Analysis of Microstructural Effects on the Thermal Conductivity of Alumina-Spinel Refractories Compared to Alumina Ceramics

Abstract

Alumina-spinel refractory bricks, composed of 82 wt.% alumina and 18 wt.% MgAl₂O₄ spinel phases, are used in steel ladles due to their ability to resist chemical attack and thermal shock. Thermal shock resistance is determined, in part, by the thermal conductivity of the material. Thermal conductivity measurements for alumina-spinel refractory, three model alumina ceramics, and single crystal sapphire were made with the laser-flash technique from 20 °C to 1000 °C. At room temperature, these gave 6.5 W m⁻¹ K⁻¹ for the refractory, 5.8 to 22 W m⁻¹ K⁻¹ for the alumina ceramics, and 36 W m⁻¹ K⁻¹ for sapphire, despite all materials containing >81 vol.% of alumina. The differences are explained by the roles of porosity, grain boundary thermal resistance, and the spinel phase (refractory). In order to estimate the thermal conductivity of alumina grains in each material, these microstructural effects are modelled with Landauer’s relation for porosity and thermal resistors in series for grains combined with grain boundaries. For two alumina ceramics, the grains yielded similar behaviour to the single crystal. By taking the spinel phase into account with a two-phase mixture relation, the alumina grains in the refractory were estimated with a value of 31 ± 2 W m⁻¹ K⁻¹, close to sapphire.

1. Introduction

The fabrication of ceramics, metals, and glasses involves high-temperature processes where refractory materials are crucial. As examples, they are used as thermal insulation or simply to contain the hot molten metal or glass. For such functions, these materials have been developed over time to achieve outstanding properties, especially regarding high-temperature resistance, mechanical and thermal properties, resistance to thermal shock, and also resistance to chemical corrosion. Today, research on refractory materials remains essential to enable them to play a key role in current societal challenges of sustainable development, such as energy efficiency and decarbonization [1].

A significant amount of the refractory market is devoted to the steelmaking industry [2,3]. The vessel used for the secondary metallurgy, called the steel ladle, is the highest consumer of refractories of the entire steel process due to the large energy input required to maintain the liquid steel above the casting temperature (T_melt > 1500 °C) [4,5]. It is in this context that the present work was made as part of the European Union HORIZON project ATHOR (Advanced THermomechanical multiscale mOdelling of Refractory linings), whose purpose was to investigate the thermomechanical behaviour of refractory linings used for steel ladles in the steelmaking industry. The consortium involved 7 European universities and 8 industrial partners [6,7]. The present study focuses on alumina-spinel bricks, which are used in the middle zone of the working lining. This region is in contact with the hot liquid steel and is also subjected to thermal shocks due to the cycling processes of filling and tapping [8,9]. The spinel phase of MgAl₂O₄ offers a good combination of physical and chemical properties, such as high refractoriness (the melting point of the spinel is around 2135 °C), high mechanical strength, and high resistance to chemical attack, which make alumina-spinel bricks suitable for contact with hot liquid metal [10]. This material is also studied for optical, nuclear, and gas sensor applications [11,12,13].

The ability of the refractory material to resist thermal shock depends on several physical characteristics: Young’s modulus, the coefficient of thermal expansion, and also the thermal conductivity. A higher value of thermal conductivity enables the material to dissipate heat more quickly and consequently minimize any thermal gradients inside the brick, reducing potential damage [14,15]. This emphasizes the importance of the thermal conductivity with respect to the application of alumina-spinel bricks in the working lining. In fact, both thermal conductivity (λ) and heat capacity (C_p) are key physical properties in the performance of a refractory material used in a high-temperature application. If a sudden temperature change (thermal shock) occurs in the external boundary conditions, the thermal response time and transient temperature distribution within the refractory are, in addition to conductivity, determined by the heat capacity. In the case of refractories and most ceramics, these thermal properties depend essentially on lattice vibrations. Variation of vibrational heat capacity with temperature is described by the well-known Debye model [16]. With respect to thermal conductivity, heat is carried by lattice vibrations across the solid when it is subjected to a thermal gradient. An increase in mutual interference between the lattice vibrations at higher temperatures explains why the thermal conductivity decreases through shortening of the mean free path. However, in contrast to the heat capacity per unit mass (specific heat), the thermal conductivity can also be strongly modulated by the microstructure. The present study is devoted to the thermal conductivity of single-phase alumina ceramics and alumina-spinel refractories with a strong focus on the influence of the microstructure. Three aspects are examined in particular: the effect of pore volume fraction, the effect of grain boundary/interface thermal resistance, and the situation where two solid phases, alumina and spinel, are present.

