9
Thermal Conductivity of ZrB₂ and HfB₂

Gregory J. K. Harrington and Greg E. Hilmas

Department of Materials Science and Engineering, Missouri University of Science and Technology, Rolla, MO, USA

9.1 Introduction

ZrB₂ and HfB₂ are proposed for use in hypersonic vehicles and reentry spacecraft where they will be used as thermal management materials for extreme thermal environments. These applications typically include sharp control surfaces (leading/trailing edges) as well as scramjet engine components [1–5]. Hypersonic control surfaces need thermal conductivities (k) that are as high as possible to conduct heat generated from atmospheric friction through the material so that it can be dissipated elsewhere [1, 3]. In addition, higher k values increase thermal shock resistance in high heat flux situations. Because engine components cannot radiate to the atmosphere, lower k refractory materials could be beneficial, depending on the cooling capabilities of the vehicle [3]. From a materials engineering perspective, the effects of processing, composition (intentional additions or from impurities), and the resulting microstructure need to be characterized to design materials with appropriate properties. The goal of this chapter is to review the state-of-the-art with respect to thermal conductivity for single-phase and composite ZrB₂ and HfB₂ ceramics.

9.2 Conductivity of ZrB₂ and HfB₂

9.2.1 Pure ZrB₂

Few nominally phase-pure ZrB₂ materials (referred to hereafter as “pure ZrB₂”) have been produced for measurement of k. As summarized in Table 9.1, k has been reported for fewer than 25 materials starting with the work by Sindeband et al. [6] in 1950, up through the publication of this chapter. This is likely due to the difficulty of densifying ZrB₂ without sintering aids and the frequency with which second phases are added to ZrB₂ to improve strength, fracture toughness, resistance to oxidation, and thermal shock performance. From this point on, reports before 1980 are referred to as “historic” while anything after is considered “current.” Most studies have evaluated k indirectly using the bulk density (ρ), thermal diffusivity (D), and the constant pressure heat capacity (C_p), along with Equation 9.1 [8, 10–18, 20–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication). However, a few studies have measured k directly using steady-state methods [7–9] or a direct dynamic method [19]. These exceptions have been noted in Table 9.1.

Table 9.1. Thermal conductivity of pure ZrB₂ with information on starting/final materials, processing, and density.

Powder processing	Purity (%)	Densification technique	Special notes(gs in µm)	ρrelative (%)	Thermal conductivity(k, W/m-K)		Ref.
					25 °C	at max temp
Fused-salt electrolysis	—	PS	—	85	24	—	[6]
—	96.3	HP	—	70	—	29 at 2200 °C	[7]
“Fluid energy milled”	99.2	HP	gs = 28.5 µm	100	—	82 at 1000 °C	[8]
“Fluid energy milled”	99.2	HP	gs = 28.5 µm	100	—	82 at 2000 °C	[8]
—	—	HP	—	100a	—	133 at 2020 °C	[9]
—	99.1 (final)	HP	gs = 11 µm	97.4	—	92 at 1143 °C	[10]
—	—	HP	gs = 13 µm	100a	—	84 at 885 °C	[11]
“Vibrogrinding”	—	CVD	Pyrolytic	100a	—	84 at 2150 °C	[12]
“Vibrogrinding”	—	PS 2100 °C	Ni sintering aid	100a	—	55 at 2200 °C	[12]
—	—	RF FZR	SX—a-direction	100	132-145	—	[13]
—	—	RF FZR	SX—c-direction	100	95-102	78-82 at 200 °C	[13]
BM w/ZrO2	99.5	HP at 2000 °C	gs = 5 µm	75	—	43 at 2000 °C	[14]
AM w/WC	—	HP at 1900 °C	gs = 6 µma	100	56	67 at 1325 °C	[15]
BM w/SiC	98	SPS at 1900 °C	gs = 3.0 µm	95.6	113	—	[16]
BM ZrH2 + B w/SiC	99 ZrH2, 95.9 B	rxn SPS at 1800 °C	gs = 5.35 µm	97	133	—	[17]
HM Zr + B	99.7 Zr, 99 B	rxn SPS	gs = 5.6 µm	92.5	—	92 at 435 °C	[18]
BM w/ZrO2	—	SPS at 2000 °C	gs = 6–11 µm	95	50	—	[19]
AM w/WC	—	HP at 1900 °C	gs = 3.3 µm	100	—	59 at 2000 °C	[21]
BM w/ZrB2	—	HP 2100 °C	gs = 22 µm	97	93	97 at 2000 °C	[20]
BM w/ZrB2	98.7	HP at 2150 °C	gs = 25, 0.2 wt%C sintering aid	100a	99	75 at 2100 °C	[22]
BM ZrH2 + B w/ZrB2	99.7 ZrH2, >99 B	HP at 1900 °C	gs = 5.5 µm	100a	107	66 at 2000 °C	(Jason Lonergan, Missouri University of Science and Technology, personal communication)
BM w/ZrB2	98.7	HP at 2150 °C	gs = 21, 0.5 wt%C sintering aid	100a	89	87 at 200 °C	[23]

AM, attrition milling; BM, ball milling; HM, hand milling; HP, hot pressing; PS, pressureless sintering; RF FZR, radio frequency float zone refinement; SPS, spark plasma sintering; gs, grain size.

^adensity has been corrected to 100% by author.

(9.1)

For studies utilizing Equation 9.1, D was evaluated by dynamic methods including xenon flash [16, 17], laser flash [8, 10, 13–15, 20–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication), or a plane temperature wave approach [12, 18]. Overall, D is the most important parameter for calculating k. Compared to C_p and ρ, D is the property that is most affected by changes in grain size and purity. In some cases, C_p values have either been measured by drop calorimetry [8], DSC [14, 15], or one of the flash techniques [13, 16, 17]. More likely, C_p values have been taken from thermochemical tables [10, 12, 18, 20–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication), historic literature by either Schick [24] or Bolgar et al. [25], or sources such as the NIST-JANAF [26] tables or HSC chemistry [28] (HSC primarily derives its ZrB₂ equations from the NIST-JANAF). Figure 9.1 shows C_p for ZrB₂ as a function of temperature from several of studies [8, 14–17] as well as the thermochemical tables [24, 26–28].

Figure 9.1. Thermochemical and experimental heat capacity (C_p) values for ZrB₂ [8, 14–17, 24, 26, 27].

Figure 9.1 shows obvious discrepancies among reported values. Despite divergence at higher temperatures, data from Schick et al., the NIST-JANAF tables, HSC, and a more recent DSC data by Zimmermann et al. [15] appear to agree between room temperature (RT) and approximately 200°C (430–550 J/kg-K). In contrast, room-temperature values reported by Guo et al. [17] and Ikegami et al. [16], which were both obtained by a comparison method using xenon flash, are approximately 20 and approximately 29% higher, respectively. The cause of the discrepancies in the C_p values is unclear, but it goes without saying that reliable C_p values are of utmost importance when calculating k. Finally, density is almost always evaluated using the Archimedes method. For cases in which density values are corrected for the effects of thermal expansion, tabulated data [20–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication) or experimental dilatometry results [7, 8, 10, 13–15, 22] were used.

Most studies have used some type of milling procedure for particle size reduction, to mix the starting components, or to blend-in sintering aids. Contamination from milling media is common and may affect the properties of the densified ceramics. Only the modern reports provide much detail on the types of media used, with WC [15, 20], SiC [16, 17], ZrO₂ [14, 19], and ZrB₂ [21–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication) being the most common media types. Pressure-assisted densification techniques such as hot pressing (HP) or spark plasma sintering (SPS; also known as field-assisted sintering (FAST) or pulsed electric current sintering (PECS)) were most often employed for the densification of ZrB₂ [7–11, 14–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication), but pressureless sintering (PS) [6, 12], chemical vapor deposition [12], and float zone refinement [13] have also been utilized. From these densification techniques, researchers were often successful at producing greater than or equal to 95% dense ceramics with several achieving full density. Figures 9.2 and 9.3 summarize thermal conductivity data from historic and current literature. All data have been presented “as published” unless otherwise noted in the figures.

Figure 9.2. Historic thermal conductivity as a function of temperature for ZrB₂. Data for Clougherty changed testing method at 1000°C [6–12].

Figure 9.3. Current thermal conductivity as a function of temperature values for ZrB₂. ^aData corrected for ρ. ^bData corrected for C_p [13–18, 20–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication).

9.2.1.1 Historic ZrB₂.

 Historically, ceramics with the lowest thermal conductivities were produced by Sindeband et al. (~25 W/m-K) [6] and Neel et al. (20–46 W/m-K) [7], likely due to low relative densities, which were 85 and 70%, respectively. The effect of density is especially evident at elevated temperatures where k begins to rise as temperature increases, presumably due to photon heat transport across pores. Despite the low density of these ceramics, research by Neel et al. was significant in that ZrB₂ was tested up to 2200°C, which has since only been equaled by Fridlender et al. [12]. In addition to high testing temperatures, Neel et al., along with Samsonov et al. [9] and Clougherty et al. [8], are the only researchers to perform direct steady-state measurements of thermal conductivity for any ZrB₂-based materials.

