such as Slack’s model, the intrinsic lattice thermal
conductivity can be predicted at different temperatures, which
is necessary to eliminate extrinsic uncertainties from experimental thermal conductivity. For a typical perfect insulation ceramic,
heat is conducted primarily by acoustic phonons because their
group velocities are much larger than those of optical phonons.
At temperatures above the Debye temperature, the interactions
among acoustic phonons themselves are influenced by anharmonic
Umklapp processes; thus, the relationship between intrinsic lattice
thermal conductivity and temperature T may be well described
by ∼1/T . Under these approximations, Slack derived the following
expression for the intrinsic lattice thermal conductivity [45]:
=A
M̄ı
3 1/3
a n
2T
(29)
where A = 3.04 × 10−7 W mol kg−1 m−2 K−3 , M̄ is the mean atomic
mass, ı3 is the average volume per atom, n is the number of atoms
per primitive cell, a is the acoustic-mode Debye temperature, T is
the absolute temperature, and is the high temperature limit of
the acoustic phonon Grüneisen parameter.
Typically, when considering new materials with complex crystal structures, phonon dispersion relationships cannot be used to
determine a . Alternatively, the acoustic-mode Debye temperature
ı (Å)
n
M (g)
2.40
2.39
2.38
2.37
2.37
2.36
2.33
2.35
32
32
32
32
32
32
32
32
53.250
54.125
54.719
55.313
55.719
56.75
57.25
35.75
lattice
(W m−1 K−1 )
1222/T
1273/T
1239/T
1274/T
1309/T
1158/T
1362/T
1533/T
a is generally calculated from the “traditional” definition of the
Debye temperature D using a = D n−1/3 . Thus, Eq. (29) can be
rewritten to the common form as:
=A
3
D
2 n2/3 T
M̄ı
(30)
This expression emphasizes that ceramics with low Debye temperatures, strong anharmonicity and high complexity tend to have
low thermal conductivities. Thus, our next goal is to calculate all of
the material parameters in Eq. (30) precisely using first-principles
calculations.
Approximate formulations to calculate the Grüneisen paramdetermined from thermal
eter , include the thermodynamic
expansion coefficient, bulk modulus, molar volume and specific
volumetric heat capacity [46], mode-specific
calculated from
the volume dependence of phonon frequency [46], and acoustic
obtained from the longitudinal and transversal sound velocities [47–49]. In this study, we calculated the acoustic to depict
anharmonic scattering among acoustic phonons. The acoustic is
rooted in the original definition of the Grüneisen parameter, =
−∂lnω/∂lnV, and formulized by equalizing the vibration energy
of acoustic phonon to the energy of elastic wave propagation in
a continuum approximation of polycrystals. Based on this approximation, the anharmonicity parameter was directly related to the
pressure of thermal motion of phonon collection, which could be
further expressed in terms of bulk modulus, density and velocity of
elastic wave propagation [49]. Previous investigations have shown
that acoustic Grüneisen parameter can be reliably calculated as
follows [47–49]:
=
1+ 3
=
2
9 l 2 − 43 s 2
2 l 2 − 2s
2
2 − 3
(31)
where l and s are longitudinal and transversal sound velocities,
respectively, and is the Poisson ratio.
Using Slack’s model, the chemical composition, crystal structure, and elastic parameters of ceramics can be quantitatively
related to their lattice thermal conductivity, and the results may
provide guidelines on how to reduce intrinsic lattice thermal conductivity efficiently by tailoring chemical composition and crystal
structure. Table 8 summarizes all of the theoretical parameters
used to calculate the intrinsic lattice thermal conductivities of
X2-RE2 SiO5 using Eq. (30). The theoretical lattice thermal conductivities lattice of X2-RE2 SiO5 are plotted in Fig. 10 and show
decreases with the RE atomic number. lattice also varies with
temperature T (in K) from 1526/T (Y2 SiO5 ) to 1158/T (Yb2 SiO5 ).
However, the experimental results notably deviate from the theoretical data at both low and high temperatures, likely because heat
conduction is only depicted by Umklapp scattering among acoustic
phonons in the theoretical prediction. X2-RE2 SiO5 ceramics were
fabricated by hot pressing, which introduces a considerable amount
of defects and impurities to the result material. The phonon mean
free paths evidently decrease by extrinsic scattering of defects,
leading to lower thermal conductivities at low temperature range.
