J. Am. Ceram. Soc., 84 [12] 2937– 46 (2001)
journal
Distributed Porosity as a Control Parameter for Oxide Thermal Barriers
Made by Physical Vapor Deposition
Tian Jian Lu
Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
Carlos G. Levi
Materials Department, University of California, Santa Barbara, California 93106-5050
Haydn N. G. Wadley
Materials Science Department, University of Virginia, Charlottesville, Virginia 22903
Anthony G. Evans*
Materials Institute, Princeton University, Princeton, New Jersey 08544
substrate, through its effect on the in-plane compliance.1 Accordingly, morphological changes used to decrease the thermal conductivity must be consistent with the retention of in-plane compliance. In practice, the possibilities are limited by diffusiondriven changes in morphology that occur at the highest
temperatures encountered during turbine operation.2
The principal objective of this article is to examine the possibilities of using pore morphology to affect the thermal conductivity that might be realizable within the scope of practical methods
for depositing the oxide. Two additional provisos are that the
morphology be stable at high temperature and that the strain
tolerance be retained. Because some degree of morphological
evolution is likely, a corollary is an assessment of the conductivity
degradation that may result from such evolution. Direct measurements of thermal conductivity are difficult to perform nondestructively. A second goal is, therefore, to establish connections
between the thermal conductivity and the elastic modulus, with the
motivation that measurements of the out-of-plane modulus can be
a simple-to-implement probe for monitoring changes in the thermal conductivity. (Note that the only existing measurements3 refer
to the in-plane modulus.) The measurement possibilities are
addressed in Section VI.
To address these objectives, the article explores a range of
morphological possibilities, by compiling results from the literature and generating new thermal resistance solutions, as appropriate. A continuum heat flow analysis is used, consistent with the
phonon mean free path for stabilized zirconia being small (⬃1 nm)
relative to the size of pores that remain stable at high temperature.
This article is organized in the following manner. Basic thermal
barrier coating (TBC) pore morphologies are described. Results for
the effects of such porosity on the thermal conductivity are
assembled and correlated with measurements. Cross-correlations
with the elastic properties are then discussed.
Thermal barrier coatings (TBCs) made by electron-beam
physical vapor deposition (EB-PVD) exhibit a thermal conductivity strongly affected by a hierarchy of pores introduced
during the deposition process. These pores are in the form of
narrow gaps, aligned spheroids, and random spheres at specific sites within the coatings. Models for the effects on thermal
conductivity of pores having these shapes and spatial arrangements are taken from the literature and combined with new
results to provide descriptors of overall relationships between
the relative density and pore morphology of a coating and its
thermal conductivity. Correlations between conductivity and
elastic modulus are also explored as a means for determining
the viability of modulus measurements as a simple-to-use
conductivity probe. For the types of pore commonly found in
EB-PVD TBCs, there appears to be a tight correlation at
typical porosity levels.
I.
Introduction
T
barrier systems used in gas turbines consist of three
layers: an external thermally insulating oxide (typically yttriastabilized zirconia (YSZ)), an oxidation protection metallic sublayer, and a thermally grown oxide interlayer.1 The benefits of
such systems are strongly influenced by the thermal conductivity
of the outer oxide layer, k៮ . Materials with low k៮ are clearly
preferred, although the specifics depend upon engine design
details. Usually, a lower k៮ enables higher engine gas temperature
and reduced fuel consumption. YSZ is usually used because of its
low intrinsic conductivity (ks ⬇ 2 W/(m䡠K)). Further reductions in
k៮ are possible upon incorporating relatively small amounts of
porosity. The magnitude is governed by the morphology of the
pores, the pore volume fraction, and the spatial distribution.
Porosity is also used to achieve strain compatibility with the
HERMAL
II.
Basic Morphologies
The emphasis is on materials processed by electron-beam
physical vapor deposition (EB-PVD). Other research has addressed related effects that occur in air-plasma-sprayed (APS)
coatings.4 – 6 Images of EB-PVD TBCs taken at several resolutions, using scanning (SEM) and transmission (TEM) electron
microscopies (Figs. 1(a) and (b)), reveal three scales of porosity.
These scales are shown schematically on Fig. 1(c). The first two
scales are revealed by SEM (Fig. 1(a)) and the third by TEM (Fig.
1(b)). Additional images representing these morphologies can be
F. W. Zok—contributing editor
Manuscript No. 187834. Received March 22, 2001; approved September 4, 2001.
Supported in part by an Office of Naval Research funded MURI program on The
Science Underpinning Prime Reliant Coatings, and by ONRIFO/ONR under Contract
No. N00014-01-1-0271.
*Member, American Ceramic Society.
2937
2938
Journal of the American Ceramic Society—Lu et al.
Vol. 84, No. 12
found elsewhere.1,2,7 At the largest scale, the coatings comprise
columnar grains (width W) perpendicular to the substrate surface,
with intercolumnar gaps (width dc). These pores are designated
type A. The primary columns exhibit a feathery structure, involving two additional scales of porosity. One is the segmentation that
defines the branches of the feather (Fig. 1(a)), which can be
envisaged as microcolumns (width Wm), inclined at ⬃50° to the
horizontal axis. This type of porosity is most evident near the
periphery of the columns, where the gaps are planar and relatively
continuous. This type extends inward to about 1/3 to 1/2 the
column radius and systematically transitions into discrete pores of
high aspect ratio, which can be approximated as a combination of
oblate and prolate spheroids on planes inclined at ⬃50° (Fig. 1(b)).
