Quantitative characterization of microstructures of plasma-sprayed coatings and their conductive and elastic properties

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1 Materials Science and Engineering A 386 (2004) Quantitative characterization of microstructures of plasma-sprayed coatings and their conductive and elastic properties I. Sevostianov a,, M. Kachanov b, J. Ruud c, P. Lorraine c, M. Dubois c a Department of Mechanical Engineering, New Mexico State University, P. O. Box 30001, MSC 3450, Las Cruces, NM , USA b Department of Mechanical Engineering, Tufts University, Medford, MA 02155, USA c General Electric Global Research, Niskayuna, NY 12309, USA Received 5 April 2004; received in revised form 6 July 2004 Abstract Quantitative characterization of microstructures of plasma-sprayed coatings that accounts for their anisotropic and irregular character is developed. Dominant microstructural features are identified. It is found, in particular, that small islands of partial contacts along microcracks produce a strong effect on the conductive and elastic properties and constitute a major microstructural feature. The role of porosity is two-fold: (1) it has only a minor effect on the conductive and elastic properties, as compared with the effect of cracks (for porosities less than 15%) and (2) higher levels of porosity appear to be an indicator of higher crack density. Effective conductive and elastic properties are given in terms of the proper microstructural parameters. Cross-property connections that interrelate the conductive and the elastic properties are given in an explicit form. They can be used for optimization of the microstructure for the combined conductive and elastic performance. The developed framework is applied to yttria-stabilized-zirconia (YSZ) coatings. It is demonstrated that the framework can be directly linked to microstructural data obtained from photomicrographs. Predictions made on the basis of the microstructural analysis were tested against experimental data on elasticity/conductivity. The agreement is generally good Elsevier B.V. All rights reserved. Keywords: Plasma-sprayed coating; TBC; Conductivity; Elasticity; Microstructure; Characterization 1. Introduction Elastic and conductive properties of plasma-sprayed coatings are determined, aside from the properties of the bulk material, by the geometry of the porous space ( microstructure ). It comprises numerous microcracks and pores and is, generally, quite complex: The orientations of pores and cracks tend to be either parallel or normal to the substrate, with an orientational scatter around these preferential orientations. This results in anisotropy of the effective elastic and conductive properties. Mixtures of diverse pore shapes (narrow, crack-like ones and relatively round ones) are typical. A complicating Corresponding author. Tel.: ; fax: address: igor@me.nmsu.edu (I. Sevostianov). factor is the irregularity of pore shapes. In addition, microcracks may have islands of partial contacts along their faces a feature of primary importance, as discussed below. Quantitative characterization of such anisotropic and irregular microstructures is a challenging task. Requirements to such a characterization are that: It should be sufficiently simple, to be able to work directly with data obtained from photomicrographs; It should reflect those microstructural features that have a dominant effect on the properties of interest the elastic and the conductive ones. This means identification of the proper microstructural parameters, in which terms the considered properties are to be expressed. They must represent individual defects in /$ see front matter 2004 Elsevier B.V. All rights reserved. doi: /j.msea

2 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) accordance with their actual impact on the properties, and to reflect defect shapes and orientations correspondingly. Such parameters are not obvious, particularly for mixtures of defects of diverse shapes. The simplest characteristics of porous microstructures is the porosity p (volume fraction of pores). However, it is clearly inadequate for plasma-sprayed coatings. Anisotropies cannot be described in its terms and microcracks that have a strong effect on the properties do not contribute to p. The present paper addresses these issues by utilizing basic facts of the micromechanics of materials, as well as recent results describing the properties of defects of irregular shapes 1,2] and of islands of partial contacts on crack faces 1,3]. We also establish an explicit cross-property connection between the elastic and the conductive properties of the coatings. It maps possible combinations of these two properties and can be used for optimization of the microstructure for the combined conductive and elastic performance. It is demonstrated using yttria-stabilized-zirconia (YSZ) air-plasma-sprayed coatings that the developed theoretical framework can directly work with microstructural data. Predictions made on the basis of analysis of photomicrographs were tested against experimental data on elasticity and conductivity. The agreement was generally quite good. 2. Quantitative characterization of cracks and pores of various shapes A mixture of pores and microcracks of diverse shapes and orientations presents a challenge for its quantitative characterization. We first consider idealized geometries spherical pores and circular or elliptical microcracks. Then we discuss more realistic, irregular geometries. Results of this section are based on previous works 1,2,4 7]. A reference volume (containing a sufficiently representative population of defects) is denoted by V Spherical pores and circular or elliptical cracks Spherical pores are characterized by porosity relative volume of pores contained in V: p = 1 V (k) (1) V where V (k) are the volumes of individual pores. Circular cracks are characterized by the crack density parameter 8]: ρ = 1 l (k)3 (2) V where l (k) is radius of the k-th microcrack. The proportionality of