Other historic k values for ZrB₂ were obtained for ceramics that were >97% dense [10], fully dense [8], or had thermal conductivities corrected for porosity [9, 11, 12]. No clear trend in thermal conductivity as a function of temperature emerges as k increases for some and decreases for others as temperature increases. The k values also vary widely, from as low as approximately 35 W/m-K at 200°C to approximately 80 W/m-K at 2000°C. Both ZrB₂ and HfB₂ should exhibit k values that decrease with increasing temperature because they behave similar to metals whereby heat is transferred by phonons and electrons, both of which scatter more at elevated temperatures due to increased thermal vibrations. Based on this, and in comparison to the other historic literature values in Figure 9.2, values reported by Samsonov et al. are likely not representative of ZrB₂ as k steadily increases from approximately 60 W/m-K at 1000°C to approximately 130 W/m-K at 2000°C.

Andrievskii et al. [11] also reported k values that increased with temperature, which was not explained. In contrast, Fridlender reported the addition of 1 wt% carbonyl nickel for densification of “sintered” ZrB₂. The presence of Ni most likely caused the relatively low k values (50–54 W/m-K) and increasing k with temperature. Fridlender et al. also investigated pyrolytic ZrB₂ produced by chemical vapor deposition (CVD) that led to a “columnar microstructure with semicoherent crystallite boundaries.” No overall temperature trend was apparent for k, but the higher conductivity (84–93 W/m-K) compared to sintered material was attributed to decreased phonon scattering from more coherent grain boundaries [12].

Finally, ZrB₂ produced by Branscomb and Hunter [10] and Clougherty et al.¹ [8] had the highest k values of the historic ceramics and had k values that decreased with temperature. This behavior is consistent with high relative density and purity. Further, Branscomb and Hunter reported the highest k at low temperature (105 W/m-K at 200°C) and the highest conductivity above 1000°C (~91 W/m-K at 1143°C, excluding Fridlender's pyrolytic ZrB₂). Branscomb and Hunter analyzed the metallic impurity content and found their ceramic to be approximately 99.1 at% pure with the largest impurities being Fe (0.7 at%), Si (0.07 at%), and Nb (0.05 at%) with other impurities <0.02 at% [10]. However, the analysis did not include Hf (common impurity in Zr), C (typically introduced during processing), or O (from native oxide layers), which were likely also present. Despite this, their ZrB₂ was one of the purest historic ceramics resulting in its high k.

9.2.1.2 Current ZrB₂

 Current studies summarized in Figure 9.3 are similar to historic studies (Fig. 9.2) as a large range of values have been reported. However, values are consistent with the main factors (i.e., density, purity, and grain size) that control k.

Thermal conductivity at the highest temperatures measured for current ZrB₂ ceramics [15, 20–22] (Jason Lonergan, Missouri University of Science and Technology, personal communication) ranged from 43 W/m-K [14] to 75 W/m-K [22]. Room-temperature k values are all >85 W/m-K [13, 16–18, 21–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication) with one exception at 56 W/m-K [15]. Despite the range of k values for ZrB₂, consensus has not been reached on the intrinsic k of ZrB₂.

To investigate ZrB₂ as a lattice-matched substrate for GaN, Kinoshita et al. produced single-crystal (SX) ZrB₂ by float zone refinement (Table 9.1). ZrB₂ is hexagonal and has an anisotropic k. Measurements ranged from 132–145 W/m-K parallel to the a-axis to 95–102 W/m-K parallel to the c-axis (Fig. 9.3) [13]. Differences can be explained by different phonon and electron scattering behavior for different directions. Phonon scattering increases as the ratio of atomic masses increases between constituent elements [29]. Therefore, phonon conduction will be highest for the direction with B–B and Zr–Zr bonds, which is the basal plane. In addition, the electron contribution to k is higher along the basal plane due to electrical conduction in the close-packed planes of Zr atoms [30].

The SX values reported by Kinoshita et al. provided insight into the theoretical limits for k of ZrB₂ ceramics. Equations 9.2 and 9.3 have been used to estimate the arithmetic-mean (k_AM, upper bounds) and harmonic-mean (k_HM, lower bounds) polycrystalline thermal conductivities from SX values [31, 32]:

(9.2)

(9.3)

Using values from Kinoshita, the theoretical k for ZrB₂ is between 117 and 131 W/m-K. These values do not account for grain boundary resistances, which are influenced by both grain size and impurities, and would likely decrease thermal conductivity. Therefore, the predicted k values should be upper limits for polycrystalline ZrB₂ ceramics.

The k of one material from Figure 9.3 has a value within the SX bounds [17], while three others are within 10 W/m-K of the lower limit [16, 18] (Jason Lonergan, Missouri University of Science and Technology, personal communication). To more thoroughly compare these materials, corrections for heat capacity and density have been applied in the present analysis. The C_p values were corrected because of the uncharacteristically high values reported by Guo et al. [17] and Ikegami et al. [16], and for the low relative densities (97.2% [17], 95.6% [16], and 92.5% [18]), which were corrected using the Maxwell–Eucken relation [33, 34]. The modified C_p values decreased k considerably, while accounting for porosity in the ceramic produced by Zhang et al. [18] increased k to 125 W/m-K at 50°C. These corrections left Zhang's as the only material with a k value that fell in the theoretical range, which makes it the best representation of the intrinsic k of polycrystalline ZrB₂.

The three highest k materials from the corrected values (after C_p and ρ corrections) were all reaction processed [17] (Jason Lonergan, Missouri University of Science and Technology, personal communication). All other current ZrB₂ was produced from commercial ZrB₂ powder. An obvious division in room-temperature k values occurs at approximately 67 W/m-K. Above this value, the reported k values decrease as temperature increases [18, 21–23] (Jason Lonergan, Missouri University of Science and Technology, personal communication). Besides the material by Loehman et al., which may be anomalous due to its low density (75%) [14], the two other materials below 67 W/m-K exhibited an increase in k with temperature [15, 20]. These ceramics contained WC impurities from milling and, in both cases the authors indicated that W from the media was a likely cause for the slope and low k values. Despite the differences in room-temperature k and the change in k with temperature [21, 22] (Jason Lonergan, Missouri University of Science and Technology, personal communication), the k values converge to a narrow range (within ~10 W/m-K) above 1300°C. This indicates that factors resulting in large changes in the k at lower temperatures may give way to a common mechanism that controls k at elevated temperatures.

Purity may be the most significant factor affecting k for current ZrB₂ ceramics. Conductivities for fully dense ZrB₂ [15, 21], or values for porous ceramics corrected to full density [22, 23] (Jason Lonergan, Missouri University of Science and Technology, personal communication), span a significant range of k values (more than 50 W/m-K) and appear to be independent of grain size. Grain sizes can be divided into two separate ranges, fine with average grain sizes below 10 µm [14–18, 20] (Jason Lonergan, Missouri University of Science and Technology, personal communication) or coarse with average grain sizes over 20 µm [21–23]. Fine grains can result in either high k (>107 W/m-K at 25°C) for reaction processed ZrB₂ [17, 18] (Jason Lonergan, Missouri University of Science and Technology, personal communication) or low k (<57 W/m-K at 25°C) for conventionally processed ZrB₂ ceramics [15, 20]. Thermal conductivities for coarse-grained materials all fall between those values [21–23]. Therefore, the difference apparent in Figure 9.3 is not significantly affected by grain size. Differences in purity, which are reported much less often, are, therefore, the most likely explanation for the observed variations. Unfortunately, none of the current studies report chemical analysis of the final ceramics, although assays of starting powders are typically provided [14, 16–18, 22, 23] (Jason Lonergan, Missouri University of Science and Technology, personal communication). As a result, analysis of the effect of purity is difficult. However, some studies have systematically evaluated the effects of solid solution (SS) additions of various species as discussed in the next section.

9.2.2 ZrB₂ with Solid Solution Additions

Thermal conductivities from three studies evaluating the effects of SS additions to ZrB₂ [21–23] are summarized in Table 9.2 and plotted in Figure 9.4. Values for reaction hot-pressed ZrB₂ ceramics reported by Zhang et al. [18] are included for comparison. These studies were self-consistent since similar processing was used for pure, baseline ZrB₂ ceramics and those with subsequent SS additions. However, if SS additions [21, 23] were compared to the corrected data for reaction-processed ZrB₂ [18] instead of the baseline material produced in each study, the effects could be misleading.

