Introduction

Thermal barrier coatings (TBCs) made of ZrO2 ceramics stabilized by Y2O3 (YSZ), produced by atmospheric plasma spraying (APS) or electron-beam physical vapor deposition (EB-PVD), are widely used for gas turbines to improve their energy conversion efficiency by enabling higher gas temperatures. Compared with bulk materials, thermally sprayed coatings exhibit much lower thermal conductivity, i.e., 1/3 to 1/5 that of bulk material, as a result of a lamellar microstructure with much more limited bonding area between lamellae, which is further divided by intersplat pores (Ref 1–3). Figure 1 shows three types of pore structure which are widely abundant in a thermally sprayed ceramic coating. Except for intersplat pores, intrasplat cracks and globular voids are also inevitably formed during splat cooling from the melting temperature to room temperature. The existence of intrasplat cracks is usually due to thermal stresses and tensile quenching stress relaxation.

Fig. 1
figure 1

Microstructure of thermally sprayed YSZ coating. (a) Three types of typical pores in fractured cross-section. (b) Plan view of intrasplat cracks in a splat

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Using a laser flash technique, Kulkarni et al. determined that the thermal conductivity of a coating is inversely related to its porosity (Ref 4). However, as pointed out in that article (Ref 4), a coating with a large number of interlamellar pores may present a lower thermal conductivity despite having a lower porosity, and a coating with a large number of splat interfaces leads to a reduction in the thermal conductivity. This indicates that the thermal conductivity may also be affected by the pore geometry as well as the porosity (Ref 4). Therefore, the porosity alone may not be a proper parameter to characterize the thermal barrier ability of a coating, since, as pointed out by Bertrand et al., the thermal conductivity cannot be well correlated with the porosity (Ref 5). In the case that the macroporosity dominates the porous structure of the coating, the thermal conductivity can be correlated to the total porosity. However, in the case where a microcrack network dominates the porous structure of the coating, the crack orientation plays an important role in lowering the thermal conductivity (Ref 5).

In fact, the thermal conductivity of thermally sprayed ceramic coatings is usually explained based on the concept of thermal contact resistance (TCR) and well correlated to microstructural aspects (Ref 3, 6) such as the intersplat bonding ratio, which is mainly associated with the intersplat pores. McPherson established a classical microstructure model for thermally sprayed coatings and reported a quantitative relationship between the thermal conductivity and microstructural parameters (Ref 3, 6). It was further reported that horizontal cracks, including intersplat pores and branching cracks, are particularly effective in reducing the thermal conductivity, as the interfaces formed by such cracks are perpendicular to the primary heat flux (Ref 7).

As mentioned above, intrasplat cracks are formed in a thermally sprayed ceramic coating besides intersplat voids. Since the intrasplat cracks are mainly oriented in the through-thickness direction, which is often parallel to the thermal diffusion direction, their effect on the thermal conductivity of coatings is usually not taken into account. Recently, an investigation into sintering of ceramic coatings argued that healing of intrasplat cracks is accompanied by an increase in the thermal conductivity of YSZ coatings in the early stage of thermal exposure (Ref 8). This fact suggests an influence of intrasplat cracks on the thermal conductivity of plasma-sprayed YSZ coatings. However, a quantitative relation between intrasplat crack features and coating thermal conductivity has not been reported yet.

In this study, a microstructural model including intrasplat cracks and intersplat voids is proposed, and the influence of intrasplat cracks on the thermal conductivity is investigated using a Monte Carlo method and computing tools to understand the thermal conduction behavior of thermally sprayed coatings.

Framework of the Model with Intrasplat Cracks

General Background

Figure 2 shows the typical idealized model of the microstructure of thermally sprayed ceramic coatings, which are composed of individual lamellar splats with thickness of 1–3 μm. The bonding between these splats is limited, the bonding ratio generally being less than 32 % for plasma-sprayed ceramic coatings without special substrate preheating (Ref 1, 6, 9, 10). Each splat, with diameter of ~100 μm, is divided into many small lamellar units by intrasplat cracks, and the size of these lamellar units is 5–15 μm. When thermal convection and radiation between lamellae are neglected, the thermal conductivity in the through-thickness direction can be determined based on the TCR concept with the localized contact (bonding) area model, which was used to describe the electric contact resistance (Ref 11). The TCR of a pair of contacted units (Fig. 2a), R, is given by (Ref 11)

$$ R = \frac{1}{2a\lambda } $$
(1)

where a is the radius of the contact area and λ is the conductivity of the bulk material. This equation is denoted the conventional thermal contact theory or conventional theory in this study.

