THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS WITH ZIG-ZAG PORE MICROSTRUCTURES

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1 Acta mater. 49 (2001) THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS WITH ZIG-ZAG PORE MICROSTRUCTURES S. GU 1,T.J.LU 1, D. D. HASS 2 and H. N. G. WADLEY 2 1 Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK and 2 Materials 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% 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: address: TJL21@eng.cam.ac.uk (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 /01/$ Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S (01)

2 2540 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. Fig. 1. Typical zig-zag pore structure of a EB-DVD coating. 2. ANALYSIS 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: How does the pore spacing affect the overall thermal conductivity? Which type of pores play a more crucial role in reducing the thermal conductivity? What is the relationship between the inclined angle of extended pore and the thermal performance? How does the overall conductivity depend on the wavelength and amplitude of zig-zag intercolumn pores? 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 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 Solid conductivity The kinetic formula for the thermal conductivity of a solid, k s is [3] k s = 1 3 C pv (1) where C p 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, q D, the dominant wavelength, l c, of the phonons is of the order of q D a l /T where a l is the lattice constant. The dominant wavelength, l c, is about A in 7YSZ. For TBCs having a characteristic microstructural length scale D (e.g., the growth column width), l c D is expected and hence quantum size

3 GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS 2541 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]) 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 k p = (1 r)k g + rk p s = k g + k c (2) where k B = JK 1 is the Boltzmann constant, d g and m g are the diameter and mass of gas molecules, and N A is Avogadro s number. Upon substitution of (4b) into (4a), one gets k 0 g = (1.66g 0.92) pd 2 gn A /C v k 1/2 BT 2m g Since T/p and k 0 g T. From (3) it is seen that T k g 1+c 1 T/d w p (5) (6) where r represents the fraction of the pore surfaces that are in good thermal contact, k g is the gaseous thermal conductivity, k p s is the intrinsic thermal conductivity of the material at a contact, and k c r(k p s k g ) is introduced to represent the effective thermal contact conductance across the pore. For simplicity, k c = k s may be assumed but k c can be measured as discussed later. Depending on the pore morphology (e.g., pore width d w ), k p varies from the gas conductivity k g in the limit r 0 (no contact) to the solid conductivity k s 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 2 k s 2.6 W/mK [1]. The conductivity of the gaseous phase trapped within the pore is a function of pore width, d w, pressure, p, and temperature, T [3, 5 7]. To a good approximation: k g = k 0 g 1 + 4g 2 a A Kn g +1 Pr a A 1 (3) where k 0 g is the gas thermal conductivity in free space, g c p /c v is the specific heat ratio, a A is the accommodation coefficient of the gaseous molecules on the solid surface of the pore (a A 1 for TBCs [8]), Pr is the gas Prandtl number, Kn = /d w 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 C p now denotes the gas specific heat per unit volume and v represents the average molecular velocity. Kinetic theory of gases then yields [3], k 0 g = (1.66g 0.92) p2 1/2 (4a) m g k B T N A /C v = k BT 2pd 2 gp (4b) where c 1 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 Kn 1, the continuum theory of heat conduction in the gas applies. When Kn 1 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 d w 0, k g 0. Furthermore, the effective conductivity of a highly porous medium (e.g., silica aerogel) can be reduced by an order of magnitude relative to k 0 g 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 W/mK, compared to W/mK of air at 1 atm [6]. Dimensional analysis dictates that the overall thermal conductivity, k, of a porous TBC depends on the associated geometrical and physical properties: k = k s h(k g /k s, k c /k s, 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 k c and k g. First, k is measured both in vacuum and in the gaseous phase of interest, and the corresponding k p 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 k c. Then to obtain the component k g, one simply subtracts k c from the total pore conductance k p measured in the gaseous phase of interest.

4 2542 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 d w, wavelength l and amplitude a; the width of the unit cell equals the inter pore spacing, d p. 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 = d p /l. (8) The zig-zag pore may be continuous (Type I and II pores, Fig. 3) or discontinuous (Type III, Fig. 4), with H p denoting the pore height (H p = l for continuous pores). The pore fraction when pores are all of the same type is given by f = (d w /d p )(H p /l). (9) Fig. 4. Computational model for EB-DVD coating system containing discontinuous pores. The porosity of a coating containing the three types of pore is given by f = III S f m (10) m = I 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 k p and k s assigned separately to the pore and the surrounding matrix material, with k s representing the effective conductivity of the matrix which may contain pores of lower class in the hierarchy (Type III Type II Type I) Effective thermal conductivity: 1-D analysis To calculate the effective thermal conductivity k ( k y) of the coating system in the thickness direction (y-axis, Fig. 3), constant temperatures, T top and T bottom, 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, q k (k = x, y), everywhere in the two-phase composite medium. Once this is accomplished, the effective thermal conductivity, k, is obtained as Fig. 3. Computational model for EB-DVD coating containing periodic zig-zag pores. k = q (T top T bottom )/l (11)

5 GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS 2543 where q q y 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 k = h w k m, f m, f m, km p s k s (12) where h is a non-dimensional function. Simple analytical solutions for k 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 k x along the local coordinate x is obtained by applying the rule of mixtures: k x = fk p +(1 f)k s (13) where k s = k s if the matrix contains no other pores. It is then straightforward to show that the effective thermal conductivity k of the model structure in the direction perpendicular to the substrate surface is given by k = [fk p +(1 f)k s] sin 2 w. (14) 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 k p k s (non-conducting pores) for each type of pore: k = k III s S (1 f m ) sin 2 w m. (15) m = I From (1) it is seen that the condition k p k s 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 (f 1); (b) the pores are continuous (H p 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- 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 Comparison of analytical and finite element solutions 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 d p 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 k p /k s ratio, the 1-D model provides a lower bound solution to k. 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, d p, 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 k p k s = 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 k p /k s : 0.01, 0.33 and 0.83 are compared with the FEA solutions.