the individual crack contributions to their radii cubed is in agreement with their actual contributions to the elastic and conductive properties, where parameter ρ will be used. The parameter ρ can be generalized to cracks of the elliptical shapes, based on the results of Budiansky and O Connell 9]. In the case of multiple elliptical cracks with random deviations from circles (random orientations of ellipses axes in their planes), ellipses can be replaced by equivalent circles, of such radii that ratios S 2 /P (where S and P are the ellipse s area and perimeter) are retained 4]. Being a scalar, the parameter ρ is adequate for randomly oriented cracks with overall isotropy. In anisotropic cases of non-random crack orientations, it should be changed to a symmetric second rank crack density tensor 10]: α = 1 V (l 3 nn) (k), or, in components, α ij = 1 V (l 3 n i n j ) (k) (3) where n (k) is a unit normal to kth microcrack. The tensorgiven in Eq. (3) is a direct generalization of the scalar crack density ρ and in the case of random orientations, α ij =(ρ/3)δ ij. Strictly speaking, the elastic (but not the conductive) properties need, in the case of non-random crack orientations, yet another parameter of crack density a fourth rank tensor. However, the dependence of the effective elastic moduli on this parameter is relatively weak, and can be neglected with sufficient accuracy. In addition, the proportionality of the individual defect contributions to their sizes cubed has practical implications. In mixtures of defects of diverse sizes, the much smaller ones can be ignored, as far as the overall conductivity and elasticity is concerned. This facilitates processing of data from photomicrographs since the detection of very small defects is unnecessary Orientational distribution of microcracks and components of crack density tensor In plasma-sprayed coatings, the orientational distribution of microcracks typically has transversely isotropic symmetry. One family of cracks tends to be parallel to the coating plane x 1 x 2 ( horizontal cracks) and another one normal to it ( vertical cracks). Both families have an orientational scatter about these preferential orientations. The scatter may produce a significant impact on the values of α 11 and α 33 (and, therefore, on the effective properties). We describe the orientational distribution by the following function, containing scatter parameter λ (that may have different values, λ h and λ v for the horizontal and vertical cracks): P λ (ϕ) = 1 2π (λ2 + 1)e λϕ + λe λπ/2 ] (4) The extreme cases of fully random and ideally parallel orientations correspond to λ = 0 and, respectively. Fig. 1 shows orientational patterns that correspond to several values of λ. In the case of transverse isotropy, the crack density tensor reduces to two components α 11 = α 22 and α 33. They reflect, in an integral way, both partial crack densities ρ h and ρ v of the horizontal and vertical cracks and their scatter. For example,

3 166 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) Fig. 1. Dependence of the orientational distribution function P λ on angle ϕ at several values of λ and the corresponding orientational patterns. an increase in the orientational scatter of the horizontal cracks reduces α 33 and increases α 11. Components α 11 and α 33 are expressed in terms of ρ h and ρ v and the scatter parameters as follows: α 11 = α 22 = f 2 (λ v )ρ v + f 1 (λ h )ρ h α 33 = f 1 (λ v )ρ v + f 2 (λ h )ρ h (5) where f 1 and f 2 are functions of the scatter parameter λ given by: f 1 = 18 λ(λ2 + 3)e λπ/2 6(λ 2 ; f 2 = (λ2 + 3)(3 + λe λπ/2 ) + 9) 3(λ 2 + 9) (6) 2.3. Applicability of results for cracks to strongly oblate, crack-like pores The effect of such pores on the elastic and the conductive properties is about the same as the one of cracks, up to aspect ratios of about ,11]. Therefore, they can be replaced by cracks as far as the effective elastic and conductive properties are concerned and characterized by the crack density parameters given in Eqs. (2) and (3). Hence, Exact information on the pore aspect ratios is unnecessary, as long as they are known to be smaller than about Porosity is not a relevant parameter for the characterization of such pores. For example, changing the pore aspect ratios from 0.01 to 0.1 increases the porosity ten times, but leaves the elastic and conductive properties almost unchanged Quantitative characterization of various irregularities of the porous space We now identify the main irregularity factors and discuss their effects. One irregularity factor the orientational scatter of microcracks is already reflected in components α 11 and α 33. Another factor the fact that cracks are not ideally thin but may be slightly open (strongly oblate pores) was also discussed above. Jagged boundaries of cracks and pores. If the jaggedness is slight (its amplitude is much smaller than the defect size), it can be ignored, as far as the effective properties are concerned, and the boundaries can be smoothed out. This follows from bounding the effect of a pore by the circumscribed and inscribed comparison shapes 12]. Moderately non-spherical pores (aspect ratios in the range ). If such pores are randomly oriented (overall isotropy), they can be replaced by the spherical ones of the same volume and characterized by porosity parameter p. In the anisotropic cases of non-random orientations, the concentration parameter is tensorial 5]. Strongly non-spherical pores that are not strongly oblate, i.e. pores with aspect ratios >1.5 (strongly prolate) or in the range (0.15, 0.7) cannot be characterized in terms of porosity alone even in the isotropic case of random orientations. They will also be functions of the average eccentricity. Proper concentration parameters for such shapes were given in references 5,13] in the contexts of elasticity and conductivity, correspondingly. It appears, though, that such pores are not typical in plasma-sprayed coatings Pores of complex shapes A rigorous estimate of their effect on the overall