These pores are designated type B. Oblate and prolate spheroids
are anticipated as transitional forms in the morphological breakdown and eventual spheroidization. However, the evolutionary
process is not well understood, and the relative proportion of these
morphologies, which have substantially different effects on the
thermal conductivity, remains to be properly characterized.7 Finally, there is a background comprising nanoscale spherical pores,
designated type C. These are more evident near the core of the
column, and some are clearly linked to the evolution of the
feathery segmentation, but others may have different origins.
For this study, because the characterization of the porosity is
incomplete, a somewhat subjective representation is used. The
premise is that the scheme on Fig. 1(c) affords a meaningful
representation of pore morphology. More objective methods, such
as those based on n-point correlation,8 will be explored in the
future. Within this context, the type A gaps are taken to be periodic
and vertical, with relative width, dc/W. For type B, the pores are
taken to be spatially random but aligned along the intercolumnar
planes and nonaligned between neighboring columns. The salient
parameters include the pore aspect ratio, dw/d, and the porosity,
␾B. However, as already noted, there is uncertainty about the
relative proportion of oblate and prolate spheroids within type B
porosity. Type C pores, which are taken to be spherical and
random, are characterized by the porosity, ␾C. The measurements
obtained in this manner, summarized in Table I, are the ones used
below as input to the models.
An alternative, zigzag, morphology can be created by alternating the position of the substrate relative to the vapor flux.9 This
approach has been combined with the emerging process of directed
vapor deposition (EB-DVD)10 to generate the structure shown in
Fig. 2(b). Once more, three distinct scales characterize the porosity. Again, a geometric simplification has been made in accordance
with Fig. 2(a), with the intent that more sophisticated methods be
used later. The three morphologies are as follows (Table II). Type
I comprises ⬃0.3 ␮m gaps that separate primary columns 10 – 60
␮m wide. Within these columns, there are smaller secondary
columns ⬃1 ␮m wide, separated by narrow ⬃0.1 ␮m gaps,
designated type II. Finally, spheroidal pores ⬃20 nm wide, type
III, are present within the secondary columns.
III.
Approach
The basic pore morphologies described in the previous section are
used to motivate calculations of the thermal conductivity and crosscorrelation with the elastic properties. As a preamble, we note that,
although the influence of pores on conductivity and modulus has been
modeled extensively,4,5,11–28 the results are not sufficiently comprehensive to address all salient factors governing the thermal characteristics of TBCs. The approach adopted in this article is to survey the
Table I. Porosity Parameters for EB-PVD
Coatings
Fig. 1. Morphologies of pores in TBCs deposited using EB-PVD: (a)
SEM image; (b) image of type B and C pores; and (c) schematic of the pore
morphologies and spatial arrangements. (Micrographs courtesy of S. G.
Terry.)
Type
A
B
C
Parameter
dc/W ⫽ 0.1
d/dw ⫽ 10, ␾B ⫽ 0.07, dw ⫽ 30 nm
␾C ⫽ 0.15
December 2001
Distributed Porosity as a Control Parameter for Oxide Thermal Barriers
2939
Table II. Coefficients Governing Thermal Conductivity of
Zigzag Structures
(A) Gas-Phase Conductivity (␻ ⴝ 60°)
c1
f
Type I
0.04
3
14
0.72
0.87
0.94
†
c2
Type II
‡
Type I
0.69
0.82
0.90
†
0.03
0.09
0.05
Type II‡
0.06
0.14
0.09
(B) Porosity (␻ ⴝ 60°)
f
kp/ks
c3
c4
0.04
1/10
1/3
1/2
1/10
1/3
1/2
1/10
1/3
1/2
0.75
0.75
0.75
0.90
0.92
0.93
0.98
0.98
0.98
0.68
0.50
0.37
0.84
0.63
0.48
0.90
0.66
0.5
0
3
14
(C) Sintering (␻ ⴝ 60°)
f
dW/dp
c5
c6
0.04
0.1
1/3
1/2
0.1
1/3
1/2
0.1
1/3
1/2
0.75
0.75
0.75
0.94
0.94
0.94
0.98
0.98
0.99
0.27
0.09
0.06
0.2
0.1
0.06
0.15
0.07
0.04
3
14
†
Fig. 2. Morphologies of zigzag pores in TBCs deposited using EB-DVD:
(a) unit cell and notations; and (b) SEM image.
available results, select the methods of analysis most appropriate to
the understanding of TBCs and, then, use these methods to perform
the supplementary calculations needed to complete the assessment.
