Acta mater. 49 (2001) 2539–2547
www.elsevier.com/locate/actamat
THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS WITH
ZIG-ZAG PORE MICROSTRUCTURES
S. GU1, T. J. LU1†, D. D. HASS2 and H. N. G. WADLEY2
1
Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK and 2Materials Science
Department, University of Virginia, Charlottesville, VA 22903, USA
( Received 11 December 2000; received in revised form 5 March 2001; accepted 8 March 2001 )
Abstract—Highly porous zirconia based thermal barrier coatings have recently been synthesised with zigzag morphology pores which appear to impede heat flow through the thickness of the coating. A combined
analytical/numerical study of heat conduction across these microstructures is presented and compared with
thermal conductivity measurements. The effects of pore volume fraction, pore type, pore orientation and pore
spacing, together with the wave length and the amplitude of zig-zag pore microstructures on overall thermal
performance are quantified. The results indicate that even a few volume percent of zig-zag inter-column
pores oriented normal to the substrate surface reduce the overall thermal conductivity of the coatings by
more than 50%.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.
Keywords: Physical vapour deposition (PVD); Thermal barrier coating; Thermal conductivity; Microstructure;
Computer simulation
1. INTRODUCTION
Interest is growing in low thermal conductivity coatings that can be applied to internally cooled components subjected to high temperature service. Initial
efforts to lower the thermal conductivity of these
coatings concentrated upon modification of the coating materials composition. More recently, attention
has turned to the control of their porosities which
have a large effect upon thermal conductivity. Both
the volume fraction and the morphology of the pores
is important. As an example of the latter, recent
experimental investigations conducted by Hass et al.
[1] found that the overall thermal conductivity of
electron beam-directed vapour deposited (EB-DVD)
7 wt% yttria stablised zirconia (7YSZ) coatings containing zig-zag primary pore microstructures can be
reduced by 50% or more compared to coatings whose
primary pores are aligned through the thickness of the
coating. Similar results have been obtained by Marijnissen et al. [2] for coatings made using electron
beam physical vapour deposition (EB-PVD).
The large thermal resistance when coupled with a
potentially high inplane compliance and erosion
resistance suggest that zig-zag thermal barrier coatings have great potential for applications in high tem-
† To whom all corrrespondence should be addressed.
Fax: +44-1223-332662.
E-mail address: [email protected] (T. J. Lu)
perature environments, such as gas turbine and diesel
engines, where they reduce the temperature and oxidation rate of metal components, and hence increase
component durability and life. The porosity in thermal barrier coatings has many length scales and morphologies that can be controlled by the conditions of
deposition and the thermal environment during subsequent use. Here, a systematic analytical/numerical
study seeks to unlock an understanding of the heat
transfer process across highly porous structures to
provide guidance for coating system design, processing and application.
The thermal barriers deposited by Hass et al. [1]
exhibited a columnar structure with primary porosity
in the form of elongated pores, all with zig-zag morphologies, that extend from the substrate to the top
of the coating. The porosity was hierarchical in nature
as illustrated in Fig. 1. Three distinct pore scales were
found to co-exist in these coatings, as shown schematically in Fig. 2. The coating structure is dominated
by the largest pores (Type I), with a width exceeding
0.3 ␮m. They separate primary growth columns that
are 10~60 ␮m in width. Intermediate columns of
0.6~2.5 ␮m width exist within the primary columns.
They are bounded by narrower Type II pores that are
~0.1 ␮m wide. Finally, Type III pores of ~20 nm
width exist between even finer growth columns
(20~80 nm in width) present within the secondary
growth columns. They are usually discontinuous. It is
believed that these zig-zag inter column pore struc-
1359-6454/01/$20.00  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.
PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 1 4 1 - 0
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GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS
element method, and the validity range of the 1-D
analytical solutions is identified. Sections 4 and 5
present results from an extensive parametric study of
pore morphology effects. The predictions are compared with experimental measurements in Section 6.
2. ANALYSIS
Fig. 1. Typical zig-zag pore structure of a EB-DVD coating.