properties is given by bounding, using circumscribed and inscribed comparison shapes 12]. However, choosing these shapes as ellipsoids (for which results are available) may lead to bounds that are overly wide, particularly for the shapes with pronounced non-convexities. Planar cracks of irregular in-plane shapes can be analyzed in several different ways (see 2] for the elastic properties which can also be extended to the conductive properties). A rigorous estimate of their effect on the overall properties is given by the upper and lower bounds generated by circumscribed and inscribed comparison shapes. If the latter are chosen as ellipses, the bounds are known. However, for substantially non-convex shapes, such bounds may be overly wide. Numerical simulations show that more efficient, although not rigorous, estimate is often provided by a circular shape, with the radius yielding the same ratio S 2 /P (where S is the area and P is the perimeter of the actual shape). Other estimates are also possible. If the shape irregularities have no systematic orientational bias, such cracks can be modeled by equivalent circular ones and the same concentration parameters given in Eqs. (2) and (3) can be used for multiple cracks of this kind. In the context of geophysical applications dealing with microcracked rocks, these issues were discussed by Sayers and Kachanov 14]. Moderately non-planar cracks ( wavy crack patterns). Such patterns are fairly frequent, particularly for the

4 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) horizontal cracks. This factor was analyzed by Sevostianov and Kachanov 2] in the context of the overall elasticity. If the non-planarity is moderate, the crack can be replaced, with satisfactory accuracy, by the planar one of the same area. Again, the same crack density parameters in Eqs. (2) and (3) can be used for cracks of this kind. Strongly non-planar cracks cannot, generally, be replaced by equivalent planar ones since for example, uniaxial loading of a such a crack would generate a noticeable lateral strain, that is absent for the planar crack and is small for a moderately non-planar one. It appears, however, that strongly nonplannar cracks are infrequent in plasma-sprayed coatings. Islands of partial contact along cracks. Such islands, even if they are very small, produce a strong effect that cannot be ignored even in rough estimates. Due to its importance, it is discussed in a separate section to follow Summary on quantitative characterization of the microstructure The two main features constituting the porous space of coatings are pores and microcracks. Each has its own irregularity factors. It appears that the majority of pores in plasma-sprayed coatings are only moderately non-spherical, and their deviations from spheres are random. Such an isotropic mixture of relatively round pores can be adequately characterized by the porosity parameter p. As far as cracks are concerned, it is remarkable that in their effect on the overall properties, they can be replaced by equivalent circular ones. They are characterized by only two parameters, α 11 and α 33. The latter reflect all the irregularity factors including orientational scatter, irregular in-plane shapes and moderate non-planarities, and the presence of islands, in an integral way and in a quantifiable manner. Naturally, this quantification requires certain information describing the microstructure as an input. The most important pieces of information concern microcracks, and they are: (1) the distribution of islands of partial contacts in terms of the average island sizes and frequency of occurrence, and (2) the extent of the orientational scatter. The challenge for experimental techniques is, therefore, to extract this information from photomicrographs or by other means. The role of porosity, p, is two-fold. Its direct effect on the overall property is relatively minor, as compared with microcracks (at least, at porosities less than 15%). On the other hand, a part of the overall porosity is carried by slightly open cracks. These openings are of no direct consequence. However, higher levels of porosity may then be indicators of higher crack densities, which are of direct major importance. 3. Islands of partial contact (cracks of annular geometries)and their significance Such microgeometries have origins in an incomplete cohesion between splats. One of the scenarios of the splat for- Fig. 2. Splat formation. Adhesion zone at center is surrounded by a debonded zone, resulting in 3-D annular crack geometry (after Kudinov 15]). Its 2-D cross-section will show two line cracks. mation is shown in Fig. 2. It produces a 3-D annular crack, with an island of partial contact in the middle. The following two issues arise in this connection. Even a very small island produces a strong effect. It drastically reduces the effect of the crack on (1) elastic compliance in the direction normal to the crack, and (2) reduction of conductivity in this direction. This is illustrated by considering an equivalent circular crack of radius R eff without an island that produces the same effect. In Fig. 3 R eff is plotted as a function of the island size 2,3]. Note an almost vertical drop on the left, to the value of Since the effect of a crack is proportional to its size cubed (and, hence, to the size of the equivalent crack cubed), a very small island reduces this effect, roughly, by a factor of 4. Possible presence of islands creates an uncertainty in interpretation of 2-D cross-sectional photomicrographs. For example, two collinear crack lines may represent two entirely different 3-D geometries: (A) annular crack with an island, or (B) two separate cracks, that may or may not be normal to the photograph plane (and the visible lines may or may not represent their largest cross-sections). Geometries (A) and (B) may produce very different effects on the overall properties. Thus, in addition to the uncertainties associated with geometry (B), there is a major uncertainty due to the possibility of geometry (A). It should be noted that the effects of an island on the elastic and on the conductive properties are quite close to each other (see Fig. 3). In the first approximation, they can be taken as identical. This has important implications for the elastic conductive cross-property connections, as discussed in the text to follow. Thus, islands of partial contact constitute a microstructural factor of major importance. For the quantitative characterization, cracks with islands can be replaced by equivalent circular cracks, and, thus, described by the same crack