The effects of random pores on thermal conductivity have been
modeled using three different approaches: (1) the noninteracting
approximation,4,5,11,12,18,19 (2) the self-consistent method,13,14,20,21
and (3) the differential scheme.13,22–26 The classical Maxwell model18 for random spherical pores is representative of noninteracting
solutions. This model has been considered limited to dilute concentrations, because it does not allow for interactions between neighboring pores. In an attempt to embrace interactions at higher porosity, the
self-consistent method has been developed and applied.20,21 The
method obtains a solution for an isolated void surrounded by an
effective matrix and then solves the problem of a representative pore
inserted into an infinite matrix with an effective conductivity k៮ (yet to
be determined). This method also has limitations at high concentrations.9 Consequently, effective medium models based on the differential scheme22–26 have been applied. In this scheme, the porosity is
increased incrementally, and the effective conductivity recalculated at
each step, with the initial condition k៮ ⫽ ks at zero porosity. However,
the self-consistent method and the differential scheme have restricted
utility for assessment of the effects of the porosity present in EB-PVD
TBCs, for two reasons: (1) The schemes are difficult to realize for
nonrandom orientations and anisotropic overall properties, because of
the absence of a fundamental solution for a pore arbitrarily oriented in
a thermally conductive, anisotropic matrix; and (2) for pore arrays
biased toward shielding or amplification, it is difficult to incorporate
information about the mutual locations of pores.11
Type I: d ⫽ 25 ␮m and dw ⫽ 2 ␮m. ‡d ⫽ 1 ␮m and dw ⫽ 0.08 ␮m.
Numerical approaches have been used in a more limited
manner.11,27,28 Cell calculations have been performed for periodic
spheres27,28 and aligned cylinders28 (Figs. 3(a) and (b)). Note that
the former are consistent with results obtained using the Maxwell
model28 (Fig. 3(a)), despite the different assumptions about spatial
distributions. Moreover, it is shown below that the calculations for
aligned cylinders are also consistent with results obtained using the
noninteracting method (see Fig. 3(b)). Results for arrays of
cracklike pores have been obtained by Kachanov et al.,11 who used
a random-number generator to insert pores into a square domain
and a numerical scheme to calculate the effective properties
(notably the elastic modulus). For the six densities and fifteen
sample arrays considered, the solutions obtained using the noninteracting method correlate most closely with the numerical results.
This finding is rationalized with the assertion that, when the pores
are random, on average, shielding and amplifying effects cancel.11
Based upon both the comparisons shown on Fig. 328 and the
conclusions of Kachanov et al.,11 the noninteracting scheme is
taken as a simple and, yet, reasonably accurate method suitable for
performing assessments on TBCs. This method is used in Sections
IV and V to generate the spectrum of results needed to evaluate the
thermal performance of EB-PVD materials and to establish their
correlation with elastic properties. According to this method, the
overall thermal conductivity tensor k៮ of a body containing pores of
given shape and volume fraction, ␾, is given by
k៮
⫽1⫺
ks
冘
Hi共kp/ks,␾,pore morphology兲
(1)
i
where Hi is the influence tensor for the ith pore, kp the effective
pore conductivity (see Appendix A), and the summation carries
over all pores. When the pores are distributed in shape and spatial
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Journal of the American Ceramic Society—Lu et al.
Vol. 84, No. 12
␥ 3 ⬁ (cylinders). The above results, when combined with Eqs.
(1) and (2), can be used to calculate the effective conductivity of
a matrix containing noninteracting spheroidal pores of arbitrary
aspect ratio, orientation, and distribution, as demonstrated below.
In the following, (x,y,z) is chosen as the global coordinate system,
such that the thickness direction of a TBC is represented by the
z-axis, with k៮ ⬅ k៮ z of primary interest.
For periodic pores, approximate analytical solutions are presented in Section IV(3) using the shear lag method developed in
Ref. 15. For specialized periodic arrangements of pores, such as
the zigzag arrays in Fig. 2, the finite-element method is most
suitable, and new results are generated using this method in
Section V.
IV.
Spheroidal Pores
(1) Randomly Oriented Pores
For insulated spherical pores (kp ⫽ 0) Eq. (3) with Eqs. (1) and
(2) results in
3␾
k៮
⫽1⫺
⫽ 1 ⫺ 2␲␾N
ks
2
(4a)
where the porosity is
␾⬅
1
¥V* 共i兲
V
(4b)
and the pore number density is
Fig. 3. Thermal conductivity as a function of porosity calculated for
noninteracting pores: (a) insulated spheres/spheroids with random orientation and spatial distribution; and (b) aligned, spatially random cylinders.
Also shown are two numerical results:27 one for (a) periodic spheres and
the other for (b) periodic cylinders. Comparisons show the utility of the
noninteracting solutions.
arrangement, probability densities replace the “pore morphology”
in Eq. (1), as elaborated in Appendix B.