A low conductivity thermal barrier coating (TBC)
with distributed pores can be viewed as an air–matrix
composite. The average pore size in practical TBCs
is in general much smaller than the macroscopic size
of the TBC so the composite can be treated as an
effective homogeneous medium with an associated
set of effective properties (thermal conductivity, elastic modulus, etc.). Heat transfer across a single pore
results from a combination of gaseous conduction and
solid–solid conduction due to contact of pore surfaces: the influence of thermal radiation is ignored in
Fig. 2. Zig-zag pores at different scale levels.
tures increase the heat transport path and so reduce
the effective out-of-plane thermal conductivity of the
coatings [1, 2].
In this paper, a combined analytical and numerical
study of the overall thermal conductivity of 7YSZ
coatings embedded with varying zig-zag pore microstructures is presented. It aims to answer the following questions:
the present investigation which seeks to understand
measurements made at room temperatures. A subsequent study incorporates radiative effects that could
be important at high temperatures. Due to the small
pore width, the effect of natural convection within the
inter column pores is always negligible and has
been ignored.
앫 How does the pore spacing affect the overall thermal conductivity?
앫 Which type of pores play a more crucial role in
reducing the thermal conductivity?
앫 What is the relationship between the inclined
angle of extended pore and the thermal performance?
앫 How does the overall conductivity depend on the
wavelength and amplitude of zig-zag intercolumn pores?
The kinetic formula for the thermal conductivity of
a solid, ks is [3]
The paper is organised as follows. In Section 2, the
governing equations for a heat transfer analysis of the
zig-zag coating system are established, and an idealised one-dimensional (1-D) model is developed. The
same problem is solved in Section 3 with the finite
2.1. Solid conductivity
1
ks = Cpv⌳
3
(1)
where Cp is the volumetric phonon specific heat, v is
the average speed of sound and ⌳ is the phonon mean
free path. At temperatures below the Debye temperature, qD, the dominant wavelength, lc, of the phonons
is of the order of qDal/T where al is the lattice constant. The dominant wavelength, lc, is about 10–20
A in 7YSZ. For TBCs having a characteristic microstructural length scale D (e.g., the growth column
width), lc⬍D is expected and hence quantum size
GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS
effects can be neglected [4]. Reliable thermal conductivity measurements for non porous yttria stabilised
zirconia with a similar yttrium content to that used
by Hass et al has been reported as a function of D
(see reference [1]).
2.2. Pore conductivity
As previously discussed, three types of pore are
observed in zig-zag coatings, Fig. 2. Each type of
pore has potentially different morphological characteristics [1]. The effective thermal conductivity of a
region of material containing a pore may be modelled
by the rule of mixture, as
kp = (1⫺r)kg + rkps = kg + kc
(2)
where r represents the fraction of the pore surfaces
that are in good thermal contact, kg is the gaseous
thermal conductivity, kps is the intrinsic thermal conductivity of the material at a contact, and kc⬅r(kps⫺
kg) is introduced to represent the effective thermal
contact conductance across the pore. For simplicity,
kc = ks may be assumed but kc can be measured as
discussed later. Depending on the pore morphology
(e.g., pore width dw), kp varies from the gas conductivity kg in the limit r→0 (no contact) to the solid
conductivity ks in the other limit r→1 (complete perfect contact). At room temperature the thermal conductivity of bulk 7YSZ is microstructure dependent,
but lies in the range 2ⱕksⱕ2.6 W/mK [1].
The conductivity of the gaseous phase trapped
within the pore is a function of pore width, dw, pressure, p, and temperature, T [3, 5–7]. To a good
approximation:
冉
kg = k0g 1 +
4g 2⫺aAKn
g + 1 aA Pr
冊
⫺1
(3)
where k0g is the gas thermal conductivity in free
space, g⬅cp/cv is the specific heat ratio, aA is the
accommodation coefficient of the gaseous molecules
on the solid surface of the pore (aA⬇1 for TBCs [8]),
Pr is the gas Prandtl number, Kn = ⌳/dw is the
Knudsen number, and ⌳ is the mean free path of
gaseous molecules. Equation (1) is still valid for gases
in free space, except that Cp now denotes the gas specific heat per unit volume and v represents the average molecular velocity. Kinetic theory of gases then
yields [3],
冉 冊
k0g = (1.66g⫺0.92)
⌳=
p2
mgkBT
kBT
√2pd2gp
1/2
⌳
NA/Cv
(4a)
(4b)
2541
where kB = 1.38×10⫺23 J K⫺1 is the Boltzmann constant, dg and mg are the diameter and mass of gas
molecules, and NA is Avogadro’s number. Upon substitution of (4b) into (4a), one gets
k0g =
冉 冊
(1.66g⫺0.92) kBT
pd2gNA/Cv 2mg
1/2
(5)
Since ⌳⬀T/p and k0g⬀√T. From (3) it is seen that
kg⬀
√T
1 + c1T/dwp
(6)
where c1 is a constant.