5 168 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) families of strongly oblate, crack-like pores that are parallel and perpendicular to the substrate. Li et al. 17] proposed a model where splats were bonded along small areas and low modulus in the deposition direction was explained by bending of the unbonded parts. Their analysis, however, does not yield the modulus in the direction parallel to the substrate, or other anisotropic elastic constants. Kroupa 18] modeled the porous space by two families of spheroidal cavities and predicted relatively small changes in the elastic moduli, whereas in reality Young s moduli may be reduced by an order of magnitude, as compared to the ones of the bulk material 19]. Sevostianov and Kachanov 7] showed that microcracks and not porosity produce the dominant effect on the elastic properties and developed a model that accounted for the orientational scatter and some other irregularity factors. Their results were in agreement with the ultrasonic data of Parthasarathi et al. 20] on the full set of anisotropic moduli Earlier work on modeling the conductivity Fig. 3. (a) Reduction of the crack compliance contribution due to an island. R eff is a radius of an equivalent circular crack (producing the same effect). An almost vertical drop on the left indicates a substantial effect of even a very small island.) (b) Similar to (a), for conductivity reduction of the crack s effect on conductivity in the direction normal to the crack due to an island. density parameters given in Eqs. (2) and (3), where crack radii l (k) should be taken as R (k) eff (shown in Fig. 3). In Section 6, we discuss the available data on the effective properties of YSZ coatings, in conjunction with the information from the photomicrographs. Our analysis indicates the presence of islands of partial contact and the strong effect produced by them. The challenge is clearly seen for the experimental techniques to extract the average island sizes and the frequency of their occurrencefrom photomicrographs. This information determines the adjustment l R eff. 4. Effective elastic and conductive properties in terms of microstructural parameters Modeling of the effective properties of coatings has been addressed in a number of earlier contributions. Their short overview is given below. We note that modeling of islands of partial contacts, of the orientational scatter and other irregularity factors seems to have been overlooked Earlier work on modeling the elastic properties Leigh and Berndt 16], as well as several other authors, identified the dominant elements of the porous space as two McPherson 21] expressed conductivity in the direction normal to the substrate in terms of loss of the contact area between layers. The drawbacks of this model are that it does not account for the orientation scatter of the contacts and, importantly, that the area of contacts may generally be an insufficient parameter. As discussed above, small islands of partial contacts surrounded by a debonded area produce a disproportionately strong effect. For an extensive review and finite element modeling of the conductivity of coatings, see Doltsinis et al. 22]. Sevostianov and Kachanov 23] showed that, similarly to the elastic properties, microcracks and not porosity produce the dominant effect on the conductivity and developed an appropriate quantitative model Present micromechanical modeling The present work utilizes results on the effective properties of materials with microcrack and pores of various shapes 4,5], as well as recent results on various irregularity factors and islands of partial contacts 2,3]. We model the porous space by a mixture of microcracks of the orientational distribution that is transversely isotropic, overall, with orientational scatter, islands and other irregularity factors mentioned above and micropores of more or less round shapes. For such a mixture, the effective properties are transversely isotropic. They are characterized by two conductivities (parallel and normal to the coating) and five elastic constants. All these effective constants can be expressed in terms of the following three microstructural parameters: Two crack density parameters α 11 and α 33 that reflect, in an integral way, cracks density and orientational scatter, islands and other irregularity factors. The effect of these factors on α 11 and α 33 is quantifiable in a straightforward way, provided the corresponding microstructural information is available.

6 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) Overall porosity of the coating p. The direct role of this parameter is relatively minor, in the sense that dependence of the effective properties on p is weak (for porosities less than 15%) However, porosity level may be an indicator of the crack density level, since microcracks may be slightly open and thus carry a part of p (see discussion of Section 6). Subscript 0 indicates, hereafter, properties of the bulk material (assumed to be isotropic). The effective conductivities are: k 11 = k 22 = k α ] 11 p 1 k 33 = k α 33 3(1 p) p 3(1 p) + 3 ] p 1 (7) 2 1 p The effective elastic constants (Young s and shear moduli and Poisson s ratios) are: E 1 = E 2 = E (1 ν2 0 ) α 11 3(2 ν 0 ) 1 p + 3(1 ν 0)(9 + 5ν 0 ) 2(7 5ν 0 ) E 3 = E (1 ν2 0 ) α 33 3(2 ν 0 ) 1 p + 3(1 ν 0)(9 + 5ν 0 ) 2(7 5ν 0 ) ] p 1 (8) 1 p p 1 p ] 1 G 12 = G (1 ν 0) α 11 3(2 ν 0 ) 1 p + 15(3 ν 0) p 2(7 5ν 0 ) 1 p G 13 = G 23 = G (1 ν 0) 3(2 ν 0 ) α 11 + α 33 1 p + 15(3 ν 0) p 2(7 5ν 0 ) 1 p ν 12 = ν 31 = ν (1 ν 0)(1 + 5ν 0 ) E 1 E 3 E 0 2ν 0 (7 5ν 0 ) ] p 1 p 4.4. Direct and indirect effects of porosity ] 1 ] 1 The results above correspond to Mori Tanaka s scheme that accounts for interactions between defects by placing each one into the average, over the solid matrix, field (elastic or thermal). This average field is raised by the factor of (1 p) 1 by porosity but is not affected by cracks. This factor in Eqs. (7) and (8) reflects the interaction effect. As seen from these formulas, the effect of porosity is twofold: indirect effect: it amplifies the effect of microcracks, by raising the average stress environment for them (factor of (1 p) 1 in the terms proportional to α 11 and α 33 ); direct effect is reflected in terms proportional to p/(1 p) Comparison of the effects of porosity and microcracks on the overall properties Data from photomicrographs indicate that typical values of crack density parameters α 11 and α 33 properly adjusted for irregularity factors in YSZ coatings are of the order of , whereas porosity p is of the order of or lower (Section 6). This implies a reduction in the elastic stiffnesses and in the conductivities by a factor of 2 5 as compared to the values for the bulk material. As seen from Eqs. (7) and (8), microcracks are the dominant factor in this reduction, with porosity playing a relatively minor role. Indeed, if the effect of porosity is fully neglected, by setting p = 0, this would change the effective elastic/conductive constants by about 20% for the typical values of parameters (as compared to the overall changes by the factor of 2 4). It should be noted that if only the terms accounting for the direct impact of porosity (proportional to p/(1 p)) are neglected, but the indirect impact terms (the factor of (1 p) 1 at α 11, α 33 ) are retained, the resulting changes are typically below 10% for the conductivities and the elastic constants. This remark has important implications for the cross-property connection discussed in the text to follow. In addition, at porosities substantially higher than 10 15%, the effect of porosity may be comparable to the one of cracks and all the terms entering Eqs. (7) and (8) should be retained Porosity as a possible indicator of crack density Examination of photomicrographs indicates that cracks are not ideally thin, but may be slightly open. This means that the overall porosity p is split between two partial porosities, p pores and p cracks, where the second term is due to cracks. Further, p cracks is split between the horizontal and vertical cracks, p H cracks and pv cracks. The split does not affect the interaction factor of (1 p) 1 in Eqs. (7) and (8) since it is the total porosity that raises the average stress in the matrix. Strictly speaking, however, it necessitates certain alterations in these formulas: Terms proportional to p V cracks and ph cracks should be added to crack densities α 11 and α 33, respectively, to account for slight opening of the cracks. This correction is typically minor. Terms p/(1 p) should be changed to p pores p pores /(1 p). These alterations could be done in a straightforward way, by utilizing results of previous works 5,11] on slightly open cracks, provided the split ratio is known.

7 170 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) The mentioned alterations are not taken into account either in Eqs. (7) and (8) or in processing of the microstructural data, for the following reasons. First, the split information (particularly in the 3-D formulation) is difficult to recover from the data obtained from photomicrographs because an attempt to do so would introduce additional uncertainty factors. Second, the corresponding changes in the predicted effective properties would be relatively minor as can be seen from hypothetical numerical examples. Although the split p = p pores + p cracks does not affect the numerical values of the effective properties, it does have an important consequence of the physical nature: the fact that cracks carry a certain porosity level, implies that higher levels of the overall porosity appear to be indicators of higher crack densities. The data of Table 2 serve as an illustration. We note that, in the absence of data on the split ratios, this indicator is a qualitative one. 5. Elastic conductive cross-property connections If the terms corresponding to the direct impact of porosity are omitted then Eq. (7) for the conductivities and Eq. (8) for the stiffnesses acquire a similar structure. The microstructural information enters both Eqs. (7) and (8) through two parameters: α 11 (1 p) 1 and α 33 (1 p) 1. They can be expressed in terms of conductivities k 11 /k 0, k 33 /k 0 from Eq. (7) and substituted into Eq. (8). This yields an explicit elastic conductive cross-property connection. In particular, Young s moduli E i in the direction x i (that is either normal or parallel to the coating) and conductivity k i in the same direction can be interrelated by the following simple formula: E 0 E i = 4(1 ν2 0 ) k 0 k i 2.14 k 0 k i (9) E i 2 ν 0 k i k i This cross-property connection is explicit and does not contain any adjustable parameters. The cross-property connection is insensitive to microstructural information, such as crack density and orientational scatter and the presence of islands. The underlying reason is that the mentioned microstructural information affects the elastic and the conductive properties in a similar way. This constitutes the power of this result one does not have to know details of microgeometries in order to use the connection. It should be noted that for coatings of higher porosities (P > 0.15), all the terms in Eqs. (7) and (8), including the direct impact terms, have to be retained. Then the crossproperty connection can still be established, but it will contain porosity p as a parameter. Its use will then require knowledge of p. Verification of the cross-property connection in Eq. (9) on YSZ coatings is discussed below. In a broader context of materials science, such connections have been experimentally verified on three material systems other than coatings metals with fatigue microcracks 24], short fiber reinforced plastics 25] and porous metals 26]. 