Many papers in the literature discuss how the increase of the pore volume fraction reduces the thermal conductivity due to the lower thermal conductivity value of the gas, usually air, compared to that of the solid phase [17,18,19,20,21]. Evidently, porous refractories, characterized by very low values of thermal conductivity, are suitable as thermal insulation [22]. A number of reviews/papers assess the ability of different models to predict the thermal conductivity of a porous solid [23,24,25]. Providing the choice of the model is well adapted to the spatial distribution of the pores (open or closed, shape) in the solid and the solid-phase thermal conductivity is known, accurate agreement with experimental results for single-phase ceramics can be achieved, even with fairly simple analytical relations such as the Maxwell–Eucken relation [26] or Landauer’s relation [27]. For alumina-silica and alumina-calcia refractories, Akiyoshi et al. developed a general expression to predict the thermal conductivity empirically as a function of chemical composition, density, and porosity [28].

The approach above can be exploited in a straightforward manner if the solid-phase thermal conductivity remains constant with variation of porosity. However, this is not always the case when significant changes in grain size occur. Two effects can be identified. When the heat path is crossed by a grain boundary, this acts as an additional localized thermal resistance (or Kapitza resistance) due to the increased atomic disorder in the interface region [29]. Smaller grain size increases the number of interfaces crossing the heat path per unit length, which increases the total grain boundary thermal resistance. A second effect concerns very small grain sizes (<300 nm), where even the crystallite boundaries parallel to the heat flow can play a role. Because of the finite size of the crystal lattice in this situation, the low-frequency long wavelength vibrations (or phonons) moving in all directions are filtered out by the grain boundaries and do not contribute to the intrinsic grain thermal conductivity. Both of these effects result in a lower thermal conductivity for a small-grain ceramic in comparison to a large-grain ceramic or single crystal. In previous work on small-grain porous alumina ceramics, a method to separate the contributions of localized grain boundary thermal resistance from the grain conductivity was proposed [30].

Finally, if the roles of porosity and of grain boundaries/interfaces can be either accounted for or neglected, the next step is to examine the thermal conductivity of a material containing two solid phases. Though numerical modelling may be considered, a first approach is similar to that used for porosity, with analytical relations chosen with respect to the spatial distribution of the two-phase mixture. For example, close agreement has been achieved between experimental values of thermal conductivity for alumina inclusions in a glass matrix and predictions with the Maxwell–Eucken relation [31].

Most refractory materials are complex, with less refined starting materials, yielding two or more solid phases, with strong variation in the grain sizes as well as porosity. Despite that, the purpose of the present paper is to apply an approach developed successfully for simpler ceramic materials (single-phase, with fairly narrow ranges of grain and pore sizes) to an industrial refractory, namely alumina-spinel bricks, in order to elucidate the different contributions to the overall thermal conductivity. The analysis will be made using analytical relations that have been simplified by attributing a value of zero to the pore thermal conductivity. Contributions of radiation heat transfer across large pores (>0.1 mm) and within semi-transparent polycrystalline ceramics or even single crystals to the effective thermal conductivity have been neglected. This is because of the smaller pore size, the presence of grain boundaries in these materials, which attenuate radiation transfer through scattering, and the temperature range of measurements [32,33]. By accounting for the pore volume fraction, grain boundary/interface thermal resistance, and in the case of alumina-spinel refractory (a mixture of two solid phases), values for the intrinsic conductivity of the alumina grains in each material have been estimated. Results will be compared for single crystal sapphire, model single-phase alumina ceramics, and the alumina-spinel refractory brick.

2. Materials and Methods

2.1. Materials

The study was devoted to alumina-spinel refractory bricks, provided by RHI Magnesita GmbH (Vienna, Austria), and single-phase alumina ceramics. In terms of stoichiometric oxides, according to XRF measurements, the alumina-spinel bricks contain 94 wt.% of Al₂O₃, 5 wt.% of MgO, and 1 wt.% of other oxides [8,9,34,35,36,37]. When magnesia and alumina are heated together during brick fabrication, they react to form an intermediate spinel phase (MgAl₂O₄). Assuming stoichiometric spinel for convenience in the calculations, this corresponds to 82 wt.% of alumina phase and 18 wt.% of spinel phase, or equivalently 81 vol.% alumina and 19 vol.% spinel based on the respective densities of each phase. Working on the same refractory bricks, Samadi found similar values with XRD analysis, obtaining 81.4 wt.% of alumina and 18.6 wt.% of spinel [9]. The SEM micrograph in Figure 1 shows the presence of alumina aggregates of different sizes, a fine matrix of spinel phase, and pores both in the grains and in the matrix.

In order to obtain thermal conductivity values for the spinel phase, specific samples with a high spinel content (87 vol.%) were made at RWTH Aachen University from an alumina-rich magnesium aluminate spinel powder denoted AR78. This powder is one of the raw materials typically used for the fabrication of alumina-spinel bricks. After uniaxial pressing of the dry AR78 powder, sintering at 1600 °C for 6 h yielded specimens with dimensions of 10 mm in diameter and 2 mm in thickness.