Fig. 2
figure 2

Idealized model of the microstructure of thermally sprayed ceramic coatings. (a) Plan view of lamellar interface and (b) cross-sectional view of the coating. 2a refers to the diameter of the splat bonding area (Ref 6)

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It is worth addressing the definition of thermal resistance, since there are different definitions for this concept in different engineering fields. In this study, the “thermal resistance” is a fundamental quantity to describe the resistance to heat conduction of a subject with three-dimensional size. It can be expressed as (Ref 12–14)

$$ R = \frac{\delta }{\lambda A} $$
(2)

where A is the area normal to the heat flow and δ is the distance parallel to the heat flow.

The normalized thermal resistance of one coating unit (being equal to the normalized thermal resistivity of the coating) is

$$ \frac{{R_{\text{C}} }}{{R_{0} }} = \frac{1/(2a\lambda )}{{\delta /(\lambda \cdot(\pi a^{2} /\alpha ))}} = \frac{{{{\pi}}a}}{2\alpha \delta } $$
(3)

where R C and R 0 are the TCR and the bulk thermal resistance of one unit, respectively, α is the intersplat bond ratio (taken as 10–40 %), and δ is the splat thickness (taken as 2 μm in this study).

Model Structure

Since this study focuses on the influence of intrasplat cracks on the thermal conductivity of the coating, thermal convection and radiation between lamellae are neglected. Figure 3 shows schematically the idealized structural model including intrasplat cracks and localized bonding areas for a thermally sprayed ceramic coating. The whole coating can be regarded as a repetitive stacking of a three-layered structure, including a typical unit layer (red, thick), an upper unit layer (black, middle), and a lower unit layer (green, thin). The thickness of each layer is consistent with that of a splat. Each layer is divided into small units by intrasplat cracks. The unit is assumed to be of square morphology with typical size of 10 × 10 μm2. Each unit is bonded to the lower layer by a circular bonding area, namely T/(L 1, L 2, L 3, L 4) (shown in red in Fig. 3). The bonding areas are assumed to be randomly distributed within the lower surface of the unit. The number of bonded areas at the interface ranges from 0 to a specific number (U 2/T, U 3/T, U 4/T, black) for a given bonding ratio, since these bonding areas are randomly located within the bottom surface of the units in the upper layer (i.e., the bottom surface of U 2–U 4). The U i /T bonding areas (for example, U 2/T and U 3/T, black) on the top surface of the typical unit do not always have a completely round shape but are often segments separated by intrasplat cracks around this typical unit.

Fig. 3
figure 3

Idealized structure model including intrasplat cracks and localized bonding region for thermally sprayed ceramic coatings

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Such a random distribution of bonding areas can be determined using numerical calculation tools. The thermal resistance of each unit is the sum of the thermal resistances of the two surfaces (bonded to the upper and lower layers) in series connection. The bonding areas on each surface are treated as in parallel connection. Therefore, the next step is to calculate the thermal resistance of each bonding segment area, which may be complete or a fraction separated by intrasplat cracks.

Compared with the ideal thermal flow line model shown in Fig. 4, some of the thermal flow paths are cut by intrasplat cracks in the present model. This leads to an increase of the effective thermal resistance surrounding the bonding area. Figure 5 shows schematically a qualitative demonstration of the effect of the cutting position of intrasplat cracks on the thermal conductivity of a bonding area. When an intrasplat crack approaches the periphery of a bonded area, the thermal resistance (R) increases due to the blocking effect by greater cutting of the thermal flow paths. However, when the intrasplat crack is located at the center of a bonded area, the thermal resistance is not influenced by the intrasplat crack due to the cylindrical symmetry of the model.