200
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
Fig. 11. Thermal expansion coefficients of X2-RE2 SiO5 at different temperatures.
At high temperatures, phonon scattering of native defects which
exerts a weak effect on the measured thermal conductivities, is
temperature-independent. However, thermal irradiation apparently enhances thermal transportation and dominates abnormal
enhancements in the experimental thermal conductivities.
We compare the parameters presented in Slack’s model to
understand the origins of the different thermal conductivities of
X2-RE2 SiO5 . X2-RE2 SiO5 species have the same number of atoms
per primitive unit cell n, and similar average volumes of one atom
in the primitive unit cell ı3 and Grüneisen parameter . However,
Y2 SiO5 shows notably different magnitudes of Debye temperature
D and mean atomic mass M. The M D 3 of Yb2 SiO5 is approximately
85% of that of Y2 SiO5 . This deviation leads to a significant difference
in thermal conductivity, as thermal conductivity is proportional to
D 3 , as shown in Eq. (30). The lower Debye temperature of Yb2 SiO5
leads to its smaller theoretical lattice thermal conductivity in comparison with that of Y2 SiO5 . is also relevant when evaluating the
intrinsic low thermal conductivity. The
is often referred to as
a temperature-dependent anharmonicity parameter that reflects
the deviation of phonon vibrations in a crystal lattice from harmonic oscillations. Anharmonicity of lattice vibrations drives the
phonon–phonon Umklapp and normal processes that limit the latvalues of
tice thermal conductivity. The Grüneisen parameter
X2-RE2 SiO5 are calculated to be 1.50–1.54, which indicates a high
degree of anharmonicity in the materials. Beyond the value of M D 3 ,
Yb2 SiO5 and Y2 SiO5 possess the largest and smallest Grüneisen
parameters , respectively, which also contribute to their extreme
low and relatively high theoretical lattice thermal conductivities.
3.6. Thermal expansion
The thermal expansion coefficients (TECs) of polycrystalline
X2-RE2 SiO5 measured by an optical dilatometer are illustrated in
Fig. 11. The lengths of the X2-RE2 SiO5 samples expand with increasing temperature up to 1473 K. Because the as-sintered samples have
no preferred orientation (Fig. 4), isotropic thermal expansion of the
X2-RE2 SiO5 bulk materials is observed. The TECs are determined as
a function of temperature during the heating process in terms of
the slopes of thermal expansions and are found to approximately
range from 6.94 × 10−6 K−1 to 8.84 × 10−6 K−1 at 1473 K. The TECs
of RE silicates are only slightly modified by the RE3+ ionic radius
but strongly depend on the crystal structures of different polymorphs. This finding can be explained by considering the two types
of bonds present in the RE silicate structure, namely Si O and RE O
bonds. Previous investigations indicate that the mean coefficient of
thermal expansion of the [SiO4 ] tetrahedron is about ∼0 K−1 [50].
Therefore, X2-RE2 SiO5 mainly expand within the [REO6/7 ] polyhedron chains. Hazen and Prewitt [51] further found that the TECs
of different polymorphs may be mainly determined by the charge
and coordination of RE cations. X2-RE2 SiO5 orthosilicates have the
same charge and coordination of RE ions, and therefore, the TECs
weakly depend on the RE elements.
The TEC is an important parameter for the property evaluation
of ETBC materials. TEC mismatch between the substrate and ETBC
presents a dominant effect on the performance of an RE orthosilicate and may result in coating cracking, and spalling because of high
thermal stress. The TECs of X2-RE2 SiO5 samples are higher than
those of SiC and SiC/SiC CMC (4.5–5.5 × 10−6 K−1 ) [5]. Two possible solutions can balance this TEC mismatch: addition of a gradient
transition layer and enhancement of the porosity of the X2-RE2 SiO5
layer. The porosity of ceramic coatings produced via the plasma
spray method typically ranges from 3% to 20% [52]. Pores provide
spaces for expansion, which lowers the TECs of X2-RE2 SiO5 and
residual stresses [53]. In addition, RE disilicates always form at the
interface between silicon-based ceramic and X2-RE2 SiO5 coatings
because of the presence of SiO2 , which can release thermal stress
[54].