From equation (3) it is seen that the detailed mechanism underlying the transfer of heat by the gas that
fills the space within the pore is controlled by the
Knudsen number, Kn. When Kn1, the continuum
theory of heat conduction in the gas applies. When
Kn1 the energy exchange involves collisions of gas
molecules and the pore surface with relatively few
intervening interatomic or intermolecular collisions.
For intermediate values of Kn, the two processes are
in transition. Thus, when the pore spacing is sufficiently small, the pore gas conductivity decreases
with increasing temperature and decreasing pressure,
in contrast to that of the gas in free space. In the limit
dw→0, kg→0. Furthermore, the effective conductivity
of a highly porous medium (e.g., silica aerogel) can
be reduced by an order of magnitude relative to k0g by
increasing the Knudsen number (i.e., by introducing
very fine pores in the system). For example, the thermal conductivity of a 94% porosity carbon-opacified
aerogel measured at 10⫺2 atm in air at room temperature is 0.0055 W/mK, compared to 0.027 W/mK of
air at 1 atm [6].
Dimensional analysis dictates that the overall thermal conductivity, k̄, of a porous TBC depends on the
associated geometrical and physical properties:
k̄ = ksh(kg/ks, kc/ks, f, pore geometry), where f is the
pore fraction and h is a dimensionless function to be
determined in the remaining sections. Once h is
found, the following procedure may be used to back
out kc and kg. First, k̄ is measured both in vacuum
and in the gaseous phase of interest, and the corresponding kp calculated from the function h. Since the
competition between heat transfer by surface asperity
contact and gaseous conduction vanishes under vacuum conditions with the gaseous phase removed, to
a good approximation, the pore thermal conductance
measured in vacuum at room temperature represents
the contact conductance kc. Then to obtain the component kg, one simply subtracts kc from the total pore
conductance kp measured in the gaseous phase of
interest.
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GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS
2.3. Unit cell
As an initial step to investigate the transport of heat
across the barrier coating, an idealised coating system
with uniformly distributed, periodic zig-zag pore
structures is analyzed. A unit cell of the model system
is sketched in Fig. 3; the whole structure of the zigzag coating system can be simplified as a collection of
these column units. Let (x, y) be the global coordinate
system and let (x*, y*) be the local coordinates rotated
clockwise by an angle 90°⫺w from (x, y), with ω
being the pore inclination angle (Fig. 3). The unit cell
contains one zig-zag pore of width dw, wavelength l
and amplitude a; the width of the unit cell equals the
inter pore spacing, dp. The pore inclination angle is
related to zig-zag wavelength and amplitude by
w = tan⫺1(l/2a)
(7)
The geometrical aspect ratio (slenderness) of the thermal unit is characterised by
f = dp/l.
(8)
The zig-zag pore may be continuous (Type I and II
pores, Fig. 3) or discontinuous (Type III, Fig. 4), with
Hp denoting the pore height (Hp = l for continuous
pores). The pore fraction when pores are all of the
same type is given by
f = (dw/dp)(Hp/l).
Fig. 4. Computational model for EB-DVD coating system containing discontinuous pores.
(9)
The porosity of a coating containing the three types
of pore is given by
III
f̄ = S fm
m=I
(10)
where m = (I, II, III) is the pore type index. Only one
type of pore is explicitly modelled by the unit cell;
however, in the presence of other types of pore, the
combined effects can be accounted for by using the
effective medium model [9]. That is, the unit cell is
treated as a two-phase composite medium, with conductivities kp and k̄s assigned separately to the pore
and the surrounding matrix material, with k̄s representing the effective conductivity of the matrix
which may contain pores of lower class in the hierarchy (Type III⬍Type II⬍Type I).