In the current work, the analysis was made by comparing the behaviour of alumina-spinel bricks (94 wt.% alumina) with the behaviour of four alumina model materials (>99 wt.% alumina) with differences in the microstructural characteristics.

Table 1. Summary of the porosity, average grain size, and sintering temperature of five alumina-based materials: one refractory material (Alumina-Spinel) and four model materials.

Sample Designation	Materials	Total Porosity (%)	Average Grain Size or Range (μm)	Sintering Temperature (°C)
Alumina-Spinel	Refractories	19	-	-
Sapphire	Model materials	0	Single crystal	-
Alumina TM-DA		3	0.15–0.5	1200
Alumina AKP30-1450		20	0.45	1450
Alumina AKP30-1300		38	0.30	1300

summarizes the major characteristics of each investigated material.

Sample densities were evaluated by dry mass measurements and an estimate for disc volume based on average diameter and thickness. The pore volume fraction (ν_p), corresponding to total porosity, is then given by the following equation:

$${\nu }_p=1-{\displaystyle \frac{\rho }{{\rho }_0}}$$

(1)

where $\rho$ is the sample (apparent) density and ${\rho }_0$ is the theoretical density. In the porous samples, an average grain diameter (ϕ) was estimated using the software measurement bar on a population of grains (>50 grains) observed by scanning electron microscopy. The number of grain boundaries or interfaces per unit length (n) is then given by the following equation:

$$n={\displaystyle \frac{1}{\varphi }}$$

(2)

The Alumina TM-DA samples were made, with powder supplied by TAIMEI chemicals (Tokyo, Japan), at NITech (Nagoya, Japan) using the pulse electric current sintering method (PECS) [38]. The samples were sintered in a furnace at 1200 °C for 5 min with a heating rate of 100 K/min and subjected to a uniaxial pressure of 100 MPa, yielding an almost fully dense material with limited grain growth. Figure 2 shows an example of the microstructure of Alumina TM-DA samples. It is possible to observe the presence of small grains (0.15 μm to 0.5 μm) homogeneously distributed without much porosity.

The Alumina AKP30-1300 (Figure 3a) and Alumina AKP30-1450 (Figure 3b) samples were prepared at IRCER by uniaxial pressing of the dry powder, supplied by Sumitomo Chemicals (Tokyo, Japan) and denoted AKP30, and then fired respectively at 1300 °C and 1450 °C for 6 min [30].

2.2. Methods

The thermal conductivity values were evaluated using the laser-flash method [39,40]. This technique, developed by Parker et al. in 1961 [41], measures the thermal diffusivity of a solid by sending a short-duration light energy pulse to impact on the front face of a sample held at a given temperature. Then, an infrared detector records the temperature evolution on the back face as a function of time. The evaluation of the thermal diffusivity (α) from the temperature–time behaviour can be performed with different models, such as those of Parker [41], Cape-Lehman [42], Degiovanni [43], or Mehling [44]. In this study, we have mainly used the Cape-Lehman model, which takes into account heat losses through the sample boundaries during experimental tests. However, for high-temperature measurements, when a direct radiative heat transfer between sample faces was observed, the Mehling analysis was used to account for this signal perturbation.

The measurements were carried out up to 1273 K with a heating rate of 5 K/min in an argon atmosphere using a NETZSCH LFA 427 Laser-Flash device (Selb, Germany). For each temperature, the values correspond to an average of three consecutive measurements, and the sample thickness value was adjusted to incorporate the effect of thermal expansion. The tests were repeated twice using two different samples, which exhibited a difference in thermal conductivity of less than 1%. The accuracy of this method is taken to be ±3% [39,40].

Once the thermal diffusivity is evaluated, the thermal conductivity (λ) can then be calculated using the following equation:

$$\lambda =\alpha ·\rho ·C_p$$

(3)

where λ is the thermal conductivity [W m⁻¹ K⁻¹], α is the thermal diffusivity [m² s⁻¹], $\rho$ is the bulk density [kg m⁻³], and C_p is the specific heat capacity [J kg⁻¹ K⁻¹]. The bulk density value was also adjusted to take thermal expansion of the material into account.

In contrast to thermal conductivity, heat capacity per unit mass does not depend on microstructural characteristics such as porosity or grain size [45]. For this reason, heat capacity values from literature were used for all alumina samples [46]. For the Alumina-Spinel sample, values were calculated using the rule of mixtures based on the composition given in Section 2.1 and heat capacity data from literature for each simple oxide [46]:

$$C_p=\sum _i{m}_i{C}_{pi}$$

(4)

where m_i is the percentage in mass and C_pi is the specific heat of each component i [47]. In earlier work on raw and calcined clays, we have shown for stable materials that Equation (4) yields results within ±5% of high-quality measurements, providing the chemical composition is known accurately [48]. Richet achieved an even better agreement to ±1% for glass compositions [49]. Direct measurements by calorimetry will not necessarily be more precise. Figure 4 shows the C_p(T) curves used for each material [46].