To obtain Hi, the fundamental problem of a pore embedded in
an infinite matrix of conductivity ks, subject to a uniform heat flux,
is solved. Because kp 3 0 for the pores relevant to electron-beamdeposited TBCs (Appendix A), most of the following results are
cited for insulating pores (cross-pore conduction and radiation are
negligible16,17,28). For a spheroidal pore, Hi in Eq. (1) has been
derived as5,22,24 –26
冉 冊再冋 冉 冊 册
冋 冉 冊
册
共i兲
Hi ⫽
V*
V
1⫺
kp
ks
1⫹
⫹ 1⫹
kp
⫺ 1 F共␥共i兲兲
ks
⫺1
共I ⫺ n共i兲n共i兲兲
kp
⫺1 (1⫺2F(␥(i))
ks
⫺1
n共i兲n共i兲
冎
(2)
where I is the unit tensor, n(i) the unit vector along the axis of
symmetry for the ith spheroid, n(i)n(i) the dyadic product of n(i), V
the reference volume of the porous medium, and V*(i) and ␥(i) ⫽
dw/d, respectively, the volume and the ratio of height (dw) to
diameter (d) of the ith spheroid. The function F(␥) is a shape
parameter, given for oblate pores (␥(i) ⬍ 1) by4
F共␥兲 ⫽
(3a)
and for prolate pores (␥
F共␥兲 ⫽
⬎ 1) by
4
1 ⫺ 关␥ /共␥ ⫺ 1兲 1/ 2 兴arctan共␥2 ⫺ 1兲1/ 2
2共1 ⫺ ␥2兲
2
冉冊
d共i兲
1
¥
V
2
3
⫽
3
␾
4␲
(4c)
where d(i) is the diameter of the ith pore.
For flat spheroidal pores (dw ⬍⬍ d), the corresponding result is
冉
冊
冉
冊
k៮
2
d
8
dw
⫽1⫺
␲⫹
␾ ⬅ 1 ⫺ 1 ⫹ ␲ ␾N
ks
3␲
dw
9
d
(5a)
with
␾N ⫽
3 d
␾
4␲ dw
(5b)
Results have been plotted on Fig. 3(a) to show variations in
conductivity with porosity for oblate spheroids having various
aspect ratios (dw/d). Note that, at fixed porosity, the thinner (low
aspect ratio) spheroids are most effective. Also plotted on Fig. 3(a)
is the numerical result for spheres placed in a simple cubic array,28
illustrating the similarity with the analytical result.
When the conductivity of the gas phase is significant, the full
solution of Eq. (2) must be used, with the gas conductivity
obtained from Appendix A (Eq. (A–3)). For example, the result for
random spherical pores is
k៮
1⫺⌸
⫽1⫺
␾
ks
1 ⫹ 共⌸ ⫺ 1兲/3
(6)
where ⌸ ⬅ kp/ks, with kp given in Appendix A.
1⫺[␥2/2(1⫺␥2)1/2]ln兵关1⫹(1⫺␥2)1/2]/[1⫺(1⫺␥2)1/2]}
2(1⫺␥2)
(i)
␾N ⬅
2
(3b)
and, for spherical pores, F(␥) ⫽ 1/3. In the limit ␥ 3 0
(penny-shaped cracks), F(␥) 3 1/2, whereas F(␥) 3 ␲/4␥ when
(2) Aligned, Spatially Random Pores
Aligned, spheroidal pores are defined as having fixed orientation and aspect ratio, but arbitrarily distributed in space and size.
When flat (Fig. 4(a)) and oriented perpendicular to the z-axis, with
k៮ ⬅ k៮ z, the pores have a substantial effect on the axial conductivity,
k៮ z, but minimal effect on the transverse conductivity, k៮ r ⬇ ks.
Consequently, results for flat pores derived for k៮ z can be used to
determine the conductivity when pores are inclined at angle ␻ to
December 2001
Distributed Porosity as a Control Parameter for Oxide Thermal Barriers
2941
such that, in both scenarios, it varies between zero and unity.
Sections through Fig. 5 yield the characteristics explicitly relevant
to TBCs, as plotted on Figs. 6 and 7. In particular, on Fig. 6, note
that there is a large effect of pore aspect ratio on the conductivity
of oblate pores at inclinations ␻ ⱕ 60°, but a negligible effect for
prolate pores. Consequently, a transition between these two morphologies has a major influence on the TBC conductivity, as
elaborated below.
(3) Periodic Pores
Paradoxically, solutions for periodic pores have not been
obtained in a rigorous analytical manner. Here, to obtain solutions,
a unit-cell approximation reminiscent of the shear lag analysis for
fiber-reinforced composites is used, as outlined in Appendix C,
leading to the result (Eq. (C–3)),
k៮ r
⫽ 1 ⫺ ␾z
ks
冋
册
k៮ z
tanh共⌿L␯/Lh兲/共⌿L␯/Lh兲
␾r
⫽ 1⫹
ks
1⫺␾r 1 ⫹ 共⌿/8兲共␾r2/␾兲共Lh/L␯兲共kp/ks)tanh共⌿L␯/Lh兲
⫺1
(9a)
where Lh is the spacing between pores along r (Fig. C–1), L␯ that
along z, and the area fractions of the pores are
冉 冊
␾r ⫽ 1 ⫺ 1 ⫺
Fig. 4. (a) Schematic of aligned, flat spheroidal pores. (b) Thermal
conductivity as a function of porosity at various aspect ratios when
orientated at ␻ ⫽ 0.
the horizontal
transformation
axis
(Fig.
k៮ ⫽ k៮ ⬘r sin2 ␻ ⫹ k៮ z⬘ cos2 ␻
4(a)),
through
the
coordinate
(7)
For flat pores (dw/d ⬍⬍ 1, e.g., d ⫽ constant and dw 3 0) oriented
perpendicular to the z-axis (i.e., ␻ ⫽ 0°), in the insulating limit,
k៮ r
⫽1⫺␾
k៮ s
k៮ z
8
⫽ 1 ⫺ ␾N
ks
3
2
(9b)
and ␾z ⫽ dw/L␯, such that the total porosity, ␾ ⫽ ␾r␾z, with ⌿ ⫽
{8/[(1 ⫺ ␾r)␾r]}1/2. For inclined arrays of periodic pores, the
solution is obtained by substituting the above results into Eq. (7).