Fig. 4
figure 4

Conventional model of thermal conductivity and the model with intrasplat cracks

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Fig. 5
figure 5

Effect of the position of intrasplat cracks on the thermal resistance

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Based on the Monte Carlo method, the patterns of the bonding areas on each unit surface can be divided into six basic cases (Fig. 6a–c, e, f, h) and two composite cases (Fig. 6d, g). The thermal resistance of each case can be calculated by different formulae.

Fig. 6
figure 6

Schematic diagram for the calculation of the thermal resistance for different basic cases and the composite cases. (a–c) Three basic cases corresponding to three critical positions of the whole circular bonding area within the unit. (d) The general composite case derived from the three basic cases of (a–c). (e, f) Two basic cases corresponding to two critical positions of a bonding area cut by only one crack of the unit. (g) The general composite case derived from the two basic cases of (e, f). (h) The basic case of a bonding area cut by two cracks perpendicular to each other

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When the bonding area is completely within a unit, three cases corresponding to three particular positions can be identified, as shown schematically in Fig. 6a–c. The case shown in Fig. 6a can be treated as the ideal thermal contact, since the thermal resistance of the bonding area located in the center of the unit is determined to have the minimum value due to little cutting of heat flow paths by the cracks (Fig. 6a):

$$ R_{{\text{Case}} {\text{-}} {\text{a}}} = \frac{1}{4a\lambda } $$
(4)

As the bonding area changes its position within the unit surface and is randomly located at another position, the thermal resistance of this area increases due to the cutting of heat flow paths by intrasplat cracks. When the circular bonding area is tangential to the crack at the middle of one crack side (edge) (Fig. 6b), the thermal resistance is doubled compared with the case where there are two circular bonding areas on both sides of the crack. However, in the above-mentioned case, the two circular bonding areas make up an irregular shape. Therefore, a shape factor [f(x) in Eq 5] should be introduced to transform the irregular bonding shape into an equivalent circular area according to electrical contact theory (Ref 11). It is assumed that the equivalent circular area and the irregular bonding shape have the same area. Consequently, the thermal resistance for the case of Fig. 6b can be given by

$$ R_{{\text{Case}} {\text{-}} {\text{b}}} = 2\frac{1}{{4a_{\text{e}} \lambda }}f\left( x \right) = \frac{1}{2\sqrt 2 a\lambda }f\left( x \right) $$
(5)

where f(x) is the shape factor, which can be obtained from a look-up table in the literature (Ref 11), and a e refers here to the radius of the equivalent circle composed of the two circles in Fig. 6b. x is the square root of the aspect ratio of the irregular shape composed of two circles.

When the bonding area is tangential to two perpendicular cracks (Fig. 6c), the thermal resistance is four times larger than the case where there are four circular bonding areas in the four quadrants, i.e.,

$$ R_{{\text{Case}} {\text{-}} {\text{c}}} = 4\frac{1}{{4a_{\text{e}} \lambda }}f\left( x \right) = \frac{1}{2a\lambda }f(x) $$
(6)

where a e refers here to the radius of the equivalent circle having the same area as the total area of the four circles in Fig. 6c.

The general case in Fig. 6d can be regarded as a composite case based on the three basic cases shown in Fig. 6a–c. The thermal resistance can be determined from the three basic cases using a linear interpolation:

$$ R_{{\text{Case}} {\text{-}} {\text{d}}} = R_{{\text{Case}} {\text{-}}{\text{a}}} + \frac{{L_{{{\text{O}}_{1} {\text{A}}}} }}{{L_{{{\text{O}}_{1} {\text{O}}_{2} }} }}\left( {R_{{\text{Case}}{\text{-}} {\text{b}}} - R_{{\text{Case}} {\text{-}} {\text{a}}} } \right) + \frac{{L_{{{\text{O}}_{2} {\text{B}}}} }}{{L_{{{\text{O}}_{2} {\text{O}}_{3} }} }}\left( {R_{{\text{Case}} {\text{-}} {\text{c}}} - R_{{\text{Case}} {\text{-}} {\text{b}}} } \right) $$
(7)

where \( L_{{{\text{AO}}_{1} }} \), \( L_{{{\text{O}}_{1} {\text{O}}_{2} }} \), \( L_{\text{AO}} \), and \( L_{{{\text{O}}_{2} {\text{O}}_{3} }} \) are the distances between the corresponding points in Fig. 6d.