3. Results and Discussions

3.1. Evolution of the Thermal Conductivity with the Temperature

Figure 5 plots the measured thermal conductivity (λ_eff) of an Alumina-Spinel sample as a function of the temperature. Repeating the run with a second sample of alumina-spinel refractory exhibited differences in values of thermal conductivity of less than 1%.

The graph shows that the thermal conductivity decreases with the increase of the temperature: λ_eff changes from 6.5 W m⁻¹ K⁻¹ at room temperature to 3 W m⁻¹ K⁻¹ at 1273 K. To explain this trend, consider that in ceramic materials, the thermal conductivity is determined by scattering of phonons through inelastic collisions. In analogy with the kinetic theory of gases following Debye’s initial approach [16], Klemens describes this property in terms of an integral over the vibrational frequencies ω [50]:

$$\lambda ={\displaystyle \frac{1}{3}}{\int }_0^{{\omega }_D}{C}_V\left(\omega \right)\nu \left(\omega \right)l\left(\omega \right)d\omega$$

(5)

where λ is the thermal conductivity, C_v the specific heat at constant volume, ν the elastic wave velocity, l the mean free path of lattice vibrations, and ${\omega }_D$ the Debye frequency.

For alumina ceramics or in the single crystal form (sapphire), heat is essentially transported by lattice vibrations. The amplitude of these vibrations relates to the number of phonons that occupy the mode ω. An increase in these vibrational amplitudes with temperature means that the probability of phonon-phonon scattering is higher. Thus, the mean free path for the lattice vibrations l(ω) decreases with the increase of temperature, and consequently, the thermal conductivity decreases.

However, in such ceramic materials, it is also important to examine the polycrystalline aspect. Phonons can interact not only with other phonons, but also with grain and pore boundaries and other defects. All these “imperfections” will act as scattering sites, reducing the mean free path. These interactions of the phonons with the microstructure can explain the differences exhibited in Figure 6. The graph shows, in fact, that for all the investigated materials, the thermal conductivity values decrease with the increase of temperature due to the increase of the phonon-phonon scattering mechanism. However, the values are quite different from each other, despite the fact that they are all alumina-based materials. At room temperature, for instance, λ_eff varies from 36 W m⁻¹ K⁻¹ in the case of Sapphire to 5.8 W m⁻¹ K⁻¹ for the Alumina AKP30-1300 sample. The Sapphire sample is a single crystal, and thus it has only external boundaries. On the contrary, all the other samples are polycrystalline materials with different porosities, different grain sizes, and consequently different numbers of grain boundaries crossing the heat path (Table 1).

3.2. Effect of the Porosity

The influence of the porosity can be studied by considering a mixture of two phases, in which the pores correspond to one of the phases: with the increase of the pore volume fraction (ν_p), the effective thermal conductivity of the material (λ_eff) decreases due to the low thermal conductivity of the gas. Figure 7 illustrates the thermal conductivity variations with porosity, based on analytical models with different assumptions for the porous phase arrangement in the solid matrix. The Maxwell–Eucken relation assumes that the pores are isolated spherical inclusions in the solid, which describes a closed porosity behaviour [26], whereas Landauer’s relation takes into account the open nature of the porosity [27] for values of ν_p > 0.15. This is revealed by the divergence of the two curves at ν_p = 0.15, a value typically considered to mark the transition between closed and open porosity in ceramics made by powder pressing. In earlier work on sintered powder compacts, good agreement within ±4% between predictions with Landauer’s relation and experimental data has been obtained on oxide ceramics with open and randomly distributed porosity up to at least 50% [17].

Consequently, a material with a higher pore volume fraction exhibits a lower thermal conductivity, providing the thermal conductivity of the solid phase does not vary. However, this is not always the case. For instance, the Alumina-Spinel brick sample has a porosity of 19%, like that of Alumina AKP30-1450 (Table 1), but the thermal conductivity values are quite different. Figure 6 shows that the Alumina-Spinel sample (black points) has a thermal conductivity closer to that of Alumina AKP30-1300 (magenta points), even if this material has almost twice the amount of porosity (38%). Therefore, the results imply that the materials have different solid-phase thermal conductivity values.