6. Experimental verification of the modeling on YSZ coatings The developed modeling was tested against data on a range of 8 wt.% yttria-stabilized-zirconia (8YSZ) coatings fabricated by an air-plasma spray process onto air-plasmasprayed bondcoats on Inconel 718 substrates. Porosity was determined from the density measured from physical dimensions and using the Archimedes method in water. Full density of 8YSZ was taken as 6.05 g/cm 3 27]. The Young s modulus parallel to the coating was measured using a surface-wave, laser ultrasound technique 28]. The generation laser was a 2 mj 532 nm doubled YAG focused as a line. The detection laser was a long pulse 1064 nm YAG laser also focused as a line. A 2 mm thick tungsten layer was sputtered onto both surfaces of the specimens for the measurements because YSZ is translucent up to about 10 m. The distance between the generation and detection laser lines on the coating surface was 6 mm. A dual cavity Fabry Perot interferometer was used, and 50 laser shots were averaged for each mesaurment. Inplane Young s modulus was determined from the arrival time of the longitudinal wave, the density of the 8YSZ coating, and the Poisson s ratio, which was measured from the arrival times of both the Rayleigh wave and the longitudinal wave. Thermal conductivity normal to the coating was calculated from measurement of the density, specific heat and thermal diffusivity. Thermal diffusivity was measured at 18 C using the laser flash method 29], using three laser shots per specimen. We selected four representative coating samples, with porosities of 10.3%, 12.1%, 12.6%, and 13.2% (Table 1). The coatings had a large range of elastic modulus (spanning a factor of 3) and thermal conductivity (spanning a factor of 2) values (Table 2). The following goals were pursued: To demonstrate that the theory can directly process the microstructural data. To affirm the ability of the model to predict the elastic and the conductive properties in terms of microstructure, as given by Eqs. (7) and (8). To verify the cross-property connection given in Eq. (9). Table 1 Microstructural characteristics of four specimens presented in Figs. 4 7 Specimen ρ 0 (g/cm 3 ) p α 11 = α 22 α 33 (A ) 5.42 ± (B ) 5.32 ± (C ) 5.29 ± (D ) 5.25 ± The data on density is used for estimation of porosity (the density of the virgin material ρ 0 = 6.05 g/cm 3 ).

8 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) Table 2 Effective properties of four specimens presented in Figs. 4 7 Specimen E 1 (GPa) k 3 (W/mk) Measured Formula (8) Cross-property formula (10) Measured Formula (7) (A ) 51.0 ± ± (B ) 44.1 ± ± (C ) 34.2 ± ± (D ) 18.6 ± ± Direct influence of porosity is ignored in calculations. To identify uncertainties in extraction of the needed microstructural information from 2-D photomicrographs. The experimental input comprised the following information: microstructural information, in the form of photomicrographs; data on the effective Young s modulus in the direction parallel to the substrate; data on the effective conductivity in the direction normal to the substrate; data on the overall porosity Verification of the elastic conductive cross-property connection We verify the cross-property connection in Eq. (9) on the basis of the available elasticity conductivity data on YSZ coatings. The elastic modulus data were in the direction parallel to the substrate (E 1 ), whereas the conductivity data were in the direction normal to the substrate (k 33 ). Therefore, a direct verification of the cross-property connection could not be done. However, an indirect verification could still be done if the ratio α 11 /α 33 (that characterizes the extent of anisotropy of microcrack orientations) could be estimated. Then, by extracting the value of α 11 from the data on E 1 we could estimate α 33 and thus predict k 33 and compare it with the directly measured value. Thus, utilizing Eqs. (9) and (7), we relate E 1 to k 33 as follows: E 1 = E (1 ν2 0 ) 2 ν 0 α 11 α 33 k 0 k 3 k 3 ] 1 (10) We estimated the ratio α 11 /α 33 from photomicrographs of Figs. 4 7, by summing up the individual crack line contributions according to Eq. (3) by interpreting the lines as radii of circular cracks. This ratio was about 1.4 for all four 6.1. Intelligent reading of the microstructure and its quantitative characterization The challenge is to characterize a complex and irregular microstructure in a sufficiently simple way for input into the model. As discussed in Section 2, this task is solved by reducing the microstructural parameters to two crack densities α 11 and α 33 and porosity p. The problem remains of determination the correct values of α 11 and α 33, by extracting these 3-D parameters from 2-D photomicrographs. This calls for an intelligent reading of the microstructure, namely: Utilization of the fact that smaller microcracks can be ignored in the processing, since the individual crack contributions to α 11 and α 33 (and, therefore, to the overall properties) are proportional to their sizes cubed. Similarly, contributions of smaller pores into the overall porosity can be ignored. Appropriate adjustment of the effective values of α 11 and α 33 for the irregularity factors, which include (1) the average size of islands and their spatial frequency and (2) estimates of the 3-D crack densities α 11 and α 33 on the basis of line crack traces in 2-D photomicrographical data. The two factors are not independent: the presence of islands affects estimation of α 11, α 33 from the 2-D data. We dealt with these uncertainties by using one of the four specimens for calibration and applying the same calibration coefficient to other three specimens. Fig. 4. Specimen (A ). (a) Representative fragment; (b) processing of the photomicrograph; (c) orientation distribution of crack density.