6 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 = Results are shown for k p /k s = 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 = 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. 4. OVERALL CONDUCTIVITY OF COATINGS CONTAINING PORES OF THE SAME TYPE 4.1. Effect of pore conductivity To further study the influence of pore conductivity, k p, upon k, 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 Pore Pore spacing Pore width inclination I µm µm II µm 0.1 µm III nm 20 nm 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 plots the normalised overall thermal conductivity of the zig-zag microstructure k /k s as a function of effective pore conductivity, k p. 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 k /k s is relatively weak. However, as the pore fraction is increased to about 0.60, the effective pore conductivity k p starts to substantially effect k /k s. 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 k because they are intrinsically discontinuous (see Section 5 later). Note that the results of Fig. 8 are less accurate in the limit k p k s 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 k p k s. Extensive FE calculations show that the influence of k p on k /k s can be well described by a linear relationship: k /k s = c 1 + c 2 (k p /k s ) (16) where c 1 and c 2 are dimensionless coefficients that are dependent upon slenderness ratio f, pore fraction f and pore inclination angle w. Table 2 lists the typi- Table 2. Thermal conductivity coefficients for equation (16) f c 1 c 2 Type I Type II Type I Type II

7 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 k /k s on l/a, as shown in Fig. 10. 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: k p /k s = 1/10, 1/3 and 1/2. cal values of c 1 and c 2 for Type I and II pores with w = 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 k /k s as a function of f for k p /k s = 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 5. OVERALL CONDUCTIVITY OF COATINGS WITH DISCONTINUOUS PORES 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 H p denoting the height of the discontinuous pore. If the pores are continuous, then H p = l. Figure 11 plots the overall thermal conductivity as a function of pore height H p /l for k p /k s = 0.01 and three different pore fractions, namely, d w /d p = 1/10, 1/3 and 1/2. The trends are well fitted by: k /k s = c 5 c 6 f (18) where c 5 and c 6 are dimensionless coefficients that are dependent upon slenderness ratio f, pore fraction d w /d p, and pore inclination angle w. Table 4 lists the values of c 5 and c 6 for w = 60 and selected geometrical parameters. k /k s = c 3 c 4 f (17) where c 3 and c 4 are dimensionless coefficients that are dependent upon slenderness ratio, f. Table 3 lists the values of c 3 and c 4 for w = 60 and selected pore conductivities 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, k p /k s = 0.01 and selected values of slenderness ratio f. Table 3. Thermal conductivity coefficients for equation (17) f c 3 c 4 k p /k s =0.10 k p /k s =0.33 k p /k s =0.50 k p /k s =0.10 k p /k s =0.33 k p /k s =

8 2546 GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS 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, k p /k s = 0.01 and three values of pore fraction: d w /d p = 1/10, 1/3 and 1/2. 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 d w /d p 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, f 1 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 k III = 0.96k s. That is, Type III pores have negligible effect on thermal transport. At the next level of Type II pores, the pore fraction is about 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 k II = 0.68k III, that is k II = 0.65k s. Finally, for Type I pores surrounded by a matrix with effective conductivity k II, the pore fraction is Consequently, from Fig. 8, the overall thermal conductivity of the model 7YSZ coating system containing all three pore types is found to be k k 1 = 0.73k II or, equivalently, k = 0.48k s. 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 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 Selection of porosity 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)] d w /d p c 5 c 6 f=14.00 f=4.00 f=0.04 f=14.00 f=4.00 f=

9 GU et al.: THERMAL CONDUCTIVITY OF ZIRCONIA COATINGS 2547 Acknowledgements This work has been supported partially by ONRIFO and ONR (Contract No. N and N ) 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, Marijnissen, G. H., Lieshout, A. H. V., Ticheler, G. J., Bons, H. J. M. and Ridder, M. L., US Patent, No. 5,876,860, Loeb, L. B., The Kinetic Theory of Gases. McGraw-Hill, New York, Chen, G. and Tien, C. L., AIAA J. Thermophysics Heat Transfer, 1993, 7, Litovsky, E. Y. and Shapiro, M., J. Am. Ceram. Soc., 1992, 75, Zeng, S. Q., Hunt, A. and Greif, R., ASME J. Heat Transfer, 1995, 117, Rohsenow, W. M. and Choi, H. Y., Heat, Mass, and Momentum Transfer. Prentice-Hall, Englewood Cliffs, NJ, McPherson, R., Thin Solid Films, 1984, 112, Willis, J. R., J. Mech. Phys. Solids, 1977, 25, 185.