Following an approach that was adopted for porous alumina ceramics [30], to verify this hypothesis, Landauer’s relation was used to estimate the thermal conductivity of the polycrystalline solid phase without the effect of the porosity [17,27]:

$${\lambda }_{eff}={\displaystyle \frac{1}{4}}\left[{\lambda }_P\left(3{\nu }_P-1\right)+{\lambda }_s\left(2-3{\nu }_P\right)\right]+{\left\{{\left[{\lambda }_P\left(3{\nu }_P-1\right)+{\lambda }_s\left(2-3{\nu }_P\right)\right]}^2+8{\lambda }_P{\lambda }_s\right\}}^{{\displaystyle \frac{1}{2}}}$$

(6)

where λ_p is the thermal conductivity of the pore phase and λ_s the thermal conductivity of the solid phase. Considering that λ_p ≪ λ_s, Equation (6) was simplified using λ_p = 0:

$${\lambda }_{eff}={\lambda }_s\left(1-{\displaystyle \frac{3}{2}}{\upsilon }_p\right)$$

(7)

Equation (7) was then re-expressed as follows:

$${\lambda }_s={\displaystyle \frac{{\mathsf{\lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\left(1-\frac{3}{2}{\mathsf{\nu }}_{\mathrm{P}}\right)}}$$

(8)

to evaluate the thermal conductivity for equivalent 100% dense ceramics. The obvious interest of calculating the thermal conductivity of the solid phase(s) is for comparison between materials of different chemical composition/crystal structure and for similar materials with different microstructures (grain size, morphology...) without the problem of porosity variation. This was first done in 1954 by Kingery et al. using Loeb’s relation, which is less precise than the Maxwell–Eucken or Landauer relations for describing the effect of porosity [51]. Justification for the accuracy of the extrapolation back to 100% dense material can be found in references [17,23]. The procedure is less reliable for materials with high pore content (>60%), but Landauer’s relation yields good results in the range 0 < ν_p < 0.6. The refractory in this work contains 20% porosity.

The results of these calculations of the thermal conductivity for the solid phases are presented in Figure 8. It can be noted that an error of 2% in the evaluation of pore volume fraction (a reasonable estimate of uncertainty) leads to an error of 4% in λ_s, but this actually partially compensates any original error in $\rho$ and hence λ_eff through Equation (3). The graph confirms that the materials have different values of thermal conductivity for the solid phase. Furthermore, the estimated values are still significantly less than that of Sapphire (36 W m⁻¹ K⁻¹). For instance, at room temperature λ_s varies from 27 W m⁻¹ K⁻¹ in the case of Alumina AKP30-1450 to 9 W m⁻¹ K⁻¹ in the case of the Alumina-Spinel sample.

3.3. Effect of the Grain Boundaries and Grain Size

Grain boundaries are disordered regions that act as scattering sites, reducing the mean free path. They can be considered as Kapitza resistances, which cause a localized temperature drop at the interface [30,52]. The overall thermal conductivity can then be described using Equation (9) as grain boundary thermal resistances in series with the grains:

$${\displaystyle \frac{1}{{\lambda }_s}}={\displaystyle \frac{1}{{\lambda }_{grain}}}+n{R}_{int}$$

(9)

where λ_s is the thermal conductivity of the polycrystalline material after removing the effect of the porosity, λ_grain is the thermal conductivity of the grains, n is the number of grain boundaries per unit length, and R_int is the average grain boundary thermal resistance. This equation handles the effect of two grain size mechanisms: (i) the presence of grain boundaries crossing the heat path given by the second term of the right-hand side in Equation (9) and (ii) the effect of finite grain size, which can alter the grain conductivity in the first term of the right-hand side in Equation (9). If the grains are small, the number (n) of grain boundaries increases, and thus, the thermal conductivity of the polycrystalline material decreases. Furthermore, if the grains are very small, the hypothesis of an ideal three-dimensional infinite lattice is no longer valid, and this cuts off all the low-frequency long wavelength phonons, which cannot contribute to the thermal conductivity of the crystallite (λ_grain) [5,30].

These effects related to grain size can explain the differences between Sapphire, Alumina TM-DA, and Alumina AKP30-1450 shown in Figure 8. We have used a method to separate the two contributions in Equation (9) and estimate the grain thermal conductivity [30] with the advantage that exact knowledge of the average grain size is not actually required. For this, phonon–phonon interactions are assumed to be the dominant mechanism, and consequently, the thermal resistivity attributed to the grains is considered to vary linearly with the temperature over the range 500 K–1000 K, as represented by the term aT in the relation:

$${\displaystyle \frac{1}{{\mathsf{\lambda }}_s}}=aT+n{R}_{int}$$

(10)

where T is temperature, and a is a constant. This corresponds to the well-known 1/T dependence of thermal conductivity, observed at medium temperatures for many ceramics, including alumina.

Equation (10) is exploited graphically to deduce first the effect of the grain boundaries as Kapitza resistances and then the grain conductivity at each temperature. Figure 9 shows that the thermal resistivity values of all the alumina-based materials behave linearly with temperature and are above the values of Sapphire. The differences shown on the y-axis by extrapolation to T = 0 K can be attributed to the total thermal resistance of the grain boundaries (nR_int), taken to be constant with temperature [30].

By subtracting this contribution from Equation (9), it is possible to evaluate the thermal conductivity of the grains (λ_grain), which is plotted in Figure 10. Figure 10a shows that the calculations of grain conductivity for Alumina TM-DA and Alumina AKP30-1450 yield values that are almost identical to those of the Sapphire single crystal (36 W m⁻¹ K⁻¹ at room temperature).