Selected finite-element calculations for the same problem have
shown that the accuracy of Eq. (9) is excellent.15 Furthermore, this
prediction, when slightly modified, has been found to agree with
the experimentally measured thermal conductivity of fiberreinforced ceramic-matrix composites containing multiple matrix
cracks.29
Comparison of these results with those for random, aligned
pores affirms the expectation that spatially dispersed pores are
more effective than periodic pores in decreasing k៮ .
V.
(8a)
with ␾N/␾ given by Eq. (5(b)). For insulated cylindrical pores
(either d ⫽ constant and dw 3 ⬁ or d 3 0 and dw ⫽ constant)
oriented perpendicular to the z-axis (horizontal cylinders, ␻ ⫽ 0°),
the corresponding result is
k៮ r
⫽1⫺␾
ks
k៮ z
⫽ 1 ⫺ 2␾
ks
d
Lh
Zigzag Pores
The thermal effect associated with zigzag pores (Fig. 2) has
been analyzed as described in Appendix D. The representative unit
cell contains an intercolumnar gap, width dw, and column inclination angle ␻, related to the wavelength ␭ and amplitude a (Fig. 8)
by
␻ ⫽ tan⫺1
␭
2a
(10a)
The geometric aspect ratio (slenderness) is characterized by
(8b)
Various results calculated using this approach are summarized on
Figs. 3–7. The variation in thermal conductivity with porosity has
been calculated for spheres/spheroids (Fig. 4(b)) and horizontal
cylinders (Fig. 3(b)). Comparison of the latter with numerical
solutions from Ref. 28 has been used to affirm the utility of the
noninteracting method. All subsequent results are determined at a
fixed porosity (␾ ⫽ 0.2), most relevant to TBCs made using
electron-beam methods (Fig. 1).
To provide further context for the presentation of the results, the
overall trends found from the calculations are illustrated on Fig. 5.
Figure 5 indicates limiting results for various morphologies as well
as trends between these limits. Here, for the purpose of presenting
the results for oblate pores with those for prolate pores, the aspect
ratio of the former is defined as dw/d, while for the latter as d/dw,
f⫽
dp
␭
(10b)
When f ⬍ 1, heat flow is (approximately) one-dimensional, and the
thermal conductivity can be determined analytically. For continuous pores (types I and II), the porosity is given by
␾ I/II ⫽
dw
dp
(10c)
where ␾I/II equals the pore fraction oriented normal to the
thickness. For discontinuous pores (type III), Eq. (10c) is replaced
by
␾ III ⫽
dw Hp
dp ␭
where Hp is the pore height.
(10d)
2942
Journal of the American Ceramic Society—Lu et al.
Vol. 84, No. 12
Fig. 5. Overall trends in thermal conductivity in the presence of aligned pores, displayed over a full spectrum of pore morphologies.
Fig. 6. Thermal conductivity of aligned, noninteracting pores as a
function of orientation at a volume fraction of ␾ ⫽ 0.2: (a) oblate; and (b)
prolate. Here dw is the length of the axis of revolution and d is twice the
radius of revolution.
Fig. 7. Thermal conductivity of aligned, noninteracting pores as a
function of aspect ratio at a volume fraction of ␾ ⫽ 0.2: (a) oblate; and (b)
prolate. Here A ⬅ dw/d for oblate spheroids and A ⬅ d/dw for prolate
spheroids.
December 2001
Distributed Porosity as a Control Parameter for Oxide Thermal Barriers
2943
where ␯s is the Poisson ratio of the solid. Accordingly, when the
pores are insulated, upon eliminating ␾N from Eqs. (4), (5), and
(13), the cross-property relationship is
冉 冊
冉 冊
Es
7 ⫺ 5␯s
k៮
⫽1⫺
⫺1
ks
共1 ⫺ ␯s兲共9 ⫹ 5␯s兲 E៮
spheres
Es
k៮
5共1 ⫺ ␯s/2兲
⫽1⫺
⫺1
ks
共1 ⫺ ␯s2兲共10 ⫺ 3␯s兲 E៮
2共4␯s2 ⫺ 5␯s ⫹ 2兲
␾
15
⫺
oblate spheroids
(14)
For aligned oblate pores, the elastic modulus in the perpendicular
direction has been determined as11,15
Fig. 8. Normalized overall thermal conductivity for zigzag coatings as a
function of pore inclination angle for selected values of normalized pore
spacing (kp/ks ⫽ 0.01 and ␾ ⫽ 0.04).