Table 9.2. Summary of solid solution effects on thermal conductivity of ZrB₂

Material (vol% or otherwise stated)	Powder processing	Densification technique	Special notes	Relative density (%)	Thermal conductivity (k, W/m-K)		Ref.
					25 °C	at max temp
ZrB2 + 2.8 mol%NbB2	BM w/ZrB2	HP at 2150 °C	gs = 7 µm, 0.5 wt%C sintering aid	100	74	76 at 200 °C	[23]
ZrB2 + 3.1 mol%W2B5	BM w/ZrB2	HP at 2150 °C	gs = 18 µm, 0.5 wt%C sintering aid	100a	35	40 at 200 °C	[23]
ZrB2 + 10%TiB2	BM w/ZrB2	HP at 2100 °C	gs = 10 µm	100a	81	72 at 2000 °C	[21]
ZrB2 + 50%TiB2	BM w/ZrB2	HP at 2100 °C	gs = 7 µm	100a	58	65 at 1900 °C	[21]

^aCorrected to 100% in this work.

Figure 9.4. Thermal conductivity as a function of temperature for ZrB₂ with solid solution additions. ^aData corrected for ρ [18, 21–23].

SS additions decrease the k values of ZrB₂ ceramics from the respective baselines, especially at room temperature. The trend in k as a function of temperature depends on both the type and amount of additive. For example, Thompson et al. showed that an SS addition of 10 vol% TiB₂ reduced k resulting in an almost linear trend with temperature. In contrast, the addition of 50 vol% TiB₂ resulted in a decrease in k of approximately 23 W/m-K at 25°C compared to the addition of 10 vol% TiB₂ [21]. Both Thompson et al. [20] and McClane et al. [23] found that smaller additions of TiB₂ (data by McClane not shown in Figure 9.4) had a minimal effect on k. In contrast, other transition metal additions such as W or Nb evaluated by McClane had a much larger effect at the same concentration [23]. McClane confirmed that SS additions of tungsten in ZrB₂ decreased k significantly (~53 W/m-K drop at 25°C for 3.1 mol% W). Evaluation of k for different SS additions was also shown to correlate to changes in lattice parameters [23].

9.2.3 Pure HfB₂

Research on k of HfB₂ is summarized in Table 9.3. Reports of k for HfB₂ are more scarce than those of ZrB₂, likely due to the higher cost and density (11.21 g/cm³ [38] vs. 6.12 g/cm³ [39]) of Hf, which may make it less attractive for aerospace applications. The k values of HfB₂, like ZrB₂, were typically calculated from D, ρ, and C_p using Equation 9.1 [10, 14, 18, 35, 37], but some direct measurements have been reported [9, 36]. Also similar to ZrB₂, C_p was either evaluated experimentally [14, 35], taken from thermochemical tables (Schick [24]) [10], or an electronic database (HSC [27]) [18, 37] and these values are plotted as a function of temperature in Figure 9.5. The tabulated data by Schick and values obtained from HSC are nearly identical, but C_p values used by Loehman et al. [14] and Gosset et al. [35] were lower and may have led to lower k values compared to studies using typical values.

Table 9.3. Thermal conductivity of pure HfB₂ with information on starting/final materials, processing, and density

Powder processing	Purity (%)	Densification technique	Special notes	Relativedensity (%)	Thermal conductivity(k, W/m-K)		Ref.
					25 °C	at max temp
—	—	HP	—	100a	—	144 at 2000 °C	[9]
—	98.2	HP	gs = 15 µm	95.2	—	74 at 1200 °C	[10]
—	99.5	HP at 2000 °C	—	95	72	—	[35]
—	—	HP at 2160 °C	—	95	104	72 at 820 °C	[36]
BM w/ZrO2	99.5 ZrB2	HP at 2000 °C	—	69	—	31 at 2000 °C	[14]
HM Hf + B	99.8 Hf, 99 B	rxn SPS	gs = 4.0 µm	95.4	107	91 at 600 °C	[37]
HM Hf + B	99.8 Hf, 99 B	rxn SPS	gs = 10.7 µm	98.1	—	101 at 420 °C	[18]
HM Hf + 1.9B	99.8 Hf, 99 B	rxn SPS	Zr rich, gs = 10 µm	98.1	—	97 at 430 °C	[18]
HM Hf + 2.1B	99.8 Hf, 99 B	rxn SPS	B rich, gs = 10.9 µm	101	—	110 at 420 °C	[18]

^aCorrected to 100% in this work.

Figure 9.5. Heat capacity as a function of temperature for HfB₂ [24, 27, 35].

Of the studies reporting powder-processing methods, hand mixing (HM) was most common [18, 37], while one study used ZrO₂ media and ball milling [14]. The two studies utilizing HM also performed reaction processing using SPS to produce HfB₂ from Hf and B. All other HfB₂ studies used commercial HfB₂ powder and densification by hot pressing [9, 14, 35, 36] or evaluated commercially sintered material [10]. Aside from HfB₂ studied by Loehman et al. [14], which had a relative density of 69%, all other reports used HfB₂ that was greater than or equal to 95% relative density.

The k values presented in Table 9.3 are plotted in Figure 9.6. Similarities are apparent between trends discussed for ZrB₂ and HfB₂. First, both display a wide variation in k values across the temperature range. Second, k should decrease with increasing temperature, porosity, and impurity content.

Figure 9.6. Thermal conductivity as a function of temperature for historic and current HfB₂. ^aData corrected for ρ [9, 10, 14, 18, 37, 39, 40].

The k values for HfB₂ reported by Samsonov et al. [9] increase with increasing temperature from 1000 to 2000°C, as did ZrB₂ reported in the same study. Again, this trend does not fit with other reports. As mentioned previously, Loehman et al. reported k for highly porous HfB₂, resulting in low values (37–31 W/m-K from 250–2000°C) [14]. Branscomb and Hunter [10] reported the only other historic HfB₂, which had a slightly negative slope of k with temperature (78 W/m-K at 200°C to 74 W/m-K at 1200°C), but k was lower compared to HfB₂ produced by Zhang et al. [18] and Gasch et al. [37] due to lower purity (98.2%) and density (95.2%).

Like the highest k for polycrystalline ZrB₂, the highest k for HfB₂ was also a result of high purity starting materials (i.e., 99.8% Hf and 99% B) [18, 37]. Zhang et al. also produced nonstoichiometric compositions of HfB_1.9 and HfB_2.1, but only HfB_2.1, was fully dense. To better evaluate these k values, and estimate a theoretical maximum k for HfB₂, reported values were adjusted for ρ (hollow shapes in Fig. 9.6). The two stoichiometric compositions had similar k values of approximately 114 W/m-K at the lowest temperatures (30–50°C). The values steadily decreased to approximately 103 W/m-K at 400°C. The similarities were likely due to the nearly identical processing procedures (i.e., HM of the constituents and densification via SPS). Unless higher purity starting materials were used, the compositions by Gasch et al. and by Zhang et al. are likely the best representation of the intrinsic k for polycrystalline HfB₂.

Interestingly, the change in stoichiometry studied by Zhang et al. had a significant effect on k. The k for the boron-rich composition was approximately 6% higher than the stoichiometric composition. In contrast, the boron deficient material had k values that were approximately 6% lower. No explanation was provided for the reported differences, but both the electron and phonon contributions to k were higher for HfB_2.1 [18].

9.2.4 Conclusions Regarding Phase-Pure ZrB₂ and HfB₂

The k values for nominally pure ZrB₂ and HfB₂ were affected more strongly by purity than grain size or porosity. Porosity decreased k; however, as these materials are mainly intended for use as structural ceramics, densities are typically maximized. Grain size had little effect on k for ZrB₂ as either higher or lower k materials were produced with fine grain sizes, with larger-grained material bridging the gap in conductivity. As these materials are all single phase, changes in purity were due to elements in SS. Reaction processing from pure elemental constituents (>99.7% metals, 99% B) has been used to produce the highest conductivity polycrystalline ZrB₂ and HfB₂. Comparisons with single-crystal ZrB₂ values (117–131 W/m-K after averaging a- and c-axis values) revealed that the k for ZrB₂ produced by reactive SPS were in good agreement with values predicted from single-crystal values, which may indicate that grain boundary resistances for ZrB₂ are low.

Due to their proposed use temperatures, many studies report k values up to 2000°C. In contrast, SX ZrB₂ was only evaluated to 200°C. Evaluation up to 2000°C would have been very valuable. Research on transition metal additions indicated the effect of SSs on k. Depending on the element (i.e., W has a much larger effect than Ti) and concentration, the k values of ZrB₂ (and HfB₂ by corollary) can change significantly. With SS additions, k drops more dramatically at room temperature compared to higher temperatures. This results in a change in slope in k versus T from negative to positive. The observed changes in k are most likely tied to changes in bonding and distortions in the lattice as they have been shown to correlate to shifts in measured lattice parameters.