The case of Alumina AKP30-1300 involves additional factors, the effects of which are revealed in

Table 2. Evaluation of the constant (a) in a linear temperature range (500 K–1000 K) using Equation (10), the average grain boundary thermal resistance (R_int) using Equation (9), and the number of grain boundaries per unit length (n) using Equation (2).

Sample Designation	a (500 K–1000 K)	Average Grain Size or Range (μm)	n (m−1)	Rint (m2 K W−1)
Alumina-Spinel	1.70 ± 0.04 × 10−4	1.0	1.0 × 106	-
		3.0 × 103	3.3 × 102	-
Sapphire	1.05 ± 0.03 × 10−4	-	-	-
Alumina TM-DA	1.05 ± 0.03 × 10−4	0.30	3.3 × 106	4.5 × 10−9
Alumina AKP30-1450	1.01 ± 0.009 × 10−4	0.45	2.2 × 106	4.5 × 10−9
Alumina AKP30-1300	1.38 ± 0.03 × 10−4	0.30	3.3 × 106	1.2 × 10−8

. This table gives the values of the slope coefficient “a” for each sample in Figure 9 and an estimate of the average grain boundary thermal resistance for the alumina ceramics (R_int). It can be seen that the samples Alumina TM-DA and Alumina AKP30-1450 exhibit values of “a” very close to the single crystal, as well as similar average grain boundary thermal resistances, implying that the microstructural aspects of grain size and pore volume fraction have been satisfactorily accounted for. Sample AKP30-1300 exhibits significantly higher values of “a” and average grain boundary thermal resistance. In fact, this sample, held for 6 min at 1300 °C, is the least advanced in the sintering process in terms of densification and grain growth. A neat explanation of the higher value of “a” is that the initial sintering mechanism of neck formation has not been completed, reducing the particle-particle contact area (or effective heat-carrying cross-section through the grains) [53]. This will increase the value of “a” [54]. The other aspect of less advanced sintering is that higher energy grain boundaries (with greater thermal resistance) have not been eliminated, consistent with the higher value of R_int for AKP30-1300 [30,53]. The smaller grain size (Table 1), approximately 0.3 μm, of this sample, through the effect of finite grain size inhibiting the grain conductivity, may also have a role. All these aspects relate to less advanced sintering for the AKP30-1300 sample.

Finally, the thermal resistivity of the Alumina-Spinel sample (equivalent to 100% dense material) is even higher with the highest value of “a” (Table 2). This might also relate to incomplete sintering of the refractory powder with a thermal treatment, say at 1400 °C. It is difficult to evaluate average grain boundary thermal resistance for two reasons. First, there is a very wide range of grain sizes in the refractory brick, from approximately 1 μm to 3 mm, as estimated from SEM micrographs. Second, the refractory powder contains two phases: alumina grains and spinel grains. Nevertheless, after correction for porosity and thermal resistance of the interfaces using the same procedure as employed for the alumina ceramics, representative values of the conductivity for the alumina-spinel grains are plotted in Figure 10b, yielding 27.5 W m⁻¹ K⁻¹ at room temperature. The aspect of a two-phase mixture is examined in the next section.

3.4. Effect of Phase Mixture

Alumina-spinel refractory bricks contain approximately 81 vol.% alumina and 19 vol.% spinel (MgAl₂O₄). To evaluate the influence of the spinel phase on the combined thermal conductivity, samples where the spinel phase is predominant (87 vol.% spinel and 13 vol.% alumina) were prepared, denoted AR78. These samples were measured with the laser-flash method up to 1273 K, revealing a steady decrease of the thermal conductivity with temperature (Figure 11, black points) attributed to the increase in phonon-phonon scattering. More refined values for the spinel phase thermal conductivity were then obtained in the following way.

The experimental results for an AR78 sample showed an overall thermal conductivity of 9.3 W m⁻¹ K⁻¹ at room temperature, including the effect of 20% porosity. The same approach previously described for the alumina ceramics and the alumina-spinel refractory bricks was then adopted to calculate the thermal conductivity of the AR78 sample without the effects of the microstructure. Firstly, Equation (8) was used to calculate the thermal conductivity of the solid phases (λ_s) equal to 13.3 W m⁻¹ K⁻¹ at room temperature. Then, two cases were analyzed: (a) the effect of the interfaces was neglected, which gave the lower values of thermal conductivity for the spinel phase; (b) the interfaces were taken into account.