冋
冋
册
冉 冊册
16共1 ⫺ ␯s2兲
E៮
⫽ 1 ⫹ ␾N
⫹ ␾4␯s共1 ⫺ ␯s兲
Es
3
⫺1
Lh
S1 L␯
E៮
⫽ 1 ⫹ D1 共1 ⫺ ␾r兲1/ 2 tanh
Es
L␯
D1 Lh
In the direction perpendicular to the substrate surface,
distributed
⫺1
periodic
(15)
k៮
⫽ 共1 ⫺ ␾I/II兲sin2 ␻
ks
(11)
Numerical results (Appendix D), plotted on Fig. 8, verify that the
analytical model is accurate when f ⬍ 1. For all other f, the thermal
conductivity is larger.
The numerical scheme has been used to explore possibilities
over a configurational range consistent with the EB-DVD process.
All of the results can be represented as linear dependencies of k៮ /ks
on one variable, with the other variables fixed. The effect of
gas-phase conductivity kp/ks ⬅ ⌸ at fixed porosity can be
expressed as
kp
k៮
⫽ c1 ⫹ c2
ks
ks
(12a)
where the dimensionless coefficients c1 and c2 depend on the
slenderness ratio and the inclination (Table II). The effect of
porosity at fixed pore conductivity is
k៮
⫽ c3 ⫺ c4␾
ks
(12b)
where c3 and c4 depend upon f, kp/ks, and ␻ (Table II). If the
columns begin to sinter and decrease the extent of the intercolumnar porosity from ␾I/II 3 ␾III (such that the zigzag pores become
discontinuous), the associated increase in conductivity can be
expressed as
k៮
⫽ c5 ⫺ c6␾III
ks
where E៮ ⬅ E៮ z, S1 ⫽ ␾r/(1 ⫺ ␾r) , with ␾r given by Eq. (9), and
the coefficient D115 is plotted as a function of ␾r on Fig. 9. The
cross-property relationships are
3/ 2
冋
册
2共1 ⫺ ␯s2兲
k៮
⫽1⫺␾ ៮
ks
共E/Es ⫺ 1兲/␾ ⫺ 4␯s共1 ⫺ ␯s兲
冉
冊
k៮
␾r
⫽ 1⫹
ks
1⫹⍀
⫺1
distributed
⫺1
periodic
(16)
where
D 1 共8/␾ r 兲 1/ 2
E S/E ⫺ 1
⍀⫽
D1共8/␾r兲1/ 2
tanh
Es/E៮ ⫺ 1
冋
册
(17)
For the two spatially random morphologies (Figs. 10(a) and (b)), it
is apparent that the correlation has minimal dependence on the
porosity. Specifically, for randomly oriented pores, at fixed modulus, the thermal conductivity decreases only slightly as the
porosity increases, because of the associated increase in the cross
section of pores normal to the heat flow direction. For aligned
pores, the conductivity increases slightly as ␾ increases. Most
importantly, at the porosity of interest for TBCs (␾ ⱕ 0.15), the
(12c)
where c5 and c6 depend upon f, dp/dw, and ␻ (Table II).
VI.
Cross-Property Correlation
Procedures based on the noninteracting scheme again have been
used to calculate the elastic modulus for random pores. Only the
results are summarized, with the objective of cross-plotting against
the thermal conductivity. For randomly oriented pores, the
Young’s modulus is given by11
冋
冋
册
E៮
3共1 ⫺ ␯s兲共9 ⫹ 5␯s兲
⫽ 1⫹␾
Es
2共7 ⫺ 5␯s兲
⫺1
spheres
册
8共1 ⫺ ␯s2兲共10 ⫺ 3␯s兲
E៮
2
⫽ 1 ⫹ ␾N
⫺ ␾ 共4␯s2 ⫺ 5␯s ⫺ 3兲
Es
45共1 ⫺ ␯s /2兲
15
oblate spheroids
⫺1
(13)
Fig. 9. Elastic modulus coefficient for periodic, aligned pores as a
function of the area fraction parameter.
2944
Journal of the American Ceramic Society—Lu et al.
correlation is quite insensitive to morphology, despite the strong
sensitivity of each property, separately, to the spatial arrangement.
This correlation suggests modulus measurement as a simple
nondestructive probe for monitoring thermal conductivity changes
attributable to porosity variations, upon deposition and during
elevated temperature exposure. Note, however, that the out-ofplane modulus needs to be measured: All existing data3 refer to the
in-plane modulus. To measure the appropriate modulus, one of
several acoustic methods is envisioned,30,31 based on a noncontact
laser ultrasonic technique that enables the time of flight for an echo
from the TBC bond coat interface to be measured. This gives the
velocity and, hence, the modulus to an accuracy determined by the
measured density and coating thickness.
Conversely, when the pores are aligned and periodic, the
correlation is quite strongly dependent on morphology. In such
cases, an elastic modulus measurement alone is not a useful probe
for the thermal conductivity. Although the pores in TBCs are not
normally of this character, this result highlights the caution that
needs to be exercised before relying only on elastic modulus
measurements as a thermal conductivity probe.
VII.