9.3 ZrB₂ and HfB₂ Composites

9.3.1 Thermal Conductivity of ZrB₂ Composites

Researchers have used additions of secondary, tertiary, and quaternary phases to improve densification and important properties (i.e., strength, thermal shock resistance, and oxidation performance) of ZrB₂ ceramics. Along with these possible improvements, the transport properties also influence the thermal management performance and further contribute to thermal shock behavior. Therefore, the k of the final ceramic is important in its own right and has been studied to understand the effects of additives. Table 9.4 (ZrB₂–SiC), Table 9.5 (other second-phase additions), and Table 9.6 (tertiary and quaternary phases) summarize the ZrB₂ composites that have been evaluated.

Table 9.4. Thermal conductivity of ZrB₂-SiC with information on starting/final materials, processing, and density

SiC vol%	Powder processing	Purity (%)	Densification technique	Special notes(gs in µm)	ρrelative	Thermal conductivity (k, W/m-K)		Ref.
						25 °C	at max temp
20%	VM w/plastic	—	HP	—	100%	—	78 at 1000 °C	[8]
20%	VM w/plastic	—	HP	—	100%	—	69 at 2000 °C	[8]
10%	BM w/SiC	98.0 ZrB2, 99.0 SiC	SPS at 1900 °C	gsZrB2 = 3.5, gsSiC = 8.9	99.4%	130	—	[16]
20%	BM w/SiC	98.0 ZrB2, 99.0 SiC	SPS at 1900 °C	gsZrB2 = 3.4, gsSiC = 8.9	99.6%	131	—	[16]
30%	BM w/SiC	98.0 ZrB2, 99.0 SiC	SPS at 1900 °C	gsZrB2 = 3.2, gsSiC = 8.9	99.6%	138	—	[16]
5%	AM w/SiC	99.5 ZrB2, 99.9 SiC	HP at 2000 °C	gsZrB2 = 2, gsSiC = 8	101%	—	82 at 2000 °C	[14]
10%	AM w/SiC	99.5 ZrB2, 99.9 SiC	HP at 2000 °C	gsZrB2 = 1, gsSiC = 4	101%	—	80 at 2000 °C	[14]
20%	AM w/SiC	99.5 ZrB2, 99.9 SiC	HP at 2000 °C	gsZrB2 = 3, gsSiC = 4	100%	—	67 at 2000 °C	[14]
20%	BM w/WC	>99.5	HP at 2000 °C	—	>98%	90	72 at 1200 °C	[41]
20%	PBM w/Si3N4	—	HP at 1900 °C	—	?	95	76 at 400 °C	[42]
14%	Coarse mixing	99.5 ZrB2, 99 SiC	Arc melting	Eutectic structure	100%	95	83 at 750 °C	[43]
31%	Coarse mixing	99.5 ZrB2, 99 SiC	Arc melting	Eutectic structure	100%	77	64 at 770 °C	[43]
50%	Coarse mixing	99.5 ZrB2, 99 SiC	Arc melting	Eutectic structure	100%	72	44 at 760 °C	[43]
20%	BM w/SiC	—	HP at 1900 °C	—	98.8%	68	56 at 1000 °C	[40]
20%	PBM w/WC	99.5 ZrB2, 99.8 SiC	SPS	—	99.3%	—	79 at 420 °C	[18]
10%	BM w/ZrO2	99 ZrB2, 99 SiCw	HP at 1800 °C	gs = 4.5, SiC whiskers	98%	—	61 at 1800 °C	[44]
20%	BM w/ZrO2	99 ZrB2, 99 SiCw	HP at 1800 °C	gs = 4, SiC whiskers	98%	—	61 at 1800 °C	[44]
30%	BM w/ZrO2	99 ZrB2, 99 SiCw	HP at 1800 °C	gs = 4, SiC whiskers	96%	—	58 at 1800 °C	[44]
20%	BM w/SiC	>96 ZrB2, >99 SiC	PS at 2100 °C	gsZrB2 = 10, gsSiC = 8	99%	94	62 at 1200 °C	[45]
30%	AM w/WC	—	HP at 1900 °C	—	100%	62	51 at 1330 °C	[15]
20%	BM w/Si3N4	98 ZrB2, 98.5 SiC	HP at 1900 °C	gsZrB2 = 2.7	>99%	94	77 at 1200 °C	[46]
20%	PBM w/SiC	—	HP at 2000 °C	gsZrB2 = 2.6, gsSiC = 1.3	99.8%	107	—	[47]

Table 9.5. Thermal conductivity of two-phase ZrB₂-based composites with information on starting/final materials, processing, and density

Material (vol% or otherwise stated)	Powder processing	Purity (%)	Densification technique	Special notes(gs in µm)	ρrelative (%)	Thermal conductivity(k, W/m-K)		Ref.
						25 °C	at max temp
26%C	—	—	HP	gsZrB2 = 20	100a	—	75 at 900 °C	[11]
50%C	VM w/plastic	—	HP	—	100	—	64.0 at 1000 °C	[8]
50%C	VM w/plastic	—	HP	—	100	—	64.0 at 2000 °C	[8]
1 wt%C	AM w/WC	—	HP at 1900 °C	gsZrB2 = 2.4	100	—	53.9 at 2000 °C	[20]
3 wt%C	AM w/WC	—	HP at 1900 °C	gsZrB2 = 1.8	100	—	63.7 at 2000 °C	[8]
10%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	88	—	[48]
20%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	83	—	[48]
30%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	81	—	[48]
40%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	76	—	[48]
2.3%MoSi2	BM w/ZrO2	—	HP at 1900 °C	gsZrB2 = 1–10	?	66	74.5 at 1200 °C	[49]
50%ZrC	—	—	HP	gsZrB2 = 10	100a	—	54.3 at 905 °C	[11]
5%ZrC	—	—	HP	gsZrB2 = 27	100a	—	79.3 at 910 °C	[11]
20%ZrC	“Vibrogrinding”	—	PS 2100 °C	gsZrB2 = 5, gsZrC = 6, 1% Ni sintering aid	100a	—	76.1 at 2200 °C	[12]
40%ZrC	“Vibrogrinding”	—	PS 2100 °C	gsZrB2 = 3, gsZrC = 4, 1% Ni sintering aid	100a	—	55.5 at 2200 °C	[12]
45%ZrC	“Vibrogrinding”	—	PS 2100 °C	gsZrB2 = 1, gsZrC = 3, 1% Ni sintering aid	100a	—	53.2 at 2150 °C	[12]
10%ZrSi2	BM w/SiC	—	PS at 1650 °C	—	100a	107	—	[50]
20%ZrSi2	BM w/SiC	—	PS at 1650 °C	—	100a	97	—	[50]
30%ZrSi2	BM w/SiC	—	PS at 1650 °C	—	100a	90	—	[50]
40%ZrSi2	BM w/SiC	—	PS at 1650 °C	—	100a	74	—	[50]

VM, vibratory mixing.

^a Corrected to 100% in this work.

Table 9.6. Thermal conductivity of three- and 4-phase ZrB₂-based composites with information on starting/final materials, processing, and density