In the first case, for a mixture of two solid phases, solving Landauer’s relation [27] for the thermal conductivity of the spinel phase (λ_spinel) gave the following:

$${\lambda }_{spinel}={\displaystyle \frac{2{\lambda }_s^2-{\lambda }_s{\lambda }_{alumina}\left(3{\nu }_{alumina}-1\right)}{{\lambda }_{alumina}+{\lambda }_s\left(2-3{\nu }_{alumina}\right)}}$$

(11)

where λ_alumina and ν_alumina are respectively the thermal conductivity and the volume fraction of the alumina grains. Using the proportion above for AR78 and a value for λ_alumina = 36 W m⁻¹ K⁻¹, Equation (11) gives λ_spinel = 11.4 W m⁻¹ K⁻¹ at room temperature (Figure 11, red points).

In the second case, Equation (9) was first used to estimate the thermal conductivity of the grains (λ_grain) equal to 22 W m⁻¹ K⁻¹ at room temperature. Then, Equation (11) was modified by replacing λ_s with λ_grain:

$${\lambda }_{spinel}={\displaystyle \frac{2{\lambda }_{grain}^2-{\lambda }_{grain}{\lambda }_{alumina}\left(3{\nu }_{alumina}-1\right)}{{\lambda }_{alumina}+{\lambda }_{grain}\left(2-3{\nu }_{alumina}\right)}}$$

(12)

By considering again the proportion above for AR78 and a value for λ_alumina = 36 W m⁻¹ K⁻¹, Equation (12) gives λ_spinel = 20.2 W m⁻¹ K⁻¹ at room temperature (Figure 11, green points).

These results are fairly similar to those in the literature: for stoichiometric spinel, Braulio et al. [55] cited values from 15 W m⁻¹ K⁻¹ at room temperature to 5 W m⁻¹ K⁻¹ at 1273 K, more or less the same as Balabanov et al. [56], who found 17 W m⁻¹ K⁻¹ at room temperature. Our values are in the same range as the second case; while Ni et al. [57] calculated, with molecular dynamics values, from 9 W m⁻¹ K⁻¹ at room temperature to 5.5 W m⁻¹ K⁻¹ at 1273 K, similar to the first case.

Next, the calculated data for λ_spinel were used to estimate the thermal conductivity of the alumina grains (λ_alumina) in the alumina-spinel refractory brick. First, a sensitivity analysis was made on the room temperature value of λ_alumina with respect to variation in λ_spinel and the choice of analytical relation (model) representing the microstructure. In a straightforward approach, the upper bound for λ_grain is given by the combination of thermal resistors in parallel:

$${\lambda }_{grain}=v_{alumina}{\lambda }_{alumina}+v_{spinel}{\lambda }_{spinel}$$

(13)

and the lower bound by thermal resistors in series:

$${\displaystyle \frac{1}{{\lambda }_{grain}}}={\displaystyle \frac{v_{alumina}}{{\lambda }_{alumina}}}+{\displaystyle \frac{v_{spinel}}{{\lambda }_{spinel}}}$$

(14)

Given that the spinel phase is observed to constitute a fine matrix around the alumina grains (Figure 1), the Maxwell–Eucken relation describes spherical grains surrounded by an insulating phase [26]:

$${\lambda }_{grain}={\lambda }_{spinel}\left[{\displaystyle \frac{{\lambda }_{alumina}+2{\lambda }_{spinel}+2v_{alumina}\left({\lambda }_{alumina}-{\lambda }_{spinel}\right)}{{\lambda }_{alumina}+2{\lambda }_{spinel}-v_{alumina}\left({\lambda }_{alumina}-{\lambda }_{spinel}\right)}}\right]$$

(15)

These relations are solved in each case for λ_alumina. If one considers that the phase of alumina grains exhibits a certain continuity (or connectivity between grains), then Landauer’s relation is pertinent, similar to Equation (12). Solving this for λ_alumina yields the following:

$${\lambda }_{alumina}={\displaystyle \frac{2{\lambda }_{grain}^2-{\lambda }_{grain}{\lambda }_{spinel}\left(3{\nu }_{spinel}-1\right)}{{\lambda }_{spinel}+{\lambda }_{grain}\left(2-3{\nu }_{spinel}\right)}}$$

(16)

The relations were tested for three values of λ_spinel = 11.4 W m⁻¹ K⁻¹, 15 W m⁻¹ K⁻¹, and 20.2 W m⁻¹ K⁻¹, corresponding to earlier calculations made with Equations (11) and (12) with and without the contribution of the interfaces. The value of λ_spinel = 15 W m⁻¹ K⁻¹ is intermediate and corresponds roughly to values given by Braulio et al. or Balabanov et al. [55,56]. The values of λ_grain now refer to the points in Figure 10b (black squares) for the Alumina-Spinel refractory sample, which have already been corrected for porosity and grain boundary effects, and at room temperature, are equal to 27.5 W m⁻¹ K⁻¹.

Table 3. Sensitivity analysis at room temperature: calculated values of thermal conductivity for the alumina grains (λ_alumina) in an alumina-spinel refractory brick with variation in λ_spinel using resistors in parallel and series as bounds, the Landauer and Maxwell–Eucken relations (λ_grain = 27.5 W m⁻¹ K⁻¹).