9 172 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) Fig. 5. Same as Fig. 4 for specimen (B ). specimens, indicating a moderate dominance of the vertical microcracks (mostly, due to large vertical cracks). It should be noted that the cross-property connection is rather sensitive to the values of the bulk material constants, E 0 and k 0. The bulk material used for the coatings had the following constants: E 0 = 210 GPa 30], v 0 = 0.3 and k 0 = 2.2 W/mK 31,32]. Verification of the cross-property connection is given in Table 2. The agreement between the data and the values predicted via the cross-property connection is quite good. The maximal error is 12% for specimen (B ) and does not exceed 8% for the other three specimens. We note that such a good agreement takes place in the case of large changes in conductivities and elastic moduli. Defects reduce them by a factor of 4 5, as compared to the bulk material Verification of the microstructure-property relationships Fig. 6. Same as Fig. 4 for specimen (C ). To verify the relations in Eqs. (7) and (8), we processed data from photomicrographs of the four specimens, with porosities varying from 10.3% to 13.2%. Theoretical predictions need, as input parameters, the values of crack densities α 11 and α 33 appropriately adjusted for the irregularity factors. These values have to be identified from the microstructures. Fig. 7. Same as Fig. 4 for specimen (D ).

10 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) The presence of islands a major microstructural factor creates an uncertainty in interpretation of the photomicrographical data. Crack lines in cross-sections can be interpreted either as traces of isolated cracks without islands or as traces of annular cracks with islands in the middle. We first demonstrate how large the extent of this uncertainty is on the example of specimen (A )(Fig. 4). Then we use this specimen for calibration and apply this calibration coefficient to the remaining three specimens. If all the crack lines in the photomicrograph of specimen (A ) are interpreted as traces of isolated cracks (without islands ) then the estimated values of crack densities are α 11 = 0.14, α 33 = Substitution of these values into Eqs. (7) and (8) yields E 1 = 111 GPa and k 3 = 1.69 W/mK. That is a very substantial overestimation of the experimental data given in Table 2. We interpret this disagreement as an indication that a significant proportion of the microcracks have islands of partial contacts. Then, crack lines in 2-D cross-sections have to be interpreted as traces of annular cracks and crack densities α 11 and α 33 have to be multiplied by a certain factor. This factor depends on the average ratio of the internal-to-external radii of the annular cracks information that could not be directly estimated from the photomicrographs. An additional uncertainty factor is related to extraction of 3-D parameters α 11 and α 33 from 2-D photomicrographs. We chose the overall correction factor of 4.6 in order to match the conductivity and elasticity data. This factor implies that the adjusted components of the crack density tensor are: α 11 = 0.64 and α 33 = 0.46, yielding E 1 = 41.5 GPa and k 3 = 0.93 W/mK. In support of the chosen value of 4.6 for the islands factor, we note that it corresponds to the ratio of the internalto-external radii of 0.5 in agreement with the data of Kudinov 15] (who estimates it to be in the interval ). We also note that both the elasticity and the conductivity data for specimen (A ) are matched by choosing this factor. We now treat this factor of 4.6 as the calibration coefficient and apply it to the other three specimens, in processing the photomicrographical data for them. The resulting predictions of the elastic and conductive properties for these specimens and comparison with the measured data are given in Tables 2 and 3. The agreement is very good for all the specimens, for both the elasticity and the conductivity data. 7. Discussion and conclusions Porous space of plasma-sprayed coatings is quite complex and irregular. It is anisotropic, and comprises a mixture of microcracks and pores of diverse shapes and orientations. Our purpose is to develop quantitative microstructure-overall property relationships. The first step in this direction is to develop a proper quantitative characterization of the microstructure. It should be sufficiently simple and reduce to a small number of microstructural parameters, so that the effective properties of interest elasticity and conductivity can be expressed in their terms. This means that the proper microstructural parameters should reflect those microstructural features that produce a dominant effect on the said properties. Moreover, they should represent individual microstructural elements in accordance with their impact on the effective properties. At the same time, the microstructural characterization should be directly linkable to the photomicrographical data. Identification of such parameters is a non-trivial task. It is addressed in the present work, by analyzing the effect of various microstructural elements on the elasticity and conductivity. Our analysis yield a relatively simple microstructural characterization in which terms the effective properties are expressed. The findings can be summarized as follows Quantitative characterization of the microstructure 1. Microcracks are microstructural elements of the dominant importance. Their quantitative characterization is complicated by various irregularity factors. Of them, the dominant ones are: Islands of partial contact along microcracks are a major microstructural feature. Their presence strongly affects the overall conductivity and the overall elasticity, even if the islands are very small. Orientational scatter of microcracks (about two preferential orientations, horizontal and vertical). Non-planarity of cracks is a somewhat less important factor, although it may produce a noticeable effect. The density of microcracks is characterized by two components, α 11 and α 33, of the crack density tensor. Our analysis quantify the effect of the above mentioned factors on their values. It appears remarkable that, in spite of the complicating factors, microcrack population can be characterized by two parameters, α 11 and α 33 that reflect, in an integral and quantifiable way, the irregularity factors and the anisotropic orientational distribution. 