Material (vol% orotherwise stated)	Powder processing	Purity (%)	Densification technique	Special notes (gs in µm)	ρrelative (%)	Thermal conductivity k (W/m-K)		Ref.
						25 °C	at max temp
5%SiC + 20%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	82	—	[48]
10%SiC + 20%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	94	—	[48]
20%SiC + 20%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	90	—	[48]
30%SiC + 20%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	98	—	[48]
5%SiC + 40%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	81	—	[48]
10%SiC + 40%MoSi2	BM w/SiC	—	HP at 1800 °C	—	100a	85	—	[48]
10.7%SiC + 8.9%B4C	BM w/B4C	—	PS, HIP at 2000 °C, 1800 °C	—	100	94	57 at 1940 °C	[48]
21.9%SiC + 7.8%B4C	BM w/B4C	—	PS, HIP at 2000 °C, 1800 °C	—	100	90	47.3 at 2000 °C	[51]
48.7%SiC + 5.1%B4C	BM w/B4C	—	PS, HIP at 2000 °C, 1800 °C	—	100	101	33.4 at 1950 °C	[51]
18%SiC + 10%C	VM w/plastic	—	HP	—	100	—	73.6 at 1000 °C	[8]
18%SiC + 10%C	VM w/plastic	—	HP	—	100	—	61.1 at 2000 °C	[8]
14%SiC + 30%C	VM w/plastic	—	HP	—	100	—	57.7 at 1000 °C	[8]
14%SiC + 30%C	VM w/plastic	—	HP	—	100	—	59.2 at 2000 °C	[8]
20%SiC + 2wt%CNT	PBM w/Si3N4	—	HP at 1900 °C	—	?	91	72.4 at 400 °C	[42]
20%SiC + 15% Cg	BM w/SiC	—	HP at 1900 °C	—	100	81	58.7 at 1000 °C	[40]
30%SiC(10%SiCw) + 10% Cg	BM w/ZrO2	99.5 ZrB2/SiCp, 99 SiCw/Cg	HP at 1800 °C	—	99.6	100	65.5 at 1200 °C	[52]
20%SiC + 10% Cf	PBM w/SiC	—	HP at 2000 °C	gsZrB2 = 2.55, gsSiC = 1.47	99.6	95	—	[47]
20%SiC + 20% Cf	PBM w/SiC	—	HP at 2000 °C	gsZrB2 = 2.79, gsSiC = 1.65	99.4	83	—	[47]
20%SiC + 30% Cf	PBM w/SiC	—	HP at 2000 °C	gsZrB2 = 2.82, gsSiC = 1.35	99.5	68	—	[47]
2%MoSi2 + 15%SiC	BM w/ZrO2	—	HP at 1820 °C	gsZrB2 = 1–5	?	62	64.5 at 1200 °C	[49]
20%SiC + 5%Si3N4 + 20%ZrC	BM w/WC	All powder >99.5	HP at 2000 °C	—	>98	82	50.0 at 1200 °C	[41]
20%SiC + 5%Si3N4	BM w/WC	All powder >99.5	HP at 2000 °C	—	>98	103	61.2 at 1200 °C	[41]
15%SiC + 2%Si3N4	BM w/SiC	—	HP at 1900 °C	—	98.1	80	58 at 1500 °C	[53]
5%SiC + 5%TaSi2	BM w/SiC	—	SPS at 1800 °C	—	>98.2	42	—	[54]
10%SiC + 5%TaSi2	BM w/SiC	—	SPS at 1800 °C	—	>98.2	40	—	[54]
20%SiC + 5%TaSi2	BM w/SiC	—	SPS at 1800 °C	—	>98.2	43	—	[54]
30%SiC + 5%TaSi2	BM w/SiC	—	SPS at 1800 °C	—	>98.2	50	—	[54]
10%SiC + 10%TaSi2	BM w/SiC	99 (all powders)	SPS at 1600 °C	—	?	43	—	[55]
20%SiC + 10%TaSi2	BM w/SiC	99 (all powders)	SPS at 1600 °C	—	?	45	—	[55]
20%SiC + 20%TaSi2	BM w/SiC	99 (all powders)	SPS at 1600 °C	—	?	45	—	[55]
30%SiC + 10%TaSi2	BM w/SiC	99 (all powders)	SPS at 1600 °C	—	?	40	—	[55]
30%SiC + 20%TaSi2	BM w/SiC	99 (all powders)	SPS at 1600 °C	—	?	46	—	[55]
10%SiC + 10%TaSi2	BM w/SiC	99 (all powders)	SPS at 1800 °C	—	?	46	—	[55]
20%SiC + 10%TaSi2	BM w/SiC	99 (all powders)	SPS at 1800 °C	—	?	47	—	[55]
20%SiC + 20%TaSi2	BM w/SiC	99 (all powders)	SPS at 1800 °C	—	?	53	—	[55]
30%SiC + 10%TaSi2	BM w/SiC	99 (all powders)	SPS at 1800 °C	—	?	47	—	[55]
30%SiC + 20%TaSi2	BM w/SiC	99 (all powders)	SPS at 1800 °C	—	?	51	—	[55]
33.3%SiC + 33.3%ZrC	BM w/SiC	—	SPS at1950°C	—	98.7	73	—	[56]
15%SiC + 15%ZrC	BM w/SiC	—	SPS at1950°C	—	98.5	86	—	[56]
30%SiC + 15%ZrC	BM w/SiC	—	SPS at1950°C	—	98.7	89	—	[56]
15%SiC + 30%ZrC	BM w/SiC	—	SPS at1950°C	—	98.8	74	—	[56]
18%SiC + 8%ZrC	BM w/ZrO2	—	SPS at 2000 °C	—	85.7	71	—	[19]
25%SiC + 12%ZrC	BM w/ZrO2	—	SPS at 2000 °C	—	81–90	71	—	[19]
33%SiC + 16%ZrC	BM w/ZrO2	—	SPS at 2000 °C	—	94–99	60	—	[19]
4%SiC + 35%ZrC	BM w/ZrO2	—	SPS at 2000 °C	—	97–99	62	—	[19]

PBM, planetary ball milling.

^a Corrected to 100% in this work.

9.3.1.1 ZrB₂-X Composites.

ZrB₂–SiC ceramics have been studied by a host of researchers (Table 9.4) [8, 14–16, 18, 46–46], along with other ZrB₂ based composites with additions of C [8, 11, 20], MoSi₂ [48, 49], ZrC [11, 12], or ZrSi₂ [50] (Table 9.5). The compositions span a large range of additive contents. However, the present review was limited to additions less than or equal to 50 vol% so the materials may be regarded as ZrB₂-based. Values of k were plotted as a function of temperature for ZrB₂–SiC (Fig. 9.7)² and ZrB₂–X (Fig. 9.8), where X is C, MoSi₂, or ZrC. All of the studies but Clougherty et al. [8] measured D and calculated k using Equation 9.1.

Figure 9.7. Thermal conductivity as a function temperature for ZrB₂ with SiC additions ranging from 5 to 50 vol%. (Note: Clougherty changed testing methods at 1000°C.) ^aData corrected for ρ [8, 14, 15, 18, 36, 41–45].

Figure 9.8. Thermal conductivity as a function of temperature for ZrB₂ with additions of carbon, MoSi₂, or ZrC. Clougherty changed testing methods at 1000°C. [8, 11, 12, 20, 21, 48].

ZrB₂–SiC

As seen in Table 9.4 and Figure 9.7, SiC additions to ZrB₂ range from 5 to 50 vol%. In general, k decreases as SiC content increases. Therefore, SiC appears to act as a lower k phase (with respect to ZrB₂). However, whether this is an expected outcome depends on the inherent conductivity of SiC in the composite.

The room-temperature k of high-purity single-crystal SiC (ppm impurity levels) can be as high as 490 W/m-K, with a theoretical upper limit predicted to be as high as 700 W/m-K. This high k is attributed to phonon transport since SiC is not electrically conductive [57]. Despite high k values at room temperature, k of SiC drops off quickly with increasing impurity content due to phonon scattering [57–62]. When N is present as an impurity, SiC can have k as low as 60 W/m-K at room temperature [57, 63]. Further, SSs with elements like Be, B, and Al, which are introduced from common sintering aids, can result in SiC ceramics with k values in the range of 250–270 W/m-K [59, 64], 100–150 W/m-K, and 80–120 W/m-K [62], respectively. In addition, k decreases with increasing amounts of each constituent [61]. Given the effect of impurities, SiC in ZrB₂–SiC composites could be below the k of ZrB_2, due to starting impurities or by picking up B from the surrounding diboride.

Other explanations are possible for observed reductions in k values for ZrB₂–SiC. For example, microcracks can form due to residual stresses from SiC particle or cluster sizes greater than approximately 11.5 µm [65]. Microcracking is similar to porosity, reducing k by decreasing the effective conduction path, thus disrupting phonon and electron transport. Depending on SiC distribution, size, morphology, and volume fraction, microcracks can be present in the final ceramic, but have rarely been discussed. Another explanation for a drop in k due to SiC additions is the presence of a second phase at the grain boundaries. An SiO₂-based phase can form at the grain boundaries in ZrB₂–SiC, but appropriate additions of B₄C and C can remove it prior to densification [66].

Loehman et al. reported one of the best examples of decreasing k with increasing SiC content in diborides. As SiC content increase from 5 vol% SiC to 20 vol% SiC, k at 250°C dropped from 112 to 100 W/m-K [14]. For their ceramic containing 5 vol% SiC, k was 112 W/m-K, which is quite high and similar to the value of 110 W/m-K reported by Zhang et al. [18] for reaction-processed ZrB₂. As the ZrB₂ comprises the majority of this material, the high conductivity may be an indication of the diboride purity.

For SiC contents of 20 vol%, a majority of k values [8, 14, 41, 46] lie in a range (highlighted in Fig. 9.7) between 88–102 W/m-K at 200°C and 66–70 W/m-K at 2000°C. Because reported k values vary for both ZrB₂ and SiC, it is difficult to define one k value for ZrB₂–20% SiC. This region is likely adequate based on the convergence of multiple research efforts into a narrow set of k values. Four ceramics containing 20 vol% SiC fell below this region [18, 40, 42, 45] and deserve further discussion. Ceramics produced by both Wang et al. [40] and Zhang et al. [45] likely had microcracking. Image analysis by both researchers showed that SiC cluster sizes were greater than or equal to 11.5 µm, indicating that both materials had SiC inclusions large enough to promote microcracking. In addition, modulus values reported were lower than expected at 480 GPa (~492 GPa if corrected for porosity) for Wang and 374 GPa (~382 GPa if corrected for ~1% porosity) for Zhang, further supporting the case for microcracking.