For the case where the bonding area is cut by only one crack, the thermal resistance can also be determined by using the shape factor f(x). Figure 6e and f show the two basic cases corresponding to two critical positions of the bonding area being cut by only one crack (the edge of the unit). The two basic cases in Fig. 6e and f are actually similar to the cases in Fig. 6b and c, respectively. The thermal resistance can be determined as

$$ R_{{\text{Case}} {\text{-}} {\text{e}}} = 2\frac{1}{{4\lambda \sqrt {{\raise0.7ex\hbox{${2S}$} \!\mathord{\left/ {\vphantom {{2S} \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} }}f\left( x \right) = \frac{1}{{2\lambda \sqrt {{\raise0.7ex\hbox{${2S}$} \!\mathord{\left/ {\vphantom {{2S} \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} }}f\left( x \right) $$
(8)
$$ R_{{\text{Case}} {\text{-}} {\text{f}}} = 4\frac{1}{{4\lambda \sqrt {{\raise0.7ex\hbox{${4S}$} \!\mathord{\left/ {\vphantom {{4S} \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} }}f\left( x \right) = \frac{1}{{2\lambda \sqrt {{\raise0.7ex\hbox{$S$} \!\mathord{\left/ {\vphantom {S \pi }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\pi $}}} }}f\left( x \right) $$
(9)

where S is the area of the bonding segment in each case.

The general case in Fig. 6g can be regarded as a composite case based on the two basic cases shown in Fig. 6e and f. The thermal resistance can be determined from the two basic cases using a linear interpolation:

$$ R_{{\text{Case}} {\text{-}} {\text{g}}} = R_{{\text{Case}} {\text{-}} {\text{e}}} + \left( {R_{{\text{Case}} {\text{-}} {\text{f}}} - R_{{\text{Case}} {\text{-}} {\text{e}}} } \right)\frac{y}{L} $$
(10)

where y is the distance between the axis of symmetry of the bonding area and the side center, and L is the maximum possible value of y.

For the last basic case in Fig. 6h, the bonding area is effectively cut by two perpendicular cracks (the edges of the unit). Similarly, the thermal resistance can be determined by using the same equation as for the case in Fig. 6f (i.e., Eq 9).

To form a dimensionless normalized thermal resistance, the results of thermal resistance obtained using the present model are divided by the thermal resistance of the conventional theory (corresponding to Eq 1).

Results and Discussion

Distribution of Bonding Segment Areas

Figure 7 shows the normalized bonding segment area (the ratio of the bonding segment area to the whole round-shaped bond area) distribution of U i /T for units with different intersplat bonding ratio. Except for some uncut areas, most of these bonding domains are divided into small bonding segments. With increasing intersplat bonding ratio, the number of uncut areas decreases, from about 40 % at intersplat bonding ratio of 10 % to less than 7 % at bonding ratio of 40 %. This fact indicates that, with increasing bonding ratio, more cracks appear in the bonding area and the bonding areas in each unit have a greater possibility of being cut. It can be noted that, even though quite a few of the bonding areas are not cut by intrasplat cracks, these cracks still change the effective thermal resistance because of the disruption of heat flow lines. With increasing bonding ratio, the distribution of normalized bonding segment area tends to shift to the left, indicating that more cut bonding segments tend to be smaller.