Application to TBC Systems
The porosity parameters summarized in Table I can be used
with the above formulas to predict the trends in thermal conductivity of EB-PVD coatings. The following protocol is used. The
overall conductivity is related to that for the columns by
k៮
k column
⫽1⫺
dc
W
(18a)
where dc/W ⫽ 0.1. The conductivity in regions containing type C
pores is (Eq. (4))
where
␾N ⫽
3 d
␾
4␲ dw B
with ␻ ⫽ 50°, d/dw ⫽ 10, and ␾B ⫽ 7 ⫻ 10⫺2 (if oblate).
Conversely, if the pores are prolate (Fig. 6), inclined at ␻ to the
horizontal axis, and surrounded by type C pores, the response is
k៮ B
⫽ 共1 ⫺ ␾B)sin2 ␻ ⫹ 共1 ⫺ 2␾B)cos2␻
k៮ C
(18b)
where ␾C ⫽ 0.15. For type B pores, the result is dependent upon
the specific shape (oblate or prolate). If the pores are oblate (see
Fig. 6) and aligned as well as inclined at ␻ to the horizontal axis
and surrounded by type C pores (Eq. (7)), then
冉
冊
8
k៮ B
2
2
៮kC ⫽ 共1 ⫺ ␾B兲sin ␻ ⫹ 1 ⫺ 3 ␾N cos ␻
(18c)
(18d)
where ␻ ⫽ 50° and ␾B ⬇ 7 ⫻ 10⫺3 (if prolate). The column
conductivity is
k column ⬅ fCk៮ C ⫹ fBk៮ B
(18e)
where the area fractions occupied by types C and B pores are fC ⫽
2/3 and fb ⫽ 1/3. Inserting the measured porosity parameters from
Table I into Eqs. (18), k៮ /ks ⫽ 0.65 for oblate type B pores and
k៮ /ks ⫽ 0.71 for prolate pores. Measured values are in the range
0.5 ⱕ k៮ /ks ⱕ 0.8.10,32 Although the models yield results in the
measured range, the uncertainties in pore shape and in the
measurements, as well as the simplicity of the representation,
indicate that a detailed experimental study is needed to complete
the assessment.
For the zigzag structures, the specific example to be analyzed
comprises insulated pores with identical inclination angle (␻ ⫽
60°). The type III pores are discontinuous with ␾III ⬇ 0.2 and f ⬍
1.10 From Eq. (12c) and Table II, the effective thermal conductivity of the pores is k៮ III ⫽ 0.96ks. For type II, ␾II ⬇ 0.08,10 such that
k៮ II ⫽ 0.68ks. Finally, for type I, ␾I ⬇ 0.04,10 such that k៮ I ⫽ 0.73ks.
Consequently, the overall thermal conductivity for all three pore
types is k៮ ⫽ 0.48ks. This compares with the measured values,10
which show that k៮ ⬇ 0.45ks.
VIII.
k៮ C
3␾C
⫽1⫺
ks
2
Vol. 84, No. 12
Concluding Remarks
The preceding analyses and the comparison with experiments
have demonstrated that models available in the literature
augmented by selected numerical results can be used to relate
the porosity introduced upon deposition of EB coatings to the
overall thermal conductivity. The limitations have been in the
paucity of experimental characterization of pore morphologies
and in the overly simplified geometric representation of the
porosity. Future application of a more objective representation,
such as that based on n-point correlation,8 would enable more
complete assessments. Given this capability, it should be
possible in the future to address some basic issues related to the
conductivity. For example, the models should have the intrinsic
ability to assess increases in the conductivity as the pore
morphology changes upon exposure to elevated temperature:2
wherein, the spheroidal pores present in the as-deposited state,
described above, become equiaxed upon exposure, degrading
their effectiveness. Moreover, given the correlation with modulus, the associated increase in conductivity should be manifest
in a corresponding change in the through-thickness Young’s
modulus, determined using an acoustic probe.30,31
Appendix A
Gas-Phase Conductivity
The conductivity of the gas phase within the pores differs from
that in free space, k⬚g ⬇ 0.027 W/(m䡠K). It is given at pressure p
and temperature T by15–17
Fig. 10. Cross-correlation between thermal conductivity and Young’s
modulus for random and aligned, nonperiodic pores, at three porosity
levels. Note that the results are weakly dependent of the aspect ratios of the
pores.
冉
k p ⫽ k°g 1 ⫹
4␩ 2 ⫺ ␣A Kn
␩ ⫹ 1 ␣A Pr
冊
⫺1
(A–1)
where ␩ ⬅ cp/c␯ is the specific heat ratio, ␣A the accommodation
coefficient (unity for TBCs17), Pr the Prandtl number, Kn ⫽ ⌳/dw
December 2001
Distributed Porosity as a Control Parameter for Oxide Thermal Barriers
the Knudsen number, and ⌳ the mean free path of gas molecules.