Microcracking does not appear to be an issue with the other two lower k compositions. Hence, purity may be the cause. Zhang et al. milled the ZrB₂ and SiC using WC media [18]. As discussed earlier, W decreases the k of ZrB₂. Finally, the material evaluated by Tian et al. used lower purity SiC (98.5%) and Si₃N₄ media for milling. Nitrogen [57], and specifically Si₃N₄, [67] contamination have been shown to decrease the conductivity of SiC.

Arc melted ZrB₂–SiC evaluated by Tu et al. may be the least practical material from a production standpoint, but it is a unique material from a scientific perspective. The material containing 50 vol% SiC had a very fine eutectic microstructure with ZrB₂ and SiC features approximately 100 nm wide [43]. The fine microstructure likely led to a higher phonon scattering (compared to other traditionally processed ZrB₂–SiC compositions), which may have resulted in a steeper slope in k with temperature.

ZrB₂-(C, MoSi₂, ZrC, and ZrSi₂).

In addition to diboride–SiC composites, k has been evaluated for two-phase ZrB₂-based compositions containing C [8, 11, 20], MoSi₂ [48, 49], ZrC [11, 12], and ZrSi₂ [50] (Table 9.5). Figure 9.8 shows k as a function of temperature for these materials. Note that the disilicides by Guo [48, 50] were only evaluated at room temperature. A broad range of k values were reported depending on the specific additive and its amount. For ZrB₂ containing 20 vol% C, the material from Clougherty et al. is the only one with k values that decreased with temperature. Like other materials evaluated by Clougherty, this one had a discontinuity at 1000°C where the test method switched from a steady-state cut bar method to calculation from D, but overall k dropped from approximately 85 W/m-K at 100°C to approximately 64 W/m-K at 2000°C [8]. The form of the carbon in the material, either before or after processing, is not discussed. Given the small decease in k at room temperature between this material and the ZrB₂ studied by Clougherty, the C is likely to be graphitic in nature. Amorphous carbon has a lower k than ZrB₂ and would have decreased k considerably more than for graphitic carbon, which has a k value about the same as ZrB₂ [68, 69].

Thompson et al. [20] and Andrievskii et al. [11] also investigated ZrB₂ with carbon additions. The k values increased with increasing temperature, indicating that the diboride was the primary influence on k. For Andrievskii et al., 26 vol% carbon was added as graphite, which reduced the conductivity approximately 10 W/m-K at all temperatures [11]. Thompson et al. added carbon in the form of phenolic resin, resulting in 1 and 3 wt% amorphous carbon after pyrolysis. After hot pressing, graphitic carbon was identified at the grain boundaries. ZrB₂ containing 1 wt% C had k values lower than pure ZrB₂. They hypothesized that graphite was aligned, with its basal plane parallel to the grain boundaries, thus disrupting phonon transport between ZrB₂ grains. However, k did not continue to decrease when C content was increased to 3 wt% C due to the segregation of W into ZrC after testing up to 2000°C. The ZrC phase, produced due to carbon additions, served as a sink for W, reducing the W content of the ZrB₂ grains and boosting the overall k of the composite [20].

The addition of ZrC to ZrB₂ typically reduces k. Andrievskii et al. evaluated both 5 and 50 vol% ZrC from 100 to 900°C, while Fridlender et al. added 20, 40, and 45 vol% ZrC and tested up to 2150 or 2200°C. The higher ZrC additions resulted in k values between 40 and 55 W/m-K, while the lower additions resulted in k values between 64 and 80 W/m-K, despite the different testing temperatures and compositions. The differences, and the decrease in k with increasing ZrC content, were probably due to the variability in k values for ZrC (20–45 W/m-K at room temperature and increasing to 30–50 W/m-K at 2000°C), which are well below the values for pure ZrB₂ [70].

Monteverde et al. evaluated ZrB₂ containing 2.3 vol% MoSi₂ as a sintering aid. The k for this material was approximately 66 W/m-K at 25°C and increased to approximately 75 W/m-K at 1200°C [49]. If it were only a second phase, this small addition would not likely decrease k of the composite much, even though the k of MoSi₂ is significantly lower than ZrB₂ [70–72]. However, Mo also dissolves into ZrB₂, which decreases k and results in k values that increase as temperature increases [23]. Therefore, even small additions of MoSi₂ have a significant effect on k.

9.3.1.2 ZrB₂ Ternary- and Quaternary-Phase Composites.

Several three-phase, and even a few four-phase, ZrB₂-based composites have been produced and evaluated (Table 9.6). One common feature of these multiphase composites is the addition of SiC, which was present in varying amounts. Once past this common addition, the tertiary and quaternary phases were MoSi₂ [48], B₄C [51], C_g [8, 40, 52], CNTs [42], carbon fiber (C_f) [47], SiC whiskers (SiC_w) [52], Si₃N₄ [41, 53], TaSi₂ [55], or ZrC [41, 56, 73]. Only two studies, [8] and [73], evaluated k other than by using Equation 9.1, and all but one study ([51] performed PS) employed HP or SPS for densification. From Table 9.6, k was evaluated beyond room temperature for 14 of these materials (Fig. 9.9) [8, 40–42, 49, 51–53].³

Figure 9.9. Thermal conductivity as a function of temperature for ZrB₂-SiC-based composites with additions of, B₄C, C (elemental, nanotubes (CNT) and graphite (Cg)), MoSi₂, Si₃N₄, SiCw (whiskers), or ZrC. Clougherty changed testing methods at 1000°C. ^aData corrected for ρ [8, 18, 36, 41, 43, 48, 60, 72].

Because of the complex effects of the addition of two, or even three, phases to ZrB₂, it is difficult to understand how each species affects k. Therefore, instead of evaluating k for each composition, more generalities will be surmised with some specific examples brought out when clear. Overall, k values presented in Figure 9.9 lie between 80 and 103 W/m-K at room temperature. Further, all of k values decrease with increasing temperature, except for ZrB₂ with 15 vol% SiC and 2 vol% MoSi₂ [49]. This is likely due to the dissolution of Mo from the MoSi₂ into ZrB₂. At the highest test temperatures (1940–2000°C), the range of reported k values increased. This is due to the low k (31 W/m-K) of ZrB₂ containing 48.7 vol% SiC and 5.1 vol% B₄C produced by Speyer et al. [51] The k of this material was likely diminished at high temperatures compared to the other materials due to a higher SiC content and the addition of B₄C, which has a k of approximately 7 W/m-K above 1700°C [74]. At lower temperatures (<1000°C), the four-phase composite containing 20 vol% SiC, 5 vol% Si₃N₄, and 20 vol% ZrC had the lowest k and a negative slope [41]. The presence of ZrC, which has an increase in k with temperature [75], may be the reason that this material does not have as negative a slope as Speyer's composition.

The highest k was reported for a composite containing 18 vol% SiC and 10 vol% C (presumably graphitic C because of the higher conductivity). At 100°C, k was approximately 115 W/m-K, which approached the value for the reaction-processed ZrB₂ by Zhang et al. [18]. However, compared to the pure ZrB₂, k of the 18% SiC–10% C composition dropped off much faster with temperature, eventually reaching 74 W/m-K at 2000°C.

9.3.2 Thermal Conductivity of HfB₂ Composites

Tables 9.7 and 9.8 summarize k values reported for HfB₂ composites. Like ZrB₂ composites (Tables 9.4 and 9.5), HfB₂ composites were primarily HfB₂-SiC (Table 9.7). [8, 14, 18, 28, 37, 41, 54, 76] The other HfB₂ composites are two-phase materials with additions of boron carbides (B₃C or B₁₂C) [78] or C (graphite) [8], or three-phase materials containing SiC and C [8], SiC and B₄C [76], or SiC and TaSi₂ [54] (Table 9.8). Densification was carried out using either HP or SPS for materials in Table 9.7 while only HP was used for those in Table 9.8. Besides the steady-state evaluation of k performed by Clougherty et al. [8] (below 1000°C), all of the other researchers evaluated D to calculate k. Plots of k as a function of temperature for HfB₂–SiC composites, and the other two- and three-phase materials, are included in Figures 9.10 and 9.11.⁴

Table 9.7. Thermal conductivity of HfB₂–SiC with information on starting/final materials, processing, and density