2. Porosity plays only a minor role in determining the overall conductivity and elasticity for porosities less than 10 15%. The following remark should be made, however. Although porosity has only a minor effect on the effective properties, the porosity level appears to be an indicator of the crack density level, since the cracks may be slightly open and thus carry a part of the overall porosity. In the absence of data on the porosity split between pores and cracks, this indicator should be viewed as only a qualitative one Microstructure-property relations. Elastic conductive cross-property connection Microstructure-property relations are given that express the anisotropic conductive and elastic properties in terms of

11 174 I. Sevostianov et al. / Materials Science and Engineering A 386 (2004) α 11 and in Eqs. (7) and (8). These relations are explicit and contain no adjustable parameters. The elastic conductive cross-property connections interrelate the elastic constants and the conductivities of the coatings. They map the possible combinations of the conductive and elastic properties, and, thus, can be used for optimization of the combined conductive and elastic performance. We mention that, applying this result to the electric (rather than thermal) conductivities may provide a control tool. Comparison of predictions of the microstructure-property relations and of the cross-property connections with the experimental data on elasticity/conductivity of YSZ coatings produces a very good agreement (Table 2) Uncertainties in interpretations of photomicrographical data The ability of the developed framework to directly work with the photomicrographical data has been demonstrated on YSZ coatings. This capability is largely due to the proper quantitative characterization of the microstructure that has been developed in the present work. Our analysis identifies uncertainty factors in interpretation of photomicrographical data. They are mostly related to the following two factors: Extraction of 3-D microstructural information from 2-D cross-sections Extraction of the islands related information such as their spatial frequency and the average island size. Uncertainties of this kind seem to be typical for the situations when the microstructural information has to be quantified. The uncertainty factors identified here for the coatings represent a challenge for experimental techniques and call for further work in this direction. Means of extraction of the islands related information are particularly needed, since the islands constitute a microstructural feature of a major importance. In spite of the uncertainties, good agreement was observed for coatings representing a broad range of elastic and thermal properties. References 1] I. Sevostianov, M. Kachanov, Int. J. Fract. 114 (2002) ] I. Sevostianov, M. Kachanov, Int. J. Fract. 114 (2002) ] I. Sevostianov, Phil. Trans. Math, Phys. Eng. Sci. 361 (2003) ] M. Kachanov, Appl. Mech. Rev. 45 (1992) ] M. Kachanov, I. Tsukrov, B. Shafiro, Appl. Mech. Rev. 47 (1994) ] I. Sevostianov, M. Kachanov, Acta Mater. 48 (2000) ] I. Sevostianov, M. Kachanov, Mater. Sci. Eng. A 297 (2001) ] J.R. Bristow, Br. J. Appl. Phys. 11 (1960) 81. 9] B. Budiansky, R.J. O Connell, Int. J. Solids Struct. 12 (1976) ] M.J. Kachanov, J. Eng. Mech. Div. 106 (1980) ] I. Sevostianov, M. Kachanov, J. Mech. Phys. Solids 50 (2002) ] C. Huet, P. Navi, P.E. Roelfstra, Continuum Models and Discrete Systems-2, 1991, p ] M. Kachanov, I. Sevostianov, B.J. Shafiro, Mech. Phys. Solids 49 (2001) 1. 14] C. Sayers, M.J. Kachanov, Geophys. Res. 100 (1995) ] V.V. Kudinov, Plasma Sprayed Coatings, Nauka, Moscow (in Russian), 1977, pp ] S.-H. Leigh, C. Berndt, Acta Mater. 47 (1999) ] C.J. Li, A. Ohmori, R. McPherson, Proceedings of the International Conference on AUSTCERAM-92, Melburn, Australia, 1992, p ] F. Kroupa, Kovove Mater. 33 (1995) 418 (in Czech). 19] Y.C. Tsui, C. Doyle, T.W. Clyne, Biomaterials 19 (1998) ] S. Parthasarathi, B.R. Tittmann, K. Sampath, E.J. Onesto, J. Therm. Spray Technol. 4 (1995) ] R. McPherson, Thin Solid Films 112 (1984) ] I. Doltsinis, J. Harding, M. Marchese, Arch. Comp. Meth. Eng. 5 (1998) ] I. Sevostianov, M. Kachanov, J. Therm. Spray Technol. 9 (2001) ] I. Sevostianov, M. Bogarapu, P. Tabakov, Int. J. Fract. 115 (2002) ] I. Sevostianov, V.E. Verijenko, M. Kachanov, Composites B 33 (2002) ] I. Sevostianov, J. Kováčik, F. Simančík, Int. J. Fract. 114 (2002) ] R.P. Ingel, D. Lewis III, J. Am. Ceram. Soc. 69 (1986) ] A.G. Avery, Meas. Sci. Tehnol. 13 (2002) ] A. Mogro-Campero, C.A. Johnson, P.J. Bednarczyk, R.B. Dinwiddie, H. Wang, Surf. Coat. Technol (1997) ] J.B. Wachtman, Mechanical Properties of Ceramics, John Wiley, New York, 1996, p ] S. Raghavan, H. Wang, R.B. Dinwiddie, W.D. Porter, M.J. Mayo, Scr. Mater. 39 (1998) ] K.S. Ravichandran, K. An, R.E. Dutton, S.L. Semiatin, J. Am. Ceram. Soc. 82 (1999) 673.