2.4. Effective thermal conductivity: 1-D analysis
To calculate the effective thermal conductivity
k̄(⬅k̄y) of the coating system in the thickness direction
(y-axis, Fig. 3), constant temperatures, Ttop and Tbottom,
are prescribed at the top and bottom surfaces of the
zig-zag thermal unit respectively. Steady-state, Fourier heat conduction is assumed. If the coating is much
wider than it is thick, both sidewalls of the unit cell
can be treated as thermally insulated. The objective
is to follow the heat conduction path from the top to
the bottom surface, and calculate the heat flux vector,
qk(k = x, y), everywhere in the two-phase composite
medium. Once this is accomplished, the effective
thermal conductivity, k̄, is obtained as
Fig. 3. Computational model for EB-DVD coating containing
periodic zig-zag pores.
k̄ =
q̄
(Ttop⫺Tbottom)/l
(11)
GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS
2543
where q̄⬅q̄y is the heat flux in the y-direction averaged in the x-direction over the top (or bottom) surface of the unit cell. Dimensional analysis dictates
that
冉
冊
kmp
k̄
= h wm, fm, fm,
ks
ks
(12)
where h is a non-dimensional function.
Simple analytical solutions for k̄ can be obtained
when the pore spacing is small relative to the zig-zag
wavelength, i.e., f⬍1. The heat transfer problem then
becomes (approximately) one-dimensional, and the
effective thermal conductivity kx along the local coordinate x is obtained by applying the rule of mixtures:
k̄x = fkp + (1⫺f)k̄s
(13)
where k̄s = ks if the matrix contains no other pores. It
is then straightforward to show that the effective thermal conductivity k̄ of the model structure in the direction perpendicular to the substrate surface is given by
k̄ = [fkp + (1⫺f)k̄s] sin2w.
Fig. 5. Finite element mesh for the zig-zag unit cell.
ers of the zig-zag unit cell was sufficient to accurately
capture the heat transfer characteristics of the coating.
(14)
3.2. Comparison of analytical and finite element solutions
The effective conductivity of a coating system simultaneously containing all three types of pore is
obtained by using equation (14) thrice. For instance,
in the limit that kpks (non-conducting pores) for
each type of pore:
In the 1-D analytical solution, equations (14) or
(15), the overall thermal conductivity is dependent on
the pore inclination. For the same inclination, it is
assumed that the only heat transport path is along the
pore inclination direction. However, in reality, as the
spacing between pores dp increases, the amount of
heat travelling in the y-direction increases and that
along the x-axis decreases, leading to a higher effective thermal conductivity in the thickness direction of
the coating. That is, for the same inclination angle at
a specified kp/ks ratio, the 1-D model provides a lower
bound solution to k̄.
The heat transfer problem for the full range of the
slenderness ratio f is solved by the finite element
method. In order to examine the geometrical effect
on thermal performance, the pore spacing, dp, and zigzag wavelength, l, are systematically varied according to the inclination. The FE results are compared
to the 1-D solution in Fig. 6 for selected values of f,
with f and kpks = 0.01. It is apparent that, there is a
geometrical limit on the validity range of the 1-D
model (Fig. 6). When the slenderness ratio, f, of the
unit cell drops below unity (which is satisfied by
nearly all EB-DVD coatings), the 1-D solution agrees
well with the numerically calculated result.
During the high temperature use of thermal barrier
coatings, calcium–magnesium–aluminum silicates
(CMAS) can infiltrate open pores resulting in the filling of the pores with a much higher thermal conductivity medium. In Fig. 7, the 1-D results for the effect
of inclination angle φ for three values of kp/ks: 0.01,
0.33 and 0.83 are compared with the FEA solutions.
III
k̄ = ks S (1⫺fm) sin2wm.
m=I
(15)
From (1) it is seen that the condition kpks is satisfied if the gaseous volume fraction in the pore r⬇1,
i.e., the contact of pore surfaces is minimal.