SiC Vol%	Powder processing	Purity (%)	Densification Technique	Special Nnotes	ρrelative (%)	Thermal Conductivity k (W/m-K)		Ref.
						25 °C	at max temp
20%	VM w/plastic	—	HP	—	100	—	62 at 1000 °C	[8]
30%	VM w/plastic	—	HP	—	100	—	58 at 1000 °C	[8]
30%	VM w/plastic	—	HP	—	100	—	60 at 2000 °C	[8]
5%	VM w/plastic	—	HP	—	100	—	59 at 1000 °C	[8]
5%	VM w/plastic	—	HP	—	100	—	68 at 2000 °C	[8]
20%	PBM or AM w/WC	98.8	HP	25 mm billet	100	42	49 at 2000 °C	[28]
20%	PBM or AM w/WC	98.8	HP	50 mm billet	100	45	51 at 2000 °C	[28]
2%	BM w/ZrO2	99.5 ZrB2, 99 SiC	HP at 2000 °C	—	101	—	77 at 2000 °C	[14]
20%	BM w/ZrO2	99.5 ZrB2, 99.9 SiC	HP at 2000 °C	—	103	—	65 at 2000 °C	[14]
5%	BM w/ZrO2	99.5 ZrB2, 99 SiC	HP at 2000 °C	—	101	—	76 at 2000 °C	[14]
20%	PBM w/WC	98.8	HP	—	99.7	79	76 at 600 °C	[37]
20%	PBM w/WC	—	SPS	—	99.1	78	73 at 600 °C	[37]
20%	BM w/ZrO2	97.6 HfB2, 99.5 SiC, 98.1 B4C	HP at 2000 °C	gsHfB2 = 6 µm	98.4	—	55 at 1800 °C	[76]
20%	—	—	HP at 2150 °C	—	99.1	—	33 at 1800 °C	[77]
5%	PBM w/WC	99.5 HfB2, 99.8 SiC	SPS	—	99	—	104 at 425 °C	[18]
5%	HM Hf + B + SiC	99.8 Hf, 99 B, 99.8 SiC	rxn SPS	—	101	—	110 at 425 °C	[18]
20%	BM w/WC	All powder >99.5	HP at 2000 °C	—	>96	141	86 at 1200 °C	[41]
20%	—	—	—	SHARP-B2 material	100	107	69 at 1800 °C	[28]

Table 9.8. Thermal conductivity of HfB₂-based composites with information on starting/final materials, processing, and density

Material (vol%)	Powder processing	Purity (%)	Densification technique	Special notes (gs in µm)	ρrelative (%)	Thermal conductivity k (W/m-K)		Ref.
						25 °C	at max temp
20%C	VM w/plastic	—	HP	—	100	—	70 at 1000 °C	[8]
20%C	VM w/plastic	—	HP	—	100	—	69 at 2000 °C	[8]
12%B12C	BM w/ZrB2	99.5 HfB2, 99% B	HP at 2000 °C	—	99.1	106	89 at 1000 °C	[78]
12%B3C	BM w/ZrB2	99.5 HfB2, 99% B	HP at 2000 °C	—	99.1	112	84 at 2000 °C	[78]
20%SiC + 10%B4C	BM w/ZrO2	—	HP at 1850 °C	gsZrB2 = 4	98.7	—	35 at 1800 °C	[76]
18%SiC + 10%C	VM w/plastic	—	HP	—	100	—	66 at 1000 °C	[8]
18%SiC + 10%C	VM w/plastic	—	HP	—	100	—	61 at 2000 °C	[8]
20%SiC + 20%Cf	PBM w/SiC	—	HP at 2100 °C	gsHfB2 = 2.6, gsSiC = 1.3	99.5	94	—	[79]
20%SiC + 30%Cf	PBM w/SiC	—	HP at 2100 °C	gsHfB2 = 2.6, gsSiC = 1.5	99.3	74	—	[79]
30%SiC + 5%TaSi2	BM w/SiC	—	SPS at 1800 °C	—	>98.2	118	—	[54]
5% SiC + 5%TaSi2	BM w/SiC	—	SPS at 1800 °C	—	>98.2	78	—	[54]
10% SiC + 5%TaSi2	BM w/SiC	—	SPS at 1800 °C	—	>98.2	90	—	[54]
20% SiC + 5%TaSi2	BM w/SiC	—	SPS at 1800 °C	—	>98.2	87	—	[54]

Figure 9.10. Thermal conductivity as a function of temperature for HfB₂–SiC with SiC contents ranging from 2 vol% to 30 vol%. Clougherty changed testing methods at 1000°C. ^aData corrected for ρ [8, 14, 18, 28, 36, 39, 78].

Figure 9.11. Thermal conductivity as a function of temperature for HfB₂ with additions of B_xC (x = 3 or 12), C, or SiC with C or B₄C. Clougherty changed testing methods at 1000°C. ^aData corrected for ρ [8, 18, 54, 75].

9.3.2.1 HfB₂–SiC.

Unlike ZrB₂–SiC (Fig. 9.7), where increasing SiC additions decreased k, HfB₂–SiC does not show the same trends (Fig. 9.10). Loehman et al. [14] and Clougherty et al. [8] evaluated SiC additions of 2, 5, and 20 vol%, and 5, 20, and 30 vol%, respectively. These systematic studies should be useful for evaluating trends. However, despite the variation in SiC content, k values did not vary considerably (≤12 W/m-K for Loehman et al. and ≤16 for Clougherty et al.). Further, no distinct trends were apparent with respect to SiC content, with both data sets exhibiting several crossover points.

Conductivities were reported for three HfB₂–20 vol% SiC composites in two separate studies ([28] and [37]) by Gasch et al. The highest k material was produced for the SHARP-B2 flight experiment. At room temperature, k was approximately 107 W/m-K and testing was performed up to 1800°C, where k dropped to 69 W/m-K. The other material in this study had significantly lower k values at room temperature and showed an increase in k with temperature (45 W/m-K at 25°C and 51 W/m-K at 1800°C) [28]. From the other Gasch study, k decreased slightly from 79 to 76 W/m-K between room temperature and 600°C [37]. Little information was given about the processing conditions for the SHARP-B2 material, so it is hard to say why its k was higher. However, powders for the lower k compositions were milled with WC media, which led to marked decreases in k for other diborides, as discussed earlier.

Similar to ZrB₂–SiC materials, microcracking appears to also be a consideration for HfB₂–SiC. The HfB₂–20 vol% SiC evaluated by Weng et al. has the lowest k of all compositions in Figure 9.10 (44–33 W/m-K from 300 to 1800°C) and the reported modulus is only 346 GPa (352 GPa, corrected for ρ), which is much lower than expected based on the composition [77]. Other researchers report modulus values of approximately 520 GPa for HfB₂ [28, 80] and 437–455 GPa for SiC. Conditions for microcracking in HfB₂–SiC composites have not been reported, but based on the modulus, and what appear to be cracks, microcracking is a likely contributing factor.

The three HfB₂–SiC composites with the highest k values are all at least as high as the pure reaction-processed HfB₂ evaluated by Zhang et al. [18] Both 5 vol% SiC compositions were characterized in the same study as the pure HfB₂, and the more conductive of the two was also produced using reaction processing. The purity of the reaction-processed diboride, along with the possibility of SiC sequestering impurities, resulted in HfB₂–5 vol% SiC with a k of 137 W/m-K at 50°C [18]. Overall, the HfB₂–20 vol% SiC presented by Mallik et al. was similar to the high k HfB₂ containing 5 vol% SiC at 25°C (141 W/m-K). However, k decreased to that of the reaction-processed HfB₂ by 200°C (~110 W/m-K) [41]. Of all the HfB₂–SiC materials, the material by Mallik is surprising based on what was observed for increasing SiC additions to ZrB₂ (Fig. 9.7) (especially given that the material was milled with WC) but no explanation is provided for the remarkably high k of this ceramic.

9.3.2.2 Other Secondary and Tertiary Additions to HfB₂.

 The k values for HfB₂ with other second and third phases are plotted in Figure 9.11. The k values vary from as low as approximately 35 W/m-K to as high as approximately 115 W/m-K. The highest k values were reported by Brown-Shaklee et al. [78]. A slight variation in the thermal conductivity of these HfB₂–B_xC composites was observed, as can be seen by the decrease in conductivity of the 12 vol% B₁₂C material between 3 and 7 W/m-K across the temperature range. The researchers mentioned that the differences in k were less than the uncertainty of the test. It was also concluded that the purity of the materials in this study led to the high k values, nearly matching the reaction-processed material by Zhang et al. [18].

Microcracking was identified in HfB₂ containing 20 vol% SiC and 10 vol% B₄C by Weng et al. [76]. Therefore, k of this composite should be higher without such flaws. The final two compositions characterized by Clougherty et al. (HfB₂–20 vol% C (presumably graphite) and HfB₂–18 vol% SiC–10 vol% C) had similar k values at 100°C (84–85 W/m-K). Their k values deviated more at 2000°C (~10 W/m-K), but based on the trend for HfB₂–20% C, and the apparent discontinuity in values from switching testing methods at 1000°C, this difference was probably not significant. These materials were HfB₂ analogues to ZrB₂-based materials evaluated within the same study [8]. A comparison with the ZrB₂ compositions (Figs. 9.7 and 9.8) shows little difference between the 20% SiC compositions. At 2000°C the 18% SiC–10% C composites were identical (~61 W/m-K) but at 100°C the k values differed by 32 W/m-K with the HfB₂ composition having the lower k. This is unexpected since ZrB₂ and HfB₂ have similar behavior and was likely due to composition (purity) and microstructure, which could be evaluated to elucidate the cause for unexpected changes.