Fig. 7
figure 7

Distribution of normalized bonding segment area of (U 1-U 4)/T for different intersplat bonding ratios: (a) 10 %, (b) 20 %, (c) 30 %, (d) 40 %

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Specific Thermal Resistance of Bonding Segments

Figure 8 shows the distribution of the specific thermal resistance of the bonding segments on the top surface. The specific thermal resistance refers to the thermal resistance of the bonding segment divided by the area of this bonding segment. (Note that the specific thermal resistance is apparently similar to, but differs from, the unit thermal resistance.) Since the thermal resistance depends on the bonding segment area, this figure shows the thermal resistance of a unit of area. Compared with the specific thermal resistance calculated according to Eq 4, most areas of the bonding segments present larger specific thermal resistance. This is due to the deviation of bonding segments from the unit center, resulting in the cutting of some heat flow paths by intrasplat cracks. Besides the deviation in the location of the bonding area, its position at an edge or corner can also increase the thermal resistance of the bonding segments. In addition, the specific thermal resistance distribution in Fig. 8 shows a right-shift tendency with increasing bonding ratio. This result can be interpreted as an effect of the aspect ratio. As shown by the left shift in Fig. 7, more bonding segments cut by intrasplat cracks are due to the position of the intersplat bondings distributed at the segment edge. Therefore, an increase of the aspect ratio eventually contributes to an increase of the specific thermal resistance of the bonding segments.

Fig. 8
figure 8

Specific thermal resistance of randomly distributed bonding segments on the unit top surface for coatings with different intersplat bonding ratios: (a) 10 %, (b) 20 %, (c) 30 %, (d) 40 %

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Normalized Thermal Resistance of Unit Top Surface

Figure 9 indicates the normalized thermal resistance of the top surface of units, showing the ratio of the resistance according to this model to that using the conventional model without considering the effect of intrasplat cracks. Values less than 1.0 indicate cases where the thermal resistance is lower than that estimated by the conventional model. The presence of normalized resistance values less than 1.0 is attributed to the fact that the total bonding segment area distributed in certain units is much larger than one circular bonding area, contributing to a higher thermal conductivity than for a circular bonding area. This phenomenon becomes more significant as the bonding ratio decreases. Moreover, most of the areas cut by intrasplat cracks lead to an increase in the thermal resistance, and this effect becomes more significant with increasing bonding ratio. The presence of units with infinitely great thermal resistance is attributed to the lack of bonding areas for a unit due to the random distribution of bonding areas. With increasing bonding ratio, such no-contact areas decrease dramatically, and when the bonding ratio is 40 %, such areas are hardly observed. The distribution also becomes more concentrated as the bonding ratio increases.

Fig. 9
figure 9

Normalized thermal resistance of unit top surface for different intersplat bonding ratios: (a) 10 %, (b) 20 %, (c) 30 %, (d) 40 %

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Thermal Resistance of the Coating Unit

Figure 10 shows the distribution of the normalized thermal conductance of one unit, taking account of the combined effect of the heat flows through the upper and lower surfaces. Since the thermal conductance is the reciprocal of the thermal resistance, the normalized thermal conductance was obtained by dividing the thermal resistance of one ideal unit (in which the bonding area is located in the unit center) by the thermal resistance of each unit. Units with normalized thermal conductance above 1.0 account for only a small fraction. As the bonding ratio increases, the proportion of units with normalized thermal conductance higher than one decreases. A unit with normalized conductance of zero corresponds to a unit without bonding for the whole interface with the top surface of the unit, resulting in complete blocking of heat flux.

Fig. 10
figure 10

Normalized thermal conductance of a unit considering the combined effect of the upper and lower surfaces

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Thermal Resistivity of the Coating

These results indicate that the presence of intrasplat cracks significantly influences the thermal resistivity (the reciprocal of the thermal conductivity) of thermally sprayed ceramic coatings, since most units present a normalized thermal conductivity less than one. A comparison of the results estimated using the present model with the results estimated using the conventional model is shown in Fig. 11a. The normalized thermal resistivity in Fig. 11a refers to the ratio of the thermal resistivity of the coating to that of the bulk material. It can be found that, based on the present model, the thermal resistivity is much higher than estimated using the conventional theory that does not take account of the effect of intrasplat cracks. To examine further the influence of intrasplat cracks on the thermal resistivity of a ceramic coating based on the present model, a thermal resistivity index is introduced, being defined as the ratio of the thermal resistivity obtained by this model to that obtained from the conventional theory shown by Eq 3. As shown in Fig. 11b, this index increases from 1.42 to 1.57 with a decrease in the bonding ratio from 40 to 10 %. It is clear that the intrasplat cracks do have a significant impact on the thermal resistivity of the thermally sprayed ceramic coating.