Thermal Conductivity Spinel λspinel (W m−1 K−1)	Parallel Model: Equation (13) λalumina (W m−1 K−1)	Landauer: Equation (16) λalumina (W m−1 K−1)	Maxwell–Eucken: Equation (15) λalumina (W m−1 K−1)	Series Model: Equation (14) λalumina (W m−1 K−1)
20.2	29.2	29.4	29.5	30.0
15	30.4	31.1	31.5	34.2
11.4	31.3	32.5	33.7	41.1

gives the estimated values for the thermal conductivity of the alumina grains (λ_alumina).

Apart from the series model, the estimates are fairly insensitive to the value of λ_spinel, yielding narrow bands of values close to 31 ± 2 W m⁻¹ K⁻¹. However, neither of the bounds, in the form of thermal resistors in series or parallel, is completely representative of the spatial distribution of the more insulating spinel phase. Both the Maxwell–Eucken and Landauer relations provide useful approximations to the dispersion of alumina grains in a spinel matrix, and yield, in fact, only small differences in their calculated values. For convenience, Landauer’s relation is chosen for the final calculation using Equation (16). The results for the two extreme cases (λ_spinel equal to 11.4 and 20.2 W m⁻¹ K⁻¹) are shown in Figure 12.

It can be pointed out that even if the calculated values for the alumina grains in the refractory brick are still apart from those of the Sapphire single crystal, they are much closer than the original measured values (Figure 5). For instance, 31 W m⁻¹ K⁻¹ at room temperature is within 14% of 36 W m⁻¹ K⁻¹ for Sapphire. Remaining differences could be assigned to the presence of other minor phases or small fractions of impurities modifying the conductivity of alumina. Popov et al. [58] demonstrated that small quantities of Cr and Ti, which substitute into the Al₂O₃ lattice, inhibit conduction. But given the number of steps and approximations describing the effects of pore volume fraction, grain boundaries/interfaces, and spatial solid phase distribution in the calculation of λ_alumina, the final set of values can be considered as plausible and satisfactory.

Finally, the same analysis procedure was applied to data in [54] for an almost pure industrial alumina refractory with 18% porosity. The measured value for the alumina refractory at room temperature was 15 W m⁻¹ K⁻¹, yielding after analysis an encouraging benchmark value of 30 W m⁻¹ K⁻¹ for the alumina grains.

4. Conclusions

Understanding the relationship between the microstructure and the material’s properties is fundamental for predicting the behaviour of refractory materials in service conditions. This paper presents a straightforward analytical approach to describe the modulation of thermal conductivity by:

i.

The pore phase using Landauer’s relation;

ii.

The grain boundary thermal resistance by assuming it acts in series with the thermal resistance of the grains;

iii.

A second solid phase using again Landauer’s relation for a mixture of two phases.

Experimentally, measurements of the macroscopic thermal conductivity from room temperature to 1273 K were made on alumina-spinel refractory containing 81 vol.% of alumina and 19 vol.% of spinel (MgAl₂O₄), revealing a decrease from 6.5 W m⁻¹ K⁻¹ to 3 W m⁻¹ K⁻¹. These values imply that, though the alumina grains are the major phase, the thermal conductivity has been strongly modulated by the microstructure and second phase. Given the complex nature of the refractory material, similar experimental measurements were made on a sapphire single crystal and model single-phase alumina ceramics with variation in the porosity and grain size. In order to deduce the thermal conductivity of the alumina grains in each material for comparison to the sapphire single crystal, the analysis first corrected for the effect of porosity with a simplified Landauer’s equation. Then the contribution of the grain boundary/interface thermal resistance was evaluated from a plot of thermal resistivity versus temperature, assuming this has a constant value. The thermal conductivity of the alumina grains in the 97% dense Alumina TM-DA sample and the 81% dense Alumina AKP30-1450 sample were shown to be very close to the Sapphire single crystal: 36 W m⁻¹ K⁻¹ at room temperature. Slightly lower values, obtained for 62% dense Alumina AKP30-1300, were attributed to less complete sintering of this sample.

For the alumina-spinel refractory, a supplementary step to this approach took the presence of the spinel phase into account, for example, with Landauer’s equation. This yielded an estimate of the thermal conductivity of the alumina grains in the refractory equal to 31 ± 2 W m⁻¹ K⁻¹ at room temperature. Fluctuations in sample thermal conductivity of the order of 5% due to measurement uncertainty or in the spinel fraction evaluation (example: 22 vol.% for an alumina-rich spinel instead of 19 vol.%) give values within the same range. Of the three steps in the analysis procedure, the estimate of the grain boundary/interface thermal resistance is considered to be most liable to errors. However, this grain boundary/interface contribution, through control of grain size distribution, offers interesting potential for increasing thermal conductivity and improving thermal shock resistance. Future work should be directed towards this aspect for both model single-phase ceramics and industrial refractory materials.