9.3.3 Conclusions Regarding Composites

Second-phase additions decrease k for diborides. The same decrease in k with temperature, as expected for the pure diborides, is observed, but the addition of second and third phases with lower k reduced k for the composites. One exception was the addition of MoSi₂, which changed the slope of k as a function of temperature, as observed for dissolution of other transition metals in diborides.

While the addition of SiC could increase the k of composites due to its high intrinsic k, all reported SiC additions lower conductivity, likely due to the purity of SiC used in the diboride systems. Additions of SiC can also be detrimental to k due to microcracking. The threshold for SiC particle or cluster size in ZrB₂ is approximately 11.5 µm. Larger sizes lead to microcracking in ZrB₂–SiC, but no research has been examined microcracking in HfB₂. Overall, the conductivities of the composites are still primarily controlled by the diboride. Whereas k may decrease with the addition of other phases, densification behavior, strength, thermal shock resistance, or oxidation resistance are often improved by these additions. Hence, understanding the factors that control k may enable design of compositions to balance the trade-offs for specific applications.

9.4 Electron and Phonon Contributions to Thermal Conductivity

The total thermal conductivity of the diborides of zirconium and hafnium, (k_t, i.e., the experimental values discussed to this point) can be separated into the electron (k_e) and phonon (k_p) contributions. Several researchers have used Equations 9.4 and 9.5 to evaluate k_t ZrB₂ [15, 18, 20, 21, 81], ZrB₂–SiC [15, 18, 46, 81], HfB₂ [18], and HfB₂–SiC [18, 81].

(9.4)

(9.5)

Equation 9.5, the Wiedemann-Franz law, was used to estimate k_e from measured electrical resistivity (ρ) at temperature (T) using the theoretical Lorenz number (L_o = 2.45e⁻⁸ W/Ω-K⁻²), which was derived for metals.

9.4.1 ZrB₂ and HfB₂

Values of k_t, k_e, and k_p for several ZrB₂ and HfB₂ ceramics are plotted as a function of temperature in Figure 9.12. Comparing k_t and k_e, the positive or negative trends with temperature for k_e are identical to those of the k_t, except for data from Tye and Clougherty⁵ [81]. On the other hand, k_p, in general, shows little variation with temperature.

Figure 9.12. Total, electron, and phonon thermal conductivities as a function of temperature for ZrB₂ and HfB₂. ^aData corrected forρ [15–17, 20, 79].

The relatively small differences between k_e and k_t (excluding Tye and Clougherty and Zimmermann et al. [15]) indicate that electrons are the dominant thermal carrier for diborides. No indication is given as to why k_p is so much higher and has a positive trend (and conversely k_e so much lower) for the material studied by Tye and Clougherty. When compared with other reports, this does not seem to be characteristic of ZrB₂. Zimmermann et al. found k_p values as high as approximately 24 W/m-K below 200°C. However, most other research [18, 21], and the ZrB₂ produced by Thompson et al. [20], indicated that k_p was less than approximately 12 W/m-K for either ZrB₂ or HfB₂.

Because k_e and k_p are estimated using Equations 9.4 and 9.5, this method may be prone to error. One assumption that permeates the studies to date is that the theoretical Lorenz number is appropriate for diborides. Calculation of negative k_p values for materials produced by Zhang et al. in Figure 9.12, indicate that a lower value for the Lorenz number may be more representative of the behavior of diborides. However, research still indicates that k_e is the dominant thermal conduction mechanism for ZrB₂ and HfB₂.

9.4.2 ZrB₂ and HfB₂ Composites with SiC

Figure 9.13 highlights the contributions of k_e and k_p to k_t for ZrB₂–SiC and HfB₂–SiC. Overall, some of the differences observed for materials with SiC additions are not explained and some trends appear to contradict prevailing theories. For example, k_e increases with increasing temperature for ZrB₂ + 20% SiC reported by Zhang et al. despite electrical resistivity values that increase with temperature for metallic conductors. Likewise, Tye and Clougherty reported significantly higher k_p values than other researchers and their k_p values increased with increasing temperature. The most important observation from Figure 9.13 is the decreasing contribution of k_e with increasing SiC content. This is expected due to the higher electrical resistivity of SiC (>1000 μΩ-cm) [82] compared to ZrB₂ (<25 μΩ-cm at 25°C) [15]. Based on this difference, electron conduction, which dominated k_t for diboride–SiC composites, is likely entirely supported by ZrB₂, with SiC acting as an electrically insulating phase.

Figure 9.13. Thermal conductivities as a function of temperature for ZrB₂–SiC and HfB₂–SiC, including separation of the electron and phonon contributions to total conductivity.^aData corrected for ρ [16, 20, 50, 84].

In general, k_p increases as SiC content increases since phonon conduction is the primary thermal transport pathway for SiC. One clear exception is ZrB₂–30 vol% SiC, which should have a higher k_p due to its higher SiC content. However, as mentioned earlier, factors other than SiC content likely control this behavior since that composition contained WC contamination from milling. Hence, a direct comparison to other ZrB₂–SiC compositions is not possible without considering purity. Like some pure diboride material, such as those evaluated by Zhang et al. [18], the k_p of HfB₂–5 vol% SiC dropped below zero, which is unrealistic. This is another indication that the theoretical Lorenz number is not appropriate for diborides.

9.4.3 Conclusions Regarding k_e and k_p Research

The total k of diborides is dominated by electron transport. Therefore, to understand and control thermal conductivity of the diborides, factors that influence k_e are most important. The electron portion of k_t is clearly altered by transition metal impurities as indicated by decreased k_e values for materials containing W. SiC, which is electrically insulating, reduces k_e and increases k_p. The key to evaluating k_e and its contribution to k_t is the Wiedemann–Franz law. In some cases, the Wiedemann–Franz law predicts k_e values that are higher than k_t, resulting in negative k_p. Not all specimens exhibit this behavior, but it is an indication that the theoretical Lorenz number may not be valid for the diborides.

9.5 Concluding Remarks

The thermal conductivity of ZrB₂- and HfB₂-based ceramics is affected by a number of factors, perhaps none more important than processing. In particular, contaminants from milling, impurities inherent to starting powders, or intentional additives almost always decrease k for ZrB₂ and HfB₂. The final microstructure, relative density, and phase distribution and content are critical and cannot be neglected in the evaluation of k. Additional phases or SSs that form during densification also strongly affect k. In particular, SSs appear to have the largest impact. Careful reporting of initial powder purity, contaminants from processing, and additives along with the resulting microstructure and phase distribution is necessary for direct comparison of k values among different materials. Further, future research on k of diborides would benefit from systematic studies designed to isolate specific compositional or processing effects rather than drawing conclusions solely based on comparison with reported values. In addition to performing more systematic research, separating electron and phonon contributions to k_t will be critical for gaining insight into what controls conductivity. The field would also benefit from experimental or computational investigation of Lorenz numbers to improve the accuracy of k_e and k_p. The electron contribution to k_t is dominant in ZrB₂ and HfB₂. Therefore, more detailed studies of factors that affect k_e would also be beneficial.

Research to date on additions of alloying elements or second phases has shown that k_t is maximized for phase pure diborides. SiC additions, despite their high k_t in single crystal form, do not increase k for either ZrB₂ or HfB₂, especially at higher temperatures. Combined with research suggesting that ZrB₂–SiC has poor oxidation resistance above approximately 1600°C [28, 83], SiC additions (as well as other silica formers) will not result in improved performance of UHTCs under the conditions they are being proposed to operate (i.e., hypersonic flight). Most other second-phase additions also decrease k_t of the diborides, so their use requires consideration of trade-offs such as improved strength or oxidation resistance for lower k_t. More research may also be useful in the area of decreasing k_t to produce ultra-high temperature insulators. SS additions such as W have been shown to impact k_p, especially at low temperatures. However, k_e dominates k_t and has the largest potential to be diminished, but is the least understood.

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Notes

1 The discontinuity at 1000°C for data by Clougherty et al. [6] is likely due to a change in testing method.

2 Values for ZrB₂ prepared by reactive SPS by Zhang et al. [18] were corrected for porosity.

3 Corrected values for rxn-SPS ZrB₂ by Zhang et al. [18] were included for comparison.

4 Corrected values for rxn-SPS HfB₂ by Zhang et al. [18] were included for comparison.

5 This reference was used for separation of the contributions to thermal conductivity in place of Clougherty et al. [8] which is the original reference for the k_t data.