Fig. 11
figure 11

Normalized thermal resistivity of the whole coating. (a) Effect of bonding ratio on the normalized thermal resistivity. (b) Thermal resistivity index showing the ratio of the thermal resistivity obtained using this model to that obtained from the conventional theory

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Correlation with Experimental Data

Intrasplat cracks are inevitably present in thermally sprayed ceramic coatings. However, it is difficult to directly reveal experimentally the effect of intrasplat cracks on the thermal conductivity of the coating, since such control of intrasplat cracks requires further work that demands a great deal of fundamental investigation on splat formation. However, it was found that cracks in thermally sprayed ceramic coatings on hot section parts will heal during high-temperature exposure in service (Ref 8, 15–17). It has been widely reported that the thermal conductivity of plasma-sprayed YSZ coatings increases with high-temperature exposure (Ref 18–22). The reason is believed to be due to an increase of lamellar bonding resulting from healing of intersplat pores. For those coatings sintered at high temperature (for example, >1,300 °C), it was consistently observed that both healing of intersplat pores and increase of thermal conductivity occurred (Ref 18). Thus, apparently the latter can be attributed to the former. However, although as reported by Trice et al. the thermal conductivity of plasma-sprayed 7YSZ coatings increased by about 40 % after sintering at 1,000 °C, significant healing of intersplat pores was not observed by transmission electron microscopy (TEM) (Ref 8). On the other hand, Ilavsky et al. reported that healing of intrasplat cracks occurred at a lower temperature than intersplat pores based on pore structure characterization by the small-angle neutron scattering method (Ref 15). Upon sintering at 1,000 °C, the void specific surface area of intrasplat pores decreased significantly, whereas that of intersplat pores changed little (Ref 15). Therefore, the increase of the thermal conductivity of YSZ coatings subjected to sintering at a relatively low temperature could be attributed to healing of intrasplat cracks. Considering the thermal conductivity index of 1.42–1.57 (for bonding ratio of 40 to 10 %) obtained in this study (Fig. 11b), the fact that healing of intrasplat cracks led to an increase of coating thermal conductivity of about 40 % as reported by Trice et al. (Ref 8) is well consistent with the result estimated using the present model.

Based on the above-mentioned results, it can be concluded that intrasplat cracks also play an important role in determining the thermal conductivity of thermally sprayed ceramic coatings. Besides the effect of healing of intersplat pores, the effect of healing of intrasplat cracks on the thermal conductivity of ceramic coatings should also be taken into account during investigation of the sintering behavior of plasma-sprayed ceramic coatings. For as-sprayed coatings, the thermal conductivity could also be tailored by controlling intrasplat cracks. Therefore, further optimization of intrasplat cracks would benefit the development of advanced TBCs with both higher thermal insulation performance and better sintering resistance.

Conclusions

A microstructure model including intrasplat cracks is proposed in this study to reveal the influence of intrasplat cracks on the thermal conductivity of thermally sprayed ceramic coatings. The results of the present model show that intrasplat cracks have a significant influence on the thermal resistivity of plasma-sprayed ceramic coatings. The deviation of each cut bonding segment from the center of the unit leads to an increase of the thermal resistance of the unit by cutting off some heat flow paths within individual splats. Using Monte Carlo simulation, it was found that the normalized thermal conductance of most units is less than one, indicating an increase of the thermal resistance compared to the thermal contact resistance resulting from intersplat pores. Using the thermal resistivity index to represent the effect of intrasplat cracks, it was revealed that the index increased from 1.42 to 1.57 with decrease of the lamellar bonding ratio from 40 to 10 %. The estimated index agrees well with the thermal conductivity change of YSZ coatings due to healing of intrasplat cracks during sintering at low temperature. Therefore, the present model qualitatively explains the contributions of intersplat pores and intrasplat cracks. Therefore, further optimization of intrasplat cracks would benefit the development of advanced TBCs with both higher thermal insulation performance and better sintering resistance.