The kinetic theory of gases gives17
冉 冊
k⬚g ⫽ 共1.66␩ ⫺ 0.92兲
p2
mgkBT
1/ 2
⌳
NA/C␯
k BT
⌳ ⫽ 1/ 2 2
2 ␲dgp
共A-2a)
(A-2b)
where kB is the Boltzmann constant, dg and mg the diameter and
mass of the gas molecules, and NA the Avogadro number. Upon
substitution of Eq. (A–2b) into (A–2a):
k⬚g ⫽
冉 冊
1.66␩ ⫺ 0.92 kBT
2mg
␲d2gNA/C␯
k៮ z
⫽1⫺
ks
冕
1
Hz共␥兲 P2共␥兲 d␥
k⬚g
1 ⫹ C1T/dwp
(B–3)
0
where Hz is the z-component of H.
Appendix C
(A–3)
where C1 ⫽ 2.5 ⫻ 10⫺5 Pa䡠m/K for air.
When Kn ⬍⬍ 1, the gas behaves as a continuum, but, when
Kn ⬎⬎ 1, the energy exchange is largely restricted to collisions of
gas molecules with the pore surface. Thus, when the pore is
sufficiently thin, the gas conductivity decreases with increasing
temperature and decreasing pressure (Fig. A1).
For the pores relevant to electron-beam deposition (width of
⬃30 nm, Fig. 1(b)), these results show that, at atmospheric
pressure, the pore conductivity is negligible: kp ⬇ 0.2kg° ⬅ 0.005
W/(m 䡠 K). Furthermore, the effect of thermal radiation is negligibly small.16,17,28 Accordingly, in much of the text, the pores are
considered to be insulating.
Fig. A1.
where ei (i ⫽ 1, 2, 3) are unit vectors along the Cartesian
coordinates xi.
For pores aligned along a specified orientation, say along z, but
having a distributed aspect ratio, P2(␥), then
1/ 2
Note that
kp ⫽
2945
Periodic Pores
To calculate the thermal conductivity for periodic pores, the
steady-state temperature field T(r,z) within a unit cell (dashed
lines, Fig. C1) has been analyzed, with q៮ z the average heat flux in
the z-direction. Symmetry and linearity dictate that the temperature
distribution on each of the transverse planes halfway between the
pores be independent of z,15 such that
k៮ ⫽ ⫺
q៮ z
共T z⫽L ␯ / 2 ⫺ Tz⫽⫺L␯ / 2兲/L␯
(C–1)
At each point along the pore surface, the local heat flow, qz ⫽
⫺ks⭸T/⭸z, is taken to be proportional to the temperature jump,
T ⫹ ⫺ T ⫺ ⫽ T(x,0⫹) ⫺ T(x,0⫺), according to
q z ⫽ ⫺hp共T ⫹ ⫺ T ⫺兲
(C–2)
with hp ⬅ kp/dw the average heat-transfer coefficient. Based on the
analogy between the variables describing elasticity and those for
steady-state heat conduction, the above boundary-value problem
Normalized pore conductivity as a function of pore width.
Appendix B
Distributed Pore Shapes and Orientations
Some results for distributed pores are summarized here for
completeness. For spheroidal pores having specified aspect ratio,
␥, but distributed orientation,
k៮
⫽1⫺
ks
冕冕
2␲
0
␲/ 2
HP1共␺,␪兲 d␺ d␪
(B–1)
0
where P1 is the probability density function characterizing the
orientations. The influence function H is given by Eq. (2), with
n ⫽ cos ␪ sin ␺e1 ⫹ sin ␪ sin ␺e2 ⫹ cos ␺e3
(B–2)
Fig. C1. (a) Schematic of the periodic pore configuration. (b) Normalized
overall thermal conductivity decrease caused by uniformly distributed, in-line
spheroids as a function of spacing.
2946
Journal of the American Ceramic Society—Lu et al.
can be solved by the shear-lag method developed in Ref. 15,
yielding
冤
冥
tanh共␺L␯ /Lh兲
k៮
␺L␯ /Lh
␾r
⫽ 1⫹
2
ks
1 ⫺ ␾r
␾
␺ r Lh kp
1⫹
tanh共␺L␯ /Lh兲
8 ␾ L␯ ks
⫺1
(C–3)
This result is plotted on Fig. C1.
Appendix D
Zigzag Pores
For zigzag pores, based on the unit cell depicted on Fig. 8,
the steady-state heat-transfer problem is solved using the
finite-element code ABAQUS. The unit cell is discretized using
the four-node linear diffusive heat-transfer element, DC2D4.
Selected calculations reveal that the use of two layers of unit
cells is sufficient to accurately capture the heat-transfer characteristics. The difference between one-layer and two-layer
models is small (⬍5%); moreover, the use of more layers of the
unit cell does not lead to further improvement in numerical
accuracy.33 A typical finite-element mesh consists of ⬃4000
elements for the solid and ⬃200 elements for the pore. Constant
temperatures T top and T bottom are prescribed at the top and
bottom surfaces. The sides are thermally insulated. The objective is to follow the heat conduction path from the bottom to the
top surface, and calculate the heat flux q k( x,z) everywhere in
the medium. Once this is accomplished, the effective thermal
conductivity k៮ is obtained as (c.f., Eq. (B–1))
k៮ ⫽
q៮
共T bottom ⫺ Ttop兲/␭
(D–1)
where q៮ ⬅ q៮ z is the heat flux in the z-direction averaged in x over
the top (or bottom) surface.
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