3. FINITE ELEMENT MODEL
3.1. Mesh design
The 1-D analytical solution is subject to two provisos: (a) the unit cell is slender (f1); (b) the pores
are continuous (Hp⬵l). When neither of the above
conditions is satisfied, the heat transfer problem can
no longer be approximated as one-dimensional. In
this and the sections that follow, the steady-state heat
transfer problem is solved with the finite element
code ABAQUS. The unit cell of Fig. 3 is analysed
with the mesh design shown in Fig. 5. A typical finite
element mesh consisted of about 4000 elements for
the solid matrix and about 200 elements for the pore.
The element type chosen is the popular 4-node linear
diffusive heat transfer element, DC2D4. Selected
numerical calculations reveal that the use of two lay-
2544
GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS
Fig. 6. Comparison between 1-D and FE predictions for the
overall thermal conductivity as a function of pore inclination
angle for Type I or II pores, volume fraction f = 0.04. Results
are shown for kp/ks = 0.01 and selected values of slenderness
ratio f.
Fig. 7. Effect of pore conductivity on overall thermal conductivity: comparison between 1-D and FEA predictions for
f = 0.04 and f = 0.04.
In these calculations the porosity was small
(f = 0.04). Notice that, for small values of porosity,
the effect of pore conductivity is well described by
the 1-D model. However, as the porosity level is
increased above 0.1, the importance of accounting for
the influence of pore conductivity also increases. This
is examined immediately below.
Fig. 8. Overall thermal conductivity plotted as a function of
effective pore conductivity for f = 0.04, w = 60° and three
values of pore fraction: f = 0.04, 0.08 and 0.60.
8 plots the normalised overall thermal conductivity of
the zig-zag microstructure k̄/ks as a function of effective pore conductivity, kp. In these plots, the pore
inclination angle is fixed at 60°; that is, the zig-zag
wavelength to amplitude ratio l/2a = √3. It is further
assumed that only Type I or II pore with a fixed pore
fraction exists in each zig-zag thermal unit. From Fig.
8 it is seen that when the level of pore fraction is
small (f = 0.04 and 0.08), the effect of pore conductivity on k̄/ks is relatively weak. However, as the pore
fraction is increased to about 0.60, the effective pore
conductivity kp starts to substantially effect k̄/ks. Note
that although the pore fraction of Type III pores may
be as high as 0.60 (cf. Table 1), the results of Fig. 8
cannot be directly used to estimate the influence of
these pores on k̄ because they are intrinsically discontinuous (see Section 5 later). Note that the results of
Fig. 8 are less accurate in the limit kp→ks because
then the sidewalls of the unit cell (Fig. 3) can no
longer be treated as thermally insulated: that is, the
heat flux is predominantly in the y-direction when
kp→ks.
Extensive FE calculations show that the influence
of kp on k̄/ks can be well described by a linear
relationship:
4. OVERALL CONDUCTIVITY OF COATINGS
CONTAINING PORES OF THE SAME TYPE
k̄/ks = c1 + c2(kp/ks)
(16)
4.1. Effect of pore conductivity
To further study the influence of pore conductivity,
kp, upon k̄, a slender zig-zag thermal unit cell
(f = 0.04) with pore fractions f = 0.04, 0.08 and 0.60
were analysed. The selected pore fractions cover the
range observed by Hass et al [1], see Table 1. Figure
Table 1. Morphological parameters for three pore types [1]
Pore type
I
II
III
Pore
inclination
Pore spacing
60°
60°
60°
10–60 µm
0.6–2.5 µm
20–80 nm
where c1 and c2 are dimensionless coefficients that
are dependent upon slenderness ratio f, pore fraction
f and pore inclination angle w. Table 2 lists the typiTable 2. Thermal conductivity coefficients for equation (16)
f
c1
c2
Pore width
Type I
0.3–0.6 µm
0.1 µm
20 nm
0.04
3
14
0.72
0.87
0.94
Type II
0.69
0.82
0.90
Type I
0.03
0.09
0.05
Type II
0.06
0.14
0.09
GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS
2545
linked to the ratio of zig-zag wavelength l to amplitude a by equation (7), the same results in Fig. 6 can
be used to evaluate the dependence of k̄/ks on l/a, as
shown in Fig. 10.
5. OVERALL CONDUCTIVITY OF COATINGS WITH
DISCONTINUOUS PORES
Fig. 9. Overall thermal conductivity plotted as a function of
Type I or II pore fraction for f = 0.04, w = 60° and three values
of normalised pore conductivity: kp/ks = 1/10, 1/3 and 1/2.
cal values of c1 and c2 for Type I and II pores with
w = 60°.