3.7. Thermal shock resistance
Rapid heating and cooling result in thermal stresses that can lead
to damage or catastrophic failure of ceramics. The TSR of a material changes with its flexural strength, elastic modulus, Poisson
ratio, and thermal expansion coefficient. Therefore, keeping these
parameters at optimal levels and carefully identifying their relationship with each other are important [52]. The TSR parameters
available in literature are “so-called figures of merit” that can facilitate the ranking and selection of materials for engineering designs
involving thermal stress fracture. Kingery defined TSR parameters
as [55]:
R=
f (1 − ␯)
E˛
(32)
where f is the flexural strength, is the Poisson ratio, E is Young’s
modulus and ˛ is the thermal expansion coefficient.
The thermal stress fracture resistance parameter represents the
reliability of a material under significant thermal stress. If the stress
exceeds the strength of a material, fracture will occur. Using the
measured thermal and mechanical properties, the TSR parameters
of X2-RE2 SiO5 are obtained and listed in Table 9, together with the
values of Al2 O3 and SiC [56] for comparison. Among the samples
surveyed, Ho2 SiO5 exhibits the best TSR at 165 K, which is even
higher than that of the engineering ceramic Al2 O3 (150 K). Such
a high TSR may be attributed to the excellent flexural strength of
Ho2 SiO5 .
In summary, the mechanical and thermal properties of X2RE2 SiO5 are systematically investigated by a combination of DFT
calculations and experimental evaluations. The main experimental
mechanical and thermal properties of X2-RE2 SiO5 are plotted in a
radar chart for comparison, as shown in Fig. 12. The radar chart facilities easy comparison of all of the attributes of interest and provides
overview information. The mechanical and thermal properties of
X2-RE2 SiO5 are normalized with the highest values among these
silicates. Therefore, every attribute is scaled between 0 and 1 for
comparison. The mechanical and thermal properties are summarized into two groups. Young’s modulus, flexural strength, and TSR
show correlations with the contraction of RE elements, while thermal conductivity, thermal expansion, and BDTT depend weakly on
the RE atomic number. Therefore, mechanical and thermal properties based on the radar chart can easily be balanced during material
selection for optimal designs of X2-RE2 SiO5 for ETBC materials.
Z. Tian et al. / Journal of the European Ceramic Society 36 (2016) 189–202
201
Table 9
Thermal shock resistance parameters of X2-RE2 SiO5 with some engineering ceramics for comparison.
Tb2 SiO5
Dy2 SiO5
Ho2 SiO5
Er2 SiO5
Tm2 SiO5
Yb2 SiO5
Lu2 SiO5
Y2 SiO5
Al2 O3
SiC
92
102
165
99
117
134
132
119
150
300
Fig. 12. Radar chart for comparison of the properties, including Young’s modulus,
flexural strength, BDTT, thermal conductivity at 300 K, thermal expansion at 1473 K
and thermal shock resistance of X2-RE2 SiO5 . Every attribute is normalized by the
highest values among all silicates and is scaled between 0 and 1.
4. Conclusion
Dense and pure X2-RE2 SiO5 ceramics are successfully synthesized using a two steps involving hot pressing sintering. The
mechanical properties of the samples, including their Young’s
modulus, shear modulus, bulk modulus, Poisson ratio, and Debye
temperature, are studied through a combination of DFT calculations and experimental investigations. The calculated values are in
agreement with the experimental results. The elastic moduli (B, G
and E) of X2-RE2 SiO5 increase from Tb2 SiO5 to Lu2 SiO5 because
of contraction of the RE3+ ionic radius. X2-RE2 SiO5 orthosilicates
exhibit anisotropic Young’s moduli. The brittle-to-ductile transitions and Debye temperatures are derived from the temperature
dependence of internal friction and Young’s modulus. In contrast
to the mechanical properties, thermal properties show weak relationships with the RE element. X2-RE2 SiO5 silicates exhibit very
low thermal conductivity; in fact, the thermal expansion coefficients of the samples are approximately 6.94–8.84 × 10−6 K−1 at
1473 K. Thermal shock resistance parameters are obtained based
on the measured properties, and Ho2 SiO5 is determined to present
the best TSR. The present work sheds light on the potential use of
X2-RE2 SiO5 orthosilicates as advanced ETBC candidates and provides guidelines for the selection or optimization of X2-RE2 SiO5 as
an ETBC material.
Acknowledgment
This work was supported by the Natural Science Foundation of
China under Grant nos. 51032006 and 51372252.
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