4.2. Effect of pore fraction
The results presented in Fig. 8 illustrate the importance of Type I or II pore fraction, f. To further quantify the effect of pore fraction on the overall thermal
conductivity, Fig. 9 plots k̄/ks as a function of f for
kp/ks = 1/10, 1/3 and 1/2; the other parameters are
identical to those used for Fig. 8. Again, it is assumed
that the coating only contains one type of Type I or
II pore. The calculations indicate a strong dependence
of the porosity upon the pore fraction. It is found that
the overall thermal conductivity of the coating
decreases linearly as the pore fraction is increased,
and can be faithfully described as
k̄/ks = c3⫺c4f
The microstructure studies in [1] revealed that discontinuous pores are present in the EB-DVD coating
systems, particularly in those of Type III due to sintering. This suggests the perfect pore column structures employed in previous FEA simulations may not
fully represent the pore morphologies of the coating
system. In this section, the pore column in the thermal
zig-zag unit is treated as discontinuous, as shown
schematically in Fig. 4, with Hp denoting the height
of the discontinuous pore. If the pores are continuous,
then Hp = l. Figure 11 plots the overall thermal conductivity as a function of pore height Hp/l for
kp/ks = 0.01 and three different pore fractions,
namely, dw/dp = 1/10, 1/3 and 1/2. The trends are well
fitted by:
k̄/ks = c5⫺c6f
(18)
where c5 and c6 are dimensionless coefficients that
are dependent upon slenderness ratio f, pore fraction
dw/dp, and pore inclination angle w. Table 4 lists the
values of c5 and c6 for w = 60° and selected geometrical parameters.
(17)
where c3 and c4 are dimensionless coefficients that
are dependent upon slenderness ratio, f. Table 3 lists
the values of c3 and c4 for w = 60° and selected
pore conductivities.
4.3. Effect of pore inclination angle
One of the distinctive characteristics of the zig-zag
microstructure is the pore inclination angle, w. During
the vapor deposition process, the pore inclination can
be changed between about 30° and 90° [1]. The comparison of 1-D and FEA models in Fig. 6 reveals that
the 1-D model is valid when the slenderness ratio
l⬍0.1. On the other hand, since the inclined angle is
Fig. 10. Overall thermal conductivity plotted as a function of
l/a for f = 0.04, kp/ks = 0.01 and selected values of slenderness
ratio f.
Table 3. Thermal conductivity coefficients for equation (17)
f
0.04
3
14
c3
c4
kp/ks=0.10
kp/ks=0.33
kp/ks=0.50
kp/ks=0.10
kp/ks=0.33
kp/ks=0.50
0.75
0.90
0.98
0.75
0.92
0.99
0.75
0.93
0.99
0.68
0.84
0.90
0.50
0.63
0.67
0.37
0.48
0.50
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GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS
Fig. 8, the overall thermal conductivity of the model
7YSZ coating system containing all three pore types
is found to be k̄⬅k̄1 = 0.73k̄II or, equivalently,
k̄ = 0.48ks. This compares well with the measured
data of Hass et al. [1].
The results described in this paper can be used to
guide the design and processing of thermal barrier
coatings with desired low thermal conductivities. The
procedure is outlined as follows:
6.1. Selection of pore inclination
Fig. 11. Overall thermal conductivity of a coating containing
discontinuous (Type III) pores plotted as a function of pore
height for f = 0.04, w = 60°, kp/ks = 0.01 and three values of
pore fraction: dw/dp = 1/10, 1/3 and 1/2.
Figure 6 shows that the route of heat transportation
increases as the pore inclination is increased. Therefore, the largest inclination angle allowable by the
vapor deposition process is desirable. This also
implies that the smallest possible zig-zag wavelength
to amplitude ratio should be selected.
6.2. Selection of porosity
6. OVERALL CONDUCTIVITY OF COATINGS
CONTAINING DIFFERENT PORE TYPES
The results presented in previous sections for a single pore type can be used to estimate the overall thermal conductivity of a zig-zag coating containing the
three types of pore present in experimentally prepared
coatings as listed in Table 1 [1]. Note that although,
for simplicity, identical inclination angle (w = 60°) is
assumed for all three types of pore in the following
discussion, the previous results can be equally applied
to treat practical zig-zag microstructures where the
inclination angle may vary from one pore type to
another [1].
For Type III pores in the specified example, the
pores are taken as discontinuous with a pore fraction
dw/dp⬇0.5 and a porosity of f⬇0.2. Since the zig-zag
pores investigated by Hass et al. [1] have low geometrical aspect ratios, f1 is selected. In the absence
of more quantitative data, insulated pores with minimal surface contact is assumed. From equation (18)
and Table 4, the effective thermal conductivity of zigzag structures containing discontinuous Type III
pores is obtained as k̄III = 0.96ks. That is, Type III
pores have negligible effect on thermal transport. At
the next level of Type II pores, the pore fraction is
about 0.08. From Fig. 8, it is seen that the coating
containing Type II pores surrounded by a matrix of
Type III pores has an effective thermal conductivity
k̄II = 0.68k̄III, that is k̄II = 0.65ks. Finally, for Type I
pores surrounded by a matrix with effective conductivity k̄II, the pore fraction is 0.04. Consequently, from
The present results suggest that the continuous
Type I and II pores with small pore fractions are more
effective than discontinuous Type III pores having
large porosities in reducing the overall conductivity.
Therefore, the porosity of Type I and II pores is perhaps the most important parameter in the processing
of a low conductivity TBC. Also, to reduce the effective pore conductivity, the contact of pore surfaces
should be avoided or minimised.
7. CONCLUSIONS
The effective thermal conductivity of EB-DVD
coating systems containing various types of periodic,
zig-zag intercolumn pores is predicted both analytically and numerically. Focus has been placed on the
influence of various morphological parameters on the
overall thermal performance, including pore type,
porosity, pore conductivity, zig-zag wavelength and
amplitude, and pore inclination. It is found that the
effective thermal conductivity is a linear function of
several morphological parameters, and that the overall thermal performance of the zig-zag microstructures is dominated by continuous Type I and II pores
even though their pore fraction is low; Type III pores,
due to their discontinuous and discrete nature, only
have a relatively small effect upon the overall conductivity. Also, to minimise the overall thermal conductivity, the pore inclination (or the zig-zag wavelength to amplitude ratio) should be kept as small as
possible, and the contact of pore surfaces should be
avoided in order to maximise the contact resistance.
Table 4. Thermal conductivity coefficients for discontinuous pores [equation (18)]
dw/dp
0.10
0.33
0.50
c5
c6
f=14.00
f=4.00
f=0.04
f=14.00
f=4.00
f=0.04
0.75
0.94
0.99
0.75
0.94
0.99
0.75
0.94
0.99
0.15
0.07
0.04
0.2
0.1
0.055
0.27
0.19
0.06
GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS
Acknowledgements—This work has been supported partially by
ONRIFO and ONR (Contract No. N00014-97-1-0106 and
N00014-01-1-0271) and partially by the Virginia Space
Grant Consortium.
REFERENCES
1. Hass, D. D., Slifka, A. J. and Wadley, H. N. G., Acta
mater., 2001, 49, 973.
2. Marijnissen, G. H., Lieshout, A. H. V., Ticheler, G. J.,
Bons, H. J. M. and Ridder, M. L., US Patent, No.
5,876,860, 1999.
2547
3. Loeb, L. B., The Kinetic Theory of Gases. McGraw-Hill,
New York, 1934.
4. Chen, G. and Tien, C. L., AIAA J. Thermophysics Heat
Transfer, 1993, 7, 311.
5. Litovsky, E. Y. and Shapiro, M., J. Am. Ceram. Soc., 1992,
75, 3425.
6. Zeng, S. Q., Hunt, A. and Greif, R., ASME J. Heat Transfer, 1995, 117, 758.
7. Rohsenow, W. M. and Choi, H. Y., Heat, Mass, and
Momentum Transfer. Prentice-Hall, Englewood Cliffs,
NJ, 1961.
8. McPherson, R., Thin Solid Films, 1984, 112, 89.
9. Willis, J. R., J. Mech. Phys. Solids